Contents

• Angels and Hopping
• Geometry and Waves
• Wave Packets and Solitons
• Particles and Potentiality
• Probability and Potential
• Collapse of the Wavepacket
• Solitons and Substantial Forms
• Impact, Force and Potential
• Appendix 1: How can an Impact be Attractive?
• Appendix 2: Feynman Path Integrals and Reality
• Bibliography

Introduction

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

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

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

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

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

[]² A  =  j
 
• []² is the d'Alembertian operator:   d²/dx²   +  d²/dy²   +  d²/dz²  -  (1/c²) d²/dt²
• A is the Minkowskian four-vector potential.
• j is the four-current (the time-like component of which is the charge density).

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  -  ( m_e c² )² ]  Y  =  0

Y  is the relativistic electron wavefunction.

m_e is the rest mass of the electron.

c    is the speed of light in a vacuum.

           H_D  Y    =      E   Y
g . ( p  - e A) Y    =      mc Y
                      p_j    =      [ hc / (2p) ] d/dx_j  ;  j = 1 ... 4
                      p₄   =   -  [ h   / (2p) ] d/dt
           g_i g_j   =   - g_j g_i    ;    i  =/= j

• H_D  is the Dirac Hamiltonian;
• p      is the four-vector operator that generates the four-momentum,
• A     is the Electromagnetic four-vector potential,
• g      is a four-vector of matrices which anticommute with each other.

                                       H_S  Y    =    E  Y
[ 1/( 2m_e )  S_j=1...3 ( p_j ) ²  +  V ]   Y    =    E  Y

"Why does "m_e" appear the electron's wave equation, but not in that of the photon?"

M_D Y_photon   =  0
M_D Y_electron  =  m_e Y_electron

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 M_M:

Y^~_electron Y_{photon     =  0}
Y^~_electron Y_{electron   =  1}
Y^~_photon Y_{photon      =  1}

M_M  =  m_e [ Y_electronY^~_electron ]

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

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

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

Two models of force can be distinguished.

• The first accounts for force in terms of influence, continuity and of fields. The existence of one thing causes a disturbance in some physical field, local to that thing. This disturbance is continued into adjacent Minkowskian volumes, as dictated by whatever differential equation the field happens to obey. Other things in these neighbourhoods then accelerate in response to the influence of this disturbance.
• The second, the famously successful "diagrammatic method" [R.P. Feynman "QED, the Strange Theory of Light and Matter" (1985)] accounts for force in terms of impulse, discontinuity and particles. The existence of one thing only affects another thing when they exist at the same Minkowskian point. Such a shared point is called a "vertex". When two things do so co-exist they exchange energy and momentum in accordance with a "vertex strength".

No Action at a Distance

Influence or Impact?

The Mental World

Appendix 1: How can an impact be attractive?

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    =  ± c  k  /  | k |

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

The Dirac Hole

Y_electron  =   S_k  a_c;k  j_c;k(r)

Y_hole       =   S_k  b_v,k  j_v;k(r)
 

• The j_j;k(r) are electron k-eigenfunctions.
• The first subscript labels the two bands: valence and conduction.
• Every a_v;k  is zero.
• Every b_c;k  is zero.
• In the absence of any electron, every a_c;k  would be zero.
• In the absence of any hole, every b_v;k  would be unity.
• Every k-component of the electron wave-packet, a_c,k, is exactly mirrored by an amplitude in the hole wave-packet.
• The only difference being that, in terms of electron amplitudes, the hole amplitude b_v,k  =  1 - a_c,k.
• The k-vectors are common, and that no "-" appears.

Recombination

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:

• Recombination is always associated with the generation of a symmetric phonon ripple.
• In the centre of mass frame, this ripple cannot propagate symmetrically.
• The atomic lattice frame momentum associated with various sectors of this ripple is biased by the velocity of the centre of mass frame relative to the atomic lattice frame.
• When the centre of mass is travelling at a suitable velocity with respect to the atomic lattice, the "bow" and "stern" waves can propagate (relative to the centre of mass frame) differently enough to carry away sufficient momentum to obviate the necessity for any asymmetric phonon production.

Electron-Positron Annihilation

Initially, for any low velocity impact on a stationary target:

p    =      m.v   +  m.0  =  m.v
E    =   2 m c²  +  ½ m v²

Finally:

p  =  h k /( 2 p )
E  =  hc k /( 2 p )

So:

                 mv       =   h   k /( 2 p )
m ( 2 c² +  ½ v² )  =   hc k /( 2 p )
    ( 2 c² +  ½ v² )  =   cv
    4 c² - 2cv +  v²  =  0
                         v  =   c [ 1 ± ( - 3 )^½ ]

Backwards Time Travel

• In particular, it must be shown how:
• a negative energy particle,
• travelling forwards in time,
• with a negative inertial mass,
• and negative electric charge,
• is equivalent both to
• a positive energy hole,
• travelling forwards in time,
• with a positive inertial mass,
• and positive electric charge,
• and to
• a positive energy particle,
• travelling backwards in time,
• with a positive inertial mass,
• and negative electric charge.

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 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

Intelligibility

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?

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?

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

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.
 

Bibliography

• Undergraduate Physics
• R.P. Feynman, R Leighton and M. Sands "Lectures on Physics" (1964)
• R.P. Feynman "QED, the Strange Theory of Light and Matter" (1985)
• Graduate Physics
• L. Schiff "Quantum Mechanics" 3rd Edition (1968)
• P. Strange "Relativistic Quantum Mechanics" (1999)
• Academic Philosophy
• Plato "Timaeus"
• K.R. Popper "Conjectures and Refutations" (1972)
• K.R. Popper "Quantum Theory and the Schism in Physics" (1982)
• Popular books on Physics, its Philosophy and related Theology
• J. Powers "Philosophy and the New Physics" (1982)
• J.C. Polkinghorne "The Quantum World" (1984)
• S. Hawking "A Brief History of Time" (1988)
• J.D. Barrow "Theories of Everything" (1990)
• P. Davies and J. Gribbin "The Matter Myth" (1991)
• P. Davies "The Mind of God" (1992)
• P. Davies "God and the New Physics" (1993)
• P. Landsburg "Seeking Ultimates: an intuitive guide to Physics" (2000)
• H. Price "Time's Arrow and Archimedes Point" (1996)