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In the 1960's and 1970's there was an explosion in development of GR. Stimulated by experimental discoveries of quasars, pulsars and cosmic background radiation, attention turned to black holes and the big bang, which culminated in the singularity theorems. These showed that spacetime singularities are inevitable both for the gravitational collapse of sufficiently massive stars and at sometime in the past during the very early phase of the universe. It was natural however to try to extend these classical results to QT and the first steps involved using quantum field theory on a purely classical background. The most important physical effect that the QT introduced, was pair particle production by a strong gravitational field. This associated a temperature (Th) and entropy (Sh) to the surface area of a black hole. Soon after the same result was rederived using a path integral formulation, which showed how the thermal character of the emission could be understood in terms of the complex geometry of the Schwarzschild metric. In particular in order that the path integral could be better defined, it was found convenient to pass (by means of a wick rotation used in QED),to a pure imaginary time coordinate by simply multiplying the time by 'i' , the periodicity of the resulting positive definite, or EUCLIDEAN metric, as opposed to Lorentzian metric, demanded that this new time coordinate be identified modulo 8piM=1/Th. It was soon realized that this periodicity in imaginary time was related to the thermal character of the resulting Green's function and was not restricted to the Schwarzschild case but could be extended generally to cover all time independent horizons including the case of the cosmological event horizon and its important connection with inflation theory. {CF "The Nature of Space & Time" (S. Hawking& R. Penrose above]. Thus from this platform, the method of " EUCLIDEAN QUANTUM GRAVITY" has been used to tackle many problems in cosmology, which would otherwise be difficult to pursue, due to our lack of a more complete theory of quantum gravity. [In this manner Einstein's equation is treated more like a (metric) 'field equation' to be quantised, rather than the traditional differential geometry notion, of a curved manifold of spacetime.]
Producing a quantum theory of gravity is particularly difficult for several reasons. Firstly quantum gravity implies quantified space time!!! Secondly gravity is intrinsically so weak it has not yet been possible to detect classical gravitational waves, let alone its graviton (it is also difficult to combine the two realms, since QT deals with very small scale phenomena, while GAR deals with large masses). Thirdly, quantum field theories are written as spinor fields, to which, the Riemannian techniques of GR not applicable Finally regarding unification, the GR group structure is non compact, while QT group is Unitary and there is a theorem which states that there is no finite unitary representation of a non compact group (this is why we have to resort to supernumbers). At first glance GR and QT look very differently mathematically, as one deals with space-time and direct observables while the other with Hilbert space and operators. One approach to synthesizing the two and providing a quantum theory of gravity, involves Topological Quantum Field Theories (TQFT). [A topologist is sometimes defined as a mathematician who cannot tell the difference between a tea cup and a doughnut, since they are diffeomorphic to each other, both having a genus of 1 ]. Quantum states are given topologies and cobordism allows a description of how quantum (gravity) states evolve i.e. TQFT maps structures in differential topology to corresponding structures in quantum theory. The state of the universe can only change when the topology of space itself changes and TQFT does not therefore presume a given fixed topology for space-time. Quantum operators are therefore related to cobordism and n-category theory (i.e. algebra of n-dimensions) is a useful advance in understanding the cobordism theory of TQFT. A topological quantum field theory will be a way to go from general relativity to quantum mechanics, i.e. given a manifold called “space”, it will spit out a Hilbert space, and given a spacetime it will spit out a linear operator. Therefore, we are looking for some kind of map between the world of manifolds and cobordisms and the world of Hilbert spaces and linear operators. This was the approach taken by Atiyah in his axiomatisation of topological quantum field theories. General relativity has taught us not only that space and time share the property of being dynamical with the rest of the physical entities, but also -more crucially- that spacetime location is relational only. [This ‘background independence’ dictates the property of diffeomorphism invariance, which itself puts constraints on any theory of quantum gravity.] Quantum mechanics has taught us that any dynamical entity is subject to Heisenberg's uncertainty at small scale. Therefore, we need a relational notion of a quantum spacetime in order to understand Planck scale physics. [N. B. Even the concept of "being at rest relative to another body" becomes ambiguous, since it depends on the choice of path , as incorporated by the Christofel connection of a Riemann manifold -- hence GR shows us that gravity is a relational diffeomorphic-invariant theory. This contrasts with SR in which if 2 vectors were equal in an original frame, they will also be equal in a translated frame. The Newtonian Principal of Equivalence states that acceleration is (locally) equivalent to gravity for all observers, who all share the same absolute space and universal time. In SR this has to be adjusted so that it applies to the relative space and time of observers moving at (constant) different velocities. Now in GR the Principal of Equivalence has to be extended further, so as to be valid for observers in different positions, even though they consequently might have different localised space-time!]
There have been two reactions to the apparent inconsistency of quantum theories with the necessary background-independence of general relativity. The first is that the geometric interpretation of GR is not fundamental, but just an emergent quality of some background-dependent theory. The opposing view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a-priori specified space-time. This geometric point of view is the one expounded in TQFT. In recent years, progress has been rapid on both fronts, leading ultimately to String Theory (which is not background independent) and Loop Quantum Gravity (LQG), which is background independent and also incorporates the diffeomorphic invariance of GR. [A special feature of the Einstein equations is that they are diffeomorphism invariant. If the equations are written down in an arbitrary coordinate system then the solutions of these coordinate equations are not uniquely determined by initial data. Applying a dffeomorphism to one solution gives another solution. In general relativistic physics, the physical objects are localized in space and time only with respect to each other. Therefore if we "displace" all dynamical objects in spacetime at once, we are not generating a different state, but an equivalent mathematical description of the same physical state. Hence, diffeomorphism invariance.] Whereas Superstring theory was initially developed as a quantum theory of the strong nuclear force which serendipitously including gravity, LQG starts with GR and seeks to incorporate QT. Topological quantum field theory provided an example of background-independent quantum theory, but usually with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity, since even in the vacuum, the metric has local degrees of freedom according to general relativity (e.g. those due to the propagation of gravity waves in empty space). The main merit of loop quantum gravity, on the other hand, is that it provides a well-defined and mathematically rigorous formulation of a background-independent, non-perturbative generally covariant quantum field theory. So far, the theory has lead to two main sets of physical results. The first is the derivation of the (Planck scale) eigenvalues of geometrical quantities such as areas and volumes. The second is the derivation of black hole entropy for ``normal'' black holes (but only up to the precise numerical factor). In LQG, physical space is a superposition of spin networks, in the same sense in which the electromagnetic field is a quantum superposition of n-photon states; in this manner space has a discrete and combinatorial character .[LQG has also predicts an inflationary effect, that is required to explain the horizon and flatness problem in cosmology]
Loop quantum gravity tries to combine general relativity and quantum theory
in a background-free theory. So, we
cannot take gravitons, strings, etc. moving on a spacetime with a pre-established
geometry as basic building blocks of
the theory. Instead, we must start with quantum states of geometry. To describe
these, we ask: What is the amplitude for a spinning test particle to come back
to the state it started in when we parallel transport it around a loop in space?
The answer doesn't depend on the starting point or the direction of the loop,
so we can ignore those. It's enough to consider spin-1=2 particles, so a state
of quantum geometry assigns to each loop an amplitude | a complex number. More
generally, a state of quantum geometry assigns an amplitude to any system of
spinning test particles tracing out paths in space, merging and splitting. These
are described by spin networks: graphs with edges labelled by spins.....together
with `intertwining operators' at vertices saying how the spins are routed. These
are described using the mathematics of spin: the representation theory of the
group SU(2). But we can also draw them!. For vertices where 3 edges meet, there's
at most one way to do this routing: For vertices where more than 3 edges meet,
we can formally `split' them to reduce the problem to the previous case: A quantum
state of the geometry of space assigns an amplitude to any spin network. So,
we can think of these states as complex linear combinations of spin networks,
with these amplitudes as coefficients: We could also use loops, but spin networks
are an orthonormal basis of states, so they are more convenient. In this theory,
the space around us is described by a huge linear combination of enormous spin
networks; a complicated `weave' that approximates the seemingly smooth geometry
we see at distances much larger than the Planck length (~ 10^-35m)
To see how this works, we need operators corresponding to interesting observables:
lengths, areas, volumes...Here we shall only consider area operators.... If
a spin network intersects a surface S transversely then this surface has a definite
area in this state, given as a sum over the spins je of the edges e poking through
S:
Area(S) = 8&(Sum)edges SqRt(j(j + 1)
in units where the Planck length is 1. In particular, the operator for area
has a discrete spectrum! Here & is a constant called the `Barbero-Immirzi
parameter'. So far we can only determine this by computing the entropy of a
black hole in loop quantum gravity and comparing the answer to Hawking's calculation.
If a surface S intersects a spin network at a vertex, we must examine the routing
to compute the area of S: To describe states with definite areas, we must split
the vertex so that the new edge intersects S transversely. This surface S' requires
a different splitting: . Different splittings give di erent bases of states.
To change from one basis to another we must use a matrix called the `6j symbols':
The area of S only has a definite value in the first basis of states, while
that of S' only has a definite value in the second basis. There is no basis
of states in which the areas of both S and S' have de nite values! In other
words, the area operators for intersecting surfaces cannot be simultaneously
diagonalized, so the uncertainty principle applies.
The main ideas underlying this approach can be summarized as follows. One begins by reformulating general relativity as a dynamical theory of connections, rather than metrics. This shift of view does not change the theory classically (although it suggests extensions of general relativity to situations in which the metric may become degenerate). However, it makes the kinematics of general relativity the same as that of SU(2) Yang-Mills theory, thereby suggesting new non-perturbative routes to quantization. Specifically, as in gauge theories, the configuration variable of general relativity is now an SU(2) connection Aia on a spatial 3-manifold and the canonically conjugate momentum Eai is analogous to the Yang- Mills ‘electric’ field. However, physically, we can now identify this electric field as a triad; it carries all the information about spatial geometry. In quantum theory, it is natural to use the gauge invariant Wilson loop functionals, P exp INT A, i.e., the path ordered exponentials of the connections around closed loops as the basic objects .The resulting framework is often called ‘loop quantum gravity’. To produce a spin network - which is the basic tool of LQG - involves the assignment of a non-zero half integer (‘spin’) –or, more precisely, a non-trivial irreducible representation of SU(2)– to each of the edges. To specify a state in Hilbert space H,~j , one only has to fix an intertwiner at each vertex which maps the incoming representations at that vertex to the outgoing ones. (For a trivalent vertex, i.e., one at which precisely three edges meet, this amounts to specifying a Clebsch-Gordon coeffcient associated with the j’s associated with the three edges.) Each resulting state is referred to as a spin network state.
A brief introduction to spin Networks
In classical general relativity, space is described by a 3-dimensional manifold with a Riemannian metric: a recipe for measuring distances and angles. In the spin network approach to quantum gravity, the geometry of space is instead described as a superposition of "spin network states". In other words, spin networks form a basis of the Hilbert space of states of quantum gravity, so we can write any state Psi as
Psi = Sum ci psii
where psii ranges over all spin networks and the coefficients ci are complex
numbers. The simplest state is the one corresponding to good old flat Euclidean
space. In this state, each coefficient ci is just the Penrose evaluation of
the corresponding spin network psii.
Actually, this interpretation wasn't fully understood until later, when Rovelli
and Smolin showed how spin networks arise naturally in the so-called "loop
representation" of quantum gravity. They also came up with a clearer picture
of the way a spin
network state corresponds to a possible geometry of space. The basic picture
is that spin network edges represent flux tubes of area: an edge labelled
with spin j contributes an area proportional to sqrt(j(j+1)) to any surface
it pierces. In quantum theory, observables are given by operators on the Hilbert
space of states of the physical system in question. You typically get these
by "quantizing" the formulas for the corresponding classical observables.
So Rovelli and Smolin took the usual formula for the area of a surface in a
3-dimensional manifold with a Riemannian metric and quantized it. Applying this
operator to a spin network state, they found the picture I just described: the
area of a surface is a sum of terms proportional to sqrt(j(j+1)), one for each
spin network edge poking through it.
The problem of finding the quantum theory of the gravitational field, and thus
understanding what is quantum spacetime, is still open. One of the most active
of the current approaches is loop quantum gravity. Loop quantum gravity is a
mathematically
well-defined, non-perturbative and background independent quantization of general
relativity, with its conventional matter couplings. The research in loop quantum
gravity forms today a vast area, ranging from mathematical foundations to physical
applications. Among the most significant results obtained are: (i) The computation
of the physical spectra of geometrical quantities such as area and volume; which
yields quantitative predictions on Planck-scale physics. (ii) A derivation of
the
Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical
picture of the microstructure of quantum physical space, characterized by a
polymer-like Planck scale discreteness. This discreteness emerges naturally
from the quantum theory
and provides a mathematically well-defined realization of Wheeler's intuition
of a spacetime "foam". Longstanding open problems within the approach
(lack of a scalar product, overcompleteness of the loop basis, implementation
of reality conditions) have been fully solved. One of the biggest problem in
loop quantum gravity is finding an adequate description of *dynamics*. This
is partially because spin networks are better suited for describing space than
spacetime. For this reason, Rovelli, Reisenberger and Baez have been trying
to describe spacetime using "spin foams" --- sort of like soap suds
with all the bubbles having faces labelled by spins. Every slice of a spin foam
is a spin network.
Now spin networks appearing in the loop representation are different from those
Penrose considered, in two important ways.
First, they can have more than 3 edges meeting at a vertex, and the vertices
must be labelled by "intertwining operators", or "intertwiners"
for short. This is a concept coming from group representation theory; what we've
been calling "spins" are really irreducible representations of
SU(2). If we orient the edges of a spin network, we should label each vertex
with an intertwiner from the tensor product of representations on the "incoming"
edges to the tensor product of representations labelling the "outgoing"
edges. When 3 edges labelled by spins j1, j2, j3 meet at a vertex, there is
at most one intertwiner
f: j1 tensor j2 -> j3,
at least up to a scalar multiple. The triangle inequality etc.are just the conditions for a nonzero intertwiner of this sort to exist. That's why Penrose didn't label his vertices with intertwiners: he considered the case where there's essentially just one way to do it! When more edges meet at a vertex, there are more intertwiners, and this extra information is physically very important. One sees this when one works out the "volume operators" in quantum gravity. Just as the spins on edges contribute *area* to surfaces they pierce, the intertwiners at vertices contribute *volume* to regions containing them!
Second, in loop quantum gravity the spin networks are *embedded* in some 3-dimensional manifold representing space. Penrose was being very radical and considering "abstract" spin networks as a purely combinatorial replacement for space, but in loop quantum gravity, one traditionally starts with general relativity on some fixed spacetime and quantizes that. Penrose's more radical approach may ultimately be the right one in this respect. The approach where we take classical physics and quantize it is very important, because we understand classical physics better, and we have to start somewhere. Ultimately, however, the world is quantum-mechanical, so it would be nice to have an approach to space based purely on quantum-mechanical concepts. Also, treating spin networks as fundamental seems like a better way to understand the "quantum fluctuations in topology"
Barbieri considers "simplicial spin networks": spin networks living in a fixed 3-dimensional manifold chopped up into tetrahedra. He only considers spin networks dual to the triangulation, that is, spin networks having one vertex in the middle of each tetrahedron and one edge intersecting each triangular face. In such a spin network there are 4 edges meeting at each vertex, and the vertex is labelled with an intertwiner of the form
f: j1 tensor j2 -> j3 tensor j4
where j1,...,j4 are the spins on these edges. If you know about the representation theory of SU(2), you know that j1 tensor j2 is a direct sum of representations of spin j5, where j5 goes from |j1 - j2| up to j1 + j2 in integer steps. So we get a basis of intertwining operators:
f: j1 tensor j2 -> j3 tensor j4
by picking one factoring through each representation j5:
j1 tensor j2 -> j5 -> j3 tensor j4
where:
a) j1 + j2 + j5 is an integer and |j1 - j2| <= j5 <= j1 + j2
b) j3 + j4 + j5 is an integer and |j3 - j4| <= j5 <= j3 + j4.
Using this, we get a basis of simplicial spin networks by labelling all the edges *and vertices* by spins satisfying the above conditions. Dually, this amounts to labelling each tetrahedron and each triangle in our manifold with a spin! Let's think of it this way, focus on a particular simplicial spin network and a particular tetrahedron. What do the spins j1,...,j5 say about the geometry of the tetrahedron? By what I said earlier, the spins j1,...,j4 describe the areas of the triangular faces: face number 1 has area proportional to sqrt(j1(j1+1)), and so on. What about j5? It also describes an area. Take the tetrahedron and hold it so that faces 1 and 2 are in front, while faces 3 and 4 are in back. Viewed this way, the outline of the tetrahedron is a figure with four edges. The midpoints of these four edges are the corners of a parallelogram, and the area of this parallelogram is proportional to sqrt(j5(j5+1)). In other words, there is an area operator corresponding to this parallelogram, and our spin network state is an eigenvector with eigenvalue proportional to sqrt(j5(j5+1)). Finally, there is also a *volume operator* corresponding to the tetrahedron, whose action on our spin network state is given by a more complicated formula involving the spins j1,...,j5.
So we see how an ordinary tetrahedron is the classical limit of a "quantum tetrahedron" whose faces have quantized areas and whose volume is also quantized. Second, we're seeing how to put together a bunch of these quantum tetrahedra to form a 3-dimensional manifold equipped with a "quantum geometry" --- which can dually be seen as a spin network. Third, all this stuff fits together in a truly elegant way, which suggests there is something good about it. The relationship between spin networks and tetrahedra connects the theory of spin networks with approaches to quantum gravity where one chops up space into tetrahedra --- like the "Regge calculus" and "dynamical triangulations" approaches.
* * * * * * * * * * * * *
Let us now turn to specifics. For simplicity, I will focus on the gravitational field; matter couplings are discussed in. The basic gravitational configuration variable is an SU(2)-connection, Ai a on a 3-manifold M representing ‘space’. As in gauge theories, the momenta are the ‘electric fields’ Ea i .10 However, in the present gravitational context, they acquire an additional meaning: they can be naturally interpreted as orthonormal triads (with density weight 1) and determine the dynamical, Riemannian geometry of M. Thus, in contrast to Wheeler’s geometrodynamics, the Riemannian structures, including the positive-definite metric on M, is now built from momentum variables. The basic kinematic objects are: i) holonomies he(A) of Ai a, which dictate how spinors are parallel transported along curves or edges e; and ii) .uxes ES,t = RS ti Ea i d2Sa of electric felds, Ea i , smeared with test fields ti on a 2-surface S. The holonomies —the raison d’ˆetre of connections— serve as the ‘elementary’ configuration variables which are to have unambiguous quantum analogs. They form an Abelian C. algebra, denoted by Cyl. Similarly, the fluxes serve as ‘elementary momentum variables’. Their Poisson brackets with holonomies define a derivation on Cyl. In this sense —as in Hamiltonian mechanics on manifolds—momenta are associated with ‘vector fields’ on the con.guration space. The first step in quantization is to use the Poisson algebra between these con.guration and momentum functions to construct an abstract .-algebra A of elementary quantum operators. This step is straightforward. The second step is to introduce a representation of this algebra by ‘concrete’ operators on a Hilbert space (which is to serve as the kinematic setup for the Dirac quantization program).
We now describe the salient features of this representation . Quantum states span a specific Hilbert space H consisting of wave functions of connections which are square integrable with respect to a natural, di.eomorphism invariant measure. This space is very large. However, it can be conveniently decomposed into a family of orthogonal, finite dimensional sub-spaces H = .,~j H.,~j , labelled by graphs ., each edge of which itself is labelled by a spin (i.e., half-integer) j . (The vector ~j stands for the collection of half-integers associated with all edges of ..) One can think of . as a ‘floating lattice’ in M —‘floating’ because its edges are arbitrary, rather than ‘rectangular’. (Indeed, since there is no back-ground metric on M, a rectangular lattice has no invariant meaning.) Mathematically, H.,~j can be regarded as the Hilbert space of a spin-system. These spaces are extremely simple to work with; this is why very explicit calculations are feasible. Elements of H.,~j are referred to as spin-network states In the quantum theory, the fundamental excitations of geometry are most conveniently expressed in terms of holonomies . They are thus one-dimensional, polymer-like and, in analogy with gauge theories, can be thought of as ‘flux lines’ of electric fields/triads.More precisely, they turn out to be flux lines of area, the simplest gauge invariant quantities constructed from the momenta Ea i : an elementary flux line deposits a quantum of area on any 2-surface S it intersects. The basic quantum operators are the holonomies ˆhe along curves or edges e in M and the .fluxes ˆ ES,t of triads ˆ Eai . Both are densely defined and self-adjoint on H .For, just as the classical Riemannian geometry of M is determined by the triads Eai; all Riemannian geometry operators —such as the area operator ˆ AS associated with a 2-surface S or the volume operator ˆ VR associated with a region R— are constructed from ˆ ES,t. In any theory without background .elds, Hamiltonian dynamics is governed by constraints. Roughly this is because in these theories di.eomorphisms correspond to gauge in the sense of Dirac. Recall that, on the Maxwell phase space, gauge transformations are generated by the functional DaEa which is constrained to vanish on physical states due to Gausslaw. Similarly, on phase spaces of background independent theories, di.eomorphisms are generated by Hamiltonians which are constrained to vanish on physical states. In the case of general relativity, there are three sets of constraints. The first set consistsof the three Gauss equations Gi := Da Ea i = 0, which, as in Yang-Mills theories, generates internal SU(2) rotations on the connection and the triad .elds. The second set consists of a co-vector (or di.eomorphism) constraint Cb := Ea i Fi ab = 0, which generates spatial di.eomorphism on M (modulo internal rotations generated by Gi). Finally, there is the key scalar (or Hamiltonian) constraint S := oijkEa i Eb jFab k + . . . = 0, which generates time-evolutions. (The . . . are extrinsic curvature terms, expressible as Poisson brackets of the connection, the total volume constructed from triads and the first term in the expression of S given above.)
Fundamentally, loop quantum gravity is a very conservative approach to quantum gravity. It starts with the equations of general relativity and attempts to apply the time-honored principles of quantization to obtain a Hilbert space of states. There are only two really new ideas in loop quantum gravity. The first is its insistence on a background-free approach. That is, unlike perturbative quantum gravity, it makes no use of a fixed ‘background’ metric on spacetime. The second is that it uses a formulation of Einstein’s equations in which parallel transport, rather than the metric, plays the main role. It is very interesting that starting from such ideas one is naturally led to describe states using spin networks! We begins by formulating general relativity in the mathematical language of connections, the basic variables of gauge theories of electro-weak and strong interactions. Thus, now the confguration variables are not metrics as in Wheeler’s geometrodynamics, but certain spin-connections; the emphasis is shifted from distances and geodesics to holonomies and Wilson loops. In ordinary quantum field theory we calculate path integrals using Feynman diagrams. [In Feynman diagrams all vector spaces are representations of some symmetry group G (e.g. Poincare group), also vertices represent morphisms between representations i.e. intertwiners. Penrose took G=SU(2), which has one irreducible representation of each dimension i.e. spin 0 (=1D) spin1/2 (=2D) spin 1 (=3D).] Copying this idea, in loop quantum gravity we may try to calculate path integrals using ‘spin foams’, which are a 2-dimensional analogue of Feynman diagrams. In general, spin networks are graphs with edges labelled by group representations and vertices labelled by intertwining operators. These reduce to Penrose’s original spin networks when the group is SU(2) and the graph is trivalent. [A spin network applies Feynmans method in a combinatorial way, producing the so called 6j symbols of a tetrahedron, which encodes the interaction of spinors in a gauge invariant (and space independent) way ] Similarly, a spin foam is a 2-dimensional complex built from vertices, edges and polygonal faces, with the faces labelled by (irreducible) group representations and the edges labelled by intertwining operators. When the group is SU(2) and three faces meet at each edge, this looks exactly like a bunch of soap suds with all the faces of the bubbles labelled by spins — hence the name ‘spin foam’. If we take a generic slice of a spin foam, we get a spin network i.e.each of its faces gives a spin network edge, and each of its edges gives a spin network vertex. Thus we can think of a spin foam as describing the geometry of spacetime, and any slice of it as describing the geometry of space at a given time. Ultimately we would like a ‘spin foam model’ of quantum gravity, in which we compute transition amplitudes between states by summming over spin foams going from one spin network to another: Spin networks serve simultaneously as a tool for calculations in representation theory and as a description of the quantum geometry of space. Spin foams extend this idea to the geometry of spacetime. [An alternative approach is to employ Lorentzian 4 dimensional spin network tetraheda i.e. 4 simplex which have 5 vertices]
Loop Quantum Gravity is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators. In LQG, the fabric of space-time is a foamy network of interacting loops mathematically described by spin networks (an evolving spin network is termed a spin foam; spin foams are to operators what spin networks are to states/bases). These loops are about 10E-35 meters in size, called the Planck scale. In previous lattice theories the field is represented by quantised tubes/strings of flux which only exist on the edges of the lattice and the field strength is given by the value of integrating around a closed loop. In LQG space and time are relational! As in GR where there are many ways of slicing a section of space time, there are many ways of slicing an evolving spin network - thus there are no things only processes! [A spin network is a graph with edges labeled by representations of some group and vertices labeled by intertwining operators. Thanks in part to the introduction of spin network techniques, we now have a mathematically rigorous and intuitively compelling picture of the kinematical aspects of loop quantum gravity.] The loops knot together forming edges, surfaces, and vertices, much as do soap bubbles joined together. In other words, space-time itself is quantized. Any attempt to divide a loop would, if successful, cause it to divide into two loops each with the original size. In LQG, spin networks represent the quantum states of the geometry of relative space-time. Looked at another way, Einstein's theory of general relativity is a classical approximation of a quantized geometry. The problem is, how do we understand the quantum geometry of space-time? The problem of time makes it difficult to apply a canonical hamiltonian approach and consequently the path integral Lagrangian is preferred. Unlike most field theories which are set on a manifold with fixed metric, the dynamics of quantum gravity are generated by 'constraints'. Quantum gravity is so hard because whereas there are tricks for quantizing Poisson manifolds (i.e. phase space) by introducing operators obeying Heisenberg's UP, the phase space of GR is infinite dimensional with singularities. From the study of Einstein's equation and QM, we arrive at (utilizing Poison brackets), . . . Gauss constraints (gauge invariance due to Yang-Mills theory, these producing Kinematic states, which do not generate dynamics) .. . .diffeomorphic constraints (due to a GR symmetry) . . . . and Hamiltonian constraints (these do produce a dynamic description but are difficult to define), which have led us to spin bundle spaces L*(A), quotient spaces L*(A/G) and bivectors that are associated with spin networks. We quantise gravity by means of a connection A and its canonical momentum conjugate E, with which we create spin networks for spinor states and from this we can produce area operators which act on space to quantise its geometry!.
Historically Regge calculus was the first attempt to quantize Riemannian gravity, producing a lattice formulation of GR by dividing up space into small (flat) 3D tetrahedral simplexes, in which the curvature is concentrated along their boundaries. In effect he applied spin networks before Penrose invented them, by analysing space in terms of tetrahedron, in which the amplitude for tetrahedra are given by 6j symbols. Once the space has been chopped up into such tetrahedra, we can find the amplitude for space-time to have a given shape by taking the products of the 6j symbols over all the tetrahedra and multiplying this by a suitable factor for each tetrahedra. From this Hilbert space of quantum tetrahedron, it was intended to produce 'Feynman propagators' for gravity and recover Einstein's field equation in the macroscopic domain of space-time. Each of the edges is associated a spin j and in one type of approach, the exponential of the action in such a configuration, is a suitable product of the the 6j symbols associated to each of the 6 edges of the 3-simplexes and a partition function is obtained by taking the sum of this products over the possible associations of the spin to the edges. Remember that angular momentum is a quantum (bi)vector and is therefore subject to Heisenberg' Uncertainty Principle. [In considering a Lorentzian (as opposed to a 3-D) spin network we employ a 4-simplex in which there are 4 tetrahedra, 5 vertices, 10 triangles and 10J symbols]. In retrospect the very first spin foam model was the Ponzano--Regge model in 3-D Riemannian quantum gravity, in which we triangulate a given 3-manifold and expresses the partition function . . .INTEGRAL Exp iS . . .as a sum over spin foams lying in the dual 2-skelaton of the triangulation. Subsequent improvements were made by invoking so called quantum groups SUq(2), which are not actually groups but a structure whose representations are j=0, 1/2, . . .k/2 where q=2pi/(k+2) (in other words we do not actually need to know the group.)
We can think of a tetrahedron in R3 with one vertex at the origin as an affine
map from T to R3 sending the vertex 0 to the origin. Such a map is given by
fixing vectors e1, e2, e3 . corresponding to the edges 01, 02, and 03. This
data is precisely a ‘cotriad’, a linear map from the tangent space
of the vertex 0 of T to R3. In the Palatini approach to general relativity one
works with a similar object, namely the ‘cotriad field’, which can
be thought of locally as an R3- valued 1-form e. But a crucial idea in modern
canonical quantum gravity is to work not with e but with the 2R3-valued 2-form
e ^e, often called the ‘densitized cotriad field’. Using this rather
than the cotriad field is the reason why areas rather than lengths are the basic
geometrical observables in the spin network approach to quantum gravity. This
suggests an alternate description of the geometry and orientation of a tetrahedron
in terms of faces rather than edges.
In this alternate description, we work not with the vectors ei but with the
bivectors ei ^ej . (A bivector in n dimensions is simply an element of 2Rn;
in the presence of a metric a bivector in three dimensions can be identified
with an ‘axial vector’, which in turn can be identified with a vector
in the presence of an orientation.) We denote these bivectors by:
E1 = e2 ^e3, . E2 = e3 ^e1, . E3 = e1^e2. Actually, rather than working directly
with a triple of bivectors, it is useful to
introduce a fourth:
E0 = -E1 - E2 - E3.
Then, if we use a metric and a orientation to think of the Ei as vectors, each
Ei is normal to the ith face of the tetrahedron (the face missing the ith vertex),
with magnitude given by the area of that face. To ‘quantize’ the tetrahedron,
Barbieri first quantizes the bivectors Ei and then imposes the closure constraint.
The notion of ‘quantized bivectors’ is implicit in Penrose’s
ground-breaking work on spin networks and ‘quantized directions’ but
quantized bivectors really date back to early quantum mechanics, where they
were introduced to describe angular momentum. Classically, the angular momentum
is a bivector. Quantum mechanically, the components of the angular momentum
no longer commute, but instead satisfy
[J1, J2] = iJ3, . . [J2, J3] = iJ1, . . . .[J3, J1] = iJ2.
The uncertainty principle thus prevents us from simultaneously measuring all
3 components with arbitrary accuracy. Moreover, each component takes on a discrete
spectrum of values, there being operators associated satisfying the above commutational
relations.Classically the area of the ith face of the tetrahedron is given by
Ai =1/2SqRt(Ei · Ei).
Quantizing these expressions, we define area operators on T by
ˆ Ai =1/2SqRt( ˆ Ei · ˆ Ei)
for i = 0, 1, 2, 3. These operators commute and are simultaneously diagonalized
on the subspaces Inv(j0.· · ·.j3); the eigenvalue of ˆ
Ai on this subspace is 1/2SqRt{ji(ji + 1)}.Similarly the volume V of a tetrahedron
is given classically in terms of the bivectors Ei by . . V = 1/6 SqRt{£ijkE1E2E3}and
we define the volume operator on T by . . V' = 1/6 SqRt{£ijkE1'E2'E3'}
Next we have Wilson loops which have been used to analyze fields in QCD by means of applying a lattice structure and integrating along closed paths. They are functions on the space of connections; at a lattice point the Wilson loop is just the trace of holonomy around the loop on the lattice, taken in some representation of the holonomic group of the gauge field. [This philosophy originated from considering the vacuum as being like the discrete lines of (magnetic) flux that is exhibited by superconductors.] Historically it was Weyl who introduced the concept of Gauge invariance, when attempting to unify GR with Electromagnetism. He suggested that the length (as well as the orientatiin of a vector) might also change when it is parallel transported along a path. Although this was found to be incorrect he was on the right track because EM actually is described by a connection that asigns a number to a closed path.in spacetime but this number describes not the 'dilation of a ruler' but the change in phase of a wavefunction in QT [this numbrer is not a change in R+ but in (U).] It was in 1918, that Weyl attempted to unify Maxwell's theory of electromagnetism with Einstein's theory of gravity in a theory where not only position and velocity but also size is relative. He introduced the term `gauge invariance' for this extra symmetry, since the idea was that comparing lengths requires a physical ruler or `gauge', which we carry from one point to another. In this theory the Lorentz group is extended to the group of all angle-preserving linear transformations of Minkowski spacetime --now called the Weyl group, which is also an inherent symmetry in string theory. The Weyl group is the product of the Lorentz group and the multiplicative group of positive numbers, which act to rescale vectors. Thus, in Weyl's theory the Levi-Civita connection is replaced by an SO(3,1) * R+ connection, , with R+ playing th e role of the electromagnetic vector potential.. A 'connection' on the lattice is simply an assignment of an element of a gauge group to each edge of the graph, representing the effect of a parallel transport along the edge. [The holonomy around such a gauge field, is a measure of the field strength, which in turn determines the value of the Feynman path integral]. Each edge of this lattice is assigned a gauge group element that represents the (holonomic) connection and the vertex is also assigned a group element that represents a gauge transformation. From this a quotient space is formed (i.e. the space of connections modulo gauge transformations), and by mapping these onto suitable(complex) irreducible spin representations, we obtain a suitable way of producing a Spin Network. [Such spin network edges represent quantized flux lines of the field]. Holonomies are a natural variable in a Yang-Mills theory [YM], in which the relevant variables do not refer to what happens at a point but rather refers to the relationship between different points connected by a line (curve). Quantum states are functions of Wilson loops over a lattice gauge theory, however over a continuous background, the space spanned by Wilson loop states is far too big for producing a basis of a QFT. However loop states are not too singular or too many in a background independent theory where spacetime is itself formed by loop states, since the position of these states is related only with respect to other loops and not to a background. Therefore the size of the space is dramatically reduced by this diffeomorphic invariance (which is itself a feature of GR). A finite linear combination of loop states are defined precisely as the spin network states of a lattice YM theory. In LQG the physical space is a quantum superposition of spin networks which predict a quantisation of area and volume, in the same sense in which, an electromagnetic field is a quantum superposition of an n-photon state where we have quantitative predictions of energy and momentum. Remember, in quantum theory, observables are given by operators on the Hilbert space of states of the physical system in question. You typically get these by "quantizing" the formulas for the corresponding classical observables. So we take the usual formula for the area of a surface in a 3-dimensional manifold with a Riemannian metric and quantized it. Applying this operator to a spin network state, they found the picture I just described: the area of a surface is a sum of terms proportional to sqrt(j(j+1)), one for each spin network edge poking through it. Also just as the spins on edges contribute *area* to surfaces they pierce, the intertwiners at vertices contribute *volume* to regions containing them!
[In an interaction between particles of different spin, if we represent each irreducible of spin j as a line formed by 2j strands, then an invariant tensor is a trivalent node where 3 such lines meet and all strands are connected across the node. The relationship between (Wilson) lines and (Penrose) strands reproduces precisely the relation between loops and spin networks. Hence in QLG we employ the holonomic (gauge) technique of loop theory and the combinotorial technique of spin networks (which gaurantees a relational aspect to space and time that is inherent in GR due to its diffeomorphic invariance). In a 3 particle (spinor) interaction there is an intertwiner between j1, j2 and j3 and we can normalise this invariant tensor, producing what is known as the 3j Wigner symbols. Now when considering more spinor interactions we can spilt these into other trivalent interactions which then defines the Wigner 6j symbols (formed by contracting four 3j symbols). These can then be topologically identified by the sides of a tetrahedron (a 3-symplex) whose invariant properties form the basis for a spin network. There is one 3j-symbol for each vertex of the tetrahedron and one representation for each edge]
Spin networks are gauge invariant and by taking suitable sums of tensor products, provide an orthonormal basis for LQG. Penrose had earlier introduced Spin Networks, in which the edges were labeled by an irreducible representation of a SU(2). A spin network is a graph in space with edges labelled by spin and vertices labelled by intertwining operators (tensors). The spin j are the quantum number of the area and the intertwiners i are the quantum numbers of the volume. They are a natural extension of Wilson loops and give gauge invariant elements of L*(A) - - in fact a basis! Rather than serving as a tool for describing the geometry of a spacetime manifold, spin networks are intended as a purely combinatorial substitute for a spacetime manifold. Lie gauge group (characterized only by its dimension d =2j+1, where j is the quantum spin number) and the vertices with intertwining operators, and it was found that such a combinatorial formalism was preferable since it produced a relational theory. Spin networks have edges which are associated with a spin j quantum of action and the number of edges intersecting at a node determines its area, while the volume is determined by the number of nodes (intertriners) in a given region. A strict connection exists between quantum tetrahedra and 4-valent vertices of SU(2) spin networks.[The connection with Regge-Ponzano method is thus evident, by which the combinotorial approach of describing space in terms of tetrahedra, which are quantised by identifying each face with an angular momentum operator.] The 4 faces of a tetrahedron are associated with 4 irreducible representations of SU(2), which are represented by a perpendicular line, the 4 of which meet at a central node of the tetrahedron (there are actually bivectors associated with each face, in keeping with constraints of GR and quantizing the bivectors/tetrahedra amounts to labelling each face with a pair of spinors). We therefore obtain a 4-valent (colour coded) spin network (each line of which represents a quantized unit of action, while the nodes behave as area operators), which can exhibit properties that are gauge invariant. The quantum bivectors allow us to construct area and volume operators which act upon the spin network basis to produce a discrete spectrum which has units of Planck length squared and cubed respectively (in the case of the area operator, the eigenvalues are 1/2sqrt (j(j+1) - - spin network edges are quantized flux tubes of area). In keeping with quantum theory we would expect these to correspond to physical observable i.e. we have a quantized space (invoking a lower size limit and thus avoiding ultraviolet divergence), which also suggests a non commuting (quantum) geometry!
* * * * *
The true physical observables in BF theory (c.f. BF theory below !!!
) are self-adjoint operators on the physical Hilbert space,
when this space is well-defined. Nonetheless it is interesting to consider operators
on the gauge invariant Hilbert space L2(A/G). These are relevant not only to
BF theory but also other gauge theories, such as 4-dimensional Lorentzian general
relativity in terms of real Ashtekar variables, where the gauge group is SU(2).
In what follows we shall use the term ‘observables’ to refer to operators
on the gauge-invariant Hilbert space. We consider observables of two kinds:
functions of
A and functions of E. Since A is analogous to the ‘position’ operator
in elementary quantum mechanics while E is analogous to the ‘momentum’,
we expect that functions of A act as multiplication operators while functions
of E act by di erentiation. As usual in quantum field theory, we need to smear
these fields — i.e., integrate them over some region of space — to
obtain operators instead of operatorvalued distributions. Since A is like a
1-form, it is tempting to smear it by integrating it over a path. Similarly,
since E is like an (n - 2)-form, it is tempting to integrate it over an (n -
2)- dimensional submanifold. This is essentially what we shall do. However,
to obtain operators on the gauge-invariant Hilbert space L2(A/G), we need to
quantize gauge-invariant functions of A and E. The simplest gauge-invariant
function of the A field is a ‘Wilson loop’: a function of the form
tr(.(T eHA)) for some loop . in S and some representation . of G. In the simplest
case, when G = U(1) and the loop . bounds a disk, we can use Stokes’ theorem
to rewrite H A as the flux of the magnetic field through this disk. In general,
a Wilson loop captures gauge-invariant information about the holonomy of the
A field around the loop. A Wilson loop is just a special case of a spin network,
and we can get an operator on L2(A/G) from any other spin network in a similar
way. As we have seen, any spin network in S defines a function . Fun(A/G). Since
Fun(A/G) is an algebra, multiplication by defines an operator on Fun(A/G). Since
is a bounded function, this operator extends to a bounded operator on L2(A/G).
We call this operator a ‘spin network observable’. Note that since
Fun(A/G) is an algebra, any product of Wilson loop observables can be written
as a finite linear combination of spin network observables. Thus spin network
observables give a way to measure correlations among the holonomies of A around
a collection of loops. When G = U(1) it is also easy to construct gauge-invariant
functions of E. We simply take any compact oriented (n - 2)-dimensional submanifold
in S, possibly with boundary, and do the integral
ZE 13
This measures the flux of the electric field through . Unfortunately, this
integral is not gauge invariant when G is nonabelian, so we need to modify the
construction slightly to handle the nonabelian case. Write
E| = e d^n-2 x
for some g-valued function e on and some (n - 2)-form dn-2x on that is compatible
with the orientation of . Then
Zphe, ei d^n-2x
is a gauge-invariant function of E. One can check that it does not depend on
how we write E as
e dn-2x. We can think of it as a precise way to define the quantity INTEGRAL
/E\ (over surface). Hence in summary,the flux through a surface is represented
by area operators for a spin network acting on surfaces described by a spin
network basis
Recall that 3-dimensional BF theory with gauge group SU(2) or SO(3) is a formulation of Riemannian general relativity in 3 dimensions. In this case is a curve, and the above quantity has a simple interpretation: it is the length of this curve. Similarly, in 4-dimensional BF theory with either of these gauge groups, is a surface, and the above quantity can be interpreted as the area of this surface. The same is true for 4-dimensional Lorentzian general relativity formulated in terms of the real Ashtekar variables. Quantizing the above function of E we obtain a self-adjoint operator E() on L2(A/G), at least when is real-analytically embedded in S.
These spin networks (formed from the above mentioned quotient gauge space), do not refer to a specific space background and we can reproduce Wilson loop calculation to imitate a quantum theory of gravity, which is relational, as in the spirit of GR. So, since spin networks form a convenient basis of kinematical states, they have largely replaced collections of loops as our basic model for 'quantum 3-geometries'. Now in order to better understand the dynamical aspects of quantum gravity, we would also like a model for 'quantum 4-geometries'. In other words, we want a truly quantum-mechanical description of the geometry of spacetime. Recently the notion of 'spin foam' has emerged as an interesting candidate; so whereas spin networks provide a language for describing the quantum geometry of space, a spin foam attempts to extend this language to describe the quantum geometry of spacetime. A spin foam is a 2-dimensional cell complex with faces labeled by representations and edges labeled by intertwining operators; generically, any slice of a spin foam gives a spin network. We calculate the amplitude for any spin foam, as a product of the face and edge amplitudes (which equate to propagators) and the vertex amplitudes (which equate to intersection amplitudes). [Abstractly, a Feynman diagram can be thought of as a graph with edges 'labelled' by a group representation and vertices labelled by intertwining operators. Feynman diagrams are 1D because they describe particle histories, while spin foams are 2D because in LQG, the gravitational field is described not in terms of point particles but as 1D spin networks. Feynman computes probability in terms of probability amplitudes for edges and vertices whereas spin foams compute probability amplitudes as a product of faces, edges, and vertices amplitudes. Like Feynman diagrams spin foams serve as a basis for quantum histories.]
Although QLG has been successful in predicting Hawking radiation and Black Hole entropy, it is restricted to the domain of quantum gravity and as yet does not offer any import on the other fundamental interactions or the possibility of unification. Unlike string theory it does however offer testable predictions, such as the variation of the speed of light at different energies. The spin foam which makes up the fabric of space-time predicts a varying refraction coefficient depending on the frequency and hence energy of the photon. It is therefore hoped that by studying gamma ray bursts from the most remote regions of the universe, this small dispersion in the arrival times of the radiation can be observed (other avenues of research also point to theories involving a variable speed of light). Some researchers believe that even the success of string theory can be explained in terms of discrete units of space that become evident on the Planck scale (which being ~ 10 E -35m, is much smaller than that of the compactified dimensions of superstring theory). Also both theories allude to a version of the Holographic principle in which entities such as black hole contain all their information in their Event Horizons - one bit for every 4 Planck areas [This arises since black holes emit Hawking radiation and therefore its mass is related to a thermodynamic temperature, hence the entropy of a black hole is proportional to its surface area while information is negentropy] However LQG emphasize the necessity to have a relational theory in which space and time are dynamic rather than fixed and the primary concept is that of processes by which information is conveyed from one part of the world to another. Hence the area of any surface in (LQG) space is nothing but the capacity of that surface as a channel of information and so the world can be construed as a network of relationships. This avenue of approach has lead to the study of a relational logic called Topos theory and non commutative geometry, in which it is impossible to determine enough information to locate a point (a point is then described by an appropriate matrix) but it can support a description of particles and fields evolving in time)
{ . { . { . { . { Regarding LQG more formerly, the Wilson loops is the trace of the holonomy of 'A' around some loop alpha taken in some (finite dimensional) representation rho of a gauge group G, and is written as
Psi(A) = tr(rho(Texp(Integral A))
Penrose introduced the notion of a spin network as an attempt to go beyond the concept of a manifold towards a more combinatorial approach to spacetime. In his definition, a spin network is a trivalent graph labeled by spins j = 0, 1 2 , 1, . . . , satisfying the rule that if edges labeled by spins j1, j2, j3 meet at a vertex, the Clebsch- Gordon condition holds: |j1 - j2| = j3 = j1 + j2. In fact, the spins should be thought of as labeling finite-dimensional irreducible representations of SL(2,C), and the Clebsch-Gordon condition is necessary and sufficient for there to be a nontrivial intertwining operator from j1*. j2 to j3. One can use the representation theory of SL(2,C) to obtain numerical invariants of such labeled graphs.
Now we want to obtain a Hilbert space L2(A) --- the space of square-integrable functions on A ---that can represent gravity and the configuration space of GR is the space of all metrics, hence we need bivectors (2-forms) to be the quantity analogous to the position (and momentum )operators. [Remember that the Einstein equation is itself a second rank tensor and since we are quantizing a metric, the analogue of the position and momentum are the fields qij and p^ij]. We need to find gauge invariant vectors in L2 and it turns out that these can be thought of as wavefunctions Psi(A) on the quotient of A by the group G and we denote this subspace as L2(A/G) and the most obvious example of vectors in L2(A/G) are Wilson loops (also any product of Wilson Loops is a gauge invariant element of L2(A)). The space L2(A/G) is then naturally isomorphic to the G invariant subspace of L2(A) More generally still we can get vectors in L2(A/G) from spin networks by labeling each edge e with a representation rho e of G. So given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P --M, there is a canonical ‘generalized measure’ on the space A/G of smooth connections on P modulo gauge transformations. This allows us to define a Hilbert space L2(A/G). We then construct a set of vectors spanning L2(A/G). These vectors are described in terms of ‘spin networks’: graphs phi embedded in M, with oriented edges labeled by irreducible unitary representations of G, and with vertices labeled by intertwining operators from the tensor product of representations labeling the incoming edges to the tensor product of representations labeling the outgoing edges. From this we can obtain an orthonormal basis of spin networks associated to any fixed graph phi. Let
He(A) = rho e (Texp(Integral A))
which can be interpreted as a matrix and by forming a tensor product of all these matrices one for each edge we obtain a huge tensor having one superscript and one subscript. Next, for each vertex v of the graph let S(v) be the set of edges having v as a source and T(v) as the of target edges. For each vertex v we choose a n intertwining operator
Iv: $ rho e ~ £ rho e . . . . . . . . . e a member of T(v) and S(v) respectively in each tensor product denoted by £
we can think of Iv as a tensor with one subscript and one superscript for each edge e a member of T(v), Next we form a tensor product of all these tensors I(v) one for each vertex to obtain a large tensor I. Then we form the tensor product of He(A) $ I , which we can contract to obtain a number which of course depends on A which is our spin network state Psi(A). Hence a Wilson loop is just a special case of a spin network with only one edge and one vertex, with the intertwining operator taken to be the identity operator. [These spin network states are gauge invariant and lie in L2(A) and what's more span the whole space of vectors in L2(A)]
It can then be shown that
L2(A) is isomorphic to Tensorproduct(SUM p*p')
and
L2(A?G) is isomorphic to SUM(TensorproductInv(v,p)) which consequently means that the Hilbert space is spanned by spin network states;
phi (p,l) = Tensorsum lv
where p is any labeling of the edges 'e' of graph phi by an irreducible representation p'e' and lv is an element of the invariant subspace Inv(v,p)
Having constructed the gauge theory on the graph and related these to spin networks we then proceed to construct the Hibert space of quantum tetrahedron T;
. . T = {psi<H4:(E0+E1+E2+E3)psi=0} . . . . . where E's are operators constructed from angular momentum operators
and from these we can obtain quantum area A# and volume V# operators needed for LQG;note that these are non commuting
A# =1/2(E1.E2)^1/2 . . . . . . . . . .each area operator having an eigenvalue of 1/2j(j+1)^1/2
V# =1/6/epsilondelta E1E2E3 \^1/2 . . .
Classically, in the metric representation a state of gravity is described by the metric on S and its first time derivative. However the suitable variables for the canonical quantization of gravity, is the phase space consisting of A and E., where A is an SU(2) connection on space and E is a densitizes frame field on space**. We can rewrite interesting functions of the metric in terms of the new variables, and then attempt to quantize them by replacing A and ~ E (which is itself formed from the soldering or vierbein field e^e) by their quantum versions in these expressions, obtaining operators on L2(A). [Recall that in the metric formulation of general relativity, the analogs of the position and momentum are the fields qij and pij on S. In the new variables, the analogs of position and momentum are instead the connection A+ and the field (e ^ e)+ restricted to S. Einstein's equation is obtainable from the action generated by e and the curvature F of the gauge connection A]. Due to the Hamiltonian constrain it is problematic finding the relevant operators but those that generate area and volume turn out to have discrete spectrum: certain multiples of Planck length squared for the area operator, and certain multiples of the Planck length cubed for the volume operator. In quantum theory, the spectrum of a self-adjoint operator corresponds to the values the corresponding observable can assume, so this is an indication that area and volume are `quantized' in the very literal sense of assuming a discrete set of values! Moreover, there is a simple geometrical reason for this fact. To speak in rather oversimplified terms, the area operator applied to a given spin network state counts the number of points at which an edge of the spin network intersects the surface in question, weighted by a factor of sqrt j(j + 1)`2 lp, where j is the spin labelling that edge. Similarly, the volume operator applied to a spin network state counts the number of vertices of the spin network contained in the region in question, weighted by some function of their labellings by intertwiners and the geometry of the incident edges.
**Suppose we have a smooth
4-manifold M and a vector bundle T . M isomorphic to the tangent bundle and
equipped with an orientation and positive-definite metric. There is a formulation
of general relativity in which the basic fields are a metric-preserving connection
A on T and a T -valued 1-form e, usually called the ‘cotetrad’ field.
The action is given by
INT (M)tr(e^ e^. F), where F is the curvature of A, the wedge product of T
-valued forms is taken in the obvious way, and the map tr:4T . R is defined
using the metric and orientation on T . The equations of motion are dA(e^. e)
= 0, . .e^. F = 0,
where dA denotes the exterior covariant derivative. When e is nondegenerate
these equations are equivalent to the vacuum Einstein equations. On the other
hand, one can modify this formulation, working not with the cotetrad field e
but with the field E = e ^e. First, note that if one takes E to be an arbitrary
2T -valued 2-form and uses the action
INT (M) tr(E ^F)
one obtains not general relativity but BF theory, whose equations of motion
are simply
dAE = 0, . F = 0.
The reason is that not every such E field can be written as e . e for some cotetrad
field e. For this to hold, it must satisfy some extra constraints. If one finds
stationary points of the above action subject to these constraints, one obtains
equations
equivalent to the vacuum Einstein equations, at least when E is nondegenerate.
Now, doing general relativity with the cotetrad field
e is very much like describing 4-simplices using vectors for edges, while doing
general relativity with the E field is very much like describing 4-simplices
using bivectors for faces. Suppose we have
a 4-simplex anely embedded in R4. We can number its vertices 0, 1, 2, 3, 4,
and translate it so that the vertex 0 is located at the origin. Then one way
to describe its geometry is by the positions e1, e2, e3, e4 of the other four
vertices. Another way is to
use the bivectors . . .Eab = ea ^eb . . . corresponding to the six triangular
faces with 0 as one of their vertices. However, not every collection of bivectors
Eab comes from a 4-simplex this way. In addition to the obvious skew-symmetry
Eab = -Eba, some extra constraints must hold. They have exactly the same form
as the extra constraints that we need to obtain general relativity from BF theory.
Barrett and Crane showed that these constraints take a particularly nice form
if we describe them in terms of bivectors corresponding to all ten triangular
faces of the 4-simplex. Quantizing these constraints, one obtains conditions
that a Spin(4) spin foam must satisfy to describe a quantum 4-geometry built
from flat quantum 4-simplices. Interestingly, these conditions also guarantee
that the quantum geometries for 3-dimensional submanifolds obtained using the
right-handed copy of SU(2) agree with those coming from the left-handed copy.}
.} .} . }
*************************************
Given a manifold M and a vector bundle V the simplest topological invariance that can be defined are the characteristic classes. The Atiyah-Singer theorem, shows that integrals over certain characteristic classes yield various index theorems, specifically an integral over the product of 2 curvature forms, one for the gravitational part and one for the gauge part (viz. the Yang Mills field Fij and a Riemann curvature tensor). For example if M has dimension 2n, the Euler class is a 2n form that can be integrated over M to give a number--- the Euler characteristic-- while the Chern class yields several indexes when integrated over characteristic polynomials of the gravitational curvature form. Topological quantum field theory can be approached from three different directions, the purely combinatorial, the category theoretical and through a path integral formulation of a quantum field theory. Spin networks enter into all three of these, and the best studied example of such a quantum field theory, is Chern-Simons theory, which is based on a 3 form derived from a gauge field A ( i.e. A is a connection one form for a gauge group G)..We will now look in more detail at some of these ideas.
!!!BF Theory: Classical Field
Equations
To set up BF theory, we take as our gauge group any Lie group G whose Lie algebra
g is equipped with an invariant nondegenerate bilinear form h·, ·i.
We take as our spacetime any n-dimensional oriented smooth manifold M, and choose
a principal G-bundle P over M. The basic fields in the theory are then:
• a connection A on P,
• an ad(P)-valued (n - 2)-form E on M.
Here ad(P) is the vector bundle associated to P via the adjoint action of G
on its Lie algebra. The curvature of A is an ad(P)-valued 2-form F on M. If
we pick a local trivialization we can think of A as a g-valued 1-form on M,
F as a g-valued 2-form, and E as a g-valued (n - 2)-form. The Lagrangian for
BF theory is:
L = tr(E . F).
Here tr(E.F) is the n-form constructed by taking the wedge product of the di
erential form parts of E and F and using the bilinear form h·, ·i
to pair their g-valued parts. The notation ‘tr’ refers to the fact
that when G is semisimple we can take this bilinear form to be the Killing form
hx, yi = tr(xy), where the trace is taken in the adjoint representation. We
obtain the field equations by setting the variation of the action to zero:
0 = d ZM L = ZM
tr(dE . F + E . dF) = ZM
tr(dE . F + E . dAdA) = ZM
tr(dE . F + (-1)n-1dAE . dA)
where dA stands for the exterior covariant derivative. Here in the second step
we used the identity dF = dAdA, while in the final step we did an integration
by parts. We see that the variation of the action vanishes for all dE and dA
if and only if the following field equations hold:
F = 0, dAE = 0.
So, we have a connection A (a gauge field) on a G bundle P over a manifold M and from this we can obtain a curvature F (a 2-form) given by F= dA + A^A. This curvature F represents the strength of the field in the same way that it represents the strength of a gravitational field in GR, (it can be easily shown that the effect of a gauge transformation 'g' on F changes it to gFg^-1, in other words when we do a gauge transformation the holonomy gets conjugated by a group element) . We can represent the electromagnetic field using the theory of forms (pfaffians) and the second order tensor Fuv (a 4D matrix,where 6 elements represent the -+electric field and 6 the -+magnetic field components). Maxwell's equations are then given by dF = 0; and d*F = J (current density), where * is the Hodge operator and d is the exterior derivative operator. [Also note that F = 1/2Fuvdxu^dxv =E^dt + B ]. Now if F is a 2-form, then *F must be a (n-2)-form, hence F^*F is an n form, which turns out to be what we require for the (gauge invariant) Lagrangian. If M is space-time this is called Yang-Mills theory and in general we do need a metric space in order to define the Hodge operator'*', however we can avoid this requirement by employing an adjoint representation if permitted as in the case of BF theory.[ An Ad(P)-valued p-form is a locally g-valued p-form which g acts on by conjugation.For example F is an Ad(P)-valued 2-form, because its a locally g-valued p-form which g maps to gFg^-1, In other words the group element g acts on the (2-form) connection F (and also the gauge A) by means of conjugation.].
So with or without a metric, if we have an Ad(P)-valued (n-2) form E, then we get a gauge invariant Lagrangian with L = tr(E^F) - . voila BE or rather BF theory! In general, we can also replace *F with any (n-2)-form, (say B) so long as it acts in the right direction under the action of 'g' and we thus obtain a gauge invariant Lagrangian L =tr(B^F). The properties of F that we needed in order to prove that the Yang–Mills Lagrangian is gauge-invariant are that: 1) it is an (n - 2)-form so L can be integrated over an n-dimensional manifold; and 2) it is Ad(P) valued, so it has the proper behaviour under gauge transformations. Accordingly, we will assume that the manifold M is equipped not only with a connection, but also with a new field E which is an Ad(P)-valued (n - 2)-form. Now we can form the lagrangian Tr(E ^ F), which gives rise to the ill-named “BF theory”.If n = 3 and G is respectively SO(3) or SO(2,1) then this 'BF theory' is respectively Riemannian of Lorentzian 3-D GR. For 4-D, then F^F , B^F and even B^B are all 4-forms, so the Lagrangian L can be built from linear combinations of all of these [Also by adding Lambda B^B^B inside the trace, we effectively add a cosmological constant to the 3-D GR]. In order to produce 4-D Riemannian gravity, we have to put a metric on R-4 in order to define the Lie algebra so(4) and we need to introduce a so called 'soldering form' 'e', which is the related to vierbein tetrad frame of the spinor field that is dragged along by the curved metric (note that Cartan has demonstrated that pure Riemannian techniques cannot be applied to Dirac's spinor equation etc.). The Lagrangian is then L = tr(e^e^F), or if we require a cosmological constant, we can add Lambda tr(e^e^e^e). Note that e is an R-4 valued form not a g-valued form. More relevant, is the Lorentzian 4-D gravity in which our basic fields are an so(3,1) connection A and an R-4 valued 1-form e. We can therefore see the correspondence between this approach and BF theory if we associate e^e with E. More precisely, given any solution of the EF equations for which B is of the form e^e, we get a solution of the equations of GR. Unfortunately we only get a special class of solution this way but it is still quite interesting and this relation is the key to the spin foam models.
Classical Phase Space
To determine the classical phase space of BF theory we assume spacetime has
the form M = R × S where the real line R represents time and S is an oriented
smooth (n - 1)-dimensional manifold representing space. This is no real loss
of generality, since any oriented hypersurface in any oriented n-dimensional
manifold has a neighborhood of this form. We can thus use the results of canonical
quantization to study the dynamics of BF theory on quite general spacetimes.
If we work in temporal gauge, where the time component of the connection A vanishes,
we see the momentum canonically conjugate to A is
.L . . A = E.
This is reminiscent of the situation in electromagnetism, where the electric
field is canonically conjugate to the vector potential. This is why we use the
notation ‘E’. Originally people used the notation ‘B’ for
this field, hence the term ‘BF theory’, which has subsequently become
ingrained. But to understand the physical meaning of the theory, it is better
to call this field ‘E’ and think of it as analogous to the electric
field. Of course, the analogy is best when G = U(1). Let P|S be the restriction
of the bundle P to the ‘time-zero’ slice {0}×S, which we identify
with S. Before we take into account the constraints imposed by the field equations,
the configuration space of BF theory is the space A of connections on P|S. The
corresponding classical phase space, which we call the ‘kinematical phase
space’, is the cotangent bundle T *A. A point in this phase space consists
of a connection A on P|S and an ad(P|S)-valued (n-2)-form E on S. The symplectic
structure on this phase space is given by
.((dA, dE), (dA' , dE')) = ZS tr(dA . dE' - dA' . dE).
This reflects the fact that A and E are canonically conjugate variables. However,
the field equations of BF theory put constraints on the initial data A and E:
B = 0, dAE = 0 where B is the curvature of the connection A . A, analogous to
the magnetic field in electromagnetism. To deal with these constraints, we should
apply symplectic reduction to T *A to obtain the physical phase space. The constraint
dAE = 0, called the Gauss law, is analogous to the equation in vacuum electromagnetism
saying that the divergence of the electric field vanishes. This constraint generates
the action of gauge transformations on T *A. Doing symplectic reduction with
respect to this constraint, we thus obtain the ‘gauge-invariant phase space’
T *(A/G), where G is the group of gauge transformations of the bundle P|S.
The constraint B = 0 is analogous to an equation requiring the magnetic field
to vanish. Of course, no such equation exists in electromagnetism; this constraint
is special to BF theory. It generates transformations of the form A 7. A, E
7. E + dA.,
so these transformations, discussed in the previous section, really are gauge
symmetries as claimed.
Doing symplectic reduction with respect to this constraint, we obtain the ‘physical
phase space’ T *(A0/G), where A0 is the space of flat connections on P|S.
Points in this phase space correspond to physical states of classical BF theory.
* * * * *
Traditionally it had been difficult to realize this hope with
any degree of rigor because the spacesA and A/G are typically infinite-dimensional,
making it difficult to define L2(A) and L2(A/G). The great achievement of loop
quantum gravity is that it gives rigorous and background-free, hence diffeomorphism-invariant,
definitions of these Hilbert spaces. It does so by breaking away from the traditional
Fock space formalism and taking holonomies along paths as the basic variables
to be quantized. The result is a picture in which the basic excitations are
not 0-dimensional particles but 1-dimensional ‘spin network edges’.
As we shall see, this eventually leads us to a picture in which 1-dimensional
Feynman diagrams are replaced by 2-dimensional ‘spin foams’.
In what follows we shall assume that the gauge group G is compact and connected
and the manifold S representing space is real-analytic. The case where S merely
smooth is considerably more complicated, but people know how to handle it. The
case where G is not connected would only require some slight modifications in
our formalism. However, nobody really knows how to handle the case where G is
noncompact! This is why, when we apply our results to quantum gravity, we consider
the quantization of the vacuum Einstein equations for Riemannian rather than
Lorentzian metrics: SO(n) is compact but SO(n, 1) is not. The Lorentzian case
is just beginning to receive the serious study that it deserves. To define L2(A),
we start with the algebra Fun(A) consisting of all functions on A of the form
(A) = f(T eR1A, . . . , T eRnA).
Here ãi is a real-analytic path in S, T eR i
A is the holonomy of A along this path, and f is a continuous complex-valued
function of finitely many such holonomies. Then we define an inner product on
Fun(A) and complete it to obtain the Hilbert space L2(A). To define this inner
product,
we need to think about graphs embedded in space:
Definition 1. A finite collection of real-analytic paths .i: [0,
1] . S form a graph in S if they are embedded and intersect, it at all, only
at their endpoints. We then call them edges and call their endpoints vertices.
Given a vertex v, we say an edge .i is outgoing from v if .i(0) = v, and we
say .i is incoming to v if .i(1) = v. Suppose we fix a collection of paths .1,
. . . , .n that form a graph in S. We can think of the holonomies along these
paths as elements of G. Using this idea one can show that the functions of the
form
(A) = f(T eR 1
A, . . . , T eR n A)
for these particular paths .i form a subalgebra of Fun(A) that is isomorphic
to the algebra of all continuous complex-valued functions on Gn. Given two functions
in this subalgebra, we can thus define their inner product by h ,i = ZGn
where the integral is done using normalized Haar measure on Gn. Moreover, given
any functions , . Fun(A) there is always some subalgebra of this form that
contains them. Thus we can always define their inner product this way. Of course
we have to check that this definition is independent of the choices involved,
but this is not too hard. Completing the space Fun(A) in the norm associated
to this inner product, we obtain the ‘kinematical Hilbert space’ L2(A).
Similarly, we may define Fun(A/G) to be the space consisting of all functions
in Fun(A) that are invariant under gauge transformations, and complete it in
the above norm to obtain the ‘gaugeinvariant Hilbert space’ L2(A/G).
This space can be described in a very concrete way: it is spanned
by ‘spin network states’.
Definition 2. A spin network in S is a triple = (., ., .) consisting of:
1. a graph . in S,
2. for each edge e of ., an irreducible representation .e of G,
3. for each vertex v of ., an intertwining operator
.v: .e1 . · · · . .en . .e'1 . .e' m
where e1, . . . , en are the edges incoming to v and e'
1, . . . e'
m are the edges outgoing from v.
In what follows we call an intertwining operator an intertwiner.
There is an easy way to get a function in Fun(A/G) from a spin network in S.
To explain how it works, it is easiest to give an example. Suppose we have a
spin network in S with three edges e1, e2, e3 and two vertices v1, v2 as follows:
We draw arrows on the edges to indicate their orientation, and
write little letters near the beginningand end of each edge. Then for any connection
A . A we define
(A) = .e1 (T eRe1A)ab .e2 (T eRe2 A)cd .e3 (T eRe3A)ef (.v1 )ace (.v2 )bdf
In other words, we take the holonomy along each edge of , think of it as a group
element, and put it into the representation labelling that edge. Picking a basis
for this representation we think of the result as a matrix with one superscript
and one subscript. We use the little letter near the beginning of the edge for
the superscript and the little letter near the end of the edge for thesubscript.
In addition, we write the intertwining operator for each vertex as a tensor.
This tensor has one superscript for each edge incoming to the vertex and one
subscript for each edge outgoing from the vertex. Note that this recipe ensures
that each letter appears once as a superscript and once as a subscript! Finally,
using the Einstein summation convention we sum over all repeatedindices and
get a number, which of course depends on the connection A. This is (A). Since
:A . C is a continuous function of finitely many holonomies, it lies in Fun(A).
Using the fact that the .v are intertwiners, one can show that this function
is gauge-invariant. We thus have . Fun(A/G). We call a ‘spin network state’.
The only hard part is to prove that spin network states span L2(A/G).
General overview of the methods of Quantum Loop Gravity
Einstein and Schrodinger had recast general relativity as a theory
of connections already in the fifties However, they used the ‘Levi-Civita
connections’ that features in the parallel transport of vectors and found
that the theory becomes rather complicated. This episode had been forgotten
and connections were re-introduced in the mid-eighties. However, now these
were ‘spin-connections’, required to parallel propagate spinors,
and they turn out to simplify Einstein’s equations considerably. For example,
with the dynamical evolution dictated by Einstein’s equations can now be
visualized simply as a geodesic motion on the space of spin-connections (with
respect to a natural metric extracted from the constraint equations). Since
general relativity is now regarded as a dynamical theory of connections, this
reincarnation of the canonical approach is called ‘connection- dynamics’.
Perhaps the most important advantage of the passage from metrics to connections
is
that the phase-space of general relativity is now the same as that of gauge
theories. Although loop quantum gravity does not provide a natural unification
of dynamics of all interactions, this program does provide a kinematical unifcation.
[At a kinematical level, there is already an unification because the quantum
configuration space of general relativity is the same as in gauge theories which
govern the strong and electro-weak interactions. But the non-trivial issue is
that of dynamics.] More precisely, in this approach one begins by formulating
general relativity in the mathematical language of connections, the basic variables
of gauge theories of electro-weak and strong interactions. Thus, now the configuration
variables are not metrics as in Wheeler’s geometrodynamics, but certain
spin-connections; the emphasis is shifted from distances and geodesics to holonomies
and Wilson loops. Consequently, the basic kinematical structures are the same
as those used in gauge theories. A key difference, however, is that while a
background space-time metric is available and crucially used in gauge theories,
there are no background fields whatsoever now. Their absence is forced upon
us by the requirement of diffeomorphism invariance (or ‘general covariance’
).
Let us now turn to specifics. The basic gravitational configuration variable is an SU(2)-connection, Ai a on a 3-manifold M representing ‘space’. As in gauge theories,the momenta are the ‘electric fields’ Eai .10 However, in the present gravitational context, they acquire an additional meaning: they can be naturally interpreted as orthonormal triads (with density weight 1) and determine the dynamical, Riemannian geometry of M. Thus, in contrast to Wheeler’s geometrodynamics, the Riemannian structures, including the positive- definite metric on M, is now built from momentum variables. The basic kinematic objects are: i) holonomies he(A) of Ai a, which dictate how spinors are parallel transported along curves or edges e; and ii) .uxes ES,t = RS ti Eai d2Sa of electric .elds, Eai , smeared with test fields ti on a 2-surface S. The holonomies —the raison d’ˆetre of connections— serve as the ‘elementary’ configuration variables which are to have unambiguous quantum analogs. They form an Abelian C. algebra, denoted by Cyl. Similarly, the fluxes serve as ‘elementary momentum variables’. Their Poisson brackets with holonomies de.ne a derivation on Cyl. In this sense —as in Hamiltonian mechanics on manifolds—momenta are associated with ‘vector .elds’ on the configuration space. The first step in quantization is to use the Poisson algebra between these configuration and momentum functions to construct an abstract *.algebra A of elementary quantum operators. This step is straightforward. The second step is to introduce a representation of this algebra by ‘concrete’ operators on a Hilbert space (which is to serve as the kinematic setup for the Dirac quantization program).
We now describe the salient features of this representation. Quantum states span a specfic Hilbert space H consisting of wave functions of connections which are square integrable with respect to a natural, diffeomorphism invariant measure. This space is very large. However, it can be conveniently decomposed into a family of orthogonal, finite dimensional sub-spaces H = .,~j H.,~j , labelled by graphs ., each edge of which itself is labelled by a spin (i.e., half-integer) j (The vector ~j stands for the collection of half-integers associated with all edges of ..) One can think of . as a ‘floating lattice’ in M —‘floating’ because its edges are arbitrary, rather than ‘rectangular’. (Indeed, since there is no background metric on M, a rectangular lattice has no invariant meaning.) Mathematically, H.,~j can be regarded as the Hilbert space of a spin-system. These spaces are extremely simple to work with; this is why very explicit calculations are feasible. Elements of H.,~j are referred to as spin-network state. In the quantum theory, the fundamental excitations of geometry are most conveniently expressed in terms of holonomies . They are thus one-dimensional, polymer-like and, in analogy with gauge theories, can be thought of as ‘flux lines’ of electric fields/triads. More precisely, they turn out to be flux lines of area, the simplest gauge invariant quantities constructed from the momenta Eai : an elementary .flux line deposits a quantum of area on any 2-surface S it intersects. Thus, if quantum geometry were to be excited along just a few flux lines, most surfaces would have zero area and the quantum state would not at all resemble a classical geometry. This state would be analogous, in Maxwell theory, to a ‘genuinely quantum mechanical state’ with just a few photons. In the Maxwell case, one must superpose photons coherently to obtain a semi-classical state that can be approximated by a classical electromagnetic field. Similarly, here, semi-classical geometries can result only if a huge number of these elementary excitations are superposed in suitable dense configurations. The state of quantum geometry around you, for example, must have so many elementary excitations that approximately 10^68 of them intersect the sheet of paper you are reading. Even in such states, the geometry is still distributional, concentrated on the underlying elementary flux lines. But if suitably coarse-grained, it can be approximated by a smooth metric. Thus, the continuum picture is only an approximation that arises from coarse braining of semi-classical states. The basic quantum operators are the holonomies ˆhe along curves or edges e in M and the fluxes ˆ ES,t of triads ˆ Eai . Both are densely defined and self-adjoint on H.
For, diffeomorphism invariance constrains the possible forms of
the final expressions severely and the detailed calculations then serve essentially
to fix numerical coefficients and other details. Let me illustrate this point
with the example of the area operators ˆ AS. Since they are associated
with 2-surfaces S while the states are 1-dimensional excitations, the diffeomorphism
covariance requires that the action of ˆ AS on a state ..,~j must be concentrated
at the intersections of S with .. The detailed expression bears out this expectation:
the action of ˆ AS on ..,~j is dictated simply by the spin labels jI attached
to those edges of. which intersect S. For all surfaces S and 3-dimensional regions
R in M, ˆ AS and ˆ VR are densely de.ned, self-adjoint operators.
All their eigenvalues are discrete. Naively, one might expect that the eigenvalues
would be uniformly spaced given by, e.g., integral multiples of the Planck area
or volume. Indeed, for area, such assumptions were routinely made in the
initial investigations of the origin of black hole entropy and, for volume,
they are made in quantum gravity approaches based on causal sets where discreteness
is postulated at the outset. In quantum Riemannian geometry, this expectation
is not borne out; the distribution of eigenvalues is quite subtle. In particular,
the eigenvalues crowd rapidly as areas and volumes increase. In the case of
area operators, the complete spectrum is known in a closed form, and the .rst
several hundred eigenvalues have been explicitly computed numerically. For a
large eigenvalue an, the separation an = an+1 - an between consecutive eigenvalues
decreases exponentially: an exp-(pan/lPl) l2 Pl! Because of such strong crowding,
the continuum approximation becomes excellent quite rapidly just a few orders
of magnitude above the Planck scale. At the Planck scale, however, there is
a precise and very specific replacement. This is the arena of quantum geometry.
The premise is that the standard perturbation theory fails because it ignores
this fundamental discreteness. There is however a further subtlety. This non-perturbative
quantization has a one parameter family of ambiguities labelled by . > 0.
This gamma is called the Barbero-Immirzi parameter and is rather similar to
the well-known .theta -parameter of QCD. In QCD, a single classical theory gives
rise to inequivalent sectors of quantum theory, labelled by .. Similarly, .
is classically irrelevant but di.erent values of . correspond to unitarily inequivalent
representations of the algebra of geometric operators. The overall mathematical
structure of all these sectors is very similar; the only di.erence is that the
eigenvalues of all geometric operators scale with .. For example, the simplest
eigenvalues of the area operator ˆ AS in the . quantum sector is given
by;
a{j} = 8p.l2Pl X pjI (jI + 1)
where {j} is a collection of 1/2-integers jI , with I = 1, . . .N for some N. Since the representations are unitarily inequivalent, as usual, one must rely on Nature to resolve this ambiguity: Just as Nature must select a specific value of theta in QCD, it must select a specific value of gamma in loop quantum gravity. With one judicious experiment —e.g., measurement of the lowest eigenvalue of the area operator ˆ AS for a 2-surface S of any given topology— we could determine the value of . and .x the theory. Unfortunately, such experiments are hard to perform! However, the Bekenstein-Hawking formula of black hole entropy provides an indirect measurement of this lowest eigenvalue of area for the 2-sphere topology and can therefore be used to fix the value of gamma at ~ 0.2735
Auxillary Details on Quantum Gravity
In the case of 2-D theory of gravity it turns out that the trace F is provides the suitable Lagrangian if the Lie group is SO(2) and this is known as the "1 st Chern form" theory. Likewise we can develop a "2 nd Chern Form theory" by using Tr(F^F) as the Lagrangian (from which we obtain the action S by integrating over M); the gauge group G = SO(3,1) but unfortunately this does not give us GR and also like 2-D gravity, does not have local degrees of freedom but does have some interesting topological degrees of freedom. The Einstein equation is a second rank tensor and the analogue of the position and momentum are the fields qij and p^ij. With new variables such as Gauge and gravitational curvature forms, the analogues of position and momentum are instead the connection A and the 2 form gravitational field (e.g. the 'soldering form' relating to the spinor vierbein). One of the most useful action that has been investigated so far, involves the Chern-Simons Lagrangian mentioned above, which is based on the 3- form L = tr(A^dA) + 2/3tr(A^A^A), with which we can define isotopic invariants of spin networks. We therefore have BF theory in which £=tr(B^F) which yield the equations of motion F=0 and d_AE=0, alternatively we have Yang-Mills theory in which £= tr(F^*F) and the equation is d_*F=0 and of course the most studied of all, the Chern-Simon theory.
Heuristically, we can study 2-D manifolds, in order to gain an insight into a possible TQFT for gravity and it is found that if we consider a cylindrical space time R x S1 (in which R is the configuration space) Maxwell's equation d*F=0 produce an electric field but no magnetic components. [More generally, in D dimensions, the electric field has d-1 components while the magnetic field has (d-1)(d-1)(d-2)/2 components, hence there are no components at all] We find that Maxwell's equation on cylindrical space are analogous to the equations of a point particle on a line viz.
position q ~ holonomy around a circle a
momentum p ~ electric field e (where e = dtAx-dxAx)
mass m ~ 1/L (where L is the circumference f the cylinder)
First of all, note that to get going, we foliated space-time into space and time. The connection on space, the A field, is the analogous to position, while the E field is analogous to momentum. This technique is applied often in lots of gauge theories: Yang-Mills theory, BF theory and various formulations of general relativity. Since the A field is analogous to position, one might think the configuration space in these gauge theories is the space of A fields -- i.e., connections on space. The idea here is that we're really interested in connections mod gauge transformations (hence the circle S1), so the configuration, space is really a space of equivalence classes: connections modulo gauge transformations. When we quantize these gauge theories, quantum states will be wavefunctions on the configuration space. Observables involving the A field get described as multiplication operators, while observables involving the E field get described as differentiation operators.
To do get anywhere in this game, we've got to understand the configuration
space: the space of connections mod gauge
transformations. In the theory we just considered, space was a circle. That
makes things very simple! To describe a connection mod gauge transformations
on the circle, all we need is its holonomy around the circle --i.e., an element
of the gauge group. If the gauge group is abelian, this is exactly right. That's
why our configuration space was R just now: space was a circle, and our gauge
group was R! If the gauge group G is nonabelian, the holonomy around a loop
isn't gauge invariant, so things get a bit more complicated. Even when space
is not a circle, it's good to describe connections mod gauge transformations
using their holonomies around loops. This is the idea behind the "loop
representation" of quantum gravity.
The point particle on the line and 2d vacuum Maxwell theory on a cylinder
are isomorphic classical systems. Since
we've already quantized the first system, we can just transfer our results over
to the second one. So let's do it. We just copy
everything we said last week, making the substitutions
q -> a
p -> e
m -> 1/L
The configuration space of the classical vacuum Maxwell equations on the cylinder
is R: the space of all values of a. So when
we quantize, we get the Hilbert space L2(R).
Besides our Hilbert space, we get various observables, which are self-adjoint
operators on this Hilbert space. The most
important observables are a and e. They work just like the position and momentum
operators for the quantum particle on the
line:
(a psi)(x) = x psi(x)
(e psi)(x) = -i psi'(x)
From these we can build fancier operators, like the Hamiltonian:
H = Le2/2
To get anywhere in this game, we've got to understand the configuration space:
the space of connections mod gauge
transformations. How did we do that just now? In the theory we just considered,
space was a circle. That makes things very
simple! To describe a connection mod gauge transformations on the circle, all
we need is its holonomy around the circle --
i.e., an element of the gauge group. If the gauge group is abelian, this is
exactly right. That's why our configuration space was
R just now; space was a circle, and our gauge group was R. If the gauge group
G is nonabelian, the holonomy around a loop
isn't gauge invariant, so things get a bit more complicated. Even when space
is not a circle, it's good to describe connections mod gauge transformations
using their holonomies around loops. This is the idea behind the "loop
representation" of quantum gravity. Effectively we are using R as the configuration
space and this gives rise to a continuous spectrum of field lines and energy,
however if we use a configuration space U(1) instead the Hilbert space L2(U(1))
yields integral electric fields/energy. [In LQG where analoges of E determine
the metric, these become quantised flux lines of area -- when they poke through
the surface they give discrete values of area] So classically R and U(1) are
locally equivalent but when quantised they are different locally, R being continuous
while U(1) is discrete.
We started this quarter by thinking about Maxwell's equations on the cylinder R x S1. We saw that in this case, the classical configuration space was just U(1). So the classical configuration space for Yang-Mills theory with some compact gauge group G will also be G. When we quantize the vacuum Maxwell theory on a cylindrical spacetime, we were treating electromagnetism as Yang-Mills theory with gauge group \R. But sometimes people think of electromagnetism as Yang-Mills with some other gauge group, namely U(1) - - - the group of unit complex numbers, or phases! this change is a bit subtle but we just need to take everything we did and modify it a tiny bit, replacing the real numbers by U(1) in all the right places. Now, just as the group R is a line, the group U(1) is a circle. When we did electromagnetism on a cylinder with gauge group R, it was isomorphic to a free point particle on the line. So when we do electromagnetism on a cylinder with gauge group U(1), we'll find that its isomorphic to a free point particle on the circle. As before, we take spacetime to be R x S1, with the obvious Lorentzian metric for which the circle has circumference equal to L. Next we put a trivial U(1) bundle on our spacetime, let A be a connection on this bundle, and let F be its curvature. We can still regard A as a 1-form and F as a 2-form. The same formula applies:
F = dA.
The vacuum Maxwell equations look just the same, too:
d*F = 0.
The difference is that now we treat A as a U(1) connection instead of an R
connection. For example, the holonomy along a
path is now an element of U(1) instead of R. If we have some curve C, the holonomy
along it used to be a real number,
namely
\int-c A.
But now it's a unit complex number, namely
exp(i \int-c A).
Another way to put it is that now we only care about \intC A modulo 2pi.
The space of connections on S1, modulo gauge transformations, will be the configuration space of our theory. Last time we saw that when the gauge group is R, this configuration space is R. Now that the gauge group is U(1), the configuration space will be.. . . U(1). The holonomy of a connection around a loop is gauge-invariant when the gauge group is abelian, so we can take the holonomy of our connection around the circle, and we an element of U(1). This element can be anything... so at least there's a map from our configuration space onto U(1). So the configuration space is now U(1) instead of R. This changes things only slightly. Our phase space is now the cotangent bundle of the circle... or in other words, the space of pairs (a,e) with a in U(1) and e in R. a is the holonomy of our connection around the circle, while e is the electric field. As before, Maxwell's equations say e is constant, while a marches along at constant velocity around the circle.
The reason we have U(1) for the configuration space of Maxwell's equations is because the only gauge-invariant function of a U(1) connection on the circle, was its holonomy all around the circle - an element of U(1). More precisely, all other gauge-invariant functions can be written as functions of this holonomy. The same logic works for any other gauge group, but now the holonomy around the circle is an element of a more general gauge group G. However if the gauge group is nonabelian, the holonomy of a connection around a loop isn't gauge-invariant.When we do a gauge transformation, the holonomy gets conjugated by a group element. The configuration space really consists of equivalence classes of elements of G, where two elements g and g' are deemed equivalent if they're conjugate:
g' = hgh-1
for some h in G. Fancy mathematicians call this action of G on itself by conjugation
the "adjoint action", so they call this space
of equivalence classes G/AdG. The configuration space of Yang-Mills theory
on the cylinder R x S1 is not really G, it's
G/AdG. Upon quantization, our Hilbert space is L2 of the classical configuration
space, so we should get L2(G/AdG).
[NB. The curvature F of the connection A, which physicists call “field
strength”. Locally, F is a g-valued 2-form, and it is a function of A:
F = d2 A = dA + A ^ A on the fundamental representation
dA + 1/2 [A,A] on the adjoint representation
When we apply a gauge transformation to A, the curvature changes as F--- F'
= gFg-1. We therefore say that F is Ad(P)-valued].
In LQG, we need to replace SU(2) by the corresponding quantum group. This has only finitely many relevant irreducible representations, corresponding to spins j = 0, 1/2, 1, ... on up to k/2 for some integer k. Then the sum over ways of labelling edges becomes a finite sum, and everything works fine. When we sum over labellings of the edges of a triangulated 3-manifold by spins, we're doing a discretized version of this path integral. The spins labelling the edges correspond to their lengths, so we are really summing over all possible geometries of spacetime. When we do quantum gravity with cosmological constant, there's a cutoff k on the spins we need to sum over -- a maximum length! That's how the infrared cutoff shows up in this discretized theory.
There are interesting connectionsbetween topological quantum field theory and loop quantum gravity. In loop quantum gravity, the preliminary Hilbert space has a basis given by `spin networks' -- roughly speaking, graphs with edges labelled by spins . We now understand quite well how a spin network describes a quantum state of the geometry of space. But spin networks are also used to describe states in TQFTs, where they arise naturally from considerations of higher-dimensional algebra. Using the relationships between 4-dimensional quantum gravity and topological quantum field theory, researchers have begun to formulate theories in which the quantum geometry of spacetime is described using `spin foams' -- roughly speaking, 2-dimensional structures made of polygons joined at their edges, with all the polygons being labelled by spins The most important part of a spin foam model is a recipe assigning an amplitude to each spin foam. Much as Feynman diagrams in ordinary quantum field theory describe processes by which one collection of particles evolves into another, spin foams describe processes by which one spin network evolves into another. Indeed, there is a category whose objects are spin networks and whose morphisms are spin foams! Like nCob, this category appears to arise very naturally from purely n-categorical considerations.
Now in general relativity spacetime becomes dynamical, so it's natural to wonder whether not only the geometry of space, but also its topology, can change with the passage of time -- and what would happen if it did! In general relativity the basic concepts are SPACE, which is an (n-1)-dimensional manifold, and also SPACETIME, which is an n-dimensional manifold going from one (n-1)-manifold to another. Technically we call M a "cobordism": an n-manifold with boundary whose boundary is the disjoint union of manifolds S and S'. We write M: S -> S' to indicate that M is a spacetime going from the space S to the space S'. Over in quantum theory, we have analogous concepts of HILBERT SPACE and OPERATOR. We describe states of a system, i.e. the different ways it can be, by unit vectors in a Hilbert space, which we call "state vectors" and we describe processes, i.e. the ways the system can change, by linear operators from one Hilbert space to another.
Analogy between general relativity and quantum theory
GENERAL RELATIVITY . . . . . . . . . . . . . . . . .. . . . . .QUANTUM THEORY
. . . .(n-1)dimensional manifold . . . . . . . . . . . . . . . . . . . . .Hilbert
space
. . . . . .(space) . . . . . . . . . . . . . .. . . . . . . . . .. . . . .
.. . . . . (states)
obordism between (n-1)-dimensional manifolds . . . . . . . . . .operator between
Hilbert spaces (process)
..(spacetime) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . Process
. . .composition of cobordisms . . . . . . . . . . . . . . . . . . . . .. .
composition of operators
. . .identity cobordism . . . . . . . . . . . . . . . . . . . .. . . . . . .
. ...identity operator
Just as sets have elements, categories have objects. Just as there are functions
between sets, there are functors between categories. Interestingly, the proper
analog of an equation between elements is not an equation between objects,
but an isomorphism. More generally, the analog of an equation between functions
is a natural isomorphism between functors. In a set, two elements are either
the same or different. In a category, two objects can be `the same in a way'
while still being different. In other words, they can be isomorphic but not
equal. Even more importantly, two objects can be the 'same' in more than one
way, since there can be different isomorphisms between them. This gives rise
to the notion of the `symmetry group'
of an object: its group of automorphisms.
: Categorification (N-dimensional algebra), is necessary for understanding the connections between quantum field theory and topology. To blend general relativity and quantum theory we need to understand how geometry and algebra are two sides of the same coin and TQFT allow us to do this. We can make this more precise using a little category theory (otherwise known as n-dimensional algebra). A category C is a gadget consisting of:
1. a collection of "objects".
2. for any pair of objects x and y, a set hom(x,y) of "morphisms"
from x to y. If f is a morphism from x to y, we
write f: x -> y. We call x the "source" of f, and call y the "target"
of f.
3. for any object x, a morphism 1x: x -> x called the "identity of x".
4. for any pair of objects x and y, a function called "composition",
taking a morphism f: x -> y and a morphism g:
y -> z to a morphism gf: x -> y.
such that these laws hold:
5. the "left and right unit laws": for any morphism f: x -> y we have 1_y f = f = f 1x.
6. the "associative law": for any morphisms f: x -> y, g: y -> z and h: x -> z we have (hg)f = h(gf).
In general, the objects of a category represent ways things can "be", while the morphisms represent ways they can "become". BEING and BECOMING: we put them on an equal footing in category theory, unlike traditional set theory, which emphasizes being and tries to make everything into sets - even functions! Now much as Feynman diagrams describe processes in which one collection of particles evolves into another, spin foams describe processes by which one spin network evolve into another. indeed there is a category whose objects are spin networks and whose morphisms are spin foams. the concept of 'space' and 'state' are completely merged into the notion of 'spin network' and likewise the concept of 'spacetime' and 'process' are merged in the notion of 'spin foam'
Consider the following alogorical story; long ago,when shepherds wanted to
see if two herds of sheep were isomorphic, they would look for a specific isomorphism.
In other words, they would line up both herds and try to match each sheep in
one herd with a sheep in the other. But one day, a shepherd invented decategorification.
She realized one could take each herd and ‘count’ it, setting up an
isomorphism between it and a set of ‘numbers’, which were nonsense
words like ‘one, two, three, . . . ’specially designed for this purpose.
By comparing the resulting numbers, she could show that two herds were isomorphic
without explicitly establishing an isomorphism! In short, the set N of natural
numbers was created by decategorifying FinSet, the category whose objects are
finite sets and whose morphisms are functions between these. Category theory
is all about objects and morphisms. For the category of sets, this means SETS
and FUNCTIONS. Of course, the usual axioms for set theory are all about SETS
and MEMBERSHIP. Thus analyzing set theory from the category-theoretic viewpoint
forces a radical change of viewpoint, which downplays membership and emphasizes
functions. [Topos theory is a new way doing logic via the categorisation
of sets]
In quantum theory we use a category namely "Hilb", where the objects
are Hilbert spaces and the morphisms are linear maps. In general relativity
we use "nCob", where the objects are (n-1)-dimensional manifolds and
the morphisms are n-dimensional
cobordisms between these. This is a nice example where the morphisms aren't
functions! A TQFT is, among other things, a functor from nCob to Hilb,
where a functor is defined as follows
Given categories C and D, a functor F: C -> D consists of:
1. a map sending any object x in C to an object F(x) in D.
2. for any pair of objects x and y, a map sending morphisms f: x -> y to morphisms F(f): F(x) -> F(y).
such that these laws hold:
3. for any object x in C, F(1x) = 1F(x).
4. for any pair of morphisms f: x -> y and g: y -> z, F(gf) = F(g)F(f).
In short: F sends objects to objects, morphisms to morphisms, and preserves sources, targets, identities and composition. Thus a category with one object, all of whose morphisms are invertible, is nothing but a group and a functor between categories of this sort is nothing but a homomorphism!
In QFT we use Feynmann diagrams to calculate inside the Category Rep(G) of a unitary representation of G, the symmetry group of our theory. Edges are labelled with unitary representations of G; 'the particles'. Vertices are labelled with the intertwining operators; 'the interactions. ' Typically G includes symmetries of spacetime and its unitary representations are described as spaces of solutions of linear partial differential equations on spacetime. In string theories we replace Feynmann diagrams with 2-D string world sheets, while in spin foam models of quantum gravity we replace Feynmann diagrams with spin foams. Both models hint at some form of 'categorification' of Feynmann diagrams but 'categorificationt' has been more developed more extensively for TQFT (such as LQG and spin foams)
********
Let's see what a 1-dimensional TQFT amounts to. It's a symmetric monoidal functor
Z: 1Cob -> Hilb
that is it sends the unit object of nCob to the unit object of Hilb.
If M is a compact oriented n-manifold, Z(M) is a complex number. We call this the partition function of M that is, " sum over states" -- which is what a partition function amounts to, in statistical mechanics .In quantum field theory, the partition function is something we calculate using a path integral: it's just the integral over all histories of the exponential of the action. All this category theory stuff is really just another way of talking about path integrals. We can therefore construct TQFTs using "state sum models", which are discretized path integrals.
Now consider a 2d TQFT:
Z: 2Cob -> Hilb.
Theorem: A 2d TQFT is a commutative Frobenius algebra.
Theorem: A 2d topological lattice field theory is a semisimple algebra.
Here are some truths about various dimension category theory. It's a fascinating fact about category theory that the hierarchy stabilizes at dimension 4: [Note that an n-dimensional TQFT is a symmetric monoidal functor from nCob to Hilb.]
Analogy between set theory and category theory
. . . . . . . .SET THEORY . . . . . . . . . .CATEGORY THEORY
. . . . . . . . elements . . . . . . . . . . . . . . . . . .objects
. . . . . . . .equations between elements . . . . isomorphisms between objects
. . . . . . . .sets . . . . . . . . . . . . . . . . . . . . . .categories
. . . . . . . .functions between sets . . . . . . . . functors between categories
. . . . . . . equations between functions . . . . natural isomorphisms between
functors
dimension 1 -- category
dimension 2 -- monoidal category
dimension 3 -- braided monoidal category
dimension 4 -- symmetric monoidal category
dimension 5 -- symmetric monoidal category
dimension 6 -- symmetric monoidal category
. .
Although we have classified TQFTs in 1 and 2 dimensions and the answers were
simple and beautiful however no comparably slick classification is known in
dimension 3. We can still construct a lot of TQFTs in dimension 3, but we can't
do it with our "bare hands" as we did in lower dimensions, starting
straight from the definitions. There are just too many objects and morphisms
in 3Cob for a brute-force approach to be fruitful but there are various tricks
for getting 3d TQFTs. One way is to build a "topological lattice field
theory". where we compute the partition function Z(M) of a closed spacetime
manifold M as follows. First we chop M up into tetrahedra (hence the relation
to spin networks). Then we label the edges somehow. Using these labellings we
compute a number for each tetrahedron, each triangle and each edge. Then we
multiply all these numbers... and finally we sum over labellings. For the theory
to be "topological", the result shouldn't depend on how we chopped
pacetime into tetrahedra.
We're building spacetimes from Feynman diagrams, (in which we relate triangulation to Feynman propagators via Poincare dualism). The metric g is what physicists would call a "propagator", while the tensor c is what they'd call a "cubic interaction". By building space-time from quantum processes we eliminate distinction between space-time and the processes that go on in space-time! In LQG, we need to replace SU(2) by the corresponding quantum group. This has only finitely many relevant irreducible representations, corresponding to spins j = 0, 1/2, 1, ... on up to k/2 for some integer k. Then the sum over ways of labelling edges becomes a finite sum, and everything works fine. When we sum over labellings of the edges of a triangulated 3-manifold by spins, we're doing a discretized version of this path integral. The spins labelling the edges correspond to their lengths, so we are really summing over all possible geometries of spacetime. When we do quantum gravity with cosmological constant, there's a cutoff k on the spins we need to sum over -- a maximum length! That's how the infrared cutoff shows up in this discretized theory.
Spin networks are combinations of representations of the Lie group SU(2) under
the tensor product in the category of representations of SU(2). Such a topological
state sum has many attractive features as a tool to describe a quantum theory
of
gravity. It occupies a position intermediate between a path integral for a continuum
theory and a lattice approximation to the theory, as a sort of magic lattice
theory which is invariant under any change of the lattice. This resonates nicely
with the old idea that it is not possible to measure the distance between two
physical points and get a value less than the Planck scale. Spin networks allow
us to carry out the analogy of Feynman diagrams but uses a combinatorial technique,
in which spin irreducible representations for input and output interacions can
be represented by certain rules that are independent of the triangulation (important
if a theory is to be relational) and these spin networks form a basis of spin
networks from one state to another (the Biedenharn-Elliot identity, 2-3 Pachner
move and the pentagon equation are equivalent ways of achieving this). By symmetry,
these also form a basis: The matrix for changing from one basis to the other
is called the "6j symbols", since it depends on 6 spins. Apart from
some puny fudge factors, the 6j symbols come from evaluating a tetrahedral spin
network -- sometimes called a "tet net". Even better, there's a deep
relation between the 6j symbols, the associator, and the tetrahedron! This is
why 3d quantum gravity works so nicely. In any quantum field theory there's
an important number called the "partition function", which depends
on the choice of spacetime. With suitable finagling, all physics can be extracted
from this. In 3d quantum gravity, to compute the partition function of a 3d
spacetime we first chop it into tetrahedra and label all the edges with spins.
This gives a bunch of tetrahedral spin networks! Then we evaluate each of these
tetrahedra using our rules and multiply all the numbers we get. Finally, we
sum over all labellings. Modulo some details I don't want to talk about yet,
this gives the partition function. This number is independent of how we chopped
spacetime into tetrahedra! Miraculously, the 6j symbols satisfy just the right
identities to make this work. In order to explain this, and explain why this
"miracle" is no accident. it is necessary to resort to category theory
to really understand the relation between 6j symbols, associators, and tetrahedra.
In spin networks we have a category where objects are representations (of a
gauge group) and the morphisms are intertwiners.
SU(2) REPRESENTATION THEORY . . . . . . . . . . . .TOPOLOGY . . . . . .. . . . . . . CATEGORY THEORY
Clebsch-Gordon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
coefficients triangle . . . . . . . . . . . . .tensor product
6j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . tetrahedron . . . . . . . . . . . . . . . . . . associator.
Biedenharn-Elliot identity . . . . . . . . . . . . . . . . . . . . . . . ..2-3
Pachner move . . . . . . . . . .. . ..pentagon equation
*****************************
The following ideas are relevant to Spin networks and
rely upon the well known fact (although difficult to prove), that any map can
be coloured by at most 4 colours.
Theorem: Any planar vector network without edge-loops evaluates to a nonzero integer.
Now, to four-color the faces of a graph like this, it turns out to be enough to three-color the edges. In other words, it's enough to label its edges with the letters i, j and k in such a way that no two edges labelled by the same letter meet at a vertex. Anyway: this means that the four-color theorem is really a theorem about vector networks! (the four colour problem was finally proved it in 1976, with the help of a computer calculation involving 1010 operations and taking 1200 hours). In short, the 4-color theorem is equivalent to this result:
Theorem: For any trivalent planar graph without edge-loops, there exists a way to label the edges by the letters i, j and k so that no two edges meeting at a vertex are labelled by the same letter.
Next, let's see why this theorem is equivalent to the
fact that any planar vector network without loops evaluates to a nonzero
integer. Think of i, j and k as the usual basis for C3, and imagine evaluating
a vector network by contracting a lot of tensors, one for each vertex. Since
the vertex of a vector network is just the cross product in disguise, the tensor
at each vertex is just
epsilon . . . . . . . .abc = 1 (a,b,c) = an even permutation
of (1,2,3) as wemarch counterclockwise around the vertex
. . . . . . . . . . . . . . . . . .-1 (a,b,c) = an odd permutation of (1,2,3)
as we march counterclockwise around the vertex
. . . . . . . . . . . . . . . . . . . . . . . . . =. 0 otherwise
The spin network is thus given by a sum over all ways
of labelling the edges by i, j, and k, where each term in this sum is a
product over all vertices of numbers 1, -1, or 0, computed using the above rule.
If there is no way to three-color the edges of our graph,
all the terms in this sum will vanish, so our network will evaluate to
zero. Thus the only thing left to check is the converse: if the network evaluates
to zero, all the terms in the sum must vanish --
so there are no ways to three-color its edges! In Penrose's paper on the subject
he omits this proof, saying there are lots of ways to do it, but none of them
very snappy.
Anyway: this means that the four-color theorem is really
a theorem about vector networks! We can also reformulate it purely
in terms of the vector cross product, as follows:
Theorem: Given two parenthesizations of the iterated cross product
v1 x v2 x ... x vn
there is a way of choosing each vector v1, ... ,vn from the set of basis vectors {i,j,k} so that both parenthesizations give the same nonzero result.
************************
MISCELLANEOUS NOTES
Now, doing general relativity with the cotetrad field e is very much like describing
4-simplices using vectors for edges, while doing general relativity with the
E field is very much like describing 4-simplices using bivectors for faces.
Suppose we have
a 4-simplex anely embedded in R4. We can number its vertices 0, 1, 2, 3, 4,
and translate it so that the vertex 0 is located at the origin. Then one way
to describe its geometry is by the positions e1, e2, e3, e4 of the other four
vertices. Another way is to
use the bivectors Eab = ea . eb . 2R46corresponding to the six triangular faces
with 0 as one of their vertices. However, not every collection of bivectors
Eab comes from a 4-simplex this way. In addition to the obvious skew-symmetry
Eab = -Eba, some extra constraints must hold. They have exactly the same form
as the extra constraints that we need to obtain general relativity from BF theory.
Barrett and Crane showed that these constraints take a particularly nice form
if we describe them in terms of bivectors corresponding to all ten triangular
faces of the 4-simplex. Quantizing these constraints, one obtains conditions
that a Spin(4) spin foam must satisfy to describe a quantum 4-geometry built
from flat quantum 4-simplices. Interestingly, these conditions also guarantee
that the quantum geometries for 3-dimensional submanifolds obtained using the
right-handed copy of SU(2) agree with those coming from the left-handed copy.
* * * * * *
The following clarifies the geometrical significance of this description of
L2(AS/GS): the space Inv(j0.j1.j2.j3) describes the states of a ‘quantum
tetrahedron’ whose ith face has area 12pji(ji + 1). First, note an important
consequence: the above equation describes L2(AS/GS) as the Hilbert space of
quantum 3-geometries of the simplicial complex . Labeling the triangles of
by spins fixes the areas of the triangles and determines a space of states
consistent with these areas for each tetrahedron in . To obtain L2(AS/GS) we
take the tensor product of all these spaces and then the direct sum over all
labelings.
To understand the quantum tetrahedron, start with a tetrahedron T having vertices
0, 1, 2, 3. We can think of a tetrahedron in R3 with one vertex at the origin
as an ane map from T to R3 sending the vertex 0 to the origin. Such a map is
given by fixing vectors e1, e2, e3 . R3 corresponding to the edges 01, 02, and
03. This data is precisely a ‘cotriad’, a linear map from the tangent
space of the vertex 0 of T to R3. In the Palatini approach to general relativity
one works with a similar object, namely the ‘cotriad field’, which
can be thought of locally as an R3-valued 1-form e. But a crucial idea in modern
canonical quantum gravity, is to work not with e but with the 2R3-valued 2-form
e . e, often called the ‘densitized cotriad field’. Using this rather
than the cotriad field is the reason why areas rather than lengths are the basic
geometrical observables in the spin network approach to quantum gravity. This
suggests an alternate description of the geometry and orientation of a tetrahedron
in terms of faces rather than edges.
In this alternate description, we work not with the vectors ei but with the
bivectors ei . ej . (A bivector in n dimensions is simply an element of 2Rn;
in the presence of a metric a bivector in three dimensions can be identified
with an ‘axial vector’, which
in turn can be identified with a vector in the presence of an orientation.)
We denote these bivectors by: E1 = e2 . e3, E2 = e3 . e1, E3 = e1 . e2.
Not every triple of bivectors Ei comes from a triple of vectors ei this way.
It is necessary and sucient that either the Ei are linearly independent and
satisfy
oIJKEI
1EJ
2 EK
3 > 0,
or that they are all multiples of a fixed bivector. The first case is the generic
one; it occurs whenever the vectors ei are linearly independent. In this case,
the Ei determine the ei up to a parity transformation ei 7. -ei. The second
case occurs when the
vectors ei lie in a plane; in this case the Ei determine the ei up to an area-preserving
(but possibly orientation-reversing) linear transformation of this plane. In
short, the map from triples ei to triples Ei is neither one-to-one nor onto.Barbieri
has suggested an interesting way around this, at least in the generic case.
This trick involves fixing an orientation on R3. If the linearly independent
triple ei is right-handed, we define the Ei as above, but if the triple ei is
left-handed, we define
the Ei with a minus sign:
E1 = -e2^e3, E2 = -e3^ e1, E3 = -e1^ e2,
* * * * *
The 3-dimensional space is triangulated by tetrahedra. Again the dual spin network is the 1-skeleton of the tetrahedra, so that there is one node for each tetrahedron and one spin network edge puncturing each of its faces This means that we now allow only four-valent spin networks. The tetrahedra are labeled by intertwiners and their faces by spins It is most intriguing that one can now propose a straightforward correspondence between spins puncturing faces of tetrahedra and the area of those faces, or the intertwiners labeling the tetrahedra and their volume, like the standard spin network results on area and volume.
In a BF theory, the B field is an so(4)-valued 2-form, which can be integrated
on a triangle to give an element of so(4), the dual of the Lie algebra so(4).
However, in Einstein gravity it is important that so(4) is identified with
R4 ^ R4, as B is constrained to be of the form B = e^e, where e is a vector
valued one-form. In this RELATIVISTIC SPIN NETWORKS AND QUANTUM GRAVITY 3 formula,
e^e is an R4^R4-valued 2-form, and is the Hodge :R4^R4 ! R4^R4.
Assuming the frame field gives a linear embedding of the triangle in R4, the
integral over this triangle, Z e ^ e
is a simple bivector, i.e., of the form f ^ g 2 R4 ^ R4. This Palatini formulation
of general relativity brings out its similarity to BF theory. In fact, if we
set E = e^ e, the Palatini Lagrangian looks exactly like the BF Lagrangian.
The big difference, of course, is that not every ad(P)-valued 2-form E is of
the form e^ e. This restricts the allowed variations of the E field when we
compute the variation of the action in general relativity. As a result, the
equations of general relativity in 4 dimensions:
e . F = 0, dAE = 0
are weaker than the BF theory equations:
F = 0, dAE = 0.
Discrete, built purely from combinatorial structures, and purely relational, so that it makes reference to no background notions of space, time or geometry1. The goal of Penrose's construction was to realize a simple model of such a system. What he posited is a system consisting of a number of \units", each of which has a total angular momentum. They interact, in ways that conserve total angular momentum. Without a background geometry, a particle can only have a total angular momentum, as there is nothing with respect to which a direction in space may be def ned. The system is then described by an arbitrary trivalent graph,
One consequence of this definition is that the value is invariant under all the identities of the theory of recoupling of angular momentum. Those identities correspond to certain graphical relationships among networks, which may be used to de ne objects such as 6j symbols completely combinatorially. In practice one uses these identities to reduce a spin network evaluation to combinations of 6j symbols, which is easier and much less prone to error than trying to keep track of all the signs and factors in 2.
We thus see that the structure of spin networks is based on the representation
theory of SU(2). This leads to the second easy generalization, which is to base
the formalism on the representation theory of any Lie Group G. In this case
a spin network is a graph. There are two cases of interest. If we are interested
in a path integral formulation in d dimensional spacetime than the con gurations
are histories.
If we are interested in a Hamiltonian formulation then the collection of all
configurations is the con guration space. In either case I will denote it C
= Qij G. That is, the configuration space consists of one copy of the group
G for each edge of the lattice. I will denote a particular configuration just
as gij .
Actually this is not the physical con guration space. What makes the theory
interesting is that there is a gauge invariance which is de ned as follows.
A gauge transformation consists of a choice of an element hi for each site of
the graph. A gauge transformation is then the map
gij ! g0ij = h
What does all of this have to do with spin networks? A great deal, for they
provide a very useful orthonormal basis of physical states. To see why, we introduce
rst the overcomplete set of states based on loops. A curve in the graph is
a list of edges ei j = e such that they each begins on the 6 node the previous
one ends on. A loop is a closed curve. Associated with each loop
= e1 e2 :::eN with N edges we may de ne the Wilson loop as
T[] = Tr[Y U(g )]
where the U(g ) are matrices in the fundamental representation of G. The space
of all T[ ] forms an overcomplete basis of C gauge. The basis is overcomplete
because of relationships between traces of products of matrices, which are called
the Mandelstam identities. However a complete basis of linearly independent
states exists and is given by the spinnets.
Flux through a surface is represented by area operators for a spin network acting on a surface described by a spin network basis.
E Psi = Int[$,S]^2 Psi
Where Int[$,S]^2 is the intersection number of loops with the surface. Electric field flux corresponds to the area of the surface Trivalent spin states are eigenvalues of volume operators; states are functions of (Wilson) loops
The true physical observables in BF theory are self-adjoint operators on the
physical Hilbert space,when this space is well-defined. Nonetheless it is interesting
to consider operators on the gaugeinvariant Hilbert space L2(A/G). These are
relevant not only to BF theory but also other gauge theories, such as 4-dimensional
Lorentzian general relativity in terms of real Ashtekar variables,where the
gauge group is SU(2). In what follows we shall use the term ‘observables’
to refer to operators on the gauge-invariant Hilbert space. We consider observables
of two kinds: functions of A and functions of E.Since A is analogous to the
‘position’ operator in elementary quantum mechanics while E is analogous
to the ‘momentum’, we expect that functions of A act as multiplication
operators while functions of E act by diherentiation. As usual in quantum field
theory, we need to smear these fields — i.e., integrate them over some
region of space — to obtain operators instead of operatorvalued distributions.
Since A is like a 1-form, it is tempting to smear it by integrating it over
a path. Similarly, since E is like an (n - 2)-form, it is tempting to integrate
it over an (n - 2)- dimensional submanifold. This is essentially what we shall
do. However, to obtain operators on the gauge-invariant Hilbert space L2(A/G),
we need to quantize gauge-invariant functions of A and E.
The simplest gauge-invariant function of the A field is a ‘Wilson loop’:
a function of the form tr(.(T eH A)) for some loop . in S and some representation
. of G. In the simplest case, when G = U(1) and the loop . bounds a disk, we
can use Stokes’ theorem to rewrite H A as the flux of the magnetic field
through this disk. In general, a Wilson loop captures gauge-invariant information
about the holonomy of the A field around the loop.
A Wilson loop is just a special case of a spin network, and we can get an operator
on L2(A/G) from any other spin network in a similar way. As we have seen, any
spin network in S defines a function . Fun(A/G). Since Fun(A/G) is an algebra,
multiplication by defines an operator on Fun(A/G). Since is a bounded function,
this operator extends to a bounded operator on L2(A/G). We call this operator
a ‘spin network observable’.[The Wilson loop may be reinterpreted
as an operator on the Hilbert space of states, and applying this operator to
the vacuum state one obtains a state in which the Yang-Mills analog of the electric
field flows around the loop gamma] Note that since Fun(A/G) is an algebra, any
product of Wilson loop observables can be written as a finite linear combination
of spin network observables. Thus spin network observables give a way to measure
correlations among the holonomies of A around a collection of loops. When G
= U(1) it is also easy to construct gauge-invariant functions of E. We simply
take any compact oriented (n - 2)-dimensional submanifold in S, possibly with
boundary, and do the integral
ZE.13
This measures the flux of the electric field through . Unfortunately, this
integral is not gaugeinvariant when G is nonabelian, so we need to modify the
construction slightly to handle the nonabelian case. Write E| = e dn-2x for
some g-valued function e on and some (n - 2)-form dn-2x on that is compatible
with the orientation of . Then Zphe, ei dn-2x is a gauge-invariant function
of E. One can check that it does not depend on how we write E as e dn-2x. We
can think of it as a precise way to define the quantity
Z |E|.
Recall that 3-dimensional BF theory with gauge group SU(2) or SO(3) is a formulation
of Riemannian general relativity in 3 dimensions. In this case is a curve,
and the above quantity has a simple interpretation: it is the length of this
curve. Similarly, in 4-dimensional BF theory with either of these gauge
groups, is a surface, and the above quantity can be interpreted as the area
of this surface. The same is true for 4-dimensional Lorentzian general relativity
formulated in terms
of the real Ashtekar variables.
Quantizing the above function of E we obtain a self-adjoint operator E() on
L2(A/G), at least when is real-analytically embedded in S. We shall not present
the quantization procedure here, but only the final result. Suppose is a spin
network in S. Generically, will intersect transversely at finitely many points,
and these points will not be vertices of :
re3
re21
reiv
2iv1
In this case we haveE() = . The irreducible representations of SU(2) correspond
to spins j = 0, 1 2 , 1 . . . , and the Casimir equals j(j + 1) in the spin-j
representation. Thus a spin network edge labelled by the spin j contributes
a length SqRt j(j + 1) to any curve it crosses transversely. As an immediate
consequence, we see that the length of a curve is not a continuously variable
quantity in 3d Riemannian quantum gravity. Instead, it has a discrete spectrum
of possible values! We also see here the di erence between using SU(2) and SO(3)
as our gauge group: only integer spins correspond to irreducible representations
of SO(3), so the spectrum of allowed lengths for curves is sparser if we use
SO(3). Of course, in a careful treatment we should also consider spin networks
intersecting nongenerically, these give the operator E() additional eigenvalues.
However, our basic qualitative conclusions here remain unchanged. Similar remarks
apply to 4-imensional BF theory with gauge group SU(2), as well as quantum gravity
in the real Ashtekar formulation. Here E() measures the area of the surface
, area is quantized, and spin network edges give area to the surfaces they
intersect! [Since the E operator vanishes unless it grasps an edge, it only
acts where spin networks intersects the surface -hence the area eigenvalue depends
on the number of times the spin network intersects the surface].This is
particularly intriguing given the Bekenstein-Hawking formula saying that the
entropy of a black hole is proportional to its area.
For a deeper understanding of BF theory with gauge group SU(2), it is helpful
to start with a classical phase space describing tetrahedron geometries and
apply geometric quantization to obtain a Hilbert space of quantum states. We
can describe a tetrahedron in R3 by specifying vectors E1, . . . ,E4 normal
to its faces, with lengths equal to the faces’ areas. We can think of these
vectors as elements of so(3)*, which has a Poisson structure familiar from the
quantum mechanics of angular momentum:
{Ja, Jb} = oabcJc.
The space of 4-tuples (E1, . . . ,E4) thus becomes a Poisson manifold. However,
a 4-tuple coming from a tetrahedron must satisfy the constraint E1 + ·
· · + E4 = 0. This constraint is the discrete analogue of the
Gauss law dAE = 0. In particular, it generates rotations, so if we take (so(3)*)4
and do Poisson reduction with respect to this constraint, we obtain a phase
space whose points correspond to tetrahedron geometries modulo rotations. If
we geometrically quantize this phase space, we obtain the ‘Hilbert space
of the quantum tetrahedron’. We can describe this Hilbert space quite explicitly
as follows. If we gometrically quantize so(3)*,we obtain the direct sum of all
the irreducible representations of SU(2):
H ~= M j=0, 1 2 ,1,... j.
Since this Hilbert space is a representation of SU(2), it has operators ˆ
Ja on it satisfying the usual angular momentum commutation relations:
[ ˆ Ja, ˆ Jb] = ioabc ˆ Jc.
We can think of H as the ‘Hilbert space of a quantum vector’ and the
operators ˆ Ja as measuring the components of this vector. If we geometrically
quantize (so(3)*).4, we obtain H.4, which is the Hilbert space for 4 quantum
vectors. There are operators on this Hilbert space corresponding to the components
of these vectors:
ˆ Ea1 = ˆ Ja . 1 . 1 . 1ˆ Ea
2 = 1 . ˆ Ja . 1 . 1ˆ Ea3 = 1 . 1 . ˆ Ja . 1ˆ Ea
4 = 1 . 1 . 1 . ˆ Ja. 21
One can show that the Hilbert space of the quantum tetrahedron is isomorphic
to
T = {. . H.4: ( ˆ E1 + ˆ E2 + ˆ E3 + ˆ E4). = 0}.
On the Hilbert space of the quantum tetrahedron there are operators
ˆ Ai = ( ˆ Ei · ˆ Ei)12
corresponding to the areas of the 4 faces of the tetrahedron, and also operators
ˆ Aij = (( ˆ Ei + ˆ Ej) · ( ˆ Ei + ˆ Ej))12
corresponding to the areas of the parallelograms. Since ˆ Aij = ˆ
Akl whenever (ijkl) is some permutation of the numbers (1234), there are really
just 3 di erent parallelogram area operators. The face area operators commute
with each other and with the parallelogram area operators, but the parallelogram
areas do not commute with each other. There is a basis of T consisting of states
that are eigenvectors of all the face area operators together with any one of
the parallelogram area
operators. If for example we pick ˆ A12 as our preferred parallelogram
area operator, any basis vector . is determined by 5 spins:
ˆ Ai. = pji(ji + 1) 1 = i = 4,
ˆ A12. = pj5(j5 + 1).
This basis vector corresponds to the intertwiner .j : j1 . j2 . j3 . j4 that
factors through the representation j5.
In 4d BF theory with gauge group SU(2), the Hilbert space L2(A/G) described
by taking the tensor product of copies of T , one for each tetrahedron in the
3-manifold S, and imposing constraints saying that when two tetrahedra share
a face their face areas must agree. This gives a clearer picture of the ‘quantum
geometry of space’ in this theory. For example, we can define
observables corresponding to the volumes of tetrahedra. The results nicely match
those of loop quantum gravity, where it has been shown that spin network vertices
give volume to the regions of space in which they lie. In loop quantum gravity
these results were derived not from BF theory, but from Lorentzian quantum gravity
formulated in terms of the real Ashtekar variables. However, these theories
differ only in their dynamics.
With regards to producing a quantum theory of gravity, Einstein's equation has energy and momentum as the source and in QT these relate to differential operators (in time and space) which obey commutation relationships. Ideally we would like to be able to formulate such quantum commutators for the conventional metric of GR but the various problems that have been encountered have led to less direct methods being developed. For example the Ashtekar (conjugate) variables A and E, where the 'momentum' parameter E is a vector density related to a 3D metric of constant time surface and the 'position' parameter A, that defines a smooth real SU(2) spin bundle connection restricted to the 3D surface. The parameter A itself consists of a part T1, that refers to the spin connection associated to the local triad (~ intrinsic curvature of the surface) and a part T2 that refers to the extrinsic curvature of the 3D surface.. These are expressed as A =T1 +&T2 where & is the Immirzi -Barbero coefficient, which in order to produce the desired result for the entropy of a black hole, has to be given the mysterious value of &= log2/(piSqRt3). In this way A and E are utilised as the relevant conjugate but mutually exclusive operators, which act upon the spin networks, producing quantised area and in particular yielding the correct 1/4 coefficient in the Bekenstein-Hawking formular.
*********
When it comes to BF theory, we have another option: we can completely eliminate
the infrared divergences by adding an extra term to the Lagrangian of our theory,
built using only the E field. This trick only works when spacetime has dimension
3 or 4. In dimension 3, the modified Lagrangian is
L = tr(E . F + 6E . E . E),
while in dimension 4 it is
L = tr(E . F + 12E . E).
For reasons that will become clear, the coupling constant is called the ‘cosmological
constant’.
We only consider the case > 0.
Adding this ‘cosmological term’ has a profound e ect on BF theory:
it changes all our calculations involving the representation theory of the gauge
group into analogous calculations involving the representation theory of the
corresponding quantum group. This gives us a well-defined and finite-dimensional
physical Hilbert space, and turns the divergent sum over spin foams into a finite
sum for the transition amplitudes between states. This process is known as ‘q-deformation’,
because the quantum group depends on a parameter q, and reduces to the original
group at q = 1. Often people think of q as a function of ~, but for us it is
a function of , and we have q = 1 when = 0.
Thus, at least in the present context, quantum groups should really be called
‘cosmological groups’ !To understand how quantum groups are related
to BF theory with a cosmological term, we need to exploit its ties to Chern-Simons
theory. This is a background-free gauge theory in 3 dimensions whose action
depends only the connection A: viz
SCS(A) =k/4p INT{M} tr(A . dA +2/3A . A . A)
It turns out that the action for 3d BF theory with cosmological term is a difference
of two Chern-Simons actions. Thus we can quantize this BF theory whenever we
can quantize Chern-Simons theory and we obtain a theory equivalent to two independent
copies of Chern-Simons theory.
* .* . * . * . *
Another problem is that it is not possible to satisfactorily incorporate gravity (which is governed by the non-compact Poincare group), in what is known as a Unitary representation (that dictates the other 3 quantum interactions), unless one resorts to supernumbers, which combines both fermions and bosons via supersymmetry. [Specifically there is a no-go theorem which states that a group that nontrivially combines both the Lorentz group and a compact Lie group cannot have a finite dimensional, unitary representation]. This concept was originally invoked in the early study of string theory but although this has received a recent surge in popularity, supersymmetry itself does not require a string formulation. [Such a Lagrangian, is that associated with the surface being swept out by a string, rather than that of a point like particle which, as it moves through time sweeps out a curve.] Supersymmetry hopes to put the quanta of both ‘particle’ and ‘field’(i.e. fermions and bosons), on an equal footing, while the latest development -- string theory, hopes to explain what symmetries are allowed, which in turn determine the conservation laws of physics. As an extra bonus, putting fermions and bosons in the same super multiparticle state, necessitates the introduction of a gravitational field (i.e. local supersymmetric transformations invoke a field that produces the localised space-time translations that is indicative of gravity ). In fact string theory alone had predicted the existence of a spin 2 Boson indicative of the graviton and the very nature of strings -- that they are extended, and not pointlike -- turns out to cure the inconsistencies that had always plagued quantum theories of gravitation. While the Feynman diagrams describing the scattering of point particles have definite points where a particle splits into two particles, the corresponding diagrams for string scattering are completely smooth . This simple fact is ultimately responsible for removing the singularities in the process of graviton that otherwise results in infinities that cannot be renormalized.
[ In string theory, tachyons arise as an excitation having negative mass-squared. From the point of view of an observer on the string itself, this excitation appeared as a state of negative energy. This was connected to a familiar fact in quantum mechanics: because of the famous uncertainty principle, the minimum energy of a localised system tends to be non-vanishing. The presence of a tachyon in string theory could be mathematically traced to the presence of this non-vanishing "zero-point energy" in typical quantum systems. It followed that if we could identify special quantum systems where the zero-point energy vanished, we would possibly be able to invent a special string without any tachyon.
Quantum systems without a zero-point energy occur in the presence of supersymmetry. We mentioned earlier that this symmetry relates bosonic and fermionic degrees of freedom. The zero-point energies associated with these two types of degrees of freedom turn out to be opposite in sign, and cancel out identically in a supersymmetric system. Although the historical route to this discovery was somewhat more complicated, our modern understanding is that this is the key to a string theory which has no tachyon -- "superstring theory". Going from string to superstring theory finally led to a proposal which seemed well-suited to describing the real world, or at least better-suited than many previous proposals. In superstring theory, the basic postulate is, as before, that vibrations of a single type of string lead to a multitude of elementary particles. But because of supersymmetry, there is no tachyon. As a bonus, the consistency condition on the number of spacetime dimensions changes when supersymmetry is introduced. Thus, instead of 26 needed for bosonic strings, superstring theory requires just 10 spacetime dimensions. Finally, the presence of a "graviton" particle is maintained in superstring theory, so it is still a theory of gravity -- in fact, of "supergravity", the supersymmetric extension of gravity. In actual fact supergravity based on point like particles has a maximum allowable dimension of 11, while bosonic strings must be 26-D but supersymmetry insists upon only 10-D. As we shall discuss later the more recent M-theory (M for Mother ? or Master or Membrane or Mystery?), unites all 5 string theories with supergravity by viewing them all from the perspective of 11 dimensions. Indeed some physicists are also considering a 12 dimensional F theory (F for Father), which is permitted if we allow 2 dimensions of time!
A string can be either open, with two end-points, or closed, like a loop. The
natural physical postulate for interactions between two open strings is that
when their end-points touch, they join together into a longer open string. However,
if the two end-points of a single open string touch, they can join together
and make a closed string. Thus, theories of open strings inevitably contain
closed strings as well The reverse is not true. A pair of closed strings can
join when any pair of points coincides, to make a single closed string. Thus
there can be theories with only closed strings and no open strings. It turns
out that the most
natural prescription leads to a single possible open string theory, called type
I superstring theory, and two distinct closed string theories, called type IIA
and type IIB string theory. A clever hybrid of the superstring with ordinary
(non-supersymmetric) string theory, called the "heterotic" string,
was also discovered and there are two such theories, making five altogether]
Briefly, a closed string has 2 sets of operators which do not interact, that is as the string propagates, it has right and left moving oscillator modes. The left moving modes are purely bosonic and exist in a 26 dimensional space, which has been compactified to 10 dimensions. The right moving modes only live in this 10 dimensional space and contain the supersymmetric part of the theory, while the compactified 16 dimensional string lives in the root lattice space of an E8*E8 isospin symmetry, which is more than large enough to contain the required spectum of particles. When the left moving half and the right moving half are put together they produce the heterotic string (meaning "hybrid vigour"). Compactification of the extra six dimensions on a Ricci flat manifold, then reduces the 10 dimensional superstring into our familiar 4 dimensional space-time. When we investigate the symmetries of these superstrings, we find that they automatically invoke a gauge 2 boson (a graviton) that is indicative of a quantized theory of gravity! we therefore need to define the relationship between the 10-dimensional world described above, (with 9 spatial and 1 time dimension) and the real 4-dimensional world (3 spatial and 1 time dimension) that we inhabit. The key requirement is that the 9 spatial dimensions that we start with should not all be physically observable. In the spirit of Kaluza and Klein, we therefore assume that 6 spatial dimensions are "curled up" on themselves, while the remaining 3 dimensions extend to infinity (or at least to very large distances). The concept of "space" is embodied in mathematics by the notion of a "manifold", something that locally looks like familiar space but may have curvature and other nontrivial properties. In particular, a manifold that is "curled up" in the way that we desire is called "compact". In the original work of Kaluza it was shown that if we start with a theory of general relativity in 5-spacetime dimensions and then curl up one of the dimensions into a circle we end up with a 4-dimensional theory of general relativity plus electromagnetism! The reason why this works is that electromagnetism is a U(1) gauge theory, and U(1) is just the group of rotations around a circle. If we assume that the electron has a degree of freedom corresponding to point on a circle, and that this point is free to vary on the circle as we move around in spacetime, we find that the theory must contain the photon and that the electron obeys the equations of motion of electromagnetism (namely Maxwell's equations). The Kaluza-Klein mechanism simply gives a geometrical explanation for this circle: it comes from an actual fifth dimension that has been curled up. In this simple example we see that even though the compact dimensions maybe too small to detect directly, they still can have profound physical implications. [Incidentally the work of Kaluza and Klein leaked over into the popular culture launching all kinds of fantasies about the "Fifth dimension"!] How would we ever really know if there were extra dimensions and how could we detect them if we had particle accelerators with high enough energies? From quantum mechanics we know that if a spatial dimension is periodic the momentum in that dimension is quantized, p = n / R (n=0,1,2,3,....), whereas if a spatial dimension is unconstrained the momentum can take on a continuum of values. As the radius of the compact dimension decreases (the circle becomes very small) then the gap between the allowed momentum values becomes very wide. Thus we have a Kaluza Klein tower of momentum states.
If we take the radius of the circle to be very large (the dimension is de-compactifying)
then the allowed values of the momentum become very closely spaced and begin
to form a continuum. These Kaluza-Klein momentum states will show up in the
mass spectrum of the uncompactifed world. In particular, a massless state in
the higher dimensional theory will show up in the lower dimensional theory as
a tower of equally spaced massive states. A particle accelerator would then
observe a set of particles with masses equally spaced from each other. Unfortunately,
we'd need a very high energy accelerator to see even the lightest massive particle.
Strings have a fascinating extra property when compactified: they can wind around
a compact dimension which leads to winding modes in the mass spectrum. A closed
string can wind around a periodic dimension an integral number of times. Similar
to the Kaluza-Klein case they contribute a momentum which goes as p = w R (w=0,1,2,...).
The crucial difference here is that this goes the other way with respect to
the radius of the compact dimension, R. So now as the compact
dimension becomes very small these winding modes are becoming very light!
Now to make contact with our 4-dimensional world we need to compactify the
10-dimensional superstring theory on a
6-dimensional compact manifold. Needless to say, the Kaluza Klein picture described
above becomes a bit more
complicated. One way could simply be to put the extra 6 dimensions on 6 circles,
which is just a 6-dimensional Torus. As it turns out this would preserve too
much supersymmetry. It is believed that some supersymmetry exists in our 4-dimensional
world at an energy scale above 1 TeV (this is the focus of much of the current
and future research at the highest energy accelerators around the word!). To
preserve the minimal amount of supersymmetry, N=1 in 4 dimensions, we need to
compactify on a special kind of 6-manifold called a Calabi-Yau manifold.
The properties of the Calabi-Yau manifold can have important implications for
low energy physics such as the types of
particles observed, their masses and quantum numbers, and the number of generations.
One of the outstanding problems in the field has been the fact that there are
many many Calabi-Yau manifolds (thousands upon thousands?) and we have no way
of knowing which one to use. In a sense we started with a virtually unique 10-dimensional
string theory and have found that possibilities for 4-dimensional physics are
far from unique, at least at the level of our current (and incomplete)
understanding. The long-standing hope of string theorists is that a detailed
knowledge of the full non-perturbative structure of the theory, will lead us
to an explanation of how and why our universe flowed from the 10-dimensional
physics that probably existed during the high energy phase of the Big Bang,
down to the low energy 4-dimensional physics that we observe today.
[Possibly we will find a unique Calabi-Yau manifold that does the trick.] It
has been shown that Calabi-Yau manifolds can be continuously connected to one
another through conifold transitions and that we can move between different
Calabi-Yau manifolds by varying parameters in the theory. This suggests the
possibility that the various 4-dimensional theories arising from different Calabi-Yau
manifolds might actually be different phases of an single underlying theory.
Thus, the most straightforward way to connect string theory to the real world is to postulate that 6 spatial dimensions form a compact manifold, whose size is so small that we are unable to detect its existence directly with the probes available to us. A class of 6-dimensional manifolds with very special properties, known as Calabi-Yau manifolds, turn out to have properties which -- when they are used as the compactification manifolds in string theory -- lead to tantalizingly realistic theories in 4 spacetime dimensions. Thedetailed particle content and dynamics of the 4-dimensional theory depend on the choice of Calabi-Yau manifold, so it is not as if the manifold is completely unobservable. In fact, this way we get to have our cake and eat it too-- the Calabi-Yau manifold would be responsible for the "zoo" of elementary particles observed in the real world, but it would not be directly observable as a collection of extra spatial dimensions -- a good thing since such extra dimensions have not been observed, at least so far.
How would we ever really know if there were extra dimensions and how could we detect them if we had particle accelerators with high enough energies? From quantum mechanics we know that if a spatial dimension is periodic the momentum in that dimension is quantized, p = n / R (n=0,1,2,3,....), whereas if a spatial dimension is unconstrained the momentum can take on a continuum of values. As the radius of the compact dimension decreases (the circle becomes very small) then the gap between the allowed momentum values becomes very wide. Thus we have a Kaluza Klein tower of momentum states.
The lightest excitations of a string theory can be described by a quantum field
theory with a certain number of elementary particles, so we will describe the
content of this field theory, temporarily ignoring the fact that there are infinitely
many string excitations of increasing energy. For the open string, the low-energy
theory is a field theory in 10 spacetime dimensions with a (massless) graviton
and a collection of "gauge" fields very much like the fields describing
photons, W and Z bosons and gluons in the real world. Thus, interactions of
the type found in nature (gravity and gauge forces) are incorporated in type
I superstring theory, albeit in 10 dimensions. Fermionic matter particles are
present too. They are charged under the gauge interactions, analogous to the
fact that electrons have electric charge or that quarks carry a "colour
charge". Also, the fermions are of a definite "handedness" or
"chirality", which means that type I theory in 10 dimensions violates
parity or left-right symmetry, just as the weak interactions do in the real
world. This is extremely encouraging, because all these particles and interactions
follow
from the simple postulate of a consistent open superstring theory!
There are several obvious negative points too. The symmetry group associated to the gauge particles of the type I string in 10 dimensions is SO(32), much larger than the expected symmetry group of the strong, weak and electromagnetic interactions in the real world, which is the product of SU(3), SU(2) and U(1). So we seem to have too many forces and their corresponding carriers. Also, all the particles appearing in the low-energy theory are exactly massless, quite different from the real-world electron or quark which has a definite mass. Finally, we are in the wrong number of spacetime dimensions -- 10 rather than 4. But none of these is a genuine obstacle. At extremely high energies, for all we know the real world might well look 10-dimensional and have lots of nearly massless particles and a very large gauge symmetry group. So one should treat 10-dimensional superstring theory in this light -- as a candidate description of the high-energy limit of the real world. To connect this to the low-energy world we have to face issues like compactification to 4 dimensions and symmetry breaking.
The superstring theories of type IIA and IIB are somewhat different. They contain a massless graviton, but do not have the "gauge" type particles that are present in type I string theory. There are fermionic matter particles, but in the absence of force carriers of the usual type, these cannot carry charges. The fermions are parity-conserving in type IIA theory and parity-violating in type IIB. Finally, there are some exotic fields called "tensor gauge fields", which are roughly like higher-spin analogues of the photon. However, the fermions and other massless particles are not charged with respect to these. All of this was initially perceived as discouraging and these theories were for many years regarded as exotic and irrelevant to the goal of finding a unified theory of nature. Recent developments have shown this view to be false. The process of compactification can actually produce gauge particles and fermions charged under the gauge interactions. This is one consequence of the duality symmetries that will be discussed in the other articles of this collection.
Finally, let us describe the hybrid "heterotic" string. This was based on the observation that excitations of a closed string look like little waves which travel one way or the opposite way around the string. These two types of waves (called "left-movers" and "right-movers" for obvious reasons) do not interact with each other even though they propagate on the same string. Hence it is meaningful to combine ordinary left-moving waves with supersymmetric right-moving waves (rather than have both types of waves be supersymmetric, as in the type II strings). The heterotic string is based on this idea.
There is a puzzle right away: we saw that ordinary strings are consistent in 26 dimensions while superstrings live in 10 dimensions. How can these two be matched and in how many dimensions does the resulting theory live? The answer comes from an adaptation of the Kaluza-Klein idea. We take 16 of the 26 dimensions to be curled up on little circles. Then the resulting 10 dimensional theory is combined into the hybrid theory described above. So the final theory is 10-dimensional, but its left-moving waves have a 26-dimensional origin, and hence there are hidden degrees of freedom corresponding to the 16 extra dimensions. Remarkably, these hidden degrees of freedom are manifested as gauge fields, and the symmetry group depends on the sizes of these compact dimensions and the shape of the "torus" formed by the 16 compact dimensions. It turns out that consistency of the resulting string theory allows only two choices for this torus. One choice leads to a gauge symmetry group of SO(32), so the resulting theory closely resembles type I (open) superstring theory. The other choice leads to something quite new: the gauge symmetry group is a product of the exceptional group E(8) with itself. This is called the E(8) x E(8) heterotic string. It is the fifth and last superstring theory in 10 dimensions. Like the type I string, both the heterotic string theories are parity-violating in 10 dimensions.
Initial attempts to compactify these five string theories to 4 dimensions strongly favoured the E(8) x E(8) heterotic string. With the SO(32) strings (heterotic and type I) the parity violating nature in 10 dimensions appeared to be destroyed by compactification, so the resulting 4-dimensional theory could not describe the real world, which is parity-violating. This was also true of the type IIB string. On the other hand, the type IIA theory is already parity-conserving in 10 dimensions, and from what was known about compactification until recently, it apparently could not induce parity-violation. The modern picture is however considerably different. All the five string theories are actually connected to each other, so in some sense they are all different limits of a single theory and recently a ‘second revolution’ has subjugated these into an 11 dimensional formulation known as M-theory which also accommodates supergravity. M theory is a recent revolution in our understanding of sting theory and has demonstrated a symmetry/duality between the 5 competing formulations, namely Type I, IIa, IIb, heterotic-O and heterotic-E as well as supergravity. This is achieved by increasing the number of dimensions to 11 and which also allows for the existence of higher dimensional 'branes' as well as the more usual strings, which represent our universe. Under this scheme it is found that the competing string theories can paired off under what is known as T-duality. For example when the extra dimension is curled up into a circle, M-theory yields the type IIA string, which when observed at low energy, can behave like type IIB theory under high energy. Similarly, if the extra dimension shrinks to a line segment, M-theory reduces to a heteroic-E string, which is conected to the heterotic-O string by dualities! [T- duality can itself also be shown to be a duality of S-duality which relates a gauge charge e, such as that of an electron with its topological soliton counterpart with charge1/e --- the magnetic monopole! In all three kinds of dualities, called S, T, and U, have been identified. It can sometimes happen that theory A with a large strength of interaction (or `strong coupling') is equivalent to theory B at weak coupling, in which case they are said to be S dual. Similarly, if theory A compactified on a space of large volume is equivalent to theory B compactified on a space of small volume, then they are called T dual. Combining these ideas, if theory A compactified on a space of large (or small) volume is equivalent to theory B at strong (or weak) coupling, they are called U dual. If theories A and B are the same, then the duality becomes a self-duality, and it can be viewed as a kind of symmetry. T duality, unlike S or U duality, can be understood perturbatively, and therefore it was discovered between the two string revolutions] Here again it becomes important to study the symmetries (and hence topologies) of the types of p-dimensional branes that can exist in such an exotic manifold. [In M-theory, a string with extra dimensions curled up into a particular Calabi-Yau space, gets translated via duality into a different string theory with extra dimensions curled up into Calabi-Yau shape (its dual) - - the physics being identical in both formulations.]
Some of the most recent suggestions/discoveries include the fact that the weakness of gravity can be shown to be a result of it leaking from a parallel membrane into our own Brane World, (it is consequently diluted by the extra dimensions that we do not directly observe). It is worth noting however that although some of these ideas may have been inspired by M-theory, they are actually independent of it. Indeed they involve a subtle paradigm shift, from considering microscopic processes on Dirichelet p-branes, to considering the whole universe as being on such a brane. [The extra dimensions and supermanifolds of M-Theory (as in 2nd generation string theory), may even be considered as merely parameters, whereas these new 'Brane Worlds' are by necessity actual large scale features of our universe.] This so called Randal-Sundrum scenario (RS1), suggests that our universe is a 3-brane embedded in a 5 dimesional space-time, with the extra dimensions of the bulk being warped. If there is another 3-brane, separated from our own by a small distance in the extra dimension and if gravity is localised to this brane, then the warp factor would ensure that gravity is weak in our brane. Thus this scenario addresses the Hierarchy problem, since it increases the natural strength of gravity and thereby reduces the Planck scale. [The hierarchy problem results from the fact that there is a huge difference, producing the so called 'Desert', between the electroweak scale and the Planck scale (~1Tev compared to ~10^16 Tev), when quantum effects should in fact drag the electroweak scale up to the Plank scale. It arises in GUT's without gravity since the weak Higgs field that is responsible for providing the mass (and short range) of the W and Z particles, is very different from strong Higgs field that provides the enormously larger mass that is needed to explain the observed stability of the proton. Yet these 2 Higgs field have to be united at high energies of the GUT's. Mass renormalization involves interactions between all particle species in the theory and it is difficult to see why the masses of similar species should not turn out to have comparable values. Indeed the large mass differences can be achieved in GUT's but only by (exceedingly) fine tuning various adjustable parameter and this, it in turn has to be altered at each energy scale.] A later version of the scenario (RS2), postulates the existence of only one brane in a warped 5-dimensional space-time. Also it has been suggested that the creation of the universe can be formulated in terms of an initial violent collision between 2 such membranes (this is called the Ekpyrotic theory). In such a Braneworld scenario, if the 2 branes are connected by quite a large extra dimension, it would imply that gravity is intrinsically much stronger than previously thought and the strings longer and therefore needing much less energy in order to vibrate, making supersymmetry particles easier to create. Indeed the observed weakness in gravity would be attributed to a leakage into the extra dimension and therefore before leakage occurred gravity would be quite strong. One of the implications of this is that the creation of a microscopic black hole would require far less energy and may be created in the collision of very energetic cosmic rays in the upper atmosphere, which may be observable as a cascade of particles (Hawking radiation) . The search continues for the perfect symmetry that is obscured by the richness of the observed universe, one that is indicated by the representational space of QT and the differential geometry of GR.
[There is also a (Maldacena), conjectured duality between physics
in the bulk of Anti-de Sitter (AdS) and a conformal field theory (CFT) on the
boundary (a de Sitter, a Minkowski and AdS universe, correspond to a positive,
zero and negative cosmological constant respectively). The AdS/CFT correspondence
is a type of duality, which states that two apparently distinct physical theories
are actually equivalent. On one side of this duality is the physics of gravity
in a spacetime known as anti-de Sitter space (AdS). while on the other is a
supersymmetric Yang-Mills gauge theory that lives on the boundary of the AdS.
A natural generalization of the work of Kaluza and Klein, shows that diffeomorphism
symmetry (reparameterisation invariance), of a higher than 4 dimensional space-time
manifold, can give rise to the gauge symmetry of a Yang-Mills theory. So by
using the Kaluza-Klein idea we may 'derive' gauge symmetries from the geometrical
symmetries of a higher dimensional space-time. In fact part of the diffeomorphism
symmetry of gravity in higher dimensions becomes a gauge symmetry in four dimensions.
In some sense, this just means that one assumed symmetry is interpreted as transformed
into another one.
Five-dimensional anti-de Sitter space has a boundary which is four-dimensional,
and in a certain limit looks like flat space-time with one time and three space
directions. The AdS/CFT correspondence states that the
physics of (string) gravity in five-dimensional anti-de Sitter space, is equivalent
(dual) to a certain supersymmetric (particle) Yang-Mills theory (a Conformal
Field Theory without gravity), which is defined on the boundary of AdS. This
Yang-Mills theory is thus a `hologram' of the physics which is happening
in five dimensions. The Yang-Mills theory has gauge group SU(N), where
N is very large, and it is said to be `supersymmetric' because it has a symmetry
which allows you to exchange bosons and fermions. In recent studies of M-Theory,
AdS black hole solutions of gauged extended supergravities can be considered
as rotating, higher dimensional membranes. We therefore embed these known lower-dimensional
black hole solutions into ten or eleven dimensions, thus allowing a higher
dimensional interpretation in terms of rotating M2, D3 and M5-branes. Consequently
we arrive at a situation in which we have CFT duals of AdS black holes, rotating
in spacetime and carrying electric charge as a consequence of the rotation of
the corresponding branes in the transverse dimensions, The AdS4 black holes
with toroidal horizon can indeed be interpreted as the near-horizon structures
of an M2 brane rotating in the extra dimensions. The four charges corresponding
to the U(1)4 Cartan subgroup are just the four angular momenta. Similarly, the
5-dimensional charged black hole with toroidal horizon corresponds to a rotating
D3-brane and the 7- dimensional charged black hole with toroidal horizon to
a rotating M5-brane. In each case, the event horizon coincides with the worldvolume
of the brane. In Kaluza-Klein theory we sometimes use to the idea that electric
charge is considered as momentum in an extra dimension, but now we can regard
the electric charges of the AdS black holes as corresponding to angular momenta
in extra dimensions.]
The Following articles are very good expositions of Quantum Loop Gravity
BRANEWORLDS
Introduction
The Higgs mechanism makes it look although the weak force symmetry is preserved at short distances (high energies) but is broken at long distances (low energies).The weak Higgs field has 2 components which are both zero when the weak symmetry is preserved, however when one of these takes on a nonzero value it breaks the weak force symmetry that interchanges the 2 Higgs fields. The symmetry is broken spontaneously because all that breaks it is the vacuum -- the actual state of the system, the non zero field in this case. However the physical laws which are unchanged, still preserve the symmetry even though the physical system does not. The symmetry transformations that act on the weak gauge bosons, also act on the quark and lepton flavours and it turns out that these transformations wont leave things the same unless they are masslees. Now because the weak force symmetry is essential at high (GUT) energies, not only is spontaneous symmetry breaking required for the gauge bosons, it's necessary for these quarks and leptons to acquire mass as well. The Higgs mechanism is the only way for all the massive fundamental particles of the standard model to acquire their masses. At high energies the internal symmetries associated with the weak force, still filters out the problematic polarization of the weak gauge bosons that would cause interactions at too high a rate.
However at low energies, where the mass is essential to reproducing the measured short range interactions of the weak force, the weak force symmetry is broken. In this mechanism, there are original 3 weak gauge bosons plus a Goldstone boson and after symmetry breaking we have the massive W+- and Z bosons plus the photon, which is able to travel in a massless mode through the Higgs vacuum unaffected, since it has no weak (flavour) charge. The problem with GUT is that although the Higgs particle has to be relatively light for weak symmetry breaking (~ 250GeV as born out by experiment,) it is partnered another particle (X), that interacts with it through the strong force and which has to be extremely heavy [This in order to explain the stability of the proton, which would otherwise decay due to the X particle allowing a quark to change into a lepton]. In other words we are left with the problem that 2 particles that are related by GUT symmetry have to have enormously different masses (the weak and strong forces have to be interchangeable at high energies). This hierarchy problem is made worse by the fact that QT requires the value of the Higgs particle to be determined by contributions of virtual particles which are of the order of the energy scale of GUT (according to the anarchic principle in which any interactions that is not forbidden by symmetry will occur). However these QT contributions (some positive some negative), which must be added to the classical value of the Higgs boson, are under GUT energy values 13 orders of magnitude greater than the weak Higgs value! The situation is even worse when we consider gravity since QT corrections now occur at the Planck scale (10^19Gev) [Newton's law states that strength is inversely proportional to the square of of the energy/mass and because gravity is so weak the Planck scale is large Another way of phrasing the hierarchy problem is to ask why gravity is so feeble]. Such a trickle down effect of QT contributions should therefore make a large quantum mass determine the ultimate mass of other particles, so that all end up rich in mass. Supersymmetry (SUSY) gives an answer to this by allowing the positive QT contributions of the bosons to be canceled by the negative contributions of the fermions. It achieves this by first pairing all the fermions and bosons with superpartners, the Higgs field then gets contributions from both particles and supersymmetric particles and because the interactions with the two are different their contribution to the Higgs particle's mass, cancel each other out.
Now in order to account for the lack of observed superpartners, it is necessary to invoke a SUSY breaking, which imbues mass to the superpartner, making it to large to to be stable or created in particle accelerators. However once supersymmetry is broken, flavour changing interactions are allowed which are not observed in nature or at least are a lot more rare than predicted. These are processes that change quarks or leptons into those of another generation (that is ones that are heavier or lighter but with the same charge). Although an electron and a slectron can interact via the weak force as can a muon and smuon, an electron would never interact directly with a smuon. If an electron were paired with a smuon or a muon with a slectron, this would allow a muon to decay into an electron and a photon, something which is never observed. However with SUSY breaking the now massive bosonic superparteners no longer have the strong sense of identity of their partner fermions, and this allows the (massive) bosonic superpartners to get all mixed up, so that not only a smuon but also a slectron would be paired with a muon for example. However this pairing of a slectron and a muon would yield all sorts of interactions that are not observed. So although SUSY can overcome the hierarchy problem it does lead to a flavour problem.The question is how do we break the SUSY, but prevent the flavour problem from occurring. One possibility for resolving the called flavour problem is to resort to large dimensional branes (without SUSY), which may also allow a remedy to the Hierarchy problem.
Branes
An alternative approach to SUSY in addressing these and other problems, is Braneworld theory. Branes originated from string theory (1989), in which D-branes (Dirichlet boundary conditions), were used as ends for strings that move in the bulk space. At around the same time p-branes were discovered as solutions to Einstein's field equation in higher dimensions. These extended infinitely far in some spatial dimensions but in the remaining dimensions they act as black holes. In some geometries they are found to give rise to new types of particles that are not accounted for in D-brane string theory. These p-branes are independent objects that can wrap around a very small curled up region of space time, an act like particles. In 1995 it was shown that D-branes and p-branes were actually the same thing and that at energies where string theory makes the same predictions as General Relativity, D-branes morph into p-branes. The way in which this equivalence is best expressed is via the important notion of duality. An important aspect of duality was revealed by Witten in 1995 when he demonstrated that a low energy version of 10D superstring theory with strong coupling was equivalent 11D supergravity with weak coupling (which could therefore be dealt with by perturbation techniques.Dualities between all the contending string theories were established bringing about a second revolution in string theory, namely M-Theory. In order to reconcile the difference in dimensions it was realised that the strings were actually membranes that extend in dimensions that were previously not recognized due to their compact size and these have been identified as the p-branes. [Eleven dimensional supergravity although not containing strings was already know to contain 2D membrane solutions] The key was therefore the realisation that rolled up dimensions are invisible at long distances or low energies, making 11D supergravity with one dimension curled up equivalent to 10D string theory [In 11D supergravity you need to know the momentum in 10D whereas in 11D superstring theory you need to specify the momentum in 9D and also the value of the charge, i.e. 10 numbers have to be specified in each case to make the particles correspond in the 2 theories. [Perhaps this is an indication that these extra dimensions should be regarded as auxilliary variables just as is charge} Ordinary uncharged strings do not pair with objects in 11D and the partners of objects in 11D theory turn out to be branes viz. charged pointlike branes called Do-branes. The 2 theories are dual because for every Do-brane of a given charge in 10D superstring theory, there is a corresponding particle with a particular 11D momentum and vice versa] It was soon realised that it was possible to formulate theories of higher dimensional branes within the context of M-theory.
Also mathematicians began to consider the possibility of a higher dimensional universe in which the particles and forces reside in a lower dmensional brane - - a Braneworld! The first example of this was the HW braneworld (Horava-witten), in which 2 branes bounding the 11th dimension, were shown to be equivalent to the heterotic string, with strong coupling (this is yet another example of duality). the new feature of braneworlds is that it allows particles/forces to exist on seperate branes and only able to communicate weakly via bulk particles such as the graviton. Although strings representing particles and forces can be trapped to branes there is no requirement to resort to them but braneworld theory does assist SUSY in solving the hierarchy problem. [The difficulty with this approach is that the SUSY needs to be broken if is to explain why we observe particles but not the (massive) superparners.] This problem results from the anarchic principle, and is due to the quantum contributions to the Higgs particles mass.
An alternative way out of this problem, is to assume that sequestrating particles on a separate brane can prevent these unwanted interactions that until now could only be restricted by symmetries. Basically by sequestrating the unwanted particles on a seperate brane the anarchic principle can be restricted. The graviton (or maybe the gauginos) are able to travel through the bulk and are responsible for communicating the SUSY breaking but since this breaking happens sufficiently far away it will have very little effect. This graviton induced communication of SUSY breaking is known as anomaly mediation. In this way the interactions of the Standard Model remain the same as in a theory with unbroken symmetry. So just as in a theory with exact SUSY unwanted flavour changing do not occur. [A refinement of this concept is that other particles responsible for flavour symmetry breaking are sequestrated on other branes, the breaking being communicated to Standard Model particles only via particles in the bulk Different flavours of quarks and leptons would be different because they interact with a different brane at a different distance (the further the distance, the smaller the mass induced), an effect termed shining] These bulk particles although originating/travelling in higher dimensions should still leave some sort of trace in our 4D world and are refered to as Kaluza-Klein KK particles (a term which is also used in the T dualities in M-theory, since in both cases, their energies a determned by the size of the extra dimensions - - branes may be curled up in M-theory ). If the extra dimensions are bigger the KK particles are lighter; if the world is higher dimensional but with no branes then all familiar particles will have KK partners. They would have exactly the same charge but carry momentum in the extra dimensions.
The next stage in development was the realization that extra dimensions rather than SUSY could explain the Standard model. This ADD idea postulates that larger dimensions might explain the (apparent) weakness of gravity and explain why the Planck scale and weak scale are so different - - thus solving the hierarchy problem [The hierarchy problem can be expressed in another way; why is the Planck scale so large when the weak scale is so small or equivalently why is the strength of gravity so weak] Essentiall the ADD model claims that the fundamental mass scale that determines gravity's strength is not the Planck scale mass but one much smaller (gravity's strength is inversely proportional to the Planck scale) In these and subsequent developements , the larger the extra dimension, the weaker the gravitational force in the effective lower dimensional universe would be. The initial spreading out in the extra dimensions would reduce the density of the force lines in lower dimensional space, so the strength of gravity experienced would be weaker. If a Tev were the typical energy of gravity, there would no longer be a hierarchy of masses in particle physics. the Planck scale mass that we measure in 4D is large only because gravity has been diluted in 'large' extra dimensions. If these ideas are true then it means that it would be possible to create black holes and KK particles at energies close to a Tev and also we may be able to detect a deviation of Newton's inverse square law at small distances.
Further developments involved warped passages connecting these branes - - the RS1 theory of Randall and Sundrum [a warped space is one in which the cross sectional dimensions have the same geometry (e.g. flat), while the extra dimensions have a variable curvature]. This comes about when we consider branes which themselves and the bulk space, have energy and can therefore induce a curvature as we move away into the extra dimension. By solving Einstein's field equation for a particular braneworld we obtain a gravitational probability function in which the strength deteriorates the further we move away from one of the branes gravity is seen to be confined on one brane (the Gravitybrane which carries positive energy ) and weakens as it heads towards the weakbrane (which carries negative energy) upon which all the standard particles reside. The warped space causes gravity to decrease at an exponential rate and in this way we do not require a large separation between the 2 branes and no contrived large number in order to explain the hierarchy. The huge ratio of the Higgs masses is expected if gravity (Planck scale), is confined near the gravitybrane, while the weak interaction is confined on the weakbrane some 16 units away (giving a reduction by 10^16)
In order to accommodate the apparent unification of the interactions that is indicated by the convergence of the strengths at higher energies, work has revolved around the application of strings which are allowed to move through the bulk These strings represent the various standard model particles and their gauge bosons. Unlike the graviton which must arise from a closed string, gauge bosons and fermions will be either stuck on a brane or free to move in the bulk. With the warped geometry model the extra dimensions are not so large and therefore these interactions are not diluted so much No longer tethered to the weakbrane they could travel anywhere in the bulk and have energies as high as the Planck scale. Only on the weakbrane does the energy have to be less than a Tev. Because these interactions would be in the bulk and therefore operate at high energies, unification of forces would be a possibility. Also the hierarchy problem in warped dimensions requires only that the Higgs particle be on a weakbrane, so that its mass will be about the same as the weak scale energy (the weak gauge bosons need not be stuck there but will automatically have the correct masses). So the weak scale would be protected (at ~1Tev), but unification could still occur at very high energies on the GUT scale, without invoking SUSY (but warped extra dimensions instead) The implications of this is that KK particles five dimensional black holes and strings could be observed at energies accessible to the new particle accelerators (tevatron and LHC)
A further developement known as RS2 involves only one brane in which the warped bulk space localises gravity via an exponential fall off. The standard model particles only act weakly with the gravity field. In this highly warped space an infinite extra dimension is possible and yet it would still appear as our 4D world
Matrix theory applies a 2 dimensional matrix property to each point in 10D stacetime and is used to explain the behaviour of Do-Branes which move through this space. Even though the theory does not explicitly contain gravity Do-Branes act like gravitons Furthermore the theory of Do-Branes mimics supergravity in 11 dimensions not 10. That is the matrix model looks as if it contains supergravity with one more dimensions than the original theory seems to describe. this has led string theorists to believe that matrix theory isxequivalent to M-theory.