The world, according to the late David Bohm, divides into objectively real, though mathematically complex-valued, quantum waves and objectively real classical particles and gauge fields. The classical particles and gauge fields always have quantum waves attached to them. For a many-particle system, the quantum wave function's domain is the system’s classical-mechanical configuration space not physical space. This is in contrast to Copenhagen-type interpretations (i.e., Bohr, Heisenberg, von-Neumann, etc. -- all are a wee bit different) in which the quantum wave is fundamental and the classical particle is not. Indeed, Bohm’s actual particle position in physical space has been called the "hidden variable". This is topsy-turvy since most practical measurements in quantum physics involve localized "particle" flashes in a detector. The quantum wave patterns build up in the statistics of the particle detections. It would be more appropriate to say that the wave is the hidden variable.

We restrict this discussion to non-relativistic quantum mechanics. Bohm’s theory appears to violate the Lorentz transformations of Einstein’s theory of special relativity at the individual quantum particle level. However, Lorentz symmetry is restored in the statistical quantum wave patterns. That is, there does appear to be a preferred rest frame in which the new nonlocal quantum forces act instantaneously. This fact is a bit ugly, but we must remember that there is a preferred global frame of reference in the standard cosmological "big bang" solution of Einstein’s general relativity field equations. The "co-moving Hubble flow" provides a new kind of covariant aether in which the cosmic photons from the big-bang are isotropic with a temperature that obeys the Planck blackbody distribution. The isotropy establishes an absolute global "rest frame", and the temperature establishes an absolute measure of global cosmic time from the big bang. Bohm conjectures that the quantum force is instantaneous in this global frame which suggests an interesting connection between a solution to classical general relativity and low-energy quantum mechanics. Special relativity is a local tangent space symmetry in general relativity. Therefore, the big bang Friedmann solution is an example of the spontaneous broken symmetry familiar in solid-state physics and in the standard model of elementary quark-lepton fermion sources and gauge boson forces. On the other hand, if we look at quantum electrodynamics we find the counter-intuitive surprising fact that the near field of virtual longitudinal and timelike photons is also instantaneous in every Lorentz frame of reference. This is because every Lorentz transformation between inertial frames in relative uniform motion induces a compensating internal phase symmetry or "gauge" transformation that re-adjusts the near field to be instantaneous in every frame. Can we impose this feature on the special-relativistic generalization of Bohm’s quantum force? This is a question for further research.

The quantum force is the negative spatial gradient of a context-dependent quantum potential that appears in the Hamilton-Jacobi equation for the particle derived from the linear Schrodinger wave equation. The quantum force’s context-dependence explains the wave-like guidance of the individual particle in the famous double slit experiment which the late Richard Feynman called the "central mystery of quantum mechanics". Context-dependence means that the guiding quantum force only depends on the "form" of the quantum potential and not its intensity or strength. This is because the quantum potential is the Laplacian of the amplitude of the wave function divided by that same amplitude. Therefore, multiplying the wave function by a constant does not change the quantum potential. Bohm calls this quantum force "active information" in which a small expenditure of energy is able to control a much larger expenditure of energy. This feature is dramatically illustrated in the quantum Carnot engine operating between a hot negative temperature and a cold positive temperature. The quantum wave is a kind of information wave. This is the same idea that David Chalmers calls for in his criteria for a post-modern physics of consciousness although he does not appear to be aware of the relevance of Bohm's theory in this context.

There is an actual point-like particle with a well defined position and momentum at each moment. There is also an actual gauge force field configuration, but in this discussion we only discuss the source particles. In fact the momentum of the particle is context-dependent determined as the gradient of the phase of the wavefunction at the actual position of the particle at every moment. This particle passes through only one of the two slits, but its attached objective wave passes through both slits. The recombination of the waves from both slits exerts a quantum force on the particle whose effects exactly reproduce the observed statistical coherent wave patterns for ensembles of particles whose initial positions are postulated to obey the Born probability rule. The conservation of probability current, derived from the linear Schrodinger equation along with the Hamilton-Jacobi equation, then ensures that the Born probability rule holds for all times. There is no conflict with the Heisenberg uncertainty principle since the statistical deviations of position and momentum measurements obey that principle. The actual particle trajectories are classically chaotic so that a small change in an initial position generally will result in an unpredictable and uncontrollable large change in trajectory over a short time. This, the simplest form of Bohm’s theory, is deterministic but not predictable. Bohm and Vigier later added a fundamental stochastic sub-quantal level which, according to Nanopoulos, originates in the virtual blackholes and super-string states of the quantum foam at the Planck scale of 10^-33 cm. In addition, the dependence of the wave of several particles on position in higher dimensional configuration space introduces the kind of "nonlocality" observed in the experiments testing Bell’s locality inequality.

We now come to "back-action" which is the main idea of this paper . The origin of this idea is Newton’s third law that for every action there is an equal and opposite reaction. We now know that this is a consequence of translational symmetry in physical space. Radiation resistance in Maxwell's electrodynamics is a kind of back-action of the field on its source. Wheeler and Feynman showed how this back-action is caused by advanced waves propagating backward in time from the future absorption of the radiation. Retarded causality then depends on the future final boundary condition of total absorption which is actually violated in the same standard big bang solution of general relativity I mentioned earlier -- as shown by Hoyle and Narlikar in the January 1995, Reviews of Modern Physics. It can also be shown that quantum spontaneous emission of real radiation by virtual zero point vacuum fluctuations can be explained as advanced wave effects from the future that are classically associated with radiation resistance. Feynman also used the term "back-action" to explain the generation of quantized vortices in superfluid helium.

The late Eugene Wigner, amplifying on von-Neumann’s "collapse" postulate in the Copenhagen-like interpretations, suggested that "consciousness" is essential for the completion of the quantum measurement. This entails a violation of the linear Schrodinger equation.

Note that the Hartree-Fock mean-field approximation for the spectra of many-electron atoms give an effective cubic nonlinear Schrodinger-like equation. The Heitler-London theory of the simple chemical bond and the Heisenberg model of ferromagnetism demand similar effective cubic nonlinearities which show up, for example, in the exchange integral. The exchange integral in the effective Lagrangian involves two electron densities or four wave function factors.The equation of motion from the action principle involves a functional derivative with respect to the wave function so the resultant nonlinearity is cubic. Cubic nonlinearities also show up in the Landau-Ginzburg equations for coherent macroscopic order parameters or giant effective quantum wavefunctions in second order phase transitions including superfluids, superconductors and lasers. The cubic nonlinearity is also found in the Higgs mechanism for mass generation of particles in the spontaneous breakdown of symmetry of the electro-weak force. The Goldstone modes from the symmetry breakdown add a new polarization component to the massless gauge bosons giving them rest mass. Fermion source particles can also acquire rest mass this way. The symmetry breakdown is from a tachyon field that undergoes Bose-Einstein condensation. The Goldstone modes are small subluminal quantized vibrations in the vacuum expectation value of the tachyon field. These nonlinearities in the Schrodinger equation which are generically induced in mean-field approximations to many-particle effects are swept away by second-quantizing the wavefunction so that it is now a creation and destruction operator of elementary excitations of collective modes in non-relativistic solid state physics. The second-quantized Schrodinger equation is now "linear" in the Fock space whose base states consist of different precise numbers of elementary excitations. Nevertheless, this is only a formal trick and the problems of the first-quantized theory are still there. The creation of antimatter at high energy has led to the idea that second-quantization is really more fundamental for relativistic theories, however, Bohm's theory which does not use second quantization may also be able to explain the creation and destruction of particles. This is also a problem for further research. Indeed Feynman's path integral formulation of relativistic quantum field theory does not use second-quantization and Bohm's picture can readily be integrated into Feynman's. Indeed, Feynman's theory is essentially an incomplete version of Bohm's without the actual trajectories. Therefore, in Bohm's theory as in Feynman's the creation and destruction of particle-antiparticle pairs at high energy correspond to actual particles turning around and moving backward in time. Feynman imposes a boundary condition which is a contour around the poles of the particle propagator in the complex-energy plane such that particles move forward in time with positive energy and backward in time with negative energy. This choice ensures microcausality. We must be prepared for exotic matter in which this boundary condition is violated. The exotic matter keeps star gate traversable wormholes from collapsing and permits the Alcubierre warp drive. One can also imagine new exotic processes in which the actual particles move in Hawking's imaginary time and also temporarily transform into tachyons on spacelike paths. These are all problems for further research. Note that the use of complex energy planes in Feynman's theory also demands the use of complex time planes since the two are related by a Fourier transform.

Wigner invoked the metaphor of Newton’s third law in the new context of the mind-matter interaction. He said that matter affects mind, but there is no corresponding reaction of mind on matter without the collapse of the quantum wave function brought about by consciousness. As shown by John Archibald Wheeler in his classic essay, "Law Without Law", this view leads to "delayed choice" actions backward-in-time (BIT) because we did not exist in the early universe and yet we observe light from the early universe. In effect, it is the future that actualizes the past in the von-Neumann-Wignerean view of quantum reality. Indeed, Roger Penrose and Henry Stapp have physics of consciousness theories based on this approach.

To summarize, Wigner uses the metaphor of back-action to say that mind acts on matter in the collapse of the wave function. Penrose and Stapp say that each qualia-event or moment of awareness requires a collapse in the wavefunction of the brain. It is assumed that there are mechanisms that preserve coherence over a time of the order of one second. The wave function is shielded from ordinary thermal fluctuations which would decohere it in times much shorter than one second.

Bohm’s theory is very different from the von Neumann-Wigner theory. First off, I identify "pre-mind" with the quantum wave attached to the relevant many-particle system. This is a new postulate that the potential for psychological "qualia" is a fundamental physical property of the universe in the sense defined by David Chalmers in his December 1995, Scientific American article. If we grant this postulate that the potential for qualia is as fundamental as charge or mass, and that this potential is identical with the quantum waves of certain particles of matter in biological and possibly other complex systems, then Bohm’s quantum force immediately explains how mind moves matter. Furthermore, in Bohm’s theory it is the quantum force that guides the particle into a given branch of the wave function in a measurement situation. This simulates Wigner’s idea that "consciousness" collapses the wavefunction. Bohm’s theory is one of "collapse without collapse". Therefore, I have given a more detailed explanation of Wigner’s idea in terms of a deeper idea due to a combination of Bohm’s and Chalmers’s views. We now have a new way of looking at the modern theories of quantum consciousness of Penrose and Stapp.

To make more contact with the standard quantum mechanics, we now see why it is that the guidance of the particle by the "mind-like" quantum force into a particular "eigenfunction" of the "observable" is, under usual conditions, unpredictable and uncontrollable. The latter two features are due to the fact that the linear Schrodinger equation has zero back-action in the sense that I mean it. Whereas Wigner’s idea of back-action was that of mind on matter, my idea is the exact reverse. The Bohmian back-action is the direct action of particulate matter on its attached guiding mind-like quantum wave whose support is in the configuration space of classical mechanics, and whose range is in Hilbert space. The result, however, is not incompatible with Wigner’s main idea and it, in fact, explains it in a deeper more elegant way.

The absence of back-action in orthodox quantum mechanics is mentioned by Bohm and Hiley in their book, The Undivided Universe.

"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. We shall in chapter 14, section 14.6, go into what it would mean to have such a dependence and we shall see that this would imply that the quantum theory is an approximation with a limited domain of validity." p.30

The relevant particles are probably the electrons whose spatial displacement controls the conformations of the protein dimers in the microtubules. It is their collective, perhaps Frohlich electric-dipole, wave form pumped far from thermal equilibrium which, I conjecture, is the physical substrate of our mental experience. All forms of life must have back-action in the sense defined above. Back-action is a necessary condition for the existence of any form of living matter in this theory.

Having come this far, things really start to get interesting. Going to Bohm and Hiley’s "section 14.6" we find:

"Other changes of this sort that might be considered would be to make the Schrodinger wave equation nonlinear and to introduce terms that would relate the Schrodinger wave function to the particle positions. … One way to make Schrodinger’s equation dependent on the particle positions (so that there would be a two-way relationship between wave and particle) can be seen by considering equation (14.1). In this equation, we regard Rn as the actual position of the nth particle. From the same arguments as apply to the GRW (Ghirardi, Rimini and Weber ) approach, it would follow that the overall wavefunction would tend to collapse towards the actual particle positions, so that in a large scale system, the empty wave packets of our interpretation would tend to disappear." pp.345-6.

It is important to realize that the idea of the mind doing a quantum computation requires that there be no collapse or decoherence whilst the computation is in progress. The GRW theories postulate that the decoherence time depends inversely as the number of particles N that form the quantum computing hardware unit. GRW introduce ad hoc two new fundamental constants of Nature of 10^-5 cm or 100 nanometers for the scale of collapse, and 10^16 sec or a billion years which divided by N gives the decoherence time beyond which quantum computing deteriorates. Nanopoulos gives a more fundamental derivations of the decoherence time in terms of the masses of the particles. If we use the Nanopoulos formulae and my hypothesis that the human quantum biocomputing unit is at the electron level rather than the proton level, then the decoherence time for the number of electrons in our bodies is of the order of 100 years or approximately one human lifetime. If we use protons as the basic unit, the decoherence time is a factor of 10^20 shorter for the same number of particles. This means one has to use a much smaller number of protons to get the same decoherence time as one would get for electrons. Using protons and a decoherence time of 1 second gives an estimate for the basic unit of our experience. The numbers for this computed by Nanopoulos and Penrose need to be compared. One can imagine that the continuity of our long-term memory that lasts a lifetime is at the electron level in our microtubules while our more immediate short-term moment-to-moment experiences are at the proton level having to do with hydrogen bonds. The coupling between short-term and long-term memories would then have to do with the interaction of hydrogen bonds with the controlling electrons in the protein dimers of the microtubules.

One last wild idea before ending this rough first draft. Let’s play with a cosmological connection in the GRW model since Nanopoulos has already linked the "quantum friction" of the fundamental decoherence time to the quantum gravity foam. Suppose, the basic decoherence time is the Hubble size of the universe divided by the speed of light of about 10 billion years presently. This means that the basic decoherence time at the big bang before inflation was the Planck time of 10^-43 sec which is not unreasonable. Let us further suppose that the spatial scale of collapse, which does not appear to be in Nanopoulos’s theory is the geometric mean between the Planck distance of 10^-33 cm and the Hubble size of the universe multiplied by the square of the dominating gauge force coupling constant. Why the square of the coupling constant? Because that is what happens in the basic Feynman diagram for any interaction force between two sources via the exchange of a virtual boson. Every thing that happens is a composite of such Feynman diagrams in standard field theory. Well if we do this, we get about 10^-5 cm for the present epoch which agrees with GRW using the fine structure constant for QED and we get the Planck scale of 10^-33 cm at the big bang since all the coupling constants grand-unify to 1 before they get to the Planck scale in the standard ideas on the subject. So first homework problem - is there anything in astronomy which would falsify this idea? Can it explain the missing-mass problem and the apparent presence of stars older than the universe? It makes a very interesting prediction. It says that, in an open universe (e.g. Freeman Dyson’s "Time Without End") things get more and more quantum mechanical as the expanding universe gets older and older. That is larger and larger numbers of particles preserve their nonlocal quantum connections to each other over longer and longer coherence times, and therefore show macroscopic quantum effects over wider and wider spatial separations in the far future of the universe even though the density of particles may be decreasing to zero asymptotically. This is a prediction for the "Mind of God" (e.g. end of Hawking’s book, A Brief History of Time) because, on the basis of my idea that the act of quantum computing requires the suspension of decoherence, eventually all the matter in the universe will be quantum mechanical on the cosmological scale. If back-action is present, direct faster-than-light (FTL) communication between distant regions of matter in the open universe can happen. Eberhard’s theorem prohibiting such communication using nonlocal quantum connectivity only works in the limit of zero back-action. In this wild idea all the matter of the open immortal universe becomes part of a Vast Active Living Intelligence (i.e. Phillip Dick’s VALIS) or cosmological brain whose quantum wave function is the "Mind of God" in Hawking’s sense. This model is different from Frank Tipler’s Omega Point which requires a closed universe. My model is consistent with I.J. Good’s notion of "GOD(D)" that he introduced about fifteen years ago. Good worked with Alan Turing breaking the Nazi War Code.

The responsibility for the attachment of "New Age" mystical thinking to quantum physics as represented in popular books like Space-Time and Beyond, The Tao of Physics and the Dancing Wu Li Masters, is ultimately Niels Bohr's transmitted by the seductive poetic prose of Wheeler to us unsuspecting students and young professors back in the 60's and the 70's. Wheeler has been Socrates to Bohr's Plato with Bohm as one of the Poets to be cast out of Athens in The Republic. It was Bohm who escaped from Bohr's Cave.

The late Richard Feynman said that the fundamental mystery of quantum mechanics is the double slit experiment with one electron going through the system at a time. Each electron appears to arrive randomly at the screen making a tiny flash. After about 20,000 electrons arrive one can begin to make out the wave-pattern of interference fringes. If the electron is a small particle it can only pass through one of the slits. The mystery is that the statistical wave pattern is what one gets when classical waves pass through both slits and coherently interfere at the screen. Each electron is somehow interfering with itself. If the electron is a tiny localized concentration of mass it must be acted upon at a distance by some sort of nonlocal connection to the slit it did not pass through. The mystery deepens because any attempt to measure which slit the electron passes through weakens the fringe contrast in the statistical pattern. Indeed, if we know for sure that the electron passed through one slit and not the other, then the fringes disappear completely. The fringe contrast is strongest when we are completely uncertain as to which slit the electron passes through. This in a nutshell is the fundamental enigma of quantum mechanics. The academic mainstream Copenhagen Interpretationists insist that this mystery is intrinsic to quantum reality. It cannot be solved even in principle. It is unthinkable. Bohm has offered a solution and that is what Peter Holland's book is about.

Each event is unpredictable, yet over time a definite and reproducible pattern is formed. It is not just arbitrary. What causes the electrons to aggregate in this way? Is there some force acting on each individual electron as it passes through the device which impels it, on the average to land in certain regions of the screen rather than others? p.xvii

Classical physics pictured the world as made out of tiny "source" particles of matter acted on by various force fields. The force field configuration is spread out all over space and changes in time according to solutions of the differential field equations. Although the field configuration is spread out it acts locally on the particles.

The Copenhagen interpretation, in the non-relativistic approximation of speeds small compared to the speed of light, throws the particles away and replaces them by their associated quantum wave functions.

The Copenhagen interpretation goes further and asserts that it is not possible, even in principle, to try to follow the motion of individual particles at the quantum level. The wave function is supposed to be a complete description of the individual particle. This is the axiom of completeness.

The word 'electron' does not actually mean anything at all - it is simply shorthand for a mathematical function. Quantum mechanics is the subject where we never know what we are talking about. p.xvii

[Bohm's] goal is a complete description of an individual real situation as it exists independently of acts of observation. p.xviii

(7) There is no reciprocal action of the particle on the wave. In classical physics there is a dialectical interplay between the particle and the field, each generating the dynamics of the other. In the pilot-wave model the dynamical connection is one way. Among the many nonclassical properties exhibited by this theory, one is that the particle does not react dynamically on the wave it is guided by. p.26

I shall attempt a brief verbal description to convey a glimpse of the divine picture of Newton's clockwork universe through the glass darkly.

Dynamical phase space combines position and momentum. For example, a single particle moving in one dimension of space has a two-dimensional phase space. The Hamiltonian (H), which is a function of phase space (p,q), is defined as a Legendre transformation of the Lagrangian. The classical mechanics equations of motion are defined from a classical action (I) which is the path integral of the Lagrangian (L) with respect to the time differential (dt) for a given trajectory determined by given position (q(t)) and velocity(dq(t)/dt) functions of the time.

Classical statistical mechanics is a theory in which motion is determinist but unpredictable.

So much for Laplace's image of an Omniscient Classical God - it is evidently an illusion. One must bear in mind in all this the tacit acceptance of the traditional idea that causes are prior to their effects. Determinism, as usually understood assumes this. One could conceive of a more general super-determinism in which what happens now has both past and future causes. The latter giving an almost mathematical definition of destiny, intent, purpose and, perhaps, meaning. Indeed, I.J. Good, in The Scientist Speculates, conjectured that the proper explanation of quantum indeterminism is future causes actualizing individual quantum events whose propensities to actualize have past causes. If one could access information directly from the future, in contrast to trying to compute the future from the past using a deterministic algorithm, then we could, in principle, defeat both classical chaos and quantum indeterminism by a kind of precognitive remote viewing or a strong form of delayed choice that requires going beyond quantum mechanics today. These wild ideas of mine, not Holland's, will be further developed below within the context defined by Holland's explication of Bohm's new paradigm.

Holland begins to introduce quantum mechanics from the Hamilton-Jacobi point of view in 2.7 on internal potentials. For the one-body problem

..in addition to the usual external potential V(q,t), a new type of energy V'(q,t) is introduced into the Hamilton-Jacobi equation which depends explicitly on S and on various orders of the derivatives of S (p.62) ... V' is not an external potential of the usual type, which is a preassigned function of the coordinates and independent of the dynamics it determines - it depends on the function S conventionally associated with the motion of the particle through the momentum (gradient of S) and the kinetic energy [square of the gradient of S divided by twice the mass m] It is therefore more appropriate to call V' an 'internal' potential ... We must suppose that there is associated with a particle pursuing a well-defined trajectory in some region of space a new type of physical field mathematically described by functions S(q,t) ... It is objectively real an always accompanies the particle in its motion - indeed it guides the latter in accordance with the laws of motion [F = ma with F = -gradient (external potential + internal potential)] ... When we speak of a 'material system' we therefore mean a composite entity comprising simultaneously real wave and corpuscular aspects. The value of the field at a spacetime point contributes as much to the definition of the 'state' of a system as does the position of the particle. ... it is better to treat V' [i.e., the quantum potential] as a new form of energy which is not reducible to kinetic or potential energies of the conventional type. ... The S-function has been elevated from a passive to an active role and now physically acts to affect particle motion, but the particle does not in turn influence the wave. p.63

One can certainly envisage a theory in which the particle and wave aspects of matter interact in a dialectical fashion but we will not discuss this possibility here since it is not necessary for an understanding of quantum mechanics. p.63

The back-reaction, not found in orthodox quantum mechanics, completes a feed-back control loop between particle and wave which explains the matter-mind connection in which mind is the quantum wave that not only moves living matter but is, in turn, moved by living matter. The latter is the basis of perception and consciousness. It is how sensory data from neuron pulses and bio-chemical messenger molecules transform their information into meaningful subjective experience (i.e., qualia).

Quantum mechanists often claim their theory is universal and applicable, in principle, to the entire universe. But biological processes, which are surely part of the same universe, lie beyond the reach of physical explantion as we currently understand it. As far as we can tell, they are not instances of quantum mechanics. p.xx

Here are other important clues that Holland leaves for Godel-jumping beyond the confines of quantum mechanics as understood in 1995.

...the appearance of derivatives of S of an order higher than the first is not something that can readily be embraced by some kind of generalization of classical mechanics which retains the basic structure of that theory viz. that matter has simply a corpuscular aspect with no active wave component. p64

The introduction of higher order derivatives of S is thus indicative of something qualitatively new in the theory of matter and motion. p.64

... the failure of quantum mechanics to reproduce the valid results of other sciences already occurs at a more basic level of macro-experience. The difficulty arises in attempts to derive classical particle and field theory from the quantum theory in cases where the former are known to be valid. The de Broglie-Bohm theory suggests that this programme is generally unrealizable; generic classical processes are inaccessible starting from quantum ones. Even in those cases where the Correspondence Principle of the de Broglie-Bohm model is obeyed (as gauged by the relative effectiveness of the quantum potential) only a subclass of admissable classical behavior may be recovered in general. p.xx

The notion of a well-defined particle trajectory is not a concept peculiar to the classical paradigm. From our new perspective we see that it has a wider applicability and indeed that classical mechanics is a particular case of quantum mechanics not in the sense of the emergence of the trajectory concept in some limit, but in the types of motion that particles can perform. p.64

.. suppose that we give rho [probability density] another, more fundamental, property - that it is an objectively real field which is, along with S, a component of a single material system. ... The, ... it is possible to assume that the probabilistic aspect of rho is a secondary property which is consistent with, and does not contradict, its role as a component of an individual system. Endowing matter with a field aspect, mathematically described by rho and S which no longer have a purely descriptive significance, enables us to avoid the paradox of an individual's properties apparently depending on an ensemble. p.65

Corresponding to this new theory of motion is a new conception of matter. The classical theory of matter, of mass points interacting via preassigned potentials whose natures are independent of and external to the points acted upon, has been replaced by a more general conception in which matter has an intrinsic field aspect, the mass points moving and interacting under the influence of the new internal energy as well as the more familiar potentials of classical dynamics. ... The generalization envisaged here is not of a statistical character and it is not tied to any particular scale (microscopic, macroscopic or something between). It is a causal determinist theory of individuals. The internal potential is an organizational or self-referential form of energy which brings about an 'inner tension' in the material system to which the mass points respond. p.65

Notes on Chapter 3

The Copenhagen interpretation is entirely pragmatic, epistemological, operational in P.W. Bridgeman's sense, logicially positivist in the sense of the Vienna Circle of the 1920's. It says that the wavefunction has only a statistical significance. There is no ultimate objective reality behind the statistical correlations we can measure. Theoretical physics in its essence is only an algorithm for correlating statistical results of sets of measurements characterized by given spacetime separations. For example, an experiment can be divided into a preparation and a detection which have a single spacetime separation. The statistical sample (ensemble) consists of a large number of individual pairs of events (preparation, detection). The experiment may be more complex like the EPR experiment on photon pairs. The sample that consists of a large number of individual triads of events (preparation, detection 1, detection 2). This is not Bohm's approach, which includes the latter but goes beyond it.

.. the characteristic distribution of spots on a screen which build up an interference pattern is evidence that the wavefunction indeed has a more potent physical role than a mere repository of information on probabilities, for how are the particles guided so that statistically they fall into such a pattern. ... not only do we entertain the notion that the wavefunction has a direct physical significance in each individual process, we go further to propose that such action is its primary feature. ... probability only enters as a subsidiary condition on a causal theory of the motion of individuals; the statistical meaning of the wavefunction is a secondary property. The additional element that we introduce apart from the wavefunction is a particle conceived in the classical sense as pursuing a definite continuous track in space and time. pp.66-67

The basic postulates of Bohm's theory in the simplest case:

(1) An individual physical system comprises a wave propagating in space and time together with a point particle which moves continuously under the guidance of the wave.

(2) The wave is mathematically described by psi(x,t), a solution to Schroedinger's wave equation.

(3) The particle motion is obtained as the solution x(t) to the equation

dx/dt = (1/m)gradS(x,t)|x=x(t) (3.1.1.)

where S is the phase of psi. To solve this equation we have to specify the initial condition

x(0) = xo

This specification constititutes the only extra information introduced by the theory that is not contained in psi(x,t) (the initial velocity is fixed once we know S). An ensemble of possible motions associated with the same wave is generated by varying xo.

(4) The probability that a particle in the ensemble lies between the points x and x+dx at time t is given by

R^2(x,t)d^3x (3.1.2)

where R^2 = |psi|^2.

This postulate has the effect of selecting from all the possible motions impled by the law (3.1.1) those that are compatible with an initial distribution R^2(,b>x,0) = Ro^2(x). It is thus a consistent subsidiary condition imposed on a causal theory of motion and has no more fundamental status than that. ... we have formulated postulate (4) in terms of the probability that a particle actually is at a precise location at time t. The usual (Copenhagen-Born) interpretation of of |psi(x,t)|^2 is that it determines the probability density of finding a particle in the volume d^3x at time t if a suitable 'measurement' is carried out. the 'probability of finding' is a special case of the 'probability of being' ..p.68

where

Therefore, the real part of the complex Schroedinger equation is

which is the quantum version of the Hamilton-Jacobi equation with the new nonlocal quantum potential

The imaginary part of the Schroedinger equation is the probability current conservation equation

Given the initial complex-valued wave function psio(x) = psi(x,0), plus continuity and finiteness on psi and grad psi (e.g., R -> 0 at infinity, So undefined at nodes of psio, psi is single-valued etc.) ensures a unique solution. Single-valuedness implies that

have the same psi. S is discontinuous only at a node of psi. These nodes can be one-dimensional lines in 3D space (for the single particle case) that either go to infinity or form closed loops at a fixed time. They can also be surfaces in space. (p.71) Wave fronts obey S(x,t) = constant. The wave crests obey S = nh (n an integer). Wave fronts can disappear on nodal lines. Consider a closed loop C in space. Do not confuse this closed loop with a nodal closed loop. C is essentially arbitrary. C cannot cross itself and generally we don't want C to intersect a nodal line or a nodal surface. The line integral of grad S around the closed loop C is nh with n an integer. If n is not zero that means that C encloses nodal lines where S is discontinuous. Michael Berry has shown that the net number of wave fronts that disappear inside C is |n|. The vector momentum field p= grad S has vanishing curl (i.e., irrotational) except along singular nodal lines which are similar to vortex lines in classical fluid dynamics. As soon as we go to the many-particle problem we are in 3N dimensional configuration space (non-relativistically - 4N relativistically with each particle having its own local time) and the role of nodes is much more complicated topologically and nonlocal in new ways.

Now something interesting can happen. Imagine there is a closed loop C in space at a fixed time. As the quantum psi wave evolves in time, its nodal lines or surfaces can cross C. This crossing through C of the quantum vortex changes the value of the integer n in the line integral of grad S on the closed loop S. (p.72). Recall, however, that C is an mathematical loop that need not have any direct objective physical meaning - but can in a particular problem. For example, imagine an electron confined to a nanoscale quantum wire loop C.

We propose to associate with the physical wave psi propagating in space a point particle of mass m which pursues a trajectory x = x(t). p.72

So the quantum Hamilton-Jacobi equation (3.2.6) gives the changing instantaneous total energy of the actual point particle moving in both the classical potential V and the quantum potential Q. That is,

Now intrepid reader pay close attention because a totally new idea of holistic context dependence is about to be introduced. Remember that Q may be primitive "mind-stuff" (Sir James Jeans) in the sense of Nick Herbert's book Elemental Mind (Dutton, 1994). Mind-stuff is not "conscious" until there is a biologically specific "back-reaction" which violates the kind of quantum mechanics we are doing here-now according to the hypothesis of Brian Josephson.

Q is not a preassigned function of the coordinates the way V is. It depends on the total quantum state ...p.74

Since the actual particle velocity is gradS/m , then (3.2.23) on the actual particle trajectory is Newton's second law with -gradQ as the wholistic context-dependent nonlocal quantum force.

where

is the classical hydrodynamic derivative describing the co-moving time rate of change seen from the instantaneous rest frame of the actual particle. (p.74)

The Bohm equations here give mathematical muscle to the detached words of Niels Bohr on the primacy of the "total experimental arrangement" encoded in Q through the global boundary conditions on psi from which Q is derived. Bohr threw the baby (particle) out with the proverbial bathwater in formulating his essentially New Agey mystical Copenhagen interpretation with Puff the Smoky Dragon. :-)

Again the only extra "hidden variable" information that must be added to the initial wavefunction is the initial position xo of the actual particle. The initial position cannot lie on a node because gradS is undefined there so we cannot compute the initial momentum of the particle from the wavefunction (p.74). The basic law of motion of the particle

is an axiom which cannot be derived from the Schroedinger equation.

The quantum probability density current j(x,t) obeys

One can also define angular momentum. (p.75)

The different members of the statistical ensemble are distinguished by their different initial positions xo.

Teleology-future to past

We do not expect that the simple-minded notion of causality employed here (that the future deterministically flows out of the past) encompasses all the phenomena encountered in experience, but it is sufficient for the quantum mechanical processes described in this book (it would not account for the development of an organism, for example, where matter apparently aggregates in certain patterns to achieve a definite end).

What about Heisenberg's uncertainty principle? It is an epistemological principle about the limits of our knowledge. It is not about the ontological condition of the actual hidden variable particle.

... our knowledge of the state of a system should not be confused with what that state actually is. p.77

The quantum theory of measurement (and the epistemological limits placed by the noncommutativity of incompatible observables) depends upon a special class of interactions between the measuring apparatus and the observed system. (p.77)

According to the quantum theory of motion the particle has a position and momentum prior to, during, and after the measurement whether this be of position, momentum or any other 'observable'. (p.77)

.. the solution psi(x,t) will in general depend on a set of parameters which characterize the environmenr that the propagating wave encounters - such as slit width, radius of scattering center etc. A different choice of these parameters will, in general imply different values of psi at each point in space and time (i.e., at all points and not just those in the vicinity of the physical objects with which the parameters are associated... We shall refer to this feature of encoding at each point information on the whole context as 'context dependence' or 'wholeness'. ... there is no 'source' of the psi-field in the conventional sense of a distinct localized entity. p.79

Bohm's theory gives a new kind of 'local' relation between particle properties and Hermitian observables (3.5) which is not found in the Copenhagen interpretation. For example, the 'local expectation value' of the Hermitian operator A^ in the state |psi> in the position representation is

where

For example:

the local expectation value of the position operator for any state evaluated along a trajectory is just that trajectory ... the local expectation value of the momentum operator evaluated along a trajectory is the particle momentum, mdx(t)/dt. It has the same form as in classical mechanics. ... The local expectation value of the Hamiltonian operator evaluated along a trajectory is the total particle energy,

(1/2)m(dx/dt)^2 + V + Q (3.5.10)

It has a similar form to the classical expression apart from the addition of Q. pp 92-4

Let us examine the role of probability in Bohm's original nonrelativistic theory. (3.6.1) In the simplest case, an individual member of the statistical ensemble consists of both a particle and its guiding wave. If we have a many-particle system the individual member of that ensemble is the collection of all of the particles and their generally entangled many-particle guiding wave. The initial conditions consist of both a given guiding wave form PSIi(x) and initial positions x (in general this is a set of 3N real numbers for an N particle system). Remember the initial speeds are determined from the guiding wave form.

.. the ensemble is 'fictitious' in the sense that we conceive of all the elements being present at the same time, each independently evolving according to the deterministic laws of motion, even though, in fact, we only ever have one system present. ... we assume that the particle and wave variables are uncorrelated ... Then the joint probability function P(x,i) factorizes

P(x,i) = pi P(x) (3.6.4)

pp 96-97

That the initial distribution of particle positions is given by the Born formula

is an empirically justified axiom of Bohm's theory. This is no better and no worse than the situation in the traditional Copenhagen interpretation.

Since probability is not intrinsic to the basic theory, an advantage of our approach is that we are able to raise the question of whether an explanation for |PSI|^2 may emerge at a deeper level, rather than accepting this as an inexplicable property of matter. p.102

The essential difference between our notion of probability and the one usually employed in quantum mechanics is that we are describing the likely state of matter as it actually is , whtever processes it may be part of.... In contrast, in the Born interpretation |psi|^2 does not represent our ignorance of an actual state but concerns rather the distribution of values found if one performs a 'measurement' .. The notion of probability used here (and in conventional treatments) requires that there exists in the real world an unlimited number of the systems under consideration ... hence that the results obtained by averaging over the ensemble are equal to those given by an average over a sequence (of actual physical processes)..

In Bohm's theory, in contrast to the Smoky Dragon theory,

.. if the system is left in an eigenstate of the operator under consideration, the actual value coincides with an eigenvalue of that operator. The key point is that the particle does not acquire these properties only when it undergoes special types of interactions ('measurements') but possesses them throughout its entire history. pp.105-6

The interference between the different terms on the RHS of this equation means that we cannot imagine that the system is actually in a definite eigenstate |a) that we are ignorant of. That would be a 'mixture'. Position in spacetime is a preferred frame you might say in Bohm's theory more fundamental than momentum and energy space which are not well-defined in the curved spacetimes of general relativity in which there are no global inertial frames. In the position representation on a given spacelike surface

The measurement process here is described by a special type of interaction Hamiltonian that is able to separate the different (x|a) so that they no longer effectively overlap because of Einstein-Podolsky-Rosen correlations to orthogonal states of the measuring system. Therefore, the initial superposition evolves into an entangled state which merges the original system with the measuring system in such a way that if we average over either the measured or the measuring system, what is left behaves like an incoherent mixture. One can in principle quantum erase the correlations if they are simple enough. Such experiments have been carried out.

Operationally, the measuring system is macroscopic and in a good measurement there is a 1-1 correspondence between the micro-property and its amplified macro-pointer reading. In practise then we average over the directly unobservable micro states (x|a) which leaves an incoherent mixture of macrostates so that Schrodinger's cat is either definitely alive or definitely dead. We then infer tha actual state of the microsystem assuming that the 1-1 correspondence is good. This is the usual procedure.

As an example, let us average over the orthogonal states of the masuring apparatus, the actual value of the observable A^ for the observed system is Bohm's new "local expectation value" (p.106)

The incoherent probability distribution over the ensemble is

3.7.2 What are the hidden variables? Let us review what the Smoky Dragon fantasy of the universe is. It asserts that the quantum wavefunction is the most complete specification of the state of an individual system. It further asserts that this wavefunction is purely epistemological having only a statistical meaning connected with our possible pragmatic knowledge of physical reality. Any further specification of the state of an individual system must then be hidden.

The problem with this point of view is that what one observes as the results of experiments are localized events: an interference pattern is built up over a period of time by, say, the blackening of definite regions of an emulsion, one after another. The confrontation between this fact and the conventional view on the meaning of PSI leads to the problems and paradoxes of the 'wave packet collapse' hypothesis. But from the point of view of the quantum theory of motion, the localized events reveal the current position x(t) of the particle (when suitably amplified through other processes). The observed position is what the actual position evolves into in the circumstances of a measurement interaction. It is therefore misleading to term x(t) a 'hidden variable' - on the contrary, it is PSI that is 'hidden' in that we only derive information about it by observing the particle ... If anything is a 'hidden-variable' interpretation of quantum mechanics, it is the conventional one that only works with PSI! p.107

Remember R = |PSI| has a dual role in Bohm's theory as an onotological nonlocal sourcesless psi-wave field of an individual particle and also as a statistical distribution of particles all having identical psi-wave fields but differing initial positions. The word "psi" is synchronistic because in my extrapolation of Bohm's theory the psi-wave is a mental quality.

Just like any classical field, the psi-wave associates a density of energy and momentum with each point in space..there may be no individual element of the ensemble which actually follows the track pursued by (x)... the expectation value of the kinetic energy and total angular momentum operators are not equal to the ensemble averages of just the kinetic energy and total angular momentum, respectively, as in classical mechanics, but also include additional quantum terms.

Holland in 3.9 treats Bohm's theory as a non-relativistic field theory. It is a new kind of field because of the lack of back-reaction. This feature may be an artifact of an approximation to a deeper theory. Holland proves that the space-integral of the Schrodinger field energy density is the mean particle energy. Similarly, for total field momentum and total field angular momentum.

What, then about the local conservation laws for each particle in the ensemble? These are readily obtained from the generalized Hamilton-Jacobi equation... Identifyiing gradS/m with dx/dt and -&S/&t with the energy E

dE/dt = &(V + Q)|x=x(t), E = (m/2) (dx/dt)^2 + Q + V (3.9.36)

[where d/dt = &/&t + (1/m)gradS. grad]

dp/dt = - grad (V + Q)| )|x=x(t) , p = m dx/dt (3.9.37)

dL/dt = - xxgrad(V + Q)|x=x(t) , L = xXp (3.9.38)

In (3.9.37 we have the ... quantum force, -gradQ, in (3.9.36) ... a quantum power term, &Q/&t, and in 3.9.38 a quantum torque -xXgradQ ... Even if the particle is classically free (V = 0), the particle energy and momentum will, in general, be variable due to the time and space dependence of the quantum potential. .... Classically conserved motion is not conserved quantum mechanically for generic solutions of the Schrodinger equation. p. 119 One might expect the conservation laws would apply to the total field plus particle system in interaction, as in classical electrodynamics. The reason they do not is that the particle does not react back on the wave...p.120

... the field satisfies its own conservation law.... the particle satisfies its own conservation law, but here the field enters as an 'external source', via Q... if we can neglect the 'environment' of the field (i.e., V = 0) then this [i.e., Schrodinger field] forms a closed system with corresponding energy, momentum and angular momentum conservation laws. The particle however, even in this case, is an open system since it receives energy and momentum from the field. .. this feature [i.e., no back-reaction] may appear as unsatisfactory, calling for a development of the theory to include a more symmetrical relation between the wave and particle. At present we have no idea how a source term for the psi-field could be consistently introduced into the dynamical equations in such a way that it does not disturb the empirically well-verified predictions of quantum theory while going beyond it to yield new testable results. p. 120 My situation at the end of the 2Oth Century is parallel to Einstein's at the end of the 19th. Einstein was disturbed by the lack of symmetry between particle and field when Maxwell's classical electromagnetic field interacting with charged particles was forced into the absolute simultaneity of Galilean relativity. He was forced to conclude that space and time separately distort in order to keep the speed of light and the four-dimensional separations between pairs of events absolutely "frame invariant" - the same for all suitable observers. Similarly, I am disturbed by the fact that the statistical predictions of quantum mechanics today demands another lack of symmetry between particle and field in which the field is the "unmoved mover" of the particle. I am forced to conclude that both sacred cows known as "causality" and "unitarity" must be distorted in order for us to exist as conscious beings. Once we let the particle be a "source" of its Schrodinger field we evade Eberhard's theorem and the use of nonlocal Einstein-Podolsky-Rosen quantum correlations as a practical communication channel becomes feasible if not inevitable. Life with this alternative communication capacity, in addition to ordinary electromagnetic signals and molecular transport etc has a definite Darwinian evolutionary advantage - which is Brian Josephson's point. Murray Gell-Mann in his book, The Quark and the Jaguar calls this "Flap Doodle - The Story Distorted". He is literally correct because the problem of biological complexity that he hopes to solve requires distortion of the statistical quantum fable of Wheeler's Smoky Dragon.

Notes on Chapter 4

Holland shows that Bohm's theory is not only statistically Galilean and gauge invariant but it is also so on

the level of individual processes that make up the ensemble. ... although gauge invariant, the quantum potential may contain more information on the [classical electromagnetic] potentials that is more detailed than that picked up by the field strengths ... the [classical] Lorentz force F does not exhaust the possibilities for an electromagnetic field to influence the motion. The quantum force may be finite in a region of space where F = 0 but [the classical vector potential] A not equal to 0. p.127

An interesting measurement would be to see if the influence of the change in magnetic flux on the fringe shift acts faster than the speed of light? The fringe shift is generally a statistical effect. The standard Bohm theory is compatible with Einstein causality on the statistical level so one would not expect to see a faster-than-light statistical effect for inanimate matter. On the other hand one can, in principle, see the fringe shift at the individual electron level. Suppose the detector is located at a bright spot of the fringe pattern if the distant magnetic flux is zero. Let there be one electron in the system at a time. Switch on the magnetic flux simultaneous with the arrival of the single electron at the detector. The flux is calibrated to produce a dark spot at the fixed location of the detector. If the signal from the detector repeatedly stops, under these conditions, we have observed a faster-than-light effect.

Section 4.1.2 is interesting. It shows that the particle energy in localized non-nodal regions of space at definite moments can be lower than the ground state energy level of the wave if the wave is in a superposition of stationary energy eigenstates of the Hamiltonian. See, in particular eqs. 4.1.17-19 on p. 139. This does not contradict the conditions of measurement in spectroscopy but may have some relevance to the idea of extracting energy from the vacuum as an alternative model of so-called cold fusion phenomena which may not have anything to do with thermonuclear fusion.

.. the particle may be conceived as having a well-defined position when the wave is a momentum eigenfunction and the 'uncertainty' relation does not have the implication normally ascribed to it. p.140 ... Just as energy [of the particle] may be conserved in non-stationary [wave] states, so momentum may be conserved when the wavefunction is not a momentum eigenstate. p.141

4.4 deals with tunnelling in one space dimension. The Bohm picture here is totally different from the Copenhagen picture. Even though there are "reflected waves" the actual particles do not turn around at a potential step when both E > V and E < V. The particle is at rest both inside and outside the classically forbidden region when E < V. p.147

Consider an energy eigenstate of an electron in the hydrogen-like atom which is a simultaneous eigenfunction of Lz and L^2.

The particle orbits the z-axis along a circle of contant radius ... with constant angular speed ... we can fit exactly |m| wavelengths into one quantum orbit ... The higher the quantum number m the faster the particle moves, and the slower the wavefronts rotate [about the z axis] .. the radius of the circle is freely specifiable ... it is independent of E or m. different from the primitive Bohr model in which an electron moves in a circle in the equatorial plane and the radius is a function of m ... The quantization here appears rather in the magnitude of the particle velocity pp. 150-152

.. the quantum force exactly balances the classical force ... This result provides the explanation according to the quantum theory of motion for the stability of matter. p.153

4.7 is diffraction at a Gaussian slit. The spreading in the front and rear of the moving Gaussian is not symmetrical.

There are many other examples where quantities of action associated with the wave or particle dynamics may be smaller than hbar. p.161

The quantum force violates Einstein's equivalence principle because it depends on the particle mass, i.e. eq. 4.8.8

This fact is important for a quantum theory of gravitation based on the motion of test particles the way it is done in Wheeler, Misner and Thorne. The classical stress energy tensor needs quantum stress terms added to it. The Guv tensor on the LHS of Einstein's equation also needs to be modified -- similarly for the constraints. The metric will depend nonlocally on wave functions not only on the classical mass distribution. Anandan has done some of this already using the Smoky Dragon . His approach should be redone using the Bohm ontology.

For a system comprising a Gaussian packet freely falling under gravity, all particles do not fall at an equal rate. p.165

The quantum equation of motion involves a net time-dependent force

d^2x/dt^2 = - 4pi^2f^2 a cos2pift 4.9.13

The single-valuedness of the wave function can be pictured as "dislocations" in the set of wave fronts caused by singularities in the phase S at the nodes of the waves. p.169 We have a dislocation of strength n where n wave crests come to an end within the Burger's circuit C in wchic the change of phase around the circuit is nh (eq. 4.11.1) This analogy between crystal defects in the solid state and quantum wave front singularities is due to Nye and Berry (1974). Fig. 4.11 is an edge dislocation from the superposition of two plane waves in which the particle executes a spiral motion at fixed radius. One can also make a screw dislocation superposition with a helical particle motion. Fig. 4.12 p. 172

Notes on Chapter 5

Interference and Tunnelling.

Only material particles not photons are considered. Division of wavefront (e.g., two-slit) and division of amplitude (e.g. semi-transparent mirror) are discussed. Experiments on former have been done with electrons, neutrons and helium atoms. Experiments on latter done with neutrons, sodium and calcium atoms. The wave aspect of matter is only ever seen as the build up of a statistical pattern of localized particle detections. Bohm posits an objective (ontological) guiding wave for each individual particle process top explain quantum statistical patterns. This is diametrically opposed to so called strictly ensemble (epistemological) interpretations of the wave function as given by Ballentine, Peres and others. One important fact is that it is not sufficient that the magnitude of the quantum potential be small compared to the classical kinetic energy not to see fringes. In fact, fringes are seen in this kind of case. Just like a classical potential, the particle responds to the space and time variations in the quantum potential, not to its absolute size.

5.1.2. Particle trajectories in the electron interferometer.

Fig. 5.1 p. 176 has a computer simulated electron path in which the effect of the quantum potential on the electron path is shown here . Of particular interest is the differential equation 5.1.9 on p. 178 for the vertical component of motion of the electron in the interferometer. A realistic numerical example is given in some detail. Bohr's Smoky Dragon prohibition against visualizing spacetime motions of real particles at the quantum level is proved wrong. There are several computer simulations Figs 5.2 -5.6 showing detailed shape for the quantum potential for two Gaussian slits as viewed from the screen. The ensemble of particle trajectories given a uniform distribution of initial positions is shown. The trajectories do not cross over a symmetry plane in the case of equal illumination of each slit.

5.1.5 Delayed-choice experiments.

According to the causal interpretation, the present merely reveals the past and has no influence upon it In this regard, at least there is no need to revise our customary conceptions of cause and effect. p.190

... something resembling Bohr's perception of the wholeness of physical phenomena reappears in the delicate dependence of the quantum potential and hence particle orbits on all the parameters ... slit width and separation, gorup velocity of packets, particle masses and so on. An alteration of any parameter, even if small, may lead to a significant change in Q. This indeed leads to a relatively simple intuitive picture of quantum wholeness. p. 190

The effect is that coherent electron beams passing on either side of a completely shielded magnetic field show a fringe shift that is a periodic function of the magnetic flux which is never in direct contact with the support of the electron wave functions. The effect is explained by the local action of the quantum force on the electron even though there is no local classical Lorentz force acting on the electron. The quantum force -gradQ depends upon a locally gauge invariant version G of the vector potential A p.196. That is,

G is invariant under the classical local gauge transformation A -> A + gradf. G is not zero when B = curl B is zero but A is not simply the the gradient of a scalar function which is the case in the AB experiment. G acts indirectly on the electron via its effect on the quantum potential Q as shown on page 196 of Holland's book.

.. electromagnetism is a gauge-invariant manifestation of the Lorentz force and the quantum force. p. 197

The terms have several distinct operational meanings that are often confused in the published peer-refereed respectable journal articles because of the inherent ambiguity of the Smoky Dragon view of quantum mechanics left to us by a befuddled Niels Bohr & Co. Indeed, John Wheeler has been a Pied Piper in this Comedy of Errors, but All's Well That End's Well even though the quantum measurement problem turns out to be Much Ado About Nothing. I have been a veritable Pinnochio in this Tale of Sound and Fury.:-)

One meaning of "nonlocal" is that the quantum force acts where there is no classical force. The basic two slit experiment for one particle at a time is nonlocal in this general sense. Holland prefers to call this "global" and to reserve the term "nonlocal" to many-particle EPR situations.

5.3 is a detailed discussion of particle trajectories in tunnelling through a potential barrier for both E less than and greater than V. Fig. 5.11 is particularly interesting. There are many subtleties about tunnelling times that cannot be settled within the Smoky Dragon picture.

5.4 is a study of division of amplitude in the neutron interferometer. 5.4.2 on beam attenutation is particularly interesting. It shows the clear difference in the fringe contrast between attenuation by a rotating perfectly absorbing beam chopper and a partially absorbing slab each giving the same effective probabilities. p.209

5.5 has new results on the meaning of time in quantum mechanics. 5.5.1 is a detailed discussion of the time of transit in tunnelling. It cannot be directly applied to the superluminal transit time experiments of Ray Chiao et-al at UCB because they were using zero rest mass photons.

It is not possible to define a Hermitian time operator acting on wave functions that does not commute with the Hamiltonian because the latter must be bounded from below in order to have a stable ground state. This is not a problem in the Bohm ontology where there is an actual particle in addition to the wave function.

... the historical problem of time in quantum theory may have been artificially created by the demand that all physical observables be associated with self-adjoint operators..p. 215

Eq. 5.5.14 defines the "mean age" in terms of the expectation of T^ and the ordinary state-independent time t. The Bohm theory here has a decisive advantage over the Copenhagen interpretation because using its new "local expectation value" of the operator T^ for an individual particle path gives equation 5.5.16.

This idea of "age" may explain the paradox that there are stars older than the universe. This may be telling us something about the wavefunction of the universe at the Big Bang because the quantum age is not the same as the classical age computed from the classical Friedmann solution of general relativity.

Notes on Chapter 6

This note is based on Chapter 6 of Peter Holland's, The Quantum Theory of Motion (Oxford, 1995).

The problem of how quantum coherent phenomena explain our ordinary world is much more subtle than is given in physics text books. In particular, it is not enough to say that quantum phenomena apply only to tiny systems or to large systems at very low temperatures near absolute zero. Indeed, quantum effects may play an important role in biology and even on a cosmological scale. If a way could be found to amplify quantum coherence on the scale of our current electronic technology it would lead to new kinds of ultra-intelligent computers and even new kinds of exotic propulsion systems. Therefore, it behooves us to take a closer look at the problem of the classical limit.

Bohm's hidden-variable theory of quantum mechanics is a misnomer because what is hidden is not the tiny matter particle, but rather its quantum wave since every actual individual quantum detection event is localized in spacetime like a particle on a point of its path. Bohr's Copenhagen interpretation, which says it is not possible to think correctly of particles on definite paths at the quantum level, dyslexically reverses the perceptual gestalt of our understanding of the actual experimental situation by focusing on the statistical wavelike patterns of particle arrivals. This may have been partly due to the primitive detectors at the turn of the century.

Bohr's school really can't make up their mind whether classical physics is prior to quantum physics or whether it emerges as some limiting case. The truth in Bohm's theory is a third alternative. Holland writes

The wavefunction, in order to satisfy an equation in which h (Planck's quantum of action), for instance, appears in the differential equations, must itself depend on h: psi = psi(x,t,h,m ....). It is not at all obvious apriori that limits such as h -> 0 will result in the classical equations, even if they are performed with due regard to the need to compare only like quantities. In fact they often do not. ... the problem of connecting quantum and classical physics is considerably more involved than the corresponding Newtonian limit of special relativistic mechanics. p.219

or (6.1.1) p.220

Use the polar form of the complex-valued wavefunction

to get the quantum version of the Hamilton-Jacobi equation

or (6.1.2)

where the context-dependent quantum potential is

The essential difference between the classical context-independent potential V and the quantum potential Q is that V is a definite function of particle position. In contrast Q depends not only on the position of the particle but also on the form of the particle's wave function as a projective ray in Hilbert space which is a fiber bundle space over spacetime. Recall, that in Bohr's "Smoky Dragon" mystical view of quantum reality, one cannot visualize the spacetime path of a particle at the quantum level. You can do it in Bohm's theory and also in Feynman's theory but Feynman does not go all the way and never really uses the particle beyond a visual aid to the imagination.

The second equation is the conservation of wavefunction current

or (6.1.3)

This equation makes sense for an individual particle and its objectively real (though mathematically complex) wavefunction field which has its own stress-energy tensor. In the many-particle theory this tensor is in configuration space not physical space. The probability interpretation appears later on with the addition of a new postulate.

What about the classical limit? It is, according to Peter Holland, bad mathematics to naively let hbar -> 0 to get the classical Hamilton-Jacobi equation that is Newton's second law, i.e.,

or (6.1.4)

In general we cannot conclude that the quantum potential Q is small when the physical actions in the system are large compared to h. This is of the utmost physical importance for new quantum computer and other technology.

To be continued.

Eq. 7.1.1 on p. 278 is the nonrelativistic Schrodinger equation for a many-particle (N) system in Galilean relativity with a common absolute time (i.e.,t ). Suppose we used the Dirac-Tomonaga idea of having a local time (i.e.,tk) for each particle. We want to do this with special relativity in mind because the Bohm theory at the individual event level is not compatible with special relativity. It is compatible with special relativity only for statistical averages. So we might try some thing like

ihbar sum over j from 1 to N [ihbar&/&tj + (hbar^2/2mj gradj^2 + V(x1,t1;x2,t2;...xN,tN)]psi = 0 (7.1.1')

The classical potential term needs some sort of delayed, and possibly advanced, action. One could redo the equation with minimal electromagnetic coupling and throw away the V term. This equation is not relativistic. It is not found in Holland's book but is of my own invention. To make it relativistic we need either a second order time derivative or we must use the Dirac spinor version. But the main point here is to put space and time on an equal footing. The wave function "psi" solution spans different spacetime events which do not lie on the same spacelike surface. The theory may not be unitary for this reason. The idea is that actual detections determine where to evaluate the solution. If there is no detection for some particle we integrate it over all spacetime. We no longer have the picture of the psi field advancing from one spacelike surface to the next. Rather the image is like an octopus with each tentacle corresponding to a particle. Each tentacle ends where an actual detection happens. The head of the octopus would be something that is outside of spacetime.

Eq. 7.1.7 on p. 279 is the equation for the actual particle trajectories guided by the psi field. These trajectories, in this approximation, do not modify the the psi field which is acting as the unguided guider. The equation is again on a spacelike surface for a common time. What might it mean if each particle has its own time? Can we use the rule that the distant particles only act from their actual detections? This does not make much sense since the Bohm theory is supposed to be independent of any detection process.

Therefore, we have a major conceptual problem on how to interpret the Bohm theory if we try to have local times for each particle as in Tomonaga's theory. How do we decide at which local time each particle influences everyt other? Is the influence continual or does some sort of Smoky Dragon-like irreversible measuring process, which violates the spirit of Bohm's idea select out specific events on the worldlines of the distant particles? Bohm avoids this problem by coupling his theory to the co-moving coordinates of the cosmological solutions of Einstein's classical general relativity equations which provides the preferred frame in cosmic time. We have to look for a globally self-consistent solution for all the trajectories. What to do in quantum gravity is another problem discussed below.

Leaving the above considerations aside, returning to Bohm's theory with a common time. We have context dependence in which the nonlocal quantum potential acting on each particle is not a preassigned function of the positions of each particle but is determined by the psi field. There are an infinite number of possible quantum potentials for each point in N-particle configuration space. Since each quantum potential determines the momentum of each particle, each quantum potential will generally give a different point in N-particle phase space.

We may say that whereas in classical dynamics the whole is the sum of its parts and their interactions, in quantum mechanics the whole is prior to its parts (particles) and its properties cannot be explained by a kind of superposition of the properties of the parts. p.282

The circumstance that the configuration space wavefunction depends on a single evolution parameter t implies that the state of the N particles is specified at a common time and that there is a nonlocal connection between them brought about by the classical and quantum potentials. p.282

... the instantaneous motion of any one particle depends on the coordinates of all the other particles at the same time . p. 282

the contribution to the total force acting on the the kth particle coming from the quantum potential does not necessarily fall off with distance and indeed the forces between particles may become stronger, even though |psi| may decrease in this limit. This is because the quantum potential Q depends on the form of psi rather than on its absolute value...the three features ... the dependence of each particle orbit on all the others, the response of the whole to localized disturbances, the extension of actions to large interparticle distances - will be referred to as 'nonlocal connection' p. 282

OK coming back to the vexing problem of how to generalize Bohm's theory so that it does not need a common time t, let's look again at Tomonaga's classic paper of 1946 section 3 in Schwinger's Dover reprint volume on Quantum Electrodynamics. In particular his equation (20). The condition for a globally self-consistent solution is his eq (21) which is N^2 conditions

must be obeyed for all pairs k and k'. A sufficient condition is that the two spacetime events for k and k' are spacelike separated. But this is not good enough. We need also to consider cauality-violating theories so that we need a necessary condition which probably involves a global constraint on the trajectories of each particle for all spacetime.

Two particle interferometry. In the first edition of The Dancing Wu Li Masters I wrote about such an experiment that would allow faster-than-light communication by causing a fringe shift at the receiver controlled from the transmitter. Larry Bartells, Henry Stapp and others showed that this could not happen because of orthogonality of the transmitter eigenfunctions. Consequently, Gary Zukav removed this part from later editions of the book. However, Holland's treatment of the problem in Bohm's perspective apparently indicates that one need not have orthogonality of the eigenfunctions at the transmitter. Note to myself, go back and work on this. There is still the general Eberhard theorem that says nom atter what you do you can't communicate via quantum nonlocality within the present rules of the game.

This note is based upon 7.3 pp.296-299 of Peter Holland's book, The Quantum Theory of Motion (Oxford University Press, 1995)

The experimental interferometer geometry is

Fig.1

/\ /\

/ \ / \

/ A\/D \

\ C/\B /

\ / \ /

\/ \/

We have four one-body wave packets (A,B,C,D) to build the entangled two-body wavefunction (modulo normalization factor) at a fixed common time

(x1,x2|psi) = (x1|A)(x2|B) + e^iphi(x1|C)(x2|D) (1) based on (7.3.1)

All of the statistical predictions for local measurements on particle 1, for example, are computed from the reduced density matrix [1]

rho(x1,x1') = Integral (psi|x1',x2)(x1,x2)d^3x2

= (A|x1')(x1|A) + (C|x1')(x1|C) + Z(C|x1')(x1|A) + Z*(A|x1')((x1|C) (2a)

Z = e^-iphi Integral(D|x2)(x2|B)d^3x2 (2b)

(2) and (3) correspond to Holland's (7.3.2) and (7.3.3) respectively.

Let Z = |Z|e^i@ and (x|j) have the polar form r(x)je^is(x)j so that

rho(x1,x1') = rA^2 + rB^2 + 2rArC|Z|cos(sA - sC + @ - phi) (3) or (7.3.4)

Holland writes

"We see that as long as |Z| =/ 0, an interference pattern will indeed be observed in the region of overlap of the particle 1 packets." p.297

Indeed the fringe contrast in the region of particle 1 is

Local Fringe Contrast = 2rArC|Z|/(rA^2 + rC^2) (4) or (7.3.5)

Notice that |Z| is apparently a nonlocal control parameter so that this looks like a communication system. In fact I proposed this very same geometry as a communication system in the first edition of The Dancing Wu Li Masters. Gary Zukav took it out of later editions. Actual experiments have been done but only for |Z|= 0 in which there is no local fringe contrast but one sees fringes in the nonlocal correlation pattern as an analog to the Aspect photon pair polarization correlation pattern.

In 7.3.2 Holland treats the approximation of plane waves and in that case |Z| = 0 and there are no local fringes. But that artificial example is not conclusive.

So the real issue here is whether or not the wave packets D and B, for example are forced by some other law like unitarity to be orthogonal under all possible experimental conditions when the packets overlap in space. If so, then this geometry cannot be used as a communication system. However, on the face of it there is no obvious physical reason why orthogonality should be demanded. Measurement theory tells us that the packets are orthogonal in a good measurement of a Hermitian observable, but there is no reason to demand such a measurement for particle 2 in the given situation.

[1] If we use the Tomonaga many-time formalism we would have rho(x1,t1,x1't1',x2t2,x2't2') but it is not clear how to define Z in this case. The integral should be over the world lines and we need to go to a path integral formalism. In any case the discussion here is non-relativistic and is not complete or conclusive for that reason. In general we have four distinct spacetime events for the fourth-order correlation function of two particles. We need to use Green's functions for the relativistic fields and the theory of multiple time wave functions of Aharanov et-al may play a significant role.

. we emphasize the role of the particle in resolving the measurement problem... The pure wave dynamics described by Schrodinger's equation does not yield any account of which result is actually realized in an individual measurement operation. ...p. 350

A pair of particles is produced moving in opposite directions along the z axis with the entangled wave function

Particle 2 is a photon in which g+(-)(x2) is an eigenfunction for linear polarization along the x(y) direction. The "receiver" for the photon consists of a pair of crossed polarizers (one i.e.,"|" oriented along x, the other i.e.,"o" oriented along y sticking out of your computer screen) followed by a photon counter as in Fig 1.

Fig 1

receiver

1 recoiling atom 2

<- -> | o >- photon counter

x-pol y-pol

There is essentially complete spatial overlap between the wavepackets g+(x2) and g-(x2) at all times. There is essentially no spatial overlap between the atomic wavepackets f+(x1) and f+(x2).

The encoding of each message bit at the "transmitter" is "0" if a position measurement X^1 of particle 1 is made, and "1" if a momentum measurement P^1 of particle 1 is made. Particle 1 is massive like an electron or a small atom. Indeed, we can envision the emission of an energetic photon from a recoiling neutral hydrogen atom. The photon cannot be too energetic because it is hard to make polarizers for x rays, but it can't be too weak because we want to be able to detect the atomic recoil.

Suppose choose to make a position measurement of the recoiling atom at the transmitter so that we encode a "0"symbol into the message string. Remember we are doing Bohm theory where there are actual particles with definite paths in addition to an objective wavefunction field in configuration space. This is not the Copenhagen interpretation in which there is only a wavefunction with no actual particles. One possibility is that the unique actual atom path enters the space region where f+(x1) has support. The effective wave function after the X^1 position measurement is then

or if the atom is in the support of f-(x1) the effective wave function is

In either case, photon 2 will not be able to pass the crossed polarizers and it is not possible for the photon counters to click. We can even increase the intensity of the source and still the counter cannot click.

Now instead, we decide to encode a "1" symbol into the message string by doing the momentum P^1 measurement on the transmitter atom. The final pair wave function is then Peter Holland's equation (11.1.8) on p. 464 which in my notation is

where p1 is the resulting eigenvalue of the momentum measurement on the atom. phi+(-)(p1) are the Fourier components of f+(-)(x1). In general phi+(p1) and phi-(p1) overlap in momentum space even when their Fourier transforms do not overlap in position space. Therefore, the effective state in equation (3) is a coherent superposition of the photon linear polarizations along x and y directions so that there is a finite probability for the photon to pass through the crossed polarizers. If we increase the intensity of the source we will get a range of possible p1 eigenvalues, but we will get a signal in the photon counter even if it has a small efficiency.

Notes on Chapter 12

There is a very interesting example on p.500 (eqs. 12.18- 12.1.10) in which a subluminal guiding wave is able to propel its particle (in one space dimension) into superluminal warp flight. That is, (in dimensionless units) the guiding wave obeys E^2 - p^2 = 1 > 0, but its particle has parts of its trajectory where po^2 - p1^2 is negative as is po when (1 - E)t + px = pi. This sort of thing is not even thinkable in the Copenhagen interpretation in which the wave without the particle is the complete description of quantum reality.

Note, the above remarks are for spin 0 matter obeying the Klein-Gordon equation. Spin 1/2 matter obeying the Dirac equation has timelike particle trajectories at least below the threshold for real pair-production. This is a problem for current research.

How is one to define localized position in orthodox relativistic quantum mechanics? We cannot do without the negative energy states. If we do not include negative energy states, the argument x of both the spin 0 Klein-Gordon and the spin 1/2 Dirac equations is not the eigenvalue of a Hermitian operator. This implies a breakdown of the Born probability density formula in this case. Newton and Wigner tried to define a position operator whose eigenfunctions have a spread equal to the Compton wave length. Their program lies uncompleted to this day.

Lorentz covariant wave equations can violate causality and permit faster-than-light signals unless one throws in ad hoc constraints. (p.502)

The spin 1/2 Dirac equation in Bohm's theory does allow us to picture slower-than-light spacetime paths (e.g.,12.2) unlike the spin zero Klein-Gordon equation. However, a relativistic theory of bound spin 1/2 particles forming spin 0 would have to be described by the Klein-Gordon equation which demands faster-than-light motions that quantum tunnel through the Einstein barrier. This is the sort of "exotic matter" needed to stabilize traversable wormholes.

Holland has a neat solution of the Klein Paradox in the Bohm theory. In the Copenhagen interpretation, a Dirac relativistic wave is incident on a potential step. As the barrier height is increased, the Dirac equation predicts the counter-intuitive result that the particle is able to get through the barrier. The Bohm picture shows that what is actually happening is that the particle motion is opposite to its wave motion in the "paradoxical" region. Therefore, the actual hidden variable particle is not incident on the barrier form the left, but is coming in from the right and simply falls "over the cliff" into the region of its "incident" wave. (p.515)

Bohm's thoery permits a visualization of the Dirac equation's "Zitterbewegung" ("Trembling movement"). This is an oscillation about the classical motion confined to a tube of diameter equal to the Compton wavelenght h/mc. (12.3.3)

So far we have focused on the classical "particle" hidden variable. But Bohm's theory also has the classical gauge fields, like the electromagnetic field, as a hidden variable. This feature is not well understood by the majority of Bohm's critics. So let's see how Bohm quantized classical field theory as a generalization of how he quantized the classical mechanics of particles.

Bohm's Version of Quantum Field Theory

Consider the neutral Lorentz scalar classical local field psi(x,t). The local classical field Lagrangian density is

The metric signature is +---. The action principle gives

The canonical conjugate field momentum is

The local classical field Hamiltonian density is

The complete Hamiltonian of the field is the 3D space integral over an entire spacelike surface whose 3D metric is gij. Unlike the particle case, the classical field Hamiltonian is a nonlocal quantity.

Now in classical Hamilton-Jacobi theory, the local canonical conjugate field momentum pi is replaced by the local Tomonaga functional derivative &S/&psi, where the nonlocal S[psi] is not a simple local function but is a nonlocal functional of the particular field configuration psi over the entire spacelike slice of 4D spacetime. The classical field Hamilton-Jacobi equation for the time evolution is

which looks like a fragment of the quantum Schrodinger equation missing a wave function.

This is re-written as the partial differential equation

Quantization is done by introducing the microcausal field commutation rules. The field commutators vanish over spacelike separations. This condition is necessary for the solution of renormalizable field theories and Holland does not question them in his book. My own program is different. I want to use microcausality-violating commutation rules which allow faster-than-light signal propagation within the spacelike surface as a description of the invisible dark matter and exotic matter. This kind of theory may eliminate the need to renormalize. But it is mathematically difficult although the use of Green's functions with new superluminal mass shells and new contours in the complex energy plane seems to be the way to go. Violations of the spin-statistics connection for dark and exotic matter are expected.

Returning to the standard quantum field theory, the classical local field psi plays the role of the "particle position" and should not be confused with the quantum wave function PSI in Hilbert space. The Schrodinger equation for the quantized field is

where &/&psi is a local functional derivative operator. This corresponds to making a small deviation of the classical field pattern psi(x,t) in a small region of the spacelike surface as described by Tomonaga . It is important to keep in mind that PSI (the quantum wave function of the quantized field) is not a local quantity, but depends upon the quantum field psi over the entire spacelike surface inside spacetime.

To get to the Bohm theory use the polar decomposition of the nonlocal complex-valued PSI

The Bohm equations for the relativistic quantum field are

where the superpotential of the quantum field is

Quantum field theory limits to classical field theory when the superpotential vanishes (Q -> 0). Note we are using hbar = 1.

Note that R^2 Dpsi is the generalized nonlocal Born probability for the field to be measured in an element of volume Dpsi about the configuration field pattern psi(x) over the entire spacelike surface.

What is important is that the nonlocal quantum wave function of the field is an active nonclassical agent strongly modifying the dynamics of the classical field.

We now introduce the assumption that at each instant t the field psi has a well defined value for all x. as in classical field theory. whatever the state PSI of the field. The time evolution may be obtained from the solution of the guidance formula

&psi(x,t)/&t = &S[psi(x),t]/&psi(x)|psi(x) = psi(x,t) (12.4.13)

(analogous to mdx/dt = grad S ) once we have specified the initial function psio(x) (analogous to xo). ...

To find the equation of motion of the field coordinate ....

(d/dt)&psi/&t = - (&/&psi)[Q + Integral dx^3 (1/2)(grad psi)^2 (12.4.14)

d/dt = &/&t + Integral dx^3 (&psi/&t) &/&psi (12.14.15)

Equation (12.4.14) is analogous to the particle equation of motion

m d^2x/dt^2 = - grad(V + Q)

Noting that

(d/dt)&psi/&t = &^2 psi/&t^2

and taking the classical 'external force' term over to the left hand side ...

D'Alembertian psi(x,t) = - &Q[psi(x),t]/&psi(x)|psi(x) = psi(x,t) (12.4.16)

... The 'quantum force' term on the right hand side is responsible for all the characteristic effects of quantum field theory. pp. 521-522

The energy of the quantum field,

E = -&S/&t is continuously variable and is not conserved in general. ...

dE/dt = &Q/&t|psi(x) = psi(x,t) (12.4.17)

.. the causal interpretation enables one to account for individual events in quantum field theory if the classical definition of an individual system, the field psi, is supplemented by the state PSI[psi]. p.522

Full Lorentz covariance is restored in Bohm's theory in the classical limit where the superpotential Q is small. This suggests a new connection of the nonlocal quantum superpotential of the local field to the global curvature parameters of the Friedmann solution.

"The breaking of Lorentz covariance is a quantum effect of individual processes" p.523

From the c-number equation (12.4.16) which is Bohm's fundamental equation for relativistic quantum field theory, it follows that "the evolution of the field is governed by a highly nonlinear and nonlocal equation that, in principle, involves the state of the field over the entire universe" p.523

Note that from the right hand side of (12.4.16) it appears that this nonlocality extends over all time as well as all space in what Kip Thorne calls a "globally self-consistent" way. Therefore, unless I am mistaken here (and I might be) the "|psi(x) = psi(x,t)" constraint implies a teleological view of the evolution of our actual universe. See below on quantum gravity and the "Mind of God" (Hawking's term) for more details.

Holland does not include the trans-temporal connections, but limits himself to spacelike surfaces - presumably the preferred surfaces of the Hubble flow in the standard Big Bang model.

This is the field-theoretic version of the holistic aspect of quantum theory ... it implies instantaneous connections between distant field elements which, in turn, implies the violation of [special] relativity in individual processes. ... This feature is characteristic of the quantum domain and vanishes in the classical limit. Nonlocality of fields is a quantum effect. .... Lorentz covariance and locality are statistical effects." pp 523-524

Although Bohm's theory has definite paths for particles of finite rest mass not so for photons of zero rest mass. (12.6). Indeed on p. 555:

...we furnish an example of a state for which the normal ordered mean energy density is negative and the current space-like. The simplest such example is superposition of the vacuum state and a 2-quantum state:

|PSI> = f|0> + f'a*k1 a*k2 e^-i(k1 + k2)t |0> (12.6.61)

... f and f' are real and positive and f^2 + f'^2 = 1. ... |k1| = |k2| = l .. let & be the angle between k1 and k2 with & =\ 0 or pi. Then

= (2kf'/V)[f' - fcos@cos^2(&/2)] (12.6.62)

where @ = 2kt - (k1 + k2).x. Clearly choosing f to be nearly unity and f' almost negligible this expression will become negative at certain spacetime points (where cos@ = 1) due to the interference term... It is this sort of thing that occurs in the Casimir effect ... where the modified vacuum state can be expanded in terms of the Fock basis appropriate to the old vacuum and the basis states interfere.

We have also ...

= V^-1(k1i + k2i)(f'^2 - ff'cos@ (12.6.63)

... in the observer's rest frame

j^uju = <:Too:>^2 - = 4k^2V^-2f'^2sin^2(&/2)(f'^2 - f^2cos^2@ cos^2(&/2)) (12.6.64)

... choose again f' very small, f large ... cos@ = -1 so that the energy density is positive.. The (12.6.64) is negative. In such states, we cannot define a subluminal flow of mean energy, even if this is positive.p.556

12.9 Beyond space-time-matter. Wavefunction of the universe

The classical analytical approach to the theory of matter assumes that a complex physical system may be broken down into a collection of subsystems obeying relatively simple laws which govern their interactions, and that the state of the whole is defined by no more than a summation of the states of the parts.

In the quantum theory of motion this procedure is turned on its head. Here the basic notion is that of an objectively real state of an individual system that lies beyond its material components (particles and fields in their classical conception), and even beyond the spacetime manifold. In the sense that its law (the Schrodinger equation) governs the law of the elements (the guidance formula), we may say that the state of the whole is prior to that of the parts (in the model studied in this book the parts are not physically determined as aspects of the whole, as they would be in a unified field theory, for instance).

Including the laws of the parts in the quantum mechanical theory allows one to overcome what we have located as the central interpretative dilemma posed by quantum mechanics the failure to provide sufficient concepts to furnish a complete description of the processes with which quantum mechanics deals in circumstances where we are sure something more could be said that is not contained in the formalism (e.g., the positions of meters).

The holistic concept represents a first step towards realizing Einstein's programme of freeing microphysics from its reliance on the classical paradigm, although not in a way that Einstein approved of. It is, perhaps paradoxi