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Cartoon proof that Stapp was correct that each eigenfunction in Hilbert space creates only one chaotic attractor in classical configuration space for a "good" measurement where the observer is not the same as the observed.
Trial by Jury by Gilbert and Sullivan
On Sept 11, 1996 lawrence barr crowell wrote:
Here might lie the core of the debate.I think the core of the present debate between me and Stapp may be explained by a few simple gifs above which actually was stimulated by one of your remarks that it was nontrivial to show the 1-1 correspondence between the branches of the wave function in quantum Hilbert space and the attractors for the motion of the hidden variable beables in classical configuration space. This prompted me to look at the problem more closely. The crucial distinction is, are we talking about the Hartree mean field approximation in which we first integrate over all Y, or are we talking about an actual individual correlation observation in which we actually look both at where the measured particle landed on the screen and what the actual pointer position was! The Hartree approximation is when we simply look at where the particle landed on the screen without looking at the actual pointer. My 4 gifs, Fig 4 for the Hartree type of observation only, show very simply for a gedankenexperiment i.e., a one-dimensional double slit in which "Which slit 1 or 2?" is measured, that one cannot make the 1-1 correspondence between branches of the wave function and the attractors IF you do a Hartree type experiment in which you do not actually measure all the measuring pointers. You can make the 1-1 correspondence that Bohm, Hiley (and Stapp) take for granted only in a correlation experiment where you observe ALL the pointers as well as the particle. Of course, for many practical experiments such a complete correlation detection is practically unattainable because the number of pointers is too large and the Hartree approximation is what is relevant.
What my gifs simply show is that, if you do NOT actually know what Yo is, the quantum force -dQ(Xo,Yo)/dX on the measured beable at Xo depends upon BOTH X-wave packets from the 2 slits which still overlap in the X configuration subspace, even though there is no overlap in the larger X-Y configuration space. This is because you must first integrate |psi(Xo,Yo)| over all Yo before computing -dQ(Xo,Yo)/dx. The total system point is not definite, but is smeared over a fuzzy line in the total configuration space in this more usual measurement situation.
Suppose that after a good von-Neumann measurement we have the un-normalized entangled pair state
Remember all of these psi's are ontological, not epistemological as in Stapp's Heisenberg potentia collapse theory. In Bohm's theory, there is no objective decay in the ZERO-backaction orthodox limit. So here we have a simulated "collapse without collapse". Bohm and Hiley argue, in The Undivided Universe, that effectively either psi(X,1)psi'(Y,1) or psi(X,2)psi'(Y,2) is active. The total system point is in either one or the other total branch. This is OK if you are only talking about a complete correlation measurement detecting every relevant pointer, but it fails at the statistical level where you have to integrate over at least one pointer.
So, if we stay at the individual level for both X and Y beables, and compute the quantum force from Q(Xo,Yo) from PSI(Xo,Yo), then it IS true that if the hidden variable Xo is in psi(X,1), then, the hidden variable Yo must be in psi'(Y,1) in order to get a non-zero quantum force -dQ(Xo,Yo)/dx on X. So the remark by Bohm and Hiley only works if you do a complete correlation type observation in which the actual positions of all pointers are recorded in the individual nonlocal quantum event.The situation is rather complex and subtle and depends very sensitively on the precise experimental question asked. Yes, Stapp is correct that you do not need back-action on this ideal complete correlation detection level that is seldom attainable in practise. My point is that you do need back-action if you want to make a similar 1-1 association of branch to attractor in the more common Hartree type of measurement in which the number of pointers is too large to be directly detected. This is probably what you were intuiting. Of course, Stapp does not have this problem in his Heisenberg potentia theory because there is a real collapse in that case.
Crowell wrote:
I have interpreted back-action as the quantization of strange attractor physics. Here the quantum hydrodynamic fluid, as given by the imaginary part the the Schrodinger equation is going to be nonconserved. There is a dissipative term that decreases its volume plus a driving term that increases its volume. The tendency will be for the QHF to become a fractal set. Similarly the particle under the infuence of the classical potential and the quantum potential will exhibit chaotic dynamics. Yet, since the two equations are coupled then for some systems the wild behavior of the particle and the QHF might act as negative feed back on each other and result in coordinated or self-controlled behavior.
OK - good.
Whether back-action results in modifications of Eberhard's theorem or reinterpretations of nonlocality is yet to be determined. As yet I see nothing that has been proven.
No, it was shown in Valentini's Phd dissertation at Cambridge under Dennis Sciama. The reference is in Brian Josephson's web pages that you can get to from http://www.hia.com/hia/pcr/z.html
Crowell
The Orch-Or idea is actually quite similar to back-action in some ways. The idea of orchestrated collapse is similar, but with the Bohr formalism you have to invoke idempotent projectors and the like that lie outsdie the domain of QM's operators. Either that or you have to have superscattering operators if you accept the idea that quantum black holes in the vacuum are responsible for this.
Date: Tue, 27 Aug 1996 11:44:40 -0700 (PDT)
From: STAPP@theorm.lbl.gov
To: sarfatti@well.com
Dear Jack,JS: Brief Preface by Sarfatti. (Aug 24)Your message of Aug 25 indicates that you have come to a better understanding of my words. To stay focused on the basic issue I first answer the question you raised at the beginning of your August 24 message, and then go onto the two key questions from the August 25 message.
Henry keeps getting back to the basic question, why is back-action required for conscious experience? In his orthodox quantum model it is not.
HPS Reply:It is important to distinguish my (Bohr/Heisenberg/von Neumann/Wigner) collapse model from the Bohm+CS model that I have described in earlier posts. This Bohm+CS model has consciousness appended to the Bohm model in the same way that it is normally appended to classical physics. It has no back-action and no collapse: it is thus just the Bohm model with consciousness appended.
As Jack says, my collapse model has no back-action. But it does have collapses, and collapses are just what Sarfatti's back-actions generate.
JS: But is it the same kind of "collapse"? I don't think so. Let's consider orthodox quantum theory. Your collapse is the traditional epistemological collapse that obeys the Born probability interpretation (i.e., probability of a branch actualizing is the squared modulus of its complex projection on the eigenfunction of the measured observable(s)). This kind of orthodox collapse is what Penrose calls "R" in his popular books. "R" is for "random". This kind of collapse is uncontrollably random. It is not the controllable "orchestrated" kind that Penrose gives the different name "orch OR" to. Bohm professed to be able to simulate the orthodox R-collapse as "collapse-without-collapse" (my Wheelerism) with his hidden-variable theory without back-action. However, I have only recently discovered in this interaction with Stapp, that Bohm only half-succeeded. His 1952 model, without back-action, does indeed reproduce all of the classic statistical predictions of orthodox quantum theory as claimed. However, as first pointed out to me by Crowell in a general way, the idea, which Stapp takes for granted, that the basins of attraction in the classical mechanical 3n configuration space, for the actual path of the material hidden-variable, or "beable", system point, is determined, one-to-one, by the eigenfunctions of the measured observable is not generally true Unfortunately, this is precisely the feature that is important for Chalmers's "hard problem" of the physics of consciousness. Stapp assumes it in his Bohm + CS model, but, unless I am mistaken, it is not mathematically true.
HPS: In my collapse model these collapses are required for conscious experiences. Analogously, Sarfatti takes the collapses that arise from his back-action to be the prerequisite of consciousness. So his model is basically the same as mine on this score: in both models a collapse is the prerequisite for consciousness.
JS: But I think we mean "collapse" in two very different meanings. You mean Penrose's "R", and I mean Penrose's "orch OR", but without his direct appeal to quantum gravity and "single graviton" emission which seems hard to swallow. I will use "controlled objective decay" i.e., "COD" instead of "collapse", or "orch OR". COD has some things in common with orch OR, but they are not identical in terms of the alleged role of quantum gravity.
HPS: But in the Bohm+CS model there is no such collapse, or collapse requirement, for consciousness.
JS: Yes, and as I shall argue below, I suspect that the Bohm+CS model is, if not inconsistent, is seriously ambiguous on a crucial matter of physical interpretation. I, tentatively, argue that it does not work. In contrast, I think the Bohm+back-action model does work. Furthermore, I believe your (Bohr/Heisenberg/von Neumann/Wigner) collapse model is internally inconsistent for different reasons, and, therefore, it also does not really work.
HPS: My question was why does Sarfatti insist on having both a classical world AND back-action/collapse, when either one alone is enough to do the job of reconciling the quantum character of nature with the classical character of our experiences of nature:
JS: Because I strongly suspect the statement you just made is not true. On the other hand, I may be wrong. Let's wait till we reach "the crisis point" below in equation (8b). There is a serious problem in the "informal language", i.e., the interpretation.
HPS: ... the classical world does the job in the Bohm+CS model, by being the reality that our thoughts are tied to, and collapses do the job in the collapse models.
JS: Your random R-type uncontrollable collapse leaves no room for active intentional consciousness able to make morally responsible freely-willed choices. Consciousness is, at best, a passive epiphenomenon in your model which is neither complete nor consistent.
HPS: Thus having both a classical world AND collapses is redundant, as far as this essential task is concerned.I shall address later the issue of the need for back action for self organization and adaptive behaviour.
In response to your remarks of Aug 25 let me say right away that the assumption in my Bohr/Heisenberg/vonNeumann/Wigner collapse interpretation is certainly that the mind/brain is in your words (Aug 25) a self-measuring system: there is no device around that is supposed to measure it, and no infinite regress.
JS: Yes, I agree that the self-measuring Godelian "strange loop" feature is required to avoid the Dennett-Ryle type of objections that you allude to in the term "infinite regress" which you explain in more detail in your book, Mind, Matter and Quantum Mechanics.
HPS: That [self-measuring] has certainly been central to my approach: the actualization of some particular pattern of brain activity is the physical side of the event whose other side, or aspect, is the experiential event. This is the vonNeumann/Wigner part of my theory.
JS: This for me is "and then a Miracle Occurs". Here we have a wavefunction world of thought-like "potentia" where all things are possible (almost) and where there are no classical material beables, hence no "particular pattern of brain activity", only to randomly and uncontrollably collapse, i.e., to actualize to "some particular pattern of brain activity". In this view the wavefunction is complete in the quantum reality of thought-like potentia. Let's contrast Stapp's purely idealistic, indeed, Hegelian epistemology with Bohm's neo-Marxist ontology.
Note: This is where an impish memory from almost forty years ago of my Cornell course with John Rawls on Hegel and Marx inserted itself. I don't often remember details of such courses and the names of professors, so Rawls obviously was a very good teacher. I believe that, like Phil Morrison, he wandered away from Ithaca to MIT. "Marx turned Hegel upside down." Compare Bohr to Hegel and Einstein to Marx. That's "Karl" not "Groucho".
HPS: The collapse models connect certain collapses associated with human brains to human conscious experiences.
JS: Fine, and this is an axiom. True, in the several Copenhagen-type purely epistemological interpretations of quantum theory, this axiom is put in at the very core of the theory, but it is still an interpretive piece of "informal language" not required by the mathematical formalism alone. We should not lose sight of this fact in comparing the competing models.
HPS: In these models the classical character of human thoughts is a consequence of the fact that the relevant collapses are to states where certain key bulk properties of the newly created brain state are `classically describable': the brain state corresponds to the observer's perceiving an alive cat, not a dead cat, or perceiving the pointer on a device to be pointing to the right, not to the left. These alternative possible perceptions, like some thinker's ideas of `to be' or `not to be', or of `freedom' or `slavery', will, according to the precepts of these collapse models, be represented by different `branches of the wave function' in the sense of Bohm's or von Neumann's theory of measurement: they will represent different brain states that are distinguishable by the fact that they correspond to different, and well separated, intervals in the values of certain macroscopic and slowly changing---on the scale of neural dynamics---variables. The separation in these variables entails that the alternative brain states are orthogonal.
JS: This is Stapp's core statement. It is complex and we must be sure we understand it. First he says: "These alternative possible perceptions, like some thinker's ideas of `to be' or `not to be', or of `freedom' or `slavery', will, according to the precepts of these collapse models, be represented by different `branches of the wave function'...". Stapp then says: "In these models the classical character of human thoughts is a consequence of the fact that the relevant collapses are to states where certain key bulk properties of the newly created brain state are `classically describable'..." This is the wonderfully poetic, romantic, Hegelian mystical notion that a thought-like quantum reality of "potentia", of coherently superposed classically disjoint possible classical realities miraculously actualizes, in a sort of downward causation, to only one classical reality that corresponds to a definite classical brain state. This brain state, on the neuronal level would be a definite configuration or firing pattern of some sort, possibly with other chemical transport parameters. What about at the microtubule level? Can we say that any given protein dimer is either open or closed? Yes, we must be able to say that like we could for any state of a classical cellular automaton.
Stapp then says "they will represent different brain states that are distinguishable by the fact that they correspond to different, and well separated, intervals in the values of certain macroscopic and slowly changing---on the scale of neural dynamics---variables. The separation in these variables entails that the alternative brain states are orthogonal." This is where the trouble begins. This is tricky, so let's hone it down some more to its bare essentials. Stapp says: "These alternative possible perceptions, ... will ... be represented by different `branches of the wave function'...". This is the idea that "perceptions", or felt-experiences, or experiences of "qualia", are in one-to-one correspondence with different "branches of the wave function".
What are these "branches"? The orthodox quantum theory of measurement assumes that "the total experimental arrangement" selects at least one observable property p that is "measured". There may also be other "compatible" properties that can be measured simultaneously with arbitrary precision. The Heisenberg uncertainty principle says that there are also incompatible properties q that cannot be so measured. The product of the precisions of p and q is a constant so as one gets better the other gets worse in inverse proportion. Remember, the "precision" is not the same as the "accuracy" of a measurement. The precision is the root mean square fluctuation when a large number of the same kind of measurement is made on a statistical ensemble of identically prepared simple systems. The observable property p, called the "eigenvalue" is represented by a Hermitian operator P. The "branches" are the "eigenfunctions" |p) of P. That is:
Each eigenfunction |p) is an "axis" or a linearly independent "dimension" or a "choice" in the quantum Hilbert space H. The complete orthonormal set of choices |p) is a quantum frame of reference or perspective in H. This is quite analogous to the Lorentz frames of spacetime. Compatible physical properties are represented by commuting operators, incompatible ones by noncommuting operators. For example:
where h is Planck's constant and i^2 = -1.
The first question Q1 is: "What is the measurement observable P for the quantum mind/classical brain problem?" In other words: "How is P selected?" This is non-trivial.
So let's review, we have the quantum state of the brain which is a coherent superposition of thought-like mutually exclusive "potentia":
where I use the summation convention over repeated indices like "p". In Stapp's orthodox "collapse" theory of the quantum mind/classical brain, the Born probability rule fits into the von Neumann projection axiom to become the statement that defines "R" in Roger Penrose's notation. That is, the probability of a "collapse" in which the coherent superposition of mutually exclusive "potentia", or in this case "qualia", miraculously actualizes to a single conscious inner private felt-experience of only one single eigenfunction or "branch" |p) is
for randomly uncontrollable R-collapse where * is the complex conjugate. Penrose's "orch OR" collapse is very different, and it is the kind of collapse that corresponds to the nonrandom controllable collapse of my back-action theory. Equation (4) is violated in "orch OR". Indeed, the degree of violation is an objective measure of the strength of "intent" or morally responsible "free will". It is this violation which transforms consciousness from a passive not responsible epiphenomenal impotent observer to the active responsible observer-participator. Violation of equation (4) is a violation of orthodox quantum mechanics. It is really new physics.
It should be understood that Bohm's 1952 pilot-wave/hidden variable quantum theory says that orthodox R-collapse is only an illusion. What is happening is that there is "collapse without collapse" in which the hidden variable or beable is controlled by only one branch |p) of the wavefunction. This is where Bohm also made a major error. In fact it is the same error Stapp makes, and it is the same error I had made. In contrast, the back-action and "orch OR" type collapse is an objective decay of the branches much like what happens in the build up of a single dominant mode of coherent electromagnetic oscillation in a laser or a Frohlich mode in a microtubule.
This inner private subjective quantum mental experience is in one-to-one correlation with the outer publically accessible objective classically material brain state in Stapp's theory.
My second question Q2 is: "How does this random R-collapse permit intention, or free will?"
A related third question Q3 is: "We never have a statistical ensemble of identically prepared complex brains. How do we make a quantum theory of individual complex systems?"
Let's back up to Stapp's remark:
"These alternative possible perceptions ... be represented by different `branches of the wave function' ... they will represent different brain states that are distinguishable by the fact that they correspond to different, and well separated, intervals in the values of certain macroscopic and slowly changing---on the scale of neural dynamics---variables. The separation in these variables entails that the alternative brain states are orthogonal."
OK, now we get to the nitty gritty. In what sense are "the alternative brain states ... orthogonal"? Now I know what it means to say that two quantum states |p) and |p') are "orthogonal" in H when p is not equal to p'. There is an inner product or "bra-ket" (...|...) defined. In this particular case orthogonality means
Suppose P is an observable of a complex system like the brain that consists of many components spread all over the physical space in the cranium. Just to make our example more concrete, let's go down to the sub-microtubule level, to the single electrons in thermally protected hydrophobic cages inside each protein dimer biocomputer switch. These single electrons control the classical shape, "open" or "closed", of each protein biological nanometer-transistor depending on their position. These electric dipole spatial displacements of charges form the collective Frohlich modes. In any case, these n electrons spread out all over the brain exist as a single "beable" system point X in a classical mechanical 3n-configuration space C3n(x). The variable x forms all the points of the space of all possible classical configurations of the complex beable. Each local point x in 3n-space is spread out over many points r in physical 3-space. Each 3n-point x includes all degrees of freedom not only the position of the electron. Therefore, x's dimension is generally larger than 3n which only includes the center of mass coordinates. For example, x could include the spin of each electron along a definite direction in physical space for example. The set of spins would form a binary quantum string of bits. Bohm's and Stapp's basic wrong idea here is that there is a one-to-one correspondence between the branches |p) of the quantum wave function |Brain) and the basins of attraction or "attractors" in C3n(x). These are the same "attractors" that appear in classical mechanical chaos theory. They can be fractal strange attractors. Classical chaos theory uses 6n-phase space which includes the momenta. Bohm's 1952 pilot-wave/beable quantum theory only needs the 3-n positional subspace because the momenta of the beables are determined by the gradient of the phase of the pilot-wave that guides their motion. Before I prove that this idea is wrong mathematically, lets see what else Stapp has to say:
HPS: Bohm's theory of measurement, like von Neumann's, is based on the idea of different branches, which correspond to different `pointer positions'. For a `good' measurement-type physical situation, the different pointer positions occupy different positions in 3-space, and, in any case, different regions in the 3n-dimensional configuration space. Hence they are orthogonal.
JS: OK, so we see that by "orthogonal" classical brain states, Stapp means "different regions in the 3n-configuration space C3n(x) of the hidden-variable beable in Bohm's theory. But what does different mean. I will show below that "different" must mean zero overlap for all subspaces of C3n(x). Only then can we properly use the word "orthogonal" in C3n(x). There is no hidden-variable in Stapp's collapse theory, but the idea is that the potentia in quantum Hilbert space H actualize such classical states in C3n(x). Each possible classical state is characterized by a basin of attraction, or simply, an "attractor" A. Stapp's (and Bohm's) key assumption is that only one measurement eigenfunction |p) in H corresponds, one-to-one, with one attractor A(p) in C3n(x). This is the error.
The word "orthogonal" has different meanings in quantum Hilbert space H and in classical configuration space C3n(x). The quantum-pilot wavefunction's domain is C3n(x) and its range is H. I am only using nonrelativistic quantum mechanics here with Galilean symmetry since the brain is nonrelativistic to a good approximation. Orthogonality of the |p) basic states in H means that the global integral over the entire C3n(x) space "/dx"
when p is different from p'. Note that (p|x) is the projection of the |p) frame of reference on the |x) frame of reference. In Dirac's language (p|x) is the "x-representative of the bra (p|." Note, that two orthogonal base states can locally overlap in C3n(x) and still be "orthogonal" in H. In other words orthogonality in Hilbert space of the mental states is a global property not a local property. In contrast, what does it mean for two classical attractors A and A' to be "orthogonal" in C3n(x)? This is a tricky issue and neither Stapp or Bohm paid enough attention to it. It is too strong to say that the basins of attraction cannot have any overlap. A weaker condition is that the actual attractor structures have no overlap in any subspace. The important idea is that classical orthogonality of the brain states is a local property of C3n(x) space. Returning to Stapp:
HPS: In the brain there are billions of such pointers. Each neuron has a huge number microchannels with, it is believed, some sort of macro-molecular gate that can be open or closed, with the state (open or closed) of these gates coordinated to the action-potential pulses flowing along the neurons.
JS: So a "pointer" is some kind of a gate or switch or "biocomputer transistor". The protein dimers in the microtubules are also "pointers" in Stapp's sense. Quantum pointers like the single electrons that control the conformation of each tubulin dimer are "Eccles gates" where quantum mind directly couples to its attached classical brain-beable substrate, using Bohm's picture. The Bohm quantum force is how quantum thought changes classical brain-beable, and back-action is how the beable changes the thought. In orthodox quantum mechanics there is no back-action, therefore, there is no direct fast way for changing classical brain states, open to external environmental signals, to directly change the thought structure. There are indirect slower ways for the brain to change its mind by first changing the Hamiltonian and the boundary conditions on the Hilbert space. Stapp thinks this is enough. I do not. Stapp continues:
HPS: Different patterns of quasi-stable neural activity will have different patterns of open and closed micro-channels, and will thus be orthogonal, so that the Bohm/von Neumann/Wigner theory will apply. If even just one such microchannel is open in one state but closed in the other then the two states will be essentially orthogonal.
JS: OK, this is the key point. What Stapp, and also Bohm, say here is true for "orthogonality" in quantum Hilbert space H as defined by equation (5b) above, but it is not true for "orthogonality" of the classical brain configurations of the "pointers" in C3n(x) space. Neither Bohm, nor Stapp, drew this distinction carefully enough. This is probably the place to show this in some detail.
First, to get properly oriented, let's look at Bohm's ontological argument on this issue from Chapter 6 of The Undivided Universe (with Basil Hiley). I have modified the notation, but not the content. The initial quantum state at t=0 of the system to be measured is
where the sum is over the repeated index p. M is the operator representing the measuring apparatus which is in the initial quantum state (y|M,0), where y is the classical system point of the "pointers". Note that M and P are different. There is no "self-measurement" here. The initial state of the combined system M+P is then
The traditional von-Neumann interaction Hamiltonian gives the time evolution
where p are the eigenvalues of P and g is the strength of the coupling of M to P. Switch the interaction off after time T. In a "good measurement" there is no overlap of the domains of the measuring eigenfunctions in the classical configuration space C3N(y) of M. We mean no overlap for all subspaces of C3N(y). Bohm and Hiley define zero back-action, i.e., orthodox "collapse-without-collapse" as follows: (p. 99)
"... when the wave packets have definitely separated, the apparatus particle y must have entered one of them, say p', in which it remains indefinitely, since the probability of being between packets is zero. From then on, the action of the quantum potential (and of course the guidance condition) on the particles will be determined only by the packet (x|p')(y - gp'T|M,0), because all the other packets (which do not overlap this one) will not contribute to it. So at least as far as the particles are concerned, we may ignore all the other packets and regard them as constituting inactive information ... this will still happen even when there is some spatial overlap between (x|p') and the remaining packets (x|p). Thus is because of the multidimensional nature of the wavefunction of the combined system, which implies that the product (x|p')(y - gp'T|M,0), and any other product, say (x|p)(y - gpT|M,0), will fail to overlap as long as one of its factors fails to overlap, even though the other factor will still have some overlap."
The last sentence of Bohm and Hiley matches Stapp's sentence "If even just one such microchannel is open in one state but closed in the other then the two states will be essentially orthogonal." Unfortunately this is all wrong at the individual beable level, though it is OK at the statistical ensemble level. It is true that any true non overlap for any part, like M, will place the combined system M+P at different points in the total configuration space C3n+3N(x,y), but that will not entirely rid the quantum force on the beable of P of overlaps of the (x|p) packets in the subspace C3n(x). The global pattern of the quantum force will, of course, change dramatically when the (y - gp'T|M,0) packets do not overlap in C3N(y), but it is not generally true that only one (x|p') packet contributes to the quantum force in that case. To see why, consider the simplest two point particle problem. The two particle wave function is (x,y|P,M,t) where n= N = 1. Each particle has mass m. Define
The Hamilton-Jacobi part of the Schrodinger equation is then
where d/dt is the partial time derivative, grad(x) and grad(y) are the local spatial gradients in the single particle subspaces. V is the local context-independent classical potential energy, and Q is the nonlocal context-dependent quantum potential defined by
The quantum force on the beable of P is -grad(x)Q. This is what is important.
Next consider the double slit experiment in which M measures which slit 1 or 2 the P-beable passes through. There are only two eigenvalues p1 and p2 of P, one for each slit. The variable x is the position on the screen beyond the two slits. Suppose, a good measurement has been made, so that for t >> T, and suspending summation convention:
Since the measuring packets (y - gp1t|M,0) and (y - gp2t|M,0) do not overlap in C3(y) there is no common support in the cross-terms that would normally make wave fringes on the screen in C3(x) space. This explains the statistical ensemble level experimental observation that measuring "Which slit?" destroys the fringes. Nevertheless, there is no actual collapse in Bohm's orthodox theory without back-action, therefore the effective R^2 is
We have now come to the crisis point of informal language, i.e., physical interpretation of the mathematics in equation 8b. Bohm and Hiley argue that the actual system point Y for the measuring apparatus in C3(y) is actually in either (y-gp1T|M,0) or (y-gp2T|M,0) which do not overlap in C3(y). Therefore, they argue that only (x|p1)(p1|P,0)(y-gp1T|M,0) or (x|p2)(p2|P,0)(y-gp2T|M,0), but not both terms in equation (8b), of the combined system is "active". One can do a quantum eraser experiment in which (y-gp1T|M,0) and (y-gp2T|M,0) are made to overlap again in C3(y) so that both packets interfere and fringes are restored. But that's no what we are interested in here. If what Bohm and Hiley, and Stapp for that matter, says here is true then there is an effective action at a distance from the M beable at Y to the P beable at X. Well, that's OK, I suppose. However, this action at a distance is a direct back-action of the M-beable on the total wavefunction and this is not supposed to happen in orthodox quantum theory. For example, Bohm and Hiley write on p.30 of The Undivided Universe:
"... it should be pointed out that unlike what happens with Maxwell's equations for example, the Schrodinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the conditions of the particles. This of course constitutes an important difference between quantum fields and other fields that have thus far been used. As we shall see, however, the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles."
Trial by Jury by Gilbert and Sullivan
By "quantum field" Bohm and Hiley mean the nonrelativistic wavefunction, not the second-quantized relativistic quantum field operator acting on Fock occupation number space. Therefore, to be consistent we are not justified in dropping one of the two terms in equation (8b). To do so involves breaking the rules of orthodox quantum theory by invoking a direct back-action effect. Furthermore, the Bohm theory sees the wavefunction as ontological, not epistemological. Therefore, to be strictly consistent, even an empty branch of M in C3(y) must allow both (x|p1) and (x|p2) to contribute to the quantum force on the P-beable if they overlap in C3(x). This is my main objection to what Stapp is proposing. There is a real ambiguity here. Here are the options.
Option 1. Integrate R^2(x,y,t) over all C3(y), using orthonormality of the M basis to get
Compare this with the situation in which there was no measurement by M in which case we would get
This "Hartree mean field" type model is physically plausible as it shows fringes when no measurement is made and no fringes when a measurement is made. The quantum force on the beable at X in C3(x) in each case is simply -grad(x)R(X,t). So we see that whether or not a measurement is made has a large effect on the pattern of the quantum force on the P-beable. Option 1 adequately accounts for all statistical experiments. So far there is no experiment detecting the quantum force pattern on a simple beable. Indeed, the beable theory is only important for complex systems because Bohm's ontological theory is a theory of individuals that does not need statistical ensembles. This is its great power for Chalmers's "hard problem" of consciousness theory. In Option 1, all overlapping branches of the initial P wavefunction contribute to the quantum force on the P-beable even though a good measurement was made. They do not contribute in the same way that they would if no measurement was made. The key point here is that, in this "option 1", which is forced upon us if we want to exclude direct back-action of the actual system point Y of M on the total pilot-wave of M+P, there is generally no 1-1 map connecting branches of the initial P-wavefunction with attractors in the C3(x) subspace of P because these branches overlap. All the overlapping branches contribute incoherently to the quantum force at X when it is in overlap regions of C3(x) when a good measurement by M is made. That is, in option 1, since there is no back-action direct influence of the actual beable locations on the pilot-wave, and since the pilot wave is ontological, i.e., it is really there like a classical field, every field point y of the shared pilot-wave instantly contributes to the quantum force on the P-beable at X. Therefore, we integrate over y. There is a preferred global rest frame in this theory at the fundamental level of individuals. This is not as bad as it seems because the Friedmann solution of general relativity, for the expanding big-bang universe, has such a frame called "the Hubble flow". Again note that M is different from P. There is no "self-measurement" in orthodox theory. Option 1 is orthodox quantum theory that corresponds to "collapse-without-collapse". It obeys the Born rule and corresponds to PenroseÆs "R" type collapse. Bohm and Hiley really do not have a consistent orthodox quantum theory of measurement as they are really using option 2 below which violates orthodox theory.
Option 2. This is really a back-action effect beyond orthodox theory, and it is, in fact, what Bohm and Hiley, and Stapp, have in mind. However, they are not aware that they are sneaking in back-action by the backdoor, if their theory is to be self-consistent. Option 2 is certainly part of the GRW theory, but that's OK since there is no pretense by GRW that they are constrained by the rules of orthodox quantum theory. In this option 2 we assume ad-hoc, in the spirit of back-action, that the actual quantum force on the P-beable is -grad(x)R(X,Y,t) where X is the actual position of the P-beable, and Y is the actual position of the M-beable. Now, it is true that only one of the branches of the initial P-wavefunction contributes. In option 2 it is true that there is a 1-1 correspondence between the branches of the initial P-wavefunction and the attractors in C3(x). This forces Stapp into the back-action camp where he does not want to be. Option 2 is PenroseÆs "OR", which is the same as GRWÆs "objective reduction". Note I have not used the word "orchestrated" yet.
HPS: Which one of the various branches generated by the Schroedinger equation is associated with conscious experience is determined, within the Bohm model, by which of these branches the trajectory of the classical-world enters, and becomes trapped within.
JS: I agree. I want to point out that Stapp implicitly assume option 2 in what follows.
HPS: The chance is essentially zero that the state (open vs closed) of *all* of the billions of microchannels channels, will ever come to be identical for the evolutions of the states corresponding to two distinct experiences. Even if all the gates *were* to become identical, account would then have to be taken of all of the particles in the brain, and eventually the entire world, with which the two initially different states have interacted. So, for all practical purposes, one can assume that the different branches never come back together again.
JS: Yes, except that the wavefunction of the brain microtubules is supposed to be shielded from thermal decoherence by the "ordered water" and possibly other mechanisms. This suggests that quantum eraser effects, in which different branches are able to coherently recombine, may play a significant role. Back-action induces an objective decay as shown in 14.6 of The Undivided Universe, so any quantum eraser effect must happen in a time short compared to the objective decay time.
HPS: The Bohm+CS model is essentially the classical epiphenomenal model corrected to incorporate, a la Bohm, the quantum corrections: one retains the classical notion of the connection of consciousness to the classically described state of the brain, but just adds an extra force, the quantum force due to the quantum potential. All quantum effects are included by just adding this extra force.
JS: Perhaps, but this "extra force" is dramatically different from any local context-independent classical electro-weak, strong or gravitational force. This quantum force is "funda-MENTAL" because it is both nonlocal and context-dependent. The context-dependence is a memory mechanism.
HPS: The theory is still deterministic, though no longer local.
JS: No, it is not deterministic for two reasons. You are forced to "option 2" which means there is a direct back-action of M at Y on the total pilot-wave of M+P. This back-action combines with the "extra force" to make a feedback control loop linking the pilot-wave to its attached beable(s). In addition there are unpredictable stochastic external classical forces on the beable(s) that I call "Darwinian natural selection pressures". You now have a highly nonlinear driven complex adaptive system, what Gell-Mann calls an "IGUS" (Information Gathering Utilization System). This is a "self-determined" system capable of surprisingly creative and intelligent behavior because of bifurcation of the attractor structure in C3n+ configuration space. The "self" is the creative feedback-control "strange loop" which permits self-measurement in which M and P are the same system. This is very important because it explains how P = M comes into being. It is self-superselected. Now the self-selected branches really do not overlap in any subspace of C3n. My scheme is globally self-consistent and dynamically changing moment to moment adapting to the Darwinian selection pressures. Think of the picture by Escher where two hands (i.e., branch in H and trajectory in C3n) "write" each other into being and becoming in mutual co-evolving self-creation. Some of this is explained in Chapter 5 of Stuart KauffmanÆs book, The Origins of Order.
Note: This is the essential idea of the Jewish Mystical Qabala as explained to Bob Toben, Fred Alan Wolf and myself by Carlos Suares in Paris around Christmas of 1973 which led to SPACE-TIME AND BEYOND.
HPS: Consciousness still seems to be an epiphenomenal sideshow, determined *by* the physical aspects, rather than determining them,
JS: No, itÆs a whole new ball-game because of the back-action which you have assumed without realizing it in your assertion of the 1-1 correspondence of branches in H to attractors in C3n.
HPS: ... although one can still argue, as I did in an earlier post, that the felt self is, in a certain sense, in control.
JS: I find that very mysterious. Please explain it again in detail.
HPS: There is a natural connection between the Bohm+CS model and the Bohr/Heisenberg/von Neumann/Wigner collapse model: the "branches" would be the same. In the Bohm+CS model the branch that will "be illuminated by the light of consciousness" will be deterministically picked out by the deterministically fixed trajectory.
JS: No, it is picked out by the freely-willed morally-responsible self-determined creative strange-loop process stochastically driven by Darwinian natural selection pressures from the environment.
HPS: In the collapse model, on the other hand, the way of selecting the branch can be controlled by a rule ...
JS: This "rule" is an axiom equivalent to the axiom in the Bohm+CS model that "the branch that will be illuminated by the light of consciousness will be self-deterministically picked out by the self-deterministically fixed trajectory." Note, that I changed StappÆs word "deterministically" to "self-deterministically" because, like the two hands in the Escher drawing, the branch and the path co-create each other in the Godelian strange loop (e.g., explained in Doug HofstadterÆs, Godel, Escher, Bach). The "extra force" from branch (in H) on path (in C3n), combined with the counter back-action force of path on branch, together form the inseparable "Self" or "I".
HPS: ... that assigns a more basic causal role to consciousness. A fundamentally random selection, based on unexplained quantum probabilities, would, of course, not give consciousness a more basic role.
JS: Oh! I thought you were arguing that the fundamentally random R-collapse of orthodox quantum theory was sufficient to give consciousness a more basic active causative role in the sense of intent and free will. Now you seem to be saying what I am saying. Breaking this "fundamentally random selection" means violating the Born rule and, therefore, going beyond orthodoxy the way Penrose does in "orch OR".
HPS: The Bohm-Sarfatti model has both a trajectory and a "back action". Your recent posts indicate that the back action produces a collapse to the "correct" branch, namely to the branch "occupied" by the trajectory.
JS: Yes, that is what I mean by "self-measurement" allowed by the self-determination of the Godelian strange loop of self-reference. Max Jammer, in The Philosophy of Quantum Mechanics mentions the related problem of the many-worlds interpretation. There is an ambiguity of basis. Into which worlds does the universe split? The universe, like the mind of living matter must self-determine this basis. P = M must be self-created in the fundamental act of observer-participation. This is WheelerÆs idea of the universe as a "self-excited circuit" translated from his Bohrian epistemological perspective to my Bohmian ontological perspective.
HPS: Thus the B-S model is a hybrid of the Bohm and collapse models: there is both the classical trajectory, which in Bohm+CS determines which branch is chosen, and also the back-action induced collapse, which is what does this same job in the collapse models.
JS: No, because, as I showed in the options 1 and 2 analysis, your theory is not self-consistent without back-action.
HPS: In the B-S model consciousness is LESS FREE than in the collapse model, since in B-S the collapse is "largely" determined by where the trajectory goes,
JS: But where the trajectory goes is determined by the structure of the branch and that is being modified by where the trajectory goes in a highly nonlinear manner like what happens in general relativity where the Weyl tensor permits curvature in the absence of an external source stress-energy tensor. You can say that the Weyl tensor structure is analogous to the "self-determinative" part of curved spacetime, while the external source stress-energy tensor is like the Darwinian natural selection pressures on the complex self-determined adaptive IGUS.
HPS: ... and where this trajectory goes is also "largely" deterministically fixed.
JS: No, because you forgot to include the self-determining creative strange-loop formed from the combined actions of the quantum force and the back-action.
HPS: The nondeterministic collapses via GRW gaussians may add some randomness, but this would not seem to inject meaningful freedom of choice.
JS: Right, the GRW model is a kind of weak-coupling limit of my more general theory. It corresponds to the Darwinian stochastic natural selection pressures over-whelming the self-determined "character structure" of the IGUS. Stuart Kauffman explains this in some detail in his book. In psychological terms the GRW limit is like a psychotic breakdown in a mental patient with complete loss of ego boundaries. It might also correspond to mystical experiences of "consciousness without object". I am not sure. The meaningful freedom of choice is in the strong-coupling limit of "robust fitness landscapes" in C3n as explained by Kauffman.
HPS: Thus the injection in this way of the classical trajectory of Bohm seems, in this context, to be a throwback to classical mechanics that needlessly limits the possibility of giving a more independent causal role to consciousness itself.
JS: No, because you only looked at the weak-coupling limit of fragile fitness landscapes.
HPS: The term basin of attraction is from Chaos Theory, which is classical mechanics, not really quantum theory.
JS: Yes, it is not in HeisenbergÆs quantum theory, but it is in Bohmian hidden-variable quantum theory. The role of the hidden variable in BohmÆs theory is played by the actualized classical state after the collapse of Heisenberg potentia to actua.
HPS: The way I use the term in the present context is to denote one of the small regions that the top level control system might migrate to: one of the quasi-stable basins into which the top level control system can fall in the brain's search for a suitable next action in the situation in which the brain/body finds itself.
JS: I think the "top level control system" is in the entangled enchanted conscious web of single electrons, one in each tubulin dimer. Each electron beable is like the key of an organ being struck by the quantum force from the thought-like macroscopic quantum pilot-wave protected against decoherence by the ordered water. The changing classical "key" configuration back-acts on the pilot-wave so it is a highly counter-intuitive creative strange loop of self-determined quantum process which is morally responsible in the strong-coupling limit. The "the brain's search for a suitable next action" is simply the dynamical evolution of the strange loop of self-determination in which P = M, indeed the basic structure of H itself, is continually morphing, bifurcating, adapting to the outside "non=self" Darwinian natural selection classical forces, which, like the Barbarians at the Gate, must be kept at bay by the strong-coupling if the organism is to survive.
HPS: This is explained in my book. These basins are, in my parlance, the same thing as the branches of the brain state discussed above.
JS: And as I explained above, this is only true in option 2 which is back-active.
HPS: The other main point is that the evolving brain/mind is a self organizing adaptive system. It has strong internal energy sources and the brain has a built-in drive to grind out sequences of events that allow the system to adapt to the situation in which it finds itself.
JS: Yes, indeed, and I have shown how above.
HPS: Even classical systems with strong internal energy sources self organize,
JS: Yes, like in PrigogineÆs models. But they lack a giant quantum wave. Therefore, they are not conscious. The important self-organizing loop is the direct one between the giant protected quantum pilot wave and its beable which Stapp calls "the top level control system" and I called "the brain substrate".
HPS: so the is no reason why the Bohm+CS system and the Bohr/Heisenberg/von Neumann/Wigner collapse systems would not both be self organizing adaptive systems.
JS: Yes, because I showed in the option 2 analysis above that you need back-action to get to where you want to go.
HPS: Due to the fatigue characteristics of the neurons that are sustaining some particular pattern, the basin of attraction will not be able to stay viable indefinitely: the basin will fade away, pushing the brain into a search for some alternative basin, etc.. There is no demonstrated need to go outside of standard quantum mechanics to get a self-organizing brain that constructs adaptive solutions to the succession of problems that face the organism.
JS to HPS: No, you are quite wrong in this last remark because of my option 2 analysis. Also, now you seem to have waffled and backtracked because you now seem to be saying that the fundamentally random R-collapse of orthodox quantum theory is enough and you said earlier that this does not permit consciousness to be more than a passive epiphenomenon. I refer here to your remark:
"HPS: ... that assigns a more basic causal role to consciousness. A fundamentally random selection, based on unexplained quantum probabilities, would, of course, not give consciousness a more basic role."
HPS: So I suggest that Occam's razor would call for the elimination of Bohm's classical world, which is not needed to account for the empirical facts (that is the conclusion of the Bohr-Einstein debate), and thus the elimination also of the back action, in favor of the Bohm/Heisenberg/vonNeumann/Wigner theory, which brings consciousness into the foundations from start, and in the context of not just the epistemological framework of Bohr, but also in the ontological framework of Heisenberg/vonNeumann/Wigner.
JS: Strange that you think of the miraculous collapse from potentia to actua as "ontological" since Heisenberg interpreted the wave function as epistemological having to do with our knowledge.
HPS: This collapse theory seems to have the capacity to do whatever the Bohm theory with back action can do, but with consciousness no longer either an ad hoc superfluous appendage, or `enslaved' by its close bondage to the nearly deterministic motion of the classical trajectory.
JS: I am glad you qualified that with "seems".
Best regards, Henry
Date: Fri, 06 Sep 1996 12:18:56 -0700
From: Jack Sarfatti
Organization: Internet Science Education Project
Subject: Stapp to Sarfatti/Sept 4
Stapp's original follows below after my brief preface. My detailed comments will follow separately. His three key points below are:
1. HPS:These key statements from your argument about my theory show that your conception of my theory is not in line with my theory itself. My P(n)'s are projection operators onto well separated "quasi-stable" regions in 3N-d configuration space, not projection operators onto single eigenstates.
JS: I think I gave a counter example in my eqs (8b) for the double slit experiment. My point here is what Stapp says IS true for the self-measuring brain, but only because of back-action. It is not true for all subspaces of 3N-d space for orthodox measurement theory when the measuring device is different from the system being measured.
2. HPS: You may be correct in saying that Penrose rejects the Born rule, and allows psychological feelings to modify this rule. Perhaps you could direct me to the passages where he says this, or give his statement to this effect. I must say that it surprises me that he should propose this since all sorts of "problems" can develop if one rejects the Born rule. Also, his idea is to derive consciousness from physics, so it would not be good to have consciousness needed to define the physics. I suspect that Penrose is too smart to make this suggestion.
JS: I could be misinterpreting Penrose, but, what then, does he mean by the word "orchestrated"? That is, the "orch". in addition to his use of the word "objective" which is the "O" in his term "orch OR". Penrose seems to think in terms of three levels of collapse i.e., "R", "OR" and "orch OR" the latter involving violation of the Born rule. When time permits I will look in his text in detail.
3. HPS: You claim that your model better allows for "free-will". It is a bit tricky to say what "free-will" is, but your model does not seem to provide much of it, because it is obtained by adding an effect of brain on mind, rather than vice versa: Bohm's model without your back action already gives the action of what you call mind on brain. And the action you propose does not look much like it is a control being exercised by mind. On the other hand, I think the Bohr/Heisenberg/von-Neumann/Wigner collapse model does have the capacity to do a good job on this score, but this is not the place to go into it.
JS: I define "free will" as the self-determinative natural process in which the Bohm force (-gradQ) of quantum pilot-wave "mind" on classical brain-substrate beable (i.e., your "top-level control system") is supplemented by a direct back-action of brain-beable on the pilot-wave. This non-orthodox direct back-action is in addition the orthodox feedbacks from external classical forces changing the Hamiltonian of the classically material top-level control system and the boundary conditions on its thought-like quantum pilot-wave. The orthodox Born rule-obeying feedbacks represent the external non-self Darwinian natural selection pressures. The non-orthodox, Born rule-violating Bohm force/back-action "creative strange loop" is the mechanism of "self-determination". As Stuart Kauffman shows there are two regimes here. Your model is the weak-coupling limit where Darwinian natural selection overwhelms the process. I am talking about the opposite strong-coupling limit where the self-determinative creative strange loop keeps the Darwinian inputs under tighter control.
From: STAPP@theorm.lbl.gov
To: sarfatti@well.com
Dear Jack,
Let me focus on the technical points.
You say:
"However, I have only recently discovered in this interaction with Stapp, that Bohm only half-succeeded. His 1952 model, without back-action, does indeed reproduce all of the classic statistical predictions of orthodox quantum theory as claimed. However, as first pointed out to me by Crowell in a general way, the idea, which Stapp takes for granted, that the basins of attraction in the classical mechanical 3n configuration space, for the actual path of the material hidden-variable, or "beable", system point, is determined, one-to-one, by the eigenfunctions of the measured observable is not generally true."Unfortunately, this is precisely the feature that is important for Chalmers's "hard problem" of the physics of consciousness.
"Stapp assumes it in his Bohm + CS model, but, unless I am mistaken, it is not mathematically true."
Your assertion is based on a misunderstanding. I use the general theory of measurement that associates each "measurement" n with a projection operator P(n) [ for any projection operator P: Psquared = P]. The collapse is:
Here P(n) projects onto one of the "branches" of the wave function that represents the state PSI(n). In Bohm's theory the different "branches" of this wave function, in 3N-dimensional configuration space, occupy non-overlapping regions, and P(n) projects onto one of these regions.
If P'(n) and P''(n) project onto different branches of PSI(n) then P'(n)P''(n)=0, and hence the branches are orthogonal: they are confined to orthogonal subspaces of the Hilbert space.
The probability that the state PSI(n) will collapse to the branch specified by the particular projection operator P'(n) is:
Taking the paradigmatic case of a pointer that will move into one or another of a set of disjoint (and well separated) intervals, to signify one or another of the different alternative possible outcomes of the measurement, we have the P'(n), P''(), ... projecting onto different intervals along the direction in 3N-d configuration space that corresponds to the location of the center of the pointer. These formulas represent the orthodox predictions of quantum theory that Bohm's theory must reproduce, and which it does indeed successfully does reproduce.
In the case of a brain that is interpreted as a "self-measuring" device the analogs of the "branches defined by different intervals in the coordinate of the center of the pointer" are the "branches defined by different regions in configuration space into which the brain might evolve". I have argued that these should correspond to regions centered around different patterns of quasi-stable neuronal activity that, like a resonating state that is being powered by feedback, sucks chemical energy available in the brain into itself. In my theory these states have the information content of say a visual scene, or some other perceivable structure. The are rather like a coherent state of a laser in that the nonlinear feedbacks have diverted a lot of energy into a highly ordered state. The different alternative possible quasi-stable configurations of this kind will be confined to different regions of configuration space, i.e., to different "branches" specified by different projection operators P'(n), P''(n), ... etc..
Notice that I never speak of eigenstates of any operator: the alternative possibilities are defined by separated intervals in the coordinate that defines the position of the center of the perceived "pointer" in the cases that Bohm considers, or the appropriate generalization in the case of a brain, namely the well separated regions in configuration space corresponding to our different possible experiences.
You say:
JS: Yes, and as I shall argue below, I suspect that the Bohm+CS model is, if not inconsistent, is seriously ambiguous on a crucial matter of physical interpretation. I, tentatively, argue that it does not work. à Furthermore, I believe your (Bohr/Heisenberg/von Neumann/Wigner) collapse model is internally inconsistent for different reasons, and, therefore, it also does not really work.
What is your argument? You say:
Stapp then says "they will represent different brain states that are distinguishable by the fact that they correspond to different, and well separated, intervals in the values of certain macroscopic and slowly changing---on the scale of neural dynamics---variables. The separation in these variables entails that the alternative brain states are orthogonal. "This is where the trouble begins.... What are these "branches"? The orthodox quantum theory of measurement assumes that "the total experimental arrangement selects at least one observable property p that is "measured".
... The observable property p, called the "eigenvalue", is represented by a Hermitian operator P. The "branches" are the "eigenfunctions" |p) of P.
... actualizes to a single conscious inner private felt-experience of only one single eigenfunction or "branch" |p)
These key statements from your argument about my theory show that your conception of my theory is not in line with my theory itself. My P(n)'s are projection operators onto well separated "quasi-stable" regions in 3N-d configuration space, not projection operators onto single eigenstates.
This mistaken perception of my theory, and of Bohm's, seems to be the basis of your claim that they are wrong, or will not work.
You say:
Equation (4) is violated in "orch OR". Indeed, the degree of violation is an objective measure of the strength of "intent" or morally responsible "free will".You may be correct in saying that Penrose rejects the Born rule, and allows psychological feelings to modify this rule. Perhaps you could direct me to the passages where he says this, or give his statement to this effect. I must say that it surprises me that he should propose this since all sorts of "problems" can develop if one rejects the Born rule. Also, his idea is to derive consciousness from physics, so it would not be good to have consciousness needed to define the physics. I suspect that Penrose is too smart to make this suggestion.Here equation (4) is a standard sort of expression of the Born rule.
You claim that your model better allows for "free-will". It is a bit tricky to say what "free-will" is, but your model does not seem to provide much of it, because it is obtained by adding an effect of brain on mind, rather than vice versa: Bohm's model without your back action already gives the action of what you call mind on brain. And the action you propose does not look much like it is a control being exercised by mind. On the other hand, I think the Bohr/Heisenberg/von-Neumann/Wigner collapse model does have the capacity to do a good job on this score, but this is not the place to go into it.
Henry
P.S. Please send this to your list. You can respond *later*, if you wish.