Smolin's "The Future of Spin Networks" is a testament to Roger Penrose's mathematical genius in the modern theoretical physics of quantum gravity, topological field theory and conformal field theory. The spin-off from his twistor theory and his early attempts at quantum geometry is significant. Penrose's original spin networks for SU(2) have been extended to any Lie group G even to categories and most importantly to the Hopf algebras of the deformed "quantum groups". If Penrose's mathematical intuition is so good, how can we doubt his physical intuition that consciousness and quantum gravity are really two aspects of the same problem?
Clearly one needs to use the deformed "quantum spin networks" rather than the original "spin networks" to get post-quantum gravitational backactivity of the space geometry on its guiding quantum computing deformed spin network. It is not obvious that the (quantum) Penrose spin network is a (post) quantum (sentient) computer. That is only my hunch at this time.
To jump ahead for a moment: Quantum spinnets come from representations of quantum groups which are Hopf algebras generated by deformed Lie Algebras. The quantum spinnets do not correspond to gauge invariant states of classical connections. (p.19) There are deformed quantum 6j symbols. The possible spins on the edges cannot exceed k + 1. No comes the most important new feature which sounds to me like my post-quantum backactivity:
"... unlike Penrose's formula for the value of a spin network, their [i.e., the deformed quantum spinnets] CAN detect information about the embedding of the network in the spatial manifold." (p.19)
This is it! The spatial manifold is the Bohm-Bell "beable" in the pilot-wave version of quantum gravity. I have defined post-quantum backactivity as the direct transfer of information from the beable to its guiding pilot wave which in this case comes from the quantum spinnet.
Penrose introduced the spin network as a model for a discrete quantum geometry. What this means is not quite clear from the Bohm point of view. Is it a beable or a pilot wave? It must be a pilot-wave since it is a quantum gravity state but perhaps it has aspects of both since it is also, apparently, the beable 3-geometry in a macroscopic limit of large networks. I shall return to this. Smolin suggests the beginnings of a unified nonperturbative theory of quantum gravity and strings.
Penrose intially introduced spin networks as a quantum pregeometry for Euclidean 3-space. Smolin has used them as the kinematical structure for quantum general relativity. They are useful in Ken Wilson's lattice gauge theory. Their "deformed" extension to "quantum spin networks" are useful in topological field theory and conformal field theory as well as in quantum general relativity with a cosmological constant.
The Penrose spin network is a discrete combinatorial structure with no reference to continuous background geometry. Indeed, the latter arises from it as a kind of classical limit. Each piece of the spin network has a total angular momentum. So we are dealing with the SU(2) group. There is nothing like a direction in space at this pregeometric level. It is a trivalent graph whose edges are labeled by integers which are twice the total angular momentum of the edge. Angular momentum is conserved at each node or vertex of the graph. The spin networks that correspond to quantum states (and histories) have open ends described by a Dirac bra | >. To take the norm, < | >, take the mirror image, tie together the corresponding open ends to get the closed network. This norm has a number called its "value". The value is invariant under all identities for the coupling of angular momenta. For example, one can define 6j symbols combinatorically. The value of the norm can be expressed in terms of 6j symbols. The three dimensions of Euclidean space comes from a definition of probability based on the value of the norm in the limit of large spin networks. The generalized spin network is defined for any Lie group G. The consequent higher valence (beyond 3) nodes require "intertwiners". There is a further generalization based on Hopf algebras in the language of monoidal categories.
Spin networks appear in the lattice gauge theory of the fundamental forces. The basic graph is a cubic lattice in d dimensions. In general the graph has nodes nj and directed edges eij linking ni to nj. The same two nodes can be connected by more than one edge. Choose a compact Lie group G. A configuration assigns each edge to an element of G. If we use the Lagrangian-based Feynman path integral then the configurations are histories. If we use the Hamiltonian then we have a configuration space C which has one copy of G for every edge in the graph. A gauge transformation consists of a choice of group G element hi for each node ni in the graph with the map
gij -> gij' = hi^-1 gij hj
The space of all these gauge transformations gives a new group g. The axiom is that all observables are invariant under g, so the physical configuration space is the quotient space c
c = C/g
In the Hamiltonian approach, the quantum states of lattice gauge theory are functions on c. There is a natural inner product to make a Dirac bra-ket using the Haar measure of G. The Penrose spin networks provide an orthonormal basis for the quantum states of the lattice. First introduce an overcomplete set of states based on loops. Note the Glauber coherent states are overcomplete. A Wilson loop is defined for each loop. (p. 7 for details). The space of all Wilson loops is an over complete basis for c. The Penrose spin networks or "spinnets" form a complete orthonormal basis. Smolin is not clear on how to get from the Wilson loops to the spinnets. Gauge invariant quantum states are constructed from the spinnets. One gets a gauge invariant state | > which is a functional of gij. When G = SU(2), | > can be expanded as a product of Wilson loops.
Smolin then discusses the use of spinnets in nonperturbative quantum gravity models. Penrose lectured on twisters when I was at Birkbeck. I sat in on his seminar though I do not recall too much. Twistor theory showed the importance of self-duality for classical spinorized gravitational field dynamics. The reduction to either self-dual (left-handed) or anti self-dual (right-handed) parts yields exact solutions to general relativty in terms of consistency conditions on certain complex manifolds. (p. 8) For example, twistor space is a complex manifold. The same trick works for classical Yang-Mills field theory where the self-dual solutions are "instantons". These duality transformations in 3+1 space-time induce chirality transformations from left to right-handed spinors. General relativity was formulated in terms of chiral structures by Sen (p.9) The Hamiltonian constraint is polynomial in terms of the self-dual (left-handed) parts of the connection for parallel transport and the curvature. Ashtekar used the self-dual part of the connection as the basic configuration variable whose canonical conjugate "momentum" is a frame field. The Hamiltonian constraint is polynomial. This technique can also be used in the Lagrangian Feynman path integral method of histories rather than configurations.
At attempt to use lattice gauge theory to do nonperurbative quantum gravity was made. The space-time connection for parallel transport was the gauge field. The physical conjecture was that perturbatively non-renormalizable models correspond to fixed points of their renormalization groups. This led nowhere. Smolin and Crane tried to make a string theory from loops independent of a background metric. They also used a "fractal spacetime" in which nonperturbative effects lowered the effective dimension of spacetime passing through the Planck scale. That is the effective dimension of space would be less than three below the Planck scale. This is not the same as the curling up of extra dimensions in the Kaluza-Klein theories. With the work of Ashketar it became clear to Smolin et-al to construct a discrete geometry from Wilson loops made from the Sen-Ashketar connection. They could not realize continuum diffeomorphism invariance using the discrete lattice. A similar problem occurs with a fixed background metric since the diffeomorphisms play the role of a gauge group. Jacobson and Smolin succeeded with a continuum theory where they got an infinite class of exact solutions of the Hamiltonian constraint. The action is concentrated at the intersections of the Wilson loops. (p. 10) The Fock space of many-particle states of conventional quantum field theory requires a fixed background metric which renders it useless for diffeomorphism-invariant non-perturbative quantum gravity. The vacuum of QCD is a superconductor with quantized fluxes of the strong force fields. Smolin et-al replaced the Fock space with a space of states spanned by an over-complete basis which was made from finite products of discrete Wilson loops [traced holonomy]. (p.11, eq. 7) The non-abelian electric field flux is quantized in the QCD case. The formula for the quantized flux is proportional to an intersection number of the loop with the surface element. The loop does not intersect itself at the surface element. These discrete states represent a discrete geometry in Smolin's language. So what is the mental pilot-wave and what is the material beable in Bohm's language is quite ambiguous. As I said befoe it could be that at this pregeometric level the split into pilot-wave and beable has not yet occurred. This corresponds to Bohm's "super-implicate order" perhaps. Smolin speaks of solving the diffeomorphism invariance on this space of states of Wilson loops. These states are labeled by diffeomorphism classes of loops that include knots, links and networks.
"Thus knot theory emerged as being important for understanding the state space of quantum gravity." p. 12
It appears that the Hilbert type of state space emerges into the classical beable geometry in an approrpriate limit.
The flux operator has a square root of an operator product in its integrand over the surface. This requires regularization and all previous regularizations required a fixed background metric. Smolin says he succeeded in regularizing in a diffeomorphic invariant way and that the quantization of the non-abelian electric flux (i.e., quark confinement) in the Yang-Mills QCD case corresponds to the quantization of the areas in his quantum gravity model. Smolin also constructed a discrete volume operator for his version of quantum gravity. The volume counts things happening at points where three or more loops meet. This volume construction requires the Penrose spinnets which form a basis for the diffeomorphic invariant quantum gravity states. Trivalent spinnets are eigenstates of Smolin's quantum gravity volume operator with discrete eigenvalues. However they encountered a problem of zero volume eigenvalues for trivalent spinnets. (p.14)
"In any case, we had finally realized that the central kinematical concept in quantum gravity is that the space of diffeomorphism invariant states is spanned by a basis in one to one correpondence with embeddings of spin networks. The transformation to the loop representation can be done directly in the spin network basis. When one modes out the spatial diffeomorphisms, one is left with a state space which has an independent basis in one to one correspondence with diffeomorphism classes of embeddings of spin networks. ... we have arrive at a kinematical basis for quantum gravity that is discrete and combinatorial ... at the level of spatial diffeomorphism invariant states the [continuous] connections have completely disappeared."
There is a natural inner product. All operators are combinatorial and topological. "The diffeomorphism invariant quantities are finite with no divergences. (p.16) Things are simple for the area operators, but more ambiguous for the regulation procedures for the volume operators and the Hamiltonian constraint. There is still a problem with the continuum limit, one cannot get long range correlations. (p.17) The extensions of the original SU(2) spinnets connect up with the deformed quantum groups.
Topological quantum field theory has three forms: combinatorial, categorical and path integral. The basic model is that of a Feynman path integral with a Chern-Simon action S on a compact 3-manifold with a connection one-form for a gauge group. The action S is invariant under small gauge transforms, but transforms as
S -> S' + 8pi^2 n
where n is an integer winding number for large transforms. The theory is formally diffeomorphic invariant. The theory is interesting only for nonlocal operators involving loops where one gets knot invariants that depend on an integer-valued coupling constant. (p.18) There are divergences that require regularization of the Wilson loop. The regularization smears the loop into a "framed" strip or ribbon of finite width. This introduces extra degrees of freedom although the width is taken to zero in the end. This Chern-Simon theory allows the computation of the expectation values of spinnets. A spinnet is a sum of products of Wilson loops. The original Penrose spinnet is "deformed" into a "quantum spinnet" at this stage of topological field theory. Note, classical spinnets describe quantum theory, but quantum spinnets describe post-quantum theory with self-organizing backactivity - that is my "Sarfatti conjecture". The deformation parameter is "q". For the Chern-Simon model with coupling constant k
q = e^pi/(k + 2)
Quantum spinnets come from representations of quantum groups which are Hopf algebras generated by deformed Lie Algebras. The quantum spinnets do not correspond to gauge invariant states of classical connections. (p.19) There are deformed quantum 6j symbols. The possible spins on the edges cannot exceed k + 1. No comes the most important new feature which sounds to me like my post-quantum backactivity:
"... unlike Penrose's formula for the value of a spin network, their [i.e., the deformed quantum spinnets] CAN detect information about the embedding of the network in the spatial manifold." (p.19)
This is it! The spatial manifold is the Bohm-Bell "beable" in the pilot-wave version of quantum gravity. I have defined post-quantum backactivity as the direct transfer of information from the beable to its guiding pilot wave which in this case comes from the quantum spinnet.
The q-spinnets can distinguish left-handed from right-handed chirality.
The category theory of Chern-Simon starts with a closed 2-surface in 3-space split in half with boundaries S punctured by edges labeled by spins. (p.19) There is a finite dimensional Hilbert space for each such closed surface with labeled punctures. The topological field theory is in the relationships between these Hilbert spaces. "Cobordism" connecting two surfaces S and S' plays a role here. (p.20) This is a 3-manifold whose boundary is the union of S and S'. The spinnet meets the boundary at the punctures where the labels agree. This gives a linear map connecting the Hilbert spaces of S and S'. This yields base states, but there is a new kind of Berry phase effect for large diffeomorphisms for the deformed quantum spinnets not found in the original Penrose spinnets. Smolin then constructs invariants of the imbeddings of the quantum spinnets in compact 3-manifolds from the inner product of this topological field theory. This gives a deformed "value" for the q-spinnet sensitive to the topology of the beable 3-geometry and the imbedding (self-organizing backactivity?). The category part is in the relationship between the topology and the representation theory. (p.21) The finite dimensional Hilbert spaces relate to conformal field theory. Smolin's approach unites QCD, topological and conformal field theories and quantum gravity into a common conceptual framework.
"This circumstance reflects a deep mathematical relationship between the representation theory of quantum groups Gq at roots of unity and the representation of the corresponding loop group at level k." (p.21)
The Chern-Simon theory is important for constructing an exact physical state Psi of quantum general relativity with a cosmological constant.
Psi = e^k Chern-Simon action/4pi
Equation (19) p. 21 shows the relation between Newton's constant G, the cosmological constant L and the Chern-Simon k, i.e.,
G^2 L = 6pi/k
This physical quantum gravity state has a good classical limit i.e.,. a DeSitter spacetime for small cosmological constant, hence large k. So when Smolin talks of spinnet based physical states their classical limit are the beable geometries. How do we interpret this in terms of Bohm's pilot-wave ideas? The beable comes from the pregeometric analog to the pilot wave. Bohm did speculate about this in The Undivided Universe. Spinnets connect algebra, representation theory and topology expressed as tensor categories. Smolin then does some model calaculations for a finite region of spacetime with a "self-dual boundary condition" in Euclidean spacetime with no causal light cone structure and also in Minkowski blackhole spacetime with the relatively tilted light cones giving event horizons. The self-dual boundary condition means that the pullback of the self-dual two form of the metric to the boundary is proportional to the pullback of the self-dual part of the curvature. The constant of proportionality is kG^2/2pi = 3/L. For both signatures there is an algebra of area operators. With k an integer one can get the finite-dimensional Hilbert spaces of"conformal blocks" mentioned above. (p.23) The eigenvector spaces of these area observables are associated with the punctures of the surface by the edges of the spinnet. See eq. 22 p. 23. He comes to a conclusion that the dimensions of the Hilbert spaces of the areas saturate the Bekenstein bound of blackhole thermodynamics in the limit of infinite k. That is, there is an upper bound to the number of q-bits that can be squeezed into the Planck scale areas.
Dim of the Hilbert space = e^ const Area/Planck Length^2 (eq. 23, p.23)
Smolin's eq. 24 on p.24 for the physical state space of non-perturbative quantum gravity with self-dual boundary conditions in 3+1 dimensions uses a direct sum of conformal blocks of the Chern-Simons theory.
So non-perturbative quantum gravity has a kinematic basis for physical states that are 1-1 with embedded deformed q-spinnets that seem to have a self-organizing backactivity built in. That is, unlike ordinary quantum theory, these deformed quantum gravity (sort of Bohm pilot-wave) spinnets sense features of the embedding, like the chiral twists, whose classical limit is the 3-geometry "beable" with lapse and shift in the ADM method. Larry Crowell has another way to describe this. The dynamics of these spin networks should be "quantum computational". If Penrose's intuition is correct, the dynamics should be "noncomputational" or "nonalgorithmic" in the classical sense. One approach that is "outside of time" is to express the Hamiltonian constraint as an operator on q-spinnets. One needs regularization and renormalization of the constraint to get a space of exact physical solutions. (p.24). The second approach "inside of time" uses a time clock matter field so that we have a proper Hamiltonian, like in ordinary quantum mechanics, rather than a constraint as in the Wheeler De Witt equation for wave functionals on Wheeler superspace. The latter is like the configuration space in Bohm's pilot-wave theory of a simple many-particle system. One can imagine a fitness landscape for basins of attraction in Wheeler superspace. Indeed, this is exactly what Smolin does in his book, The Life of the Cosmos. Self-organizing post-quantum backactivity means that the fitness landscape shifts with the self-organizing emerging actual path of the 3-geometry beable. Baby universes mean that this actual path has a branching treelike fractal structure rather than a single curve. There is a third approach using the Feynman path inregral. (p.25) The continuous paths of Feynman histories are replaced by sums over 4d spinnets rather than the 3d spinnets that Smolin has used up to now. This, so far, is Euclidean signature i.e., Hawking's "imaginary time" with no causal light cones. Smolin has still another paper on how to introduce the causal structure into the pre-geometry.
Smolin's vision.
1) Degrees of freedom other than the metric, e.g. strings, should have a geometric intepretation.
2) Any discrete pregeometry has a critical phase transition, it could be like a non-equilibrium self-organized sandpile rather than a second order equilibrium, which explains why the classical limit is so many orders of magnitude larger than the Planck scale. There is no reason to require general relativity in the small only in the large. Yet Smolin depends on diffeomorphism invariance in the small which comes from the large Einstein classical equivalence principle.
3) There is a post-quantum theory X that is a pure algebra with no background metric, but with a classical limit that is 3+1 general relativity + matter fields
4) Perturbations around the classical limit of X gives string theory.
5) The kinematics of X is a representation of a deformed Lie algebra, i.e., a Hopf algebra for quantum groups deformed from the groups of perturbative string theory. The natural language for X is that of tensor categories.
6) X obeys the Susskind "holographic universe" idea and the Beckenstein information bound which come from dividing up the undivided universe using the category theory of topological quantum field theory. (p.27)
7) Classical geometry comes from a non-perturbative critical point of X. The basic pre-geometric post-quantum observables are areas and volumes with discrete eigenvalues and eigenvectors that are built from deformed extensions of Penrose's spinnets. The non-perturbative math is from that which allows a conformal field theory to give a perturbative string theory. The idea here, I suspect, is something like the 3+1 classical geometry in the large from the critical point arises from a 1+1 string pre-geometry at the Planck scale and below.
(8) The deformation mathematics in which the quantum-computing spinnets can sense their embeddings results in a self-organized criticality that I have described as post-quantum sentient backactivity. Smolin's idea is that Euclidean 4d-space is a "dead" equilibrium phase transition like that of a ferromagnet, while the Minkowski light coned causal spacetime is a non-equilibrium self-organized critical point transition. Something like this also explains the inner felt experience in our streams of consciousness - though Smolin does not say that explicitly. Penrose does say this implicitly.