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This paper is the second of a series of three [in need of much correction and editing], in which I explore the relationships between: Impact, Force and Potential; The Principal of Least Action; Efficient and Final Causes; Singularities in Newtonian Mechanics; Quantum Indeterminacy; and human Free-Will. These papers are much more physics based and mathematical than any I have previously posted on my Web Site. I beg the indulgence of those readers for which these fields are foreign. I shall attempt to make the issues intelligible. I have appended a Bibliography so that those who are interested in acquiring a deeper familiarity with the issues here discussed may have some idea how to set about doing so. I wish to acknowledge helpful discussions with my Physicist/Neuroscientist friend: Paul Miller. Of course, he should not be held in any way accountable for the speculations and proposals made here!

In the first paper of this series, I sketched out the philosophical basis and conceptual structure of space and time: now I shall attempt to give an account of Quantum Mechanics.

The discussion so far has taken it for granted that the idea of a thing is applicable to the real world: moreover that a thing exists at one point at any one time, and at an adjacent point at the next moment of time. Unfortunately, it is impossible to sustain such a picture.

Angels and Hopping

Empirically, both momentum and Mass-Energy is found to be conserved. For a single thing in isolation this would be equivalent to the conservation of its Minkowskian four-velocity. This is Newton's First Law of Motion: isolated things travel in straight lines. Now, it is very difficult to conceive how this conservation law could be be compatible with the theory we have constructed up to now. This is because the only trajectories that have a clear rule for joining up points are those that lie exactly along one of the N lattice directions. For these directions, and these directions alone, trajectories can be prolonged from point to point by mapping each inter-point connection to its opposite, which is already presumed to be well defined.

Any trajectory that does not lie along a lattice direction has to be decomposed into a sequence of inter-point hops. Each one of these hops can only lie along a specific lattice direction. Other trajectories have to be constructed from patterns of hops in various lattice directions. For velocity to be conserved, the long term average of the N frequencies with which each of the lattice hops is adopted must be stationary. Either the hopping pattern is random: but with an unmoving macroscopic average, or it conforms to a specific repeating rule. In either case, some extrinsic guiding principle must determine the trajectory of any thing. In effect, an angel is required to steer each and every particle: save those that happen to be cruising down one of the "Interstate Highways" that parallel the lattice directions.

Geometry and Waves

This conclusion can be avoided by postulating that particles, as such, do not exist. Instead, the lattice grid is used to substantiate a wave equation. Each point is then seen not as a potential location of a thing, but as an actual thing itself. The vacuum is not empty, waiting to be filled. Space is utterly full of, and in fact constituted as a network of things [Parmenides]. Each point thing can vibrate, in some sense. More accurately, a set of parameters exists for each pointing. The values of these parameters at each point influence the values at its nearest neighbours, so a continuously varying field is set up over the lattice grid. Fluctuations in this field spread out from each point in accordance with what is known as the Green Function for the wave (or field) equation. On a macroscopic scale, this spreading out will be isotropic. Asymptotically, the expanding spherical surface generated by the Green Function specifies every possible plane wave direction and hence straight line trajectory.

The most familiar physical fields are those of Maxwellian Electromagnetism and Newtonian Gravity. Relativity Theory has succeeded in identifying the Gravitational field with the curvature of the Metric Tensor. It is possible that all the field parameters are nothing more than descriptors of the distortions of the connections that each point has with its nearest neighbours. In which case, all Physics can be reduced to geometry [Pythagarus, Plato "Timaeus", Euclid, Galileo, Descartes, K.R. Popper "Conjectures and Refutations" (1972), 75-93], as Einstein had hoped but never managed to demonstrate [K.R. Popper "Quantum Theory and the Schism in Physics" (1982), 160-173]. Unfortunately, while the Maxwellian field is entirely compatible with relativity: most obviously when expressed in terms of the four-vector potential, A, it has not yet proven possible to reduce Electromagnetism to geometry and so unite it with Gravity. This is one of the central problems of contemporary physics.

Wave Packets and Matter Waves

Mesoscopically, a wave front is a coherent pattern of point disturbances or impulses in the field. The set of isotropic Green Functions or Impulse Responses that result then add up: interfering constructively or destructively at neighbouring points. They do so in such a way as to propagate the wave front onwards. In the case of a plane wave front travelling in an isotropic medium (such as either a cubic lattice or a glass), the propagation is exactly orthogonal to the wave front, whatever that direction might be: even if it is not one of the N lattice directions.

On this hypothesis, particles are viewed as being in some manner constituted from waves: in a linear theory as wave packets, in a non-linear theory as solitons [P. Strange "Relativistic Quantum Mechanics"].

Particles and Potentialities

This wave based hypothesis can be re-presented as a particle based hypothesis if particles are conceived as taking not one route through the lattice, but rather exploring every possibility. A particle is no longer represented in Minkowskian space as a single life-line, but as a set of potentialities or amplitudes defined at each and every point and extending over the whole of the Cosmos. The amplitudes depend on the interactions of the particle with all others, and the overall pattern of amplitudes is fixed by minimizing the "Action". The amplitude of a particle at a point is the strength with which it can interact with another particle at that point.

Note that on either version of this hypothesis (which was invoked only to give an account of Newton's First Law!) every particle or wave is utterly delocalized: filling the whole of Minkowskian Space-Time. Knowledge that is gained of its potentiality in one spatio-temporal neighbourhood immediately affects our estimate of its potentiality at remote places and times. This is the basis of the "super-luminal collapse of the wavepacket" [K.R. Popper "Quantum Theory and the Schism in Physics" (1982), 74-79, quoting W. Heisenburg "The Physical Principles of Quantum Theory" (1930), 39].

Probability and Potential

A simple relationship exists between the Maxwellian field and the Wavefunction of Quantum Mechanics. Maxwell's equations can be written in the extraordinarily (and deceptively) compact form: In the vacuum (free space),  j = 0. The seemingly trivial equation that results is known as the Klein-Gordan equation [Schiff "Quantum Mechanics"]. It is the Relativistic Wave Equation that describes photons. There is no good reason, therefore, to distinguish between the four-vector that is the Electromagnetic Potential and the wave function for particles of light. Universally, however, the latter is taken to be two independent scalar fields (one for left-hand circularly polarized light and another for right-hand circularly polarized light), rather than a four-vector with two vanishing components.

The Klein-Gordan equation has to be modified slightly to deal with particles with non-zero rest mass: archetypically the electron. Following the pattern required by relativity theory, the following equation is inevitable:

[ []2  -  ( me c2 )2Y  =  0
  • Y  is the relativistic electron wavefunction.
  • me is the rest mass of the electron.
  • c    is the speed of light in a vacuum.
  • Conventionally, this equation is factored to reveal its physical significance. The Dirac equation of Relativistic Quantum Mechanics results [Schiff "Quantum Mechanics", Strange "Relativistic Quantum Mechanics", R.P. Feynman "QED, the Strange Theory of Light and Matter"].
               HD  Y    =      E   Y
    g . ( p  - e A) Y    =      mc Y
                          pj    =      [ hc / (2p) ] d/dx ;  j = 1 ... 4
                          p4   =   -  [ h   / (2p) ] d/dt
               gi gj   =   - gj gi    ;    i  =/= j
    Where: The Schrodinger Equation of non-relativistic Quantum Mechanics [R.P. Feynman, R Leighton and M. Sands "Lectures on Physics, Volume III"] can be obtained as a low energy approximation to the Dirac Equation [Schiff "Quantum Mechanics"].
                                           HS  Y    =    E  Y
    [ 1/( 2meSj=1...3 ( pj ) +  V ]   Y    =    E  Y
    A question arises at this point, that to the best of my knowledge has no good answer as yet:
    "Why does "me" appear the electron's wave equation, but not in that of the photon?"
    Whereas it is possible to replace the number m with a differential operator MD :
    MD Yphoton   =  0
    MD Yelectron  =  me Yelectron
    and an explicit form be given for MD: it remains unclear why the Y should adopt the internal symmetry (structure or form) that is in fact equivalent to either a zero mass or the rest mass of the electron, and no others. The problem is compounded by the existence of other particles that seem to be heavier versions of the electron. As yet, we have no theory for the mass of the electron, or any other elementary particle. This is a major deficiency in Quantum Mechanics.

    There is a slight complication here. The four components of  Y , unlike those of A, are not equivalent to three of space and one of time, but rather relate to the mysterious properties "spin" and "electric charge". This difficulty can be eliminated by elevating the Wavefunction (and so also the vector potential) to the status of a four-Tensor: Y. This has sixteen components, rather than four. The conventional Y and A are then revealed as the amplitudes of certain modes of vibration of this tensor field. Moreover, other modes of this field can be identified with neutrinos and particles associated with the so-called Weak Interaction. Room can now be made for the mass of the electron, by constructing a matrix operator MM:

    Y~electron Yphoton     =  0
    Y~electron Yelectron   =  1
    Y~photon Yphoton      =  1

    MM  =  me [ YelectronY~electron ]

    This can be extended to deal with muo and tau mesons: the heavy electrons. Nevertheless, it is an ad hoc expedient. While it describes the various rest masses, it does not explain their origin. Even so, it is a singular success of Relativistic Quantum Mechanics that Electromagnetism has been unified with the Weak interaction: and subsequently with the Strong nuclear force.

    What all this means is that the mysterious Y that features prominently in Quantum Mechanics texts is a version of the more familiar (but I suppose no less mysterious) Electromagnetic field.

    Collapse of the wave-packet

    The account of the Cosmos given here does not envisage things that exist at particular places at particular times, that is: have definite life-lines. Rather, particles are envisaged as either being composed of waves: essentially extended and liable to disperse, or as existing everywhere and nowhere: more like ghosts than real objects. In every day experience, as also in the Physics Laboratory, particles are found to be very real. Whereas the likelihoods of various events are accurately predicted by the formulae of Quantum Mechanics, based on the account of reality that I have sketched out above: the events themselves neither feature in the formulae no in the account of reality that underlies them. What we experience is a scintillation: a brief flash of light emanating from a singular, definite and particular place at a singular, definite and particular time. What our metaphysics knows of is a distribution of potentialities for such a scintillation to occur, and this is what the equations of Quantum Mechanics deal with. This tension is at the heart of Quantum Mechanics.

    In passing from one style of thought to the other, one has to either throw away or cobble together a wave. In describing experimental results, the wave is reinterpreted in terms of a probability distribution for discrete events: it is then said to have "collapsed". This terminology reflects the fact that only one singular and definite outcome of the many possible actually occurs. The ubiquitous wave is thrown away and replaced with a definite singular local event. In predicting what happens next, this singular event is then used as a specification of initial conditions for the wave equation. In effect the Dirac delta function that describes it is decomposed in terms of a complete set of expansion functions foreign to the event that has just been observed but natural to the next process according to which the wave will propagate.

    Our experiences are each our own, subjective; particular and partisan: the Cosmos seen from a definite temporal point of view of some one thing, within its confines. The picture that I have sketched is objective: the Cosmos seen from the eternal point of view of the Deity who is no thing, without its confines. To be consistent, the experience of each observer should itself be represented as internal configuration states of the subsystem that is the observer's mind. If this is done, at first sight it seems that the mind must itself loose definiteness and uniqueness of experience. Pursuing this direction of analysis further hazards a metaphysic akin to Everett's "MultiVerse" hypothesis: in which everything that could happen does happen in "parallel". However, this does not help to explain the fact that my experience is unique and specific, not multiple and diffuse. Neither does it give any account of my experience of time as a sequence of spatial events.

    Solitons and Substantial Forms

    These difficulties would vanish if particles turned out to be solitonic excitations of the metric, as I understand string and N-brane theory would have it. Particles would then turn out to be nothing other than the abstract, but definite, mathematical centres of the core of each solitonic disturbance.

    Where two solitons are far apart, their centres would be well defined and distinct. They would trace out unique Minkowskian life-lines. When two or more solitons occupy the same region of Minkowskian four-space, the shape or form of their cores would be distorted and their centres be less clear. The particle positions would have to be defined in terms of something like a correlation integral between the single combined actual form of the interacting solitonic waves and the various ideal substantial forms of each separate solitonic wave as it would have existed in isolation. The interacting bulk would participate to various degrees in the forms of various particles. These would not just be the forms of the two or more incoming particles that existed prior to the start of the interaction, and have life-lines that extend into times before its Minkowskian neighbourhood. Additionally, the actual form would participate in the ideal substantial forms of the outgoing particles that exist when the interaction is complete, and have life-lines that extend into times after its Minkowskian neighbourhood. Moreover, for high energy interactions, various transient forms might be required to exhaustively expand and express the actual form of the interacting mass. In fact, such transient forms are observed and are called resonances.

    In the most extreme interactions, the actual form would cease to have any significant resemblance to any of the incoming substantial forms whose collision gave rise to it. What particles such interacting matter is conceived as being made up from is largely a matter of taste. Particles only exist in isolation. To the degree that they interact, they loose their self-identity. In catastrophic encounters, any convenient accounting of matter in terms of long-lived particles would break down, and its base nature: that of an anonymous vibration of the metric or space lattice, be revealed.

    The basic wave nature of matter serves to entangle remote particles. After all, each and every electron is composed of the very same waves. Hence, each tends to be continuous and coherent with every other, no matter how remote in space and time.

    Impact, Force and Potential

    So far, the discussion has focused on the Cosmos as composed of Space-Time and either OneThing existing within this Cosmos, or a number of things existing independently. In fact, things (or distinct appearances of the OneThing) inter-act. They do not generally obey Newton's First Law. They do not travel at constant speed in straight lines. They are subject to forces. A forces is the efficient cause of acceleration: deviation from uniform rectilinear motion.

    Two models of force can be distinguished.

    No Action at a Distance

    In both of these models, the necessity for "Action at a Distance" to be contemplated is removed. This is a huge philosophical advantage, as it is very difficult to envisage how one thing can affect another thing where it is not. After all, what could "where it is" mean if not "where it acts"?

    Influence or Impact?

    The success of the second model does not indicate that it is correct and that the former is false. Its success is computational: its formulae are eminently appropriate for numerical evaluation. Its success does not imply that the only form of inter-action is that of collision, but rather only that the physical signature of any small scale details of all physical interactions rapidly decay to insignificance. In order to implement the diagrammatic method, an infinite number of "virtual particles" have to be considered. These are particles additional to those that are ever in fact observed. They have only the most transitory of existence: such that the product of their life-times and their rest-mass energies is less than Plank's Constant. Such a plenitude of virtual particles is arguably nothing more than a convenient expansion of the continuous field that really exists.

    The Mental World

    I intend in my next paper to address the issues of:  The Principal of Least Action; Efficient and Final Causes; Singularities in Newtonian Mechanics; Quantum Indeterminacy; and human Free-Will.

    Appendix 1: How can an impact be attractive?

    The conceptual difficulty with representing all interactions as collisions is that it would seem that all interactions must be repulsive. If two electrons exchange a photon, then the electron that emits the photon is impelled away from the electron that absorbs the photon: because the trajectory and hence (it would seem) the momentum of the photon must be directed from the first electron towards the second. Similarly, the electron that absorbs the photon is knocked away from the electron that emitted it. This might serve as a good picture for electrostatic repulsion between two electrons, but how can it account for attraction between an electron and a positron?

    The first part of the answer to this question lies in the realization that a photon can have either a positive or a negative energy:

    Y    =    exp [  i ( ±w t   +  k · r ) ]
    Y    =    exp [   ±i ( w t   ± k · r ) ]
    E2   =    m0c2  +  ( cp )2
    E    =     hw/(2p)  =  ± cp
    w    =   ± c | k |
    u    =  ±k  /  | k |
    In the negative case, its momentum,  p  =  hk/(2p)  is directed contrary to the group velocity,  u  =  dw/dk. The group velocity governs the motion of the photon, so if a pair of electrons were to exchange a negative energy photon, they would be deflected towards each other: the interaction force they experience would be attractive.

    A single electron interacts with the vacuum (that is, it emits and absorbs photons) whether or not a second electron is present. This does not affect its trajectory. It is not scattered from uniform motion in a straight line: it absorbs every photon that it emits just as that photon is emitted: before it has had any chance to move! Nevertheless, the fact that the photons can (and are being emitted and instantly re-absorbed) means that the electron's inertial mass is hugely modified. Surrounding each electron is a stable halo of positive energy evanescent photons, constituting the potential energy of the electrostatic field. This is the quantum mechanical version of the problematic classical "self-interaction" energy [R.P. Feynman, R Leighton and M. Sands "Lectures on Physics" (1964)], and which leads on to "renormalization" [Feynman: "Q.E.D." (1985)].

    When an electron absorbs a positive energy photon from the halo of another electron (which it can do, because the composite halo arising from the two electrons is no longer in equilibrium with either one of them) it must be repelled away from the other electron, as we have already seen. Moreover, the total kinetic energy of the two electrons is increased and the potential energy of the electric field is decreased. All of these conclusions are in harmony with classical electrostatics. For momentum to be conserved, we must have the constraint that whenever one electron absorbs a photon of momentum  p, an other electron must absorb a photon of momentum  - p.  It is important to realize that these two absorption processes are not counterbalanced by any matching emission processes. Instead, the potential energy of the electric field decreases as the total number of evanescent photons is reduced.

    Positrons can, I have already said, be thought of as electrons travelling backwards in time. The interaction of two positrons is therefore symmetric with that of two electrons, and is equally repulsive and productive of fermionic kinetic energy.

    When a positron absorbs a positive energy photon from the edge of the halo of an electron, it is attracted towards that electron: because it has a negative inertial mass. Moreover, the photon halo of the electron is depleted. Simultaneously, the electron must absorb a negative energy photon from the halo of the positron: causing it to be attracted towards the positron and depleting the positron's halo. All of these conclusions are in harmony with classical electrostatics.

    It is now appropriate to review the relationship between momentum and velocity for massive particles. The issues are most stark in relation to the electron-positron annihilation process, and its inverse.

    Electron-Hole Recombination

    Before considering the phenomenon of electron-positron annihilation, I find it helpful to reflect on the "solid-state" electron-hole recombination process. This is analogous to electron-positron annihilation, if Dirac's picture of positrons as holes in an infinite sea of negative energy electrons is taken seriously.
    The Dirac Hole
    Let the electron and hole be described by equivalent wave packets travelling with equal and opposite group velocities (with respect to the atomic lattice) along the same straight line. The reference frame is that of the "centre of mass" of the colliding electron and hole and the initial total momentum is therefore zero. Because the energies of the hole-band electrons are negative, the group velocity of the hole is directed in opposition to the sense of the momentum of the electrons that carry it. This is what is required, for the hole to be approaching the electron. The momentum of the hole is not opposed to its group velocity, because the hole is a lack of electrons and its total momentum is therefore minus that of the electron wave-packet. Hence both its group velocity and momentum are reversed even though it has exactly the same set of k-amplitudes as the electron. If the hole is to be attributed a single representative khole, this must clearly be -<kelectron>, in order to preserve the relationship <p> = h<k>/(2p). Equally, it has a positive inertial mass: as it accelerates the "missing energy" gets more negative, so the excitation energy gets more positive. Finally, it has a positive electric charge: the hole is a lack of negative charge, so its characteristic charge is positive.
    If this electron-hole pair recombines, both a photon and a phonon have to be emitted. This is because the initial momentum is exactly zero and the momentum of the emitted photon (which otherwise would be the only momentum carrier after the collision), although small, cannot be zero. If the initial velocities were not exactly equal and opposite, the initial momentum might be such as to allow for a recombination process that does not involve a phonon, but this would only be fortuitous.

    Any electron-hole pair travelling on an intercept course may recombine when their tracks cross. It is always possible to transform to the centre of mass reference frame, but in the case of solid state physics this is not a trivial enterprise because the atomic lattice constitutes an absolute frame of reference. When the centre of mass frame coincides with that of the atomic lattice, any symmetric local distortion of the lattice must be made up of a symmetric set of  k-amplitudes, and carry no net momentum. It will propagate away from the point of recombination symmetrically. Hence a directed phonon has to be created in such recombination processes. It will be counter aligned to the wave vector of the photon.

    When the centre of mass frame differs from that of the atomic lattice, the situation is very different. Although any symmetric local distortion of the lattice left behind when the electron and hole recombine is launched at the velocity of the "initial centre of mass" onto the atomic lattice, it will experience drag from the crystal and propagate symmetrically, relative to the atomic lattice.  In effect this introduces a mechanism for the lattice to abstract momentum from the recombination process in addition to standard phonon production.

    An alternative account of this phenomenon is that:

    To say that "direct" recombination processes, where the lattice does not take up any significant momentum: "do not involve phonons" is overly simplistic.

    Electron-Positron Annihilation

    The situation is entirely analogous. In fact what I have previously described is in some regards closer to Dirac Annihilation rather than Recombination, because I have made no mention of effective masses or any deviations from parabolic bands. The only difference between the mechanism described in the last section and Dirac Annihilation is that the vacuum has no "atomic lattice" in the sense of something rigid that provides an absolute frame of reference. Hence, given that phonons do not exist in a vacuum (in effect, photons are the phonons of the Minkowskian Lattice), an electron cannot "recombine" with a hole to produce a single photon in the absence of some additional massive object which can act as a source or sink of momentum.
    Initially, for any low velocity impact on a stationary target:
    p    =      m.v   +  m.0  =  m.v
    E    =   2 m c2  +  ½ m v2
    p  =  h k /( 2 p )
    E  =  hc k /( 2 p )
                     mv       =   h   k /( 2 p )
    m ( 2 c2 +  ½ v2 )  =   hc k /( 2 p )
        ( 2 c2 +  ½ v2 )  =   cv
        4 c2 - 2cv +  v=  0
                             v  =   c [ 1 ± ( - 3 )½ ]
    Which is nonsense, involving a complex initial electron velocity.  For  v = 0, the outgoing photon has a non vanishing momentum. As v is increased, the situation gets worse: the extra kinetic energy increases the momentum of the outgoing photon more rapidly than it increases that of the incoming electron.

    Backwards Time Travel

    I have previously stated that a positron can be construed as a positive energy electron moving backwards in time. I subsequently used this picture in explaining how impact forces might be attractive. This picture merits further discussion and should be related to the "Dirac hole" picture already used to make sense of electron-positron annihilation. The first task has already been accomplished. I shall now proceed to the second.

    The negative energy of a positronic solution to the Dirac equation can be made positive if the minus sign in the phase factor (the only dynamical part of the solution) is transferred to the time variable. This means that incrementing the "next" counter for this particle carries it to a previous moment in history: its "proper time" flow is reversed compared to that of neighbouring particles, the local frame of reference.

    A particle with positive inertial mass travelling backwards in time will appear to react to forces in the sense opposite to that which its charge would suggest. In unit proper time it will change its momentum by the force acting F, but to an external observer the sense of this momentum change will be negated, as the significance of "start" and "finish" will be exchanged.

    The velocity of an electron travelling backwards in time is defined by the orientation of its life-line with respect to the temporal unit vector: in exactly the same way as for an electron travelling forwards in time; or for a photon, which experiences no change in proper time whatsoever. Given that its mass is not negated by time reversal, it follows that the momentum of a backwards in time travelling electron remains aligned with its velocity: as does that of a Dirac hole, but by a double negation. This means that at any moment of time reversal, k  must also reverse.

    Just before the proper instant of time reversal, let the wave be travelling to the right. Instants that are subsequent in proper time are associated at any position co-ordinate with a more positive phase. This increment of phase is compensated for by an increase in position co-ordinate. Because proper time and local time have the same sense, this constitutes a velocity to the right.

    Just after the proper instant of time reversal, the wave is travelling to the left. Instants that are subsequent in proper time are associated at any position co-ordinate with a less positive phase. Without a reversal of  k, this increment of phase would be compensated for by a decrease in position co-ordinate. Because proper time and local time now have the opposite sense, this would constitute a velocity to the right. This conclusion is incorrect, and indicates that k must reverse at the proper instant of time reversal, just as was concluded for the Dirac hole.

    This makes sense for the following reasons:

    1. Time reversal is not by itself a Minkowskian operation. It is part of the four parity reversal operation. If this is used, instead of time reversal alone, then k also must also be reversed in order to compensate for the unwanted spatial inversion.
    2. Just before the proper instant of time reversal, both the energy and momentum of the electron are very large and positive: it has been accelerating towards the positron that it is about to annihilate with. At the moment of annihilation they both become singular. The pole in the energy as a function of proper time is resolved by negating the flow of local time. The pole in the momentum as a function of proper time is not so resolved, and manifests itself with a reversal of sign: just as would normally be expected.
    This discussion establishes the coherence of the second picture of antiparticles.

    This second picture makes it possible to dispense with Dirac's extravagant idea of an infinitely deep sea of negative energy electrons: replacing it with a more or less empty vacuum (save for "zero point energy") and a few positive energy electrons travelling backwards in time. The cost involved in this process is that the last semblance of "causality" being associated with "sequence in local time" is lost.

    It would also seem that whenever an electron gets close to some massive charged object (such as a nucleus) it might emit a photon and start travelling backwards in time as a positron! This is a version of the very Dirac Catastrophe that the electron sea was invented to avoid.

    On the other hand, how can an electron be attracted towards a positron that it is about to become, if in fact it doesn't become that positron?  If it isn't so attracted then it certainly won't become the positron, so it would seem that the simple process of intrinsic annihilation might be excluded, and only annihilations resulting from the accidental coincidence of independent life-lines be allowed.

    Appendix 2 : Feynman Path Integrals and Reality

    Richard Feynman was one of the great physicists of the Twentieth Century. Not only did he justly receive a Nobel Prize for his research work in the field of Quantum Electrodynamics, but he was one of the great educators. His three volume "Lectures on Physics" set the standard for undergraduate education in the 1970's. Of course, these great achievements and virtues did not make him infallible!


    His book "Q.E.D. the Strange Theory of Light and Matter" is a case in point. In this populist review of his own work he sets about presenting a Copenhaganist view of Quantum Mechanics while insinuating that his treatment eliminates the problem of "The Collapse of the Wavefunction" and all its concomitants. He proposes that Physical Theory has no business being anything more than a set of calculational prescriptions: that it requires no underpinning rationale and that the attempt to understand what is going on is to misunderstand the problem situation. This is pure logical positivism. For Feynman, to attempt to explain quantum phenomena in terms of the behaviour of more familiar macroscopic objects is fundamentally flawed. The right direction of explanation is, for him, only from the incomprehensible mystery of quantum mechanics to the intelligibility of classical physics.

    Undoubtedly, causality is directed from fundamental microscopic reality upwards to the macroscopic world with which we are familiar. Undoubtedly, we have no rational expectation that microscopic reality will conform in any way whatsoever to our macroscopic preconceptions. Undoubtedly, we have no valid a priori reason to believe that the basis of reality will be accessible to human imagination. This far, I am content to travel with Feynman. However, I am not content to say that all that there is is a set of rules, with no form to make them intelligible.

    The Question Why?

    The whole basis of Physics is the pursuit of the question "Why". To accept that in the most definitive realm of Physics, this quest for intelligibility must be rejected and replaced by mindless computation is to invalidate the whole process by which this conclusion was  - supposedly - arrived at. It may be that the quest for ultimate intelligibility is forlorn, but it cannot be reneged upon without invalidating the whole of physical science.

    Feynman shows his sensitivity to this issue by claiming throughout his book that his treatment avoids all the problems associated with every other account of Quantum Mechanics. He seeks to convince his reader that he has found a route between Cylla and Charybdis. In fact he has more accurately stopped his ears so that he cannot hear the siren call characteristic of our incomplete understanding.

    How Strange is Strange?

    Most of the rules that Feynman presents as being "strange" in his book, are in fact not strange at all. They are entirely intelligible on a wave mechanical account of Quantum Mechanics. All his arrows and pocket watches and the rules for generating and combining them can be understood as a technique for calculating the impulse response of the system in question. The story he tells would explain the propagation of sound just as well  as it does that of light: with the difference that sound is a macroscopic wave which does not collapse.

    The only truly strange bit of his system is the rule that the quadratic norm of the final amplitude is the probability of an event. This rule is, in Professor Feynman's presentation, lost as a needle in a pile of straws: but it is exactly this rule that is characteristic of all the ontological and epistemological problems in Quantum Mechanics. All that Feynman has done is to camouflage the real problem by compounding it with many other unproblems. His claim to have solved the difficulties of quantum theory amounts to no more than a removal of the conceptual foundation that gave rise to his own theory.  Instead of throwing light on the subject he has darkened it: rendering it radically incomprehensible.

    Junking the Scaffolding

    Now there is a partial precedent for this kind of process. The conceptual scaffolding that is necessary to the evolution of a theory in the first place may not be necessary to sustain it once its structure is fully conceived. Indeed, the scaffolding may then become a hindrance to understanding rather than a help and be best relegated to the status of a purely historical curiosity. The obvious example of such a redundant theoretical scaffolding is the mechanical basis that James Clark Maxwell employed in constructing his laws of Electromagnetism.

    It was a great advance in Electromagnetic Theory when the fields themselves were elevated to the status of objective reality.
    Although we have no intuition of the Electromagnetic Field in itself, we have no difficulty in understanding its behaviour. It has comprehensible properties of "continuity", "locality", "curvature" and "momentum density" and "energy density". In fact it is no more mysterious (though less familiar) than mass and energy or being and existence: not that I wish to suggest that these are not mysterious!  The properties of the Electromagnetic field are such that one would not be surprised if it was eventually understood as being itself the behaviour of some deeper reality.

    The removal of Maxwell's mechanical scaffolding for his theory does not compare with Feynman's removal of his. As we have seen, the equations of Electromagnetism are still comprehensible as an expression of the Einsteinium mass-energy relationship (which is itself comprehensible as a result in Minkowskian geometry), and also as a kind of conservation or continuity law. Maxwell's equations speak of a field that they govern, in just the way that Schrodinger's equation (and even more so, Dirac's equation: from which it can be derived) speak of a field that they too govern. What Feynman does is to reject the field and keep only the equations: using them to calculate numbers while failing to note that the way in which those numbers are subsequently used is contradictory of how they are first obtained.


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