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the Future of Spin Networks
by Dr. Jack Sarfatti
New Physics Reviews, Volume 1, Number 2, Version 0.1
Lee 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 2 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 back-activity 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 back-activity:
"... 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 back-activity" 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.
1
Penrose initially 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 pre-geometric 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 and 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 3 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 2 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 over-complete set of states based on loops. Note the Glauber coherent states are
over-complete. 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 although I do not recall too much.
2
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 spacetime 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 spacetime 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 space-time" in which nonperturbative effects lowered the effective dimension
of space-time passing through the Planck scale. That is the effective dimension of space would be less
than 3 below the Planck scale. This is not the same as the curling up of extra dimensions in the KaluzaKlein 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 nonperturbative quantum gravity.
The vacuum of QCD is a superconductor with quantized fluxes of the strong force fields. Smolin etal 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 before, 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)
3
It appears that the Hilbert type of state space emerges into the classical beable geometry in an
appropriate 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 nonabelian 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 3-ormore 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 arrived 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 3 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' + 8 π2 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 "spinet" 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 that
classical spinnets describe quantum theory, but quantum spinnets describe post-quantum theory with
self-organizing backactivity (i.e., that is my "Sarfatti conjecture"). The deformation parameter is "q".
For the Chern-Simon model with coupling constant k
4
q = e^ π /(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. Now comes the most important new feature (which sounds to me like my postquantum back-activity):
"... 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 back-activity 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 2 surfaces S and S' plays a role her3 (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/4π
Equation (19) p. 21 shows the relation between Newton's constant G, the cosmological constant L,
and the Chern-Simon k -- i.e.,
G2 L = 6 π /k
5
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 spinet-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 calculations for a finite region of space-time with a
"self-dual boundary condition" in Euclidean space-time with no causal light cone structure, and also in
Minkowski blackhole space-time 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 kG2/2π = 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 selfdual 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 back-activity 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 qspinnets. 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 4-D spinnets rather than the 3-D
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.
6
Smolin's vision.
1) Degrees of freedom other than the metric (e.g. strings, should have a geometric intepretation).
2) Any discrete pre-geometry 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 back-activity". Smolin's idea is that Euclidean 4-D 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, 1987)
Why John Bell preferred 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.
7
The elimination of this shifty boundary has always been for me 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 have 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 realize (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 space-time 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 speedof-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 oneway 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 1 part-in-100,000 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 EinsteinPodolsky-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
8
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. 14
According to Bell's definition of "locality", it doesn't matter if its violation is by a direct space-like
quantum action at a distance outside the local light cones of the detection events, or whether there is a
time-like or light-like "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" space-like and the backward-in-time pictures are
operationally equivalent. In his book The Unconscious Quantum, Stenger 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. Nut it is not a necessary condition. Postquantum mechanics is a nonlocal hidden variable theory with "nonlocal communication" as defined by
Stenger.
9
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. 20
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 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. 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 2-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 (i.e., 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.
10
"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
"back-activity" 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 measurements
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
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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 non-Relativistic 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 1 of the 2 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).
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Bell points to a surprising relationship between Everett's 'Many-Worlds' idea and the de BroglieBohm 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 Broglie's "determinism" does not survive the extension from Quantum to Post-Quantum
Mechanics because the "back-activity" 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 feedbackcontrol loop.
"Now these trajectories of de Broglie, innocent .... in configuration space, are really
very peculiar as regards locality in ordinary 3-space."
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