24 May 2009

1 May 2009

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.

Jack Sarfatti's reading notes on John S. Bell's "Speakable and unspeakable in quantum mechanics" (Cambridge, 19

Why John Bell prefered Bohm's ontological pilot-wave interpretation of orthodox quantum mechanics over Bohr's epistemological "Copenhagen interpretation""... despite numerous solutions of the [measurement] problem 'for all practical purposes' [i.e., 'FAPP'] , a problem of principle remains. It is that of locating precisely the boundary between what must be described by wavy quantum states on the one hand, and in Bohr's 'classical terms' on the other. The elimination of this shifty boundary has for me always been the main attraction of the [Bohm] 'pilot-wave' picture. ... all students should be introduced to it, for it encourages flexibility and precision of thought. In particular, it illustrates very explicitly Bohr's insight that the result of a 'measurement' does not in general reveal some preexisting property of the 'system', but is a product of both 'system' and 'apparatus'. It seems to me that full appreciation of this would have aborted most of the 'impossibility proofs', and most of 'quantum logic'" p.viii

Evidently Bell did not think much of David Finkelstein's "quantum logic" approach, discussed in Gary Zukav's The Dancing Wu Li Masters, which does not seem, now in hindsight, to of yielded much new physics. What happens in a self-measurement where the 'system' is the 'apparatus'? This is creative novelty where something that did not exist before comes into actuality. Indeed, this may be how our experience of time itself emerges in our stream of consciousness.

Continuing with Bell's thoughts on Bohm's pilot-wave/hidden-variable (beable) theory:

"While the usual predictions are obtained for experimental tests of special relativity, it is lamented that a preferred frame of reference is involved behind the phenomena .... Many students never reaize, it seems to me, that this primitive attitude, admitting a special system of reference which is experimentally inaccessible, is consistent .. if unsophisticated."

While the special relativity of globally flat is hostile to preferred reference frames, the globally curved, but locally flat, spacetime of general relativity is not so hostile to them. The Michelson-Morely experiment showed that the motion of the earth though the ether was undetectable. That is, the speed of light in vacuum is an absolute speed limit being the same number for all ordinary observers moving uniformly relative to each other. The equations of both special and general relativity obey this local speed limit. Special relativity can be pictured as a field of parallel invariant light cones. General relativity introduces nonparallel relative tilting of neighboring light cones giving things like the one-way horizons of black holes. However, the basic Big Bang expanding universe cosmological solution of the globally generally covariant and locally Lorentz-invariant field equations of general relativity does have a globally preferred frame of reference called the Hubble flow. This has operational meaning. The globally preferred rest frame of the universe is detected by the isotropy, to one part in one hundred thousand, of the cosmic blackbody radiation whose current temperature is a few degrees above absolute zero. Bohm has suggested that this is the frame in which the quantum potential acts instantaneously. This is adhoc, and the final understanding needs a proper theory of quantum gravity.

"Any study of the pilot-wave theory, when more than one particle is considered, leads quickly to the question of action at a distance, or 'nonlocality', and the Einstein-Podolsky-Rosen corrrelations..." p. ix

Bell rejects the many-worlds theory as well as the quantum logic theory as explanations of the meaning of quantum physics.

"My attitude to the Everett-de Witt 'many worlds' interpretation, a rather negative one ..." p. ix

Contrary to Victor Stenger's position in The Unconscious Quantum and to Murray Gell-Mann's position in The Quark and the Jaguar, who both, for different reasons, think that nonlocality is "the story distorted", Bell writing on the Einstein-Podolsky-Rosen (EPR) paradox says:"It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on the distant system with which it has interacted in the past, that creates the essential difficulty." p. 14According to Bell's definition of "locality" it doesn't matter if its violation is by a direct spacelike quantum action at a distance outside the local light cones of the detection events, or whether there is a timelike or lightlike "advanced" backward propagation of information from the future detection events to the past source pair emission event. Stenger prefers the later picture, however, what he does not understand is that both the "faster-than-light" spacelike and the backward-in-time pictures are operationally equivalent. Stenger, in his book, The Unconscious Quantum, also splits some verbal hairs between "locality", "separablility" and "completeness" which do not add any new understanding to Bell's more elegant presentation of the real physics problem.Bell summarizes the logic of the "incompleteness" argument of the original EPR paper of 1935 in the simpler Bohm "singlet spin" version in the following way. Assuming locality: "Since we can predict in advance the result of measuring any chosen component of sigma2 [i.e., the spin of particle 2], by previously measuring the same component of sigma 1 [i.e., the spin of particle 1 of the same individual pair], it follows [from locality] that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state." p. 15

The EPR argument shows that locality leads to a violation of the Heisenberg uncertainty principle for the twin particle in the pair that is not directly measured if we assume "counter factual definiteness" (CFD). That is, a measurement that could have been made but wasn't, would have had a definite result if it had been made. To see a popular discussion of how nonlocality for entangled quantum states is required to preserve the uncertainty principle see Heinz Pagels's The Cosmic Code. For a popular discussion on "counter factuals" and new experiments that confirm CFD see Roger Penrose's The Small, the Large, and the Human Mind. Note, that Bell shows that any local, i.e., "predetermined", hidden variable theory will violate the Heisenberg uncertainty principle which is a constraint on statistical fluctuations of incompatible observables in an ensemble of identical measurements. Therefore, any local hidden variable theory will violate the statistical predictions of orthodox quantum mechanics. Note that the post-quantum mechanics of consciousness that I profess does violate the statistical predictions of orthodox quantum mechanics but for entirely different reasons. That is locality is a sufficient condition to violate the statistical predictions of orthodox quantum mechanics, but it is not a necessary condition. Post-quantum mechanics is a nonlocal hidden variable theory with "nonlocal communication" as defined by Stenger.

Bell on "hidden variables": "In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant." p. 2

To balance out Victor Stenger's premature skeptical certitude in his book, The Unconscious Quantum, I include relevant remarks by John Bell, with Michael Nauenberg who I knew at Cornell and UCSC, on the role of consciousness, the universe and the impossibility of self-measurement in orthodox quantum mechanics:

"this assumes that the intermediate evolution ... is governed entirely by the Schrodinger equation, and therefore that the pointer position is not looked at until after the final interaction. If the pointer position is observed just after each interaction then the moral process comes into play... from the theorist's point of view ... the experiment may be said to start with the printed proposal and to end with the issue of the report. For him the laboratory, the experimenter, the administration, and the editorial staff of the Physical Review, are all just part of the instrumentation. The incorporation of (presumably) conscious experimenters and editors into the equipment raises a very intriguing question. For they know the results before the theorist reads the report, and the question is whether their knowledge is incompatible with the sort of interference phenomena discussed ... If the interference is destroyed, then the Schrodinger equation is incorrect for systems containing consciousness. If the interference is not destroyed the quantum mechanical description is revealed as not wrong but certainly incomplete. We have something analogous to a two-slit interference experiment where the 'particle' in any particular instance has gone through only one of the slits (and knows it!) and yet there are interference terms depending on the wave having gone through both slits. Thus we have both waves and particle trajectories as in the de Broglie-Bohm 'pilot wave' or 'hidden parameter' interpretations of quantum mechanics .... It is easy to imagine a state vector for the whole universe, quietly pursuing its linear evolution though all of time and containing somehow all possible worlds. But the usual interpretive axioms of quantum mechanics come into play only when the system interacts with something else, is 'observed'. For the universe there is nothing else, and quantum mechanics in its traditional form has nothing to say. It gives no way of, indeed no meaning in picking out from the wave of possibility the single unique thread of history.

These considerations, in our opinion, lead inescapably to the conclusion that quantum mechanics is, at best, incomplete. We look forward to a new theory which can refer meaningfully to events in a given system without requiring 'observation' by another system. The critical test cases requiring this conclusion are systems containing consciousness and the universe as a whole. Actually, the writers share with most physicists a degree of embarrassment at consciousness being dragged into physics ... It remains a logical possibility that it is an act of consciousness which is ultimately responsible for the reduction of the wavepacket ... What is more likely is that the new way of seeing things will involve an imaginative leap that will astonish us..." pp. 25-27

Henry Stapp has a post-quantum ontological collapse of the mental quantum wave function of the brain which is not caused by consciousness the way Wigner meant it, but, rather, which explains the inner experience of consciousness. What is important about Stapp's picture is that it is a self-measurement. Similarly for Penrose's "orchestrated self objective collapse". I have a "neural network" way of looking at this, using a post-Bohmian picture, where the self-organizational aspect is more obvious than in Stapp's or Penrose's picture. The big change from Bohm's picture is that there are no empty branches of the wavefunction in the self-measurement. This is a point emphasized to me by Stapp and it is crucial. The self-organizing loop consisting of "backactivity" with the Bohm force, self-consistently determines the momentary observable and its actual eigenfunction that forms a basin of attraction in configuration space at each moment of consciousness for the self-measuring "isolated" (e.g. Penrose) conscious mind-brain system pumped by sensory input. This measurement is happening inside the system consistent with the introspective nature of our, so far, private inner experiences. It is no accident that Bell in the above quote lumps quantum cosmology and conscious systems together, for an analogous self-measurement is happening in quantum cosmology. See Lee Smolin's The Life of the Cosmos on the latter problem.

Bell characterizes the Copenhagen interpretation of quantum reality, with only an epistemological wavefunction for statistical ensembles of identical simple systems, as "subjective". The classical description is "objective". p. 29 He finds the fuzziness of the "Von Neumann cut" boundary between quantum and classical realities to be "surely of a provisional nature". Bell has several reasons for the existence of "hidden variables".

"A possibility is that we find exactly where the boundary lies. More plausible to me is that we will find that there is no boundary. It is hard for me to envisage discourse about a world with no classical part - no base of given events, be they only mental events in a single consciousness to be correlated. On the other hand, it is easy to imagine that the classical domain could be extended to cover the whole. The wavefunctions would prove to be a provisional or incomplete description of the quantum mechanical part, of which an objective account would become possible. It is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called 'hidden variable' possibility. ... A second motivation .. it can be conjectured that the seemingly random statistical fluctuations are determined by the extra 'hidden variables' ... we ... because at this stage ... we certainly cannot control them ... the possibility of determinism is less compelling than the possibility of having one world instead of two ... A third motivation is .... the famous argument of Einstein, Podolsky and Rosen .... Thus we can know in advance the result of measuring any component of sigma2 [spin] by previously, and possibly at a very distant place, measuring the corresponding component of sigma1. This strongly suggests that the outcomes of such measurments, along arbitrary directions, are actually determined in advance, by variables over which we have no control ... There need then be no temptation to regard the performance of one measurement as a causal influence on the result of the second, distant, measurement. The description of the situation could be manifestly 'local' ... We will find, in fact, that no local deterministic hidden-variable can reproduce all the experimental predictions of quantum mechanics." pp. 30-31

Bell wrote the above in 1971. Since that time Eberhard et-al have removed the restriction to "determinism" a fact that Stenger seems not to have noted in his recent book, The Unconscious Quantum -- unless I am mistaken?

Bell considers nonrelativistic particles with "spin". The particle's position is a hidden variable.

"We have here a picture in which although the wave has two components, the particle only has position ... The particle does not 'spin', although the experimental phenomena associated with spin are reproduced. Thus the picture resulting from a hidden-variable account ... need not much resemble the traditional classical picture ... The electron need not turn out to be a small spinning yellow sphere.

A second way in which the scheme is instructive is in the explicit picture of the very essential role of the apparatus. The result of a 'spin measurement', for example, depends in a very complicated way on the initial position .. of the particle and on the strength and geometry of the magnetic field. Thus the result of the measurement does not actually tell us about some property previously possessed by the system, but about something that has come into being in the combination of system and apparatus. ... the present 'quantum theory of measurement' in which the quantum and classical levels interact only fitfully ... should be replaced by an interaction of a continuous, if variable, character..." pp. 35-36

Looking at the many-particle problem in Bohm's pilot-wave model of quantum reality "one sees that the behavior of a given [hidden] variable ... is determined not only by the conditions in the immediate neighborhood (in ordinary three-space) but also by what is happening at all the other positions ... That is to say, that although the system of equations is 'local' in an obvious sense in the 3n-dimensional space, it is not at all local in ordinary three-space. As applied to the Einstein-Podolsky-Rosen situation, we find that this scheme provides an explicity causal mechanism by which the operations on one of the two measuring devices can influence the response of the distant device. This is quite the reverse of the resolution hoped for by EPR, who envisaged that the first device could serve only to reveal the character of information already stored in space, and propagating in an undisturbed way towards the other equipment." p. 36

This objective nonlocality is crucial to the understanding of how the quantum thought field organizes and synchronizes separated parts of the brain, and indeed, the whole living body IMHO. The Eccles gates linking mind to matter in the brain appear to be the web of isolated control electrons that couple to the conformations of the individual protein dimers in the microtubule infrastructure. In ordinary quantum mechanics there is no possibility of nonlocal communication between these spatially separated electrons even though their collective behavior is nicely globally coordinated above and beyond classical signalling mechanisms. This is because there is no nonrandom self-organizing stability mechanism in ordinary quantum mechanics. There is such a stability mechanism in post-quantum mechanics which is predicted to come into play when the Eccles gate control electrons are sufficiently isolated from external random decoherence as discussed, for example, by Roger Penrose in The Large, the Small and the Human Mind.

Bell points to a surprising relationship between Everett's many-worlds idea and the de Broglie-Bohm pilot-wave: "the elimination of arbitrary and inessential elements from Everett's theory leads back to, and throws new light on, the concepts of de Broglie." p. 93

"... there are infinitely many different expansions ... corresponding to the infinitely many complete sets ... Is there then an additional multiplicity of universes... ? I think (I am not sure) the answer is no, and that Everett confines his interpretation to a particular expansion ... Everett's structure is based on an expansion in which instrument readings R ... are diagonalized. This preference ... is not dictated by the mathematical structure ... It is just added ... to make the model reflect human experience. The existence of such a preferred set of variables is one of the elements in the close correspondence between Everett's theory and de Broglie's -- where the positions of particles have a particular role." p. 96

"(1) Whereas Everett's special variables are the vaguely anthropocentric instrument readings, de Broglie's are related to an assumed microscopic structure of the world ....

(2) Whereas Everett assumes that all configurations of his special variables are realized at any time, each in the appropriate branch universe, the de Broglie world has a particular configuration. I do not myself see that anything useful is achieved by the assumed existence of the other branches of which I am not aware. ...

(3) Whereas Everett makes no attempt, or only a half-hearted one, to link successive configurations of the world into continuous trajectories, de Broglie does just this in a perfectly deterministic way..." p. 98

De Broglis "determinism" does not survive the extension from quantum to post-quantum mechanics because the "backactivity" from the sufficiently isolated classical "beable" to its attached pilot-wave introduces the qualitatively new feature of adaptive self-determination, or self-organization, which is non-computable in Penrose's sense. Even though the beable is isolated from random environmental decoherence enabling it to quantum compute, there are non-random secular changes from the I/0 sensory devices feeding information into the self-organizing sentient post-quantum feedback-control loop.

"Now these trajectories of de Broglie, innocent .... in configuration space, are really very peculiar as regards locality in ordinary three-space."

1 May 2009

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.

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