Spacetime Mediation of Quantum Interactions

9.10 Spacetime Mediation of Quantum Interactions

We were on the steamer from America to Japan, and I liked to take part in the social life on the steamer and, so, for instance, I took part in the dances in the evening. Paul, somehow, didn't like that too much but he would sit in a chair and look at the dances. Once I came back from a dance and took the chair beside him and he asked me, "Heisenberg, why do you dance?" I said, "Well, when there are nice girls, it is a pleasure to dance." He thought for a long time about it, and after about five minutes he said, "Heisenberg, how do you know beforehand that the girls are nice?"

W. Heisenberg

Most natural philosophers from Aristotle to Descartes held that material entities can influence each other only by coming into direct contact, i.e., “an object cannot act where it is not”. However, Newton’s theory of gravity undermined confidence in the doctrine of “direct contact”, because in Newton’s theory gravity is represented as an instantaneous universal force of attraction between every pair of objects, regardless of the distance between them, and regardless of whether the space between them contains any material substance. Admittedly Newton himself (in private writings) asserted that gravity must ultimately be attributable to some kind of process or condition in the intervening spaces between objects, but he was skeptical of any material mechanism for gravity.

During the century following the publication of Newton’s Principia the theory of gravitational force acting at a distance proved to be so successful that “force at a distance” became the standard model of physical interaction, and it was applied to the formulation of the laws of electricity and magnetism. Even after Oersted’s discovery of the link between these two forces, and the discovery of the dependence of these forces on the relative states of motion of the objects (rather than just their relative positions), Ampere was still able to formulate electrodynamics in terms of inverse-square forces at a distance, accounting for all the phenomena known at that time. The distant action interpretation of electrodynamics was carried on into the next century by Gauss, Weber, Lorenz, Lienard, Wiechert, and many others. However, at the same time, Faraday and Maxwell argued against the concept of distant action, and instead sought to represent electrodynamics in terms of direct local actions of an electromagnetic field – with mechanical properties – permeating all of space. One implication of this approach was that electromagnetic effects must require some time to propagate (through a succession of local actions) from place to place. This reinforced Maxwell’s belief in the necessity of an intervening medium, to embody the energy and momentum of action between emission and absorption. The detailed process of propagation (consistent with the known facts of electromagnetism) turned out to coincide with a model for the propagation of light. Hertz’s experimental verification of electromagnetic waves in 1887 was taken by many scientists as a conclusive demonstration of the field concept, and as a refutation of the distant action theories, which were then largely abandoned.

However, despite the success of the field interpretation, some scientists continued to work on developing the “distant action” approach. They noted that one of Maxwell’s original objections to the distant action theories – that any velocity-dependent force must violate conservation of energy – was unfounded, because the velocity-dependent potential of Weber is actually consistent with energy conservation – at least for velocities less than the speed of light. Also, the concepts of delayed propagation and “radiation” effects were not really as incompatible with distant-action theories as it might seem. Indeed in 1867 Ludwig Lorenz had proposed a distant action theory based on retarded potentials, formally identical to the Newtonian gravitational potential, but with a light-speed delay to account for the finite propagation speed, and this theory predicts all the same “radiation” effects as does Maxwell’s theory. The electromagnetic potentials affecting a charged particle at any given event are expressed as simple integrals of the charge density r and velocity v over the volume of the past light cone of the event, as follows

where r is the spatial distance from the event in question (with respect to any specified inertial coordinate system). The integrals are evaluated over the past light cone of the event, so the charge density and velocity appearing in these integrands are functions of position and time r(r, -r/c) and v(r, -r/c), with the understanding that we have set t = r = 0 at the event in question. It’s notable that, in this formulation, the effects of electromagnetic “radiation” are actually just direct (albeit distant and retarded) actions between charged particles. Also, notice that the potential at a given particle depends only on the retarded positions of other particles, so there is no “self-energy”, a troublesome concept for field theories. Despite the elegance of this “distant action” formulation, and the fact that it accounts for all the same phenomena as Maxwell’s field theory, this approach continued to be regarded as of secondary importance, largely because the precepts of contiguous local action and local conservation of energy and momentum seem more compatible with field theories than with distant action theories.

Nevertheless, one feature of the retarded potential formulation prompted a re-consideration of a very fundamental issue, namely, the temporal symmetry of the basic equations governing electromagnetism versus the apparent asymmetry of phenomena such as radiation. It can be shown that the scalar potential satisfies the equation

and this is equally well solved by either of the two functions

where f₁ is called the retarded potential and f₂ the advanced potential. The usual presumption was that only the retarded potential was physically relevant, but the justification for excluding the advanced potential was unclear. In 1902 Karl Schwarzschild proposed a fully time-symmetrical formulation of electrodynamics, essentially making use of both the advanced and retarded potentials. It attracted little attention at the time, but the same formulation was later (independently) developed by Fokker and Tetrode in 1922. Expressed in modern notation, they asserted that the trajectories of a set of N particles with masses m_j and charges e_j are such as the make stationary a certain action quantity A, so the law of motion is dA = 0 where

with

and

As mentioned previously, an action equivalent to this was originally formulated in 1902 by Karl Schwarzschild, prior to the development of special relativity, and yet it invokes what we would today call the invariant spacetime I_i,j between two particles. At the time, scientists still believed in the absolute significance of spatial and temporal intervals, so this formulation could only seem even further removed from the requirement of “local contact”, because it entailed a force between particles separated not only in space but also in time. For this reason, the “action at a distance” formulation was unappealing. However, the situation appears quite different in the context of Minkowski spacetime. The spacetime interval is the only physically invariant measure of the “distance” between two particles. The appearance of the delta function of the squared spacetime interval in the Schwarzschild action implies that the influence between two particles propagates entirely along null intervals. Therefore, one can argue that “action at a distance” is a misnomer when applied to this theory, because the propagation of action is restricted to null spacetime distances. In this way, special relativity reconciles the two seemingly contradictory historical traditions of Descartes and Newton, i.e., of direct contact and action at a distance.

Notice that the first term of A reduces to

so the variational principle for a single particle gives the classical law of inertia

It’s also worth noting that the summation in the second term of A excludes i = j, which is to say, a particle does not act on itself. This was considered to be one of the appealing features of this approach, because it avoids the troublesome infinite self-energy entailed by field theories (for discrete point-like charges). On the other hand, one might imagine that the first term in A represents the i = j case that is “missing” from the second term. In fact it is possible (with certain provisos) to modify the expression for A in such a way that the first term is eliminated and the summation in the second term include the self-action term i = j, signifying that the inertia of a charged particle is due to electromagnetic self-action. However, this can only be done at the expense of strict relativistic invariance, so it is not entirely satisfactory, and of course it does not account for quantum effects.

Another noteworthy feature of the Schwarzschild action is that it is temporally symmetrical, making equal use of both the advanced and retarded potentials, which is to say, the interactions are conveyed along all the null intervals between the worldlines of two particles, on both the forward and past lightcones of each event on those worldlines. This can be seen explicitly from the following identity

which also shows how the action A is related to the usual Coulomb potential e²/r between two particles of charge e at a distance r. Thus the Schwarzchild action also offers a resolution of the debate about temporal symmetry and the physical significance of advanced potentials. The symmetry of interactions according to this formulation of electromagnetism was stressed by Hugo Tetrode, who commented evocatively in 1922 on the implications of this interpretation:

The sun would not radiate if it were alone in space and no other bodies could absorb its radiation… If for example I observed in my telescope yesterday evening that star which let us say is 100 light years away… the star or individual atoms of it knew already 100 years ago that I, who then did not even exist, would view it yesterday evening at such and such a time.

Subsequently the same approach to classical electromagnetism was developed by Feynman and Wheeler in the 1940s. To account for the apparently asymmetric “radiation reaction” experienced by an accelerating charge (i.e., in terms of a field theory, the reaction on a charged particle to the radiation it emits when it is accelerated) without assuming that a particle acts on itself, they proposed that every ray along all the future light cone of a given particle eventually terminates at some other particle in the far distant future. Consequently, every acceleration of the given particle corresponds to slight accelerations of all those distant particles in the idea future absorber, and the given particle is at the intersection of the past light cones of all those distant particles, and hence is subject to a reaction, which can be shown to exactly “mimic” the radiation reaction force of the conventional field theory with purely retarded waves. In essence the asymmetry is attributed to asymmetric boundary conditions by postulating complete absorption in the future. Whether or not this postulate is consistent with observation is questionable, and it has also been challenged on other grounds, but it does at least illustrate how phenomena that might seem to require a field theory may actually be modeled in terms of a direct action theory. Feynman once commented on these alternative points of view:

In the customary view, things are discussed as a function of time in very great detail. For example, you have the field at this moment, a differential equation gives you the field at the next moment and so on – a method which I shall call the Hamilton method, i.e., the time differential method. On the other hand, we have [in the expression for A] a thing that describes the character of the path throughout all of space and time. The behavior of nature is determined by saying her whole spacetime path has a certain character. For an action like A, the equations obtained by variation of the X(t) functions are no longer at all easy to get back into Hamiltonian form. If you wish to use as variables only the coordinates of particles, then you can talk about the property of the paths - but the path of one particle at a given time is affected by the path of another at a different time. If you try to describe, therefore, things differentially… you need a lot of bookkeeping variables to keep track of what the particle did in the past. These are called field variables…. From the overall space-time view of the least action principle, the field disappears as nothing but bookkeeping variables insisted on by the Hamiltonian method.

Of course, by the time Feynman worked on this subject, it was known that classical physics was inadequate to account for all the phenomena, so the real objective in studying classical electrodynamics was to find a secure basis for extrapolating a viable quantum theory. The original theory of quantum mechanics replaced the classical concept of a discrete particle possessing a definite continuous path through space and time with a quantum field representing the amplitude of the probability that the particle would be found in any given region of state space. The early simplified version of quantum mechanics was not relativistically invariant, but a relativistic theory could be achieved by essentially repeating the quantization at a higher level, effectively quantizing the quantum field of the particle. Thus quantum field theory originated in efforts to reconcile quantum mechanics with special relativity. The first quantization gave a description of a particle in terms of a field representing the probabilities of all possible configurations of the particle, whereas the so-called “second quantization” described a meta-field representing the probabilities of all possible first-order fields of particles. According to one interpretation, different particles come in and out of existence, provided only that all conservation requirements are met. This “second quantization” was described by Pias as “the end of the game of marbles”, because it was no longer possible to represent events purely in terms of a fixed set of permanent particles. The Dirac equation, discussed in Section 9.4, was the first relativistically invariant description of a quantum particle – or rather, of the wave function of a particle – and it led immediately to the realization that anti-particles must also exist, and that particles can be created and annihilated. This might seem to render obsolete the particle as a suitable elementary entity, but Wheeler and Feynman proposed the ingenious idea that anti-particles are ordinary particles moving backwards in time. This seemed at least as plausible as Dirac’s hole theory, with its infinite sea of negative-energy electrons, and it allowed Feynman to pursue his particle-based approach.

Making use of many of the ideas he had explored in his re-formulation of classical electrodynamics as a direct action theory, Feynman went on to develop what he called the spacetime view of quantum field theory. Dirac had previously described how a probability amplitude could be associated with the entire path of a particle, rather than with just the particle at one specific time and place. The step analogous to “second quantization” was to consider that the overall probability amplitude for a particle to propagate from one given event in spacetime to another is simply the “sum over histories”, i.e., the integral of the probability amplitudes over all possible paths between those two events. This is called the path integral approach. For example, consider two electrons, denoted by a and b, initially at locations 1 and 2 respectively, and let K(3,4;1,2) denote the probability amplitude for these electrons to arrive at locations 2 and 4 respectively, taking into account the possibility of an exchange of one quantum of action (i.e., a photon, representing the first-order Coulomb interaction) between them, emitted and absorbed at the intermediate locations 5 and 6 along their respective paths. By reasoning very similar to that used in the Schwarzschild formulation of classical electrodynamics, Feynman developed the expression

where e is the elementary charge, g_a_m and g_b_m are the Dirac matrices (see Section 9.4) representing the two electrons, and s₅₆² is the squared magnitude of the spacetime interval between events 5 and 6. Just as in the Schwarzschild action for classical electrodynamics, we see that the action propagates along null intervals, as indicated by the delta function. However, the “+” subscript on the delta function signifies that Feynman is taking only the “positive frequency” part of the delta function. Recall that the delta function can be expressed (somewhat loosely) as

Strictly speaking, this integral diverges, so this is not a well-defined expression, but if we cut off the integration at large but finite limits, we do get a function that converges on the delta function as the limits increase. The integral is taken over all frequencies w, positive and negative, but Feynman argued that we must restrict ourselves to just the positive frequencies, which amounts to assuming that a photon is emitted only into the forward light cone of a particle, and absorbed only from the past light cone – thereby sacrificing the perfect temporal symmetry of the classical Schwarzschild action. In summary, Feynman’s objective was to derive an expression for the amplitude of a particular interaction between two charged particles, corresponding to the Coulomb potential e²/r where e is the electric charge on each particle and r is the distance between the particles. In his notation, the interaction is “turned on” when one particle is at a time and place denoted by event 5, and the other is at the time and place denoted by event 6. He begins with a double integral over the path parameters dt₅ and dt₆ , and to accept contributions only when t₅ = t₆ he multiplies the integrand by the delta function d(t₅ – t₆), treating the Coulomb interaction non-relativistically as if it acted instantaneously. Thus his expression included a factor of the form

But then he notes that, relativistically, the interaction is not instantaneous, but is retarded by the light-speed delay in propagating from one particle to the other. Thus, if we define the symbols t₅₆ = t₅ – t₆ and r₅₆ = r₅ – r₆, Feynman proposes to modify the formula, substituting in place of d(t₅₆) the delta function d(t₅₆ – r₅₆), which signifies that the integral will accept contributions when the particle at 5 is on the forward light cone of the particle at 6. (Note that we are using units such that the speed of light has the value 1.) But now he says “this turns out to be not quite right, for when this interaction is represented by photons, they must be of only positive energy, while the Fourier transform of d(t₅₆ – r₅₆) contains frequencies of both signs”. He is referring to the Fourier representation of the delta function noted above, in which the integral extends over values of the frequency parameter w from negative to positive infinity. (Oddly enough, he doesn’t mention that this representation doesn’t actually converge.) To remedy this, he defines a new type of delta function, denoted by d₊(x), consisting of only the positive frequency parts of the usual delta function, i.e.,

Here the integral ranges from 0 to positive infinite, and we have multiplied by 2 to normalize the result so that the integral of d₊(x) over all x is 1. Also, Feynman wants to account for the reverse interaction, when the particle at 6 is on the forward light cone of the particle at 5, which happens when t₅₆ + r₅₆ vanishes. He says we need to average the two possibilities, so he replaces d(t₅ – t₆) in equation (1) with the average of d₊(t₅₆ – r₅₆) and d₊(t₅₆ + r₅₆), which we’ve seen is equal to d₊(t₅₆² – r₅₆²). Thus we arrive at a relativistically invariant factor. Even this, however, turned out to be “not quite right” if we are to avoid infinities in the calculations, so Feynman proposed to replace the delta function with a function that is not perfectly sharp, so that most, but not all, of the contribution occurs when the argument is zero. He found that the results were insensitive to the precise value of the cutoff. (This corresponds to the fact that the Fourier representation of the delta function does not converge, but it can be approximated with arbitrary precision by specifying an arbitrarily large cutoff frequency.)

It’s interesting that all of classical mechanics and electrodynamics can be expressed by the simple action A described above, but this formulation does not include gravity. Considering the formal similarities between the electric force and the gravitational force, one might expect that gravitation would be most naturally incorporated into A by adding an interactive term similar to the summation representing electromagnetic interactions, with the gravitational “charge” m in place of the electric charge e. However, the masses of the particles already appear in the first term in A, which we’ve seen corresponds to the inertial behavior of objects, i.e.,

This variational formula originally pre-supposed a fixed Minkowskian background metric, but in general relativity the spacetime metric is allowed to deviate from flatness (in accord with the field equations in the presence of mass-energy), and with this allowance the very same variational equation entails all the effects of the gravitational interactions between objects. It’s ironic that, prior to general relativity, many efforts had been made to absorb the first term of A into the second, representing inertia as a aspect of electromagnetism, whereas Einstein’s general relativity emerged from his focus on interpreting gravitation as an aspect of inertia, separate and distinct from electromagnetism. Of course, the variational expression for the path of a particle represents only the passive response, whereas the entire gravitational interaction must explain how the particle actively affects the metric. The conventional bifurcation of phenomena into separate active and passive aspects seems to work surprisingly well, even though it is difficult to rigorously justify. This dubious aspect of the conventional treatment is mentioned even in traditional texts such as Jackson’s “Classical Electrodynamics”, albeit not until the final chapter.

…the problems of electrodynamics have been divided into two classes: one in which the sources of charge and current are specified and the resulting electromagnetic fields are calculated, and the other in which the external electromagnetic fields are specified and the motions of charged particles or currents are calculated… this manner of handling problems in electrodynamics can be of only approximate validity… A correct treatment must include the reaction of the radiation on the motion of the sources... [but] a completely satisfactory classical treatment of the reactive effects of radiation does not exist. The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of an elementary particle… the basic problem remains unsolved.

If we take the view that radiation effects on a given charged particle are actually just interactions (along null intervals) with other charged particles, we might say the difficulties presented by radiation reaction touch not only on the nature of elementary particles, but also on the inadequacy of attempts to represent local physical phenomena in isolation from all the other entities in the universe. In this sense, the "advantage" that is often claimed for the field interpretation in macroscopic applications – i.e., that it (seemingly) enables us to base our treatments on the state of matter and fields within a limited region – may actually be misleading.

Just as, in classical Maxwellian electrodynamics, the fields determine the motions of charges in spacetime while the charges determine the fields in spacetime, according to general relativity the shape of spacetime determines the motions of objects while those objects determine (or at least influence) the shape of spacetime. This dualistic structure naturally arises when we replace action-at-a-distance with purely local influences in such a way that the interactions between "separate" objects are mediated by an entity extending between them. We must then determine the dynamical attributes of this mediating entity, e.g., the electromagnetic field in electrodynamics, or spacetime itself in general relativity.

However, many common conceptions regarding the nature and extension of these mediating entities are called into question by the apparently "non-local" correlations in quantum mechanics, as highlighted by EPR experiments. The apparent non-locality of these phenomena arises from the fact that although we regard spacetime as metrically Minkowskian, we continue to regard it as topologically Euclidean. As discussed in the preceding sections, the observed phenomena are more consistent with a completely Minkowskian spacetime, in which physical locality is directly induced by the pseudo-metric of spacetime. According to this view, spacetime operates on matter via interactions, and matter defines for spacetime the set of allowable interactions, i.e., consistent with conservation laws. A quantum interaction is considered to originate on (or be "mediated" by) the locus of spacetime points that are null-separated from each of the interacting sites. In general this locus is a quadratic surface in spacetime, and its surface area is inversely proportional to the mass of the transferred particle.

For two timelike-separated events A and B the mediating locus is a closed surface as illustrated below (with one of the spatial dimensions suppressed)

The mediating surface is shown here as a dotted circle, but in 4D spacetime it's actually a closed surface, spherical and purely spacelike relative to the frame of the interval AB. This type of interaction corresponds to the transit of massive real particles. Of course, relative to a frame in which A and B are in different spatial locations, the locus of intersection has both timelike and spacelike extent, and is an ellipse (or rather an ellipsoidal surface in 4D) as illustrated below

The surface is purely spacelike and isotropic only when evaluated relative to its rest frame (i.e., the frame of the interval AB), whereas this surface maps to a spatial ellipsoid, consisting of points that are no longer simultaneous, relative to any other co-moving frame. The directionally asymmetric aspects of the surface area correspond precisely to the "relativistic mass" components of the corresponding particles as a function of the relative velocity of the frames.

The propagation of a free massive particle along a timelike path through spacetime can be regarded as involving a series of surfaces, from which emanate inward-going "waves" along the nullcones in both the forward and backward direction, deducting the particle from the past focal point and adding it to the future focal point, as shown below for particles with different masses.

Recall that the frequency n of the de Broglie matter wave of a particle of mass m is

where p_x, p_y, p_z are the components of momentum in the three directions. For a (relatively) stationary particle the momentums vanish and the frequency is just n = (mc2)/h sec-1. Hence the time per cycle is inversely proportional to the mass. So, since each cycle consists of an advanced and a retarded cone, the surface of intersection is a sphere (for a stationary mass particle) of radius r = h/mc, because this is how far along the null cones the wave propagates during one cycle. Of course, h/mc is just the Compton scattering wavelength of a particle of mass m, which characterizes the spatial expanse over which a particle tends to "scatter" incident photons in a characteristic way. This can be regarded as the effective size of a particle when "viewed" by means of gamma-rays. We may conceive of this effect being due to a high-energy photon getting close enough to the nominal worldline of the massive particle to interfere with the null surfaces of propagation, upsetting the phase coherence of the null waves and thereby diverting the particle from it's original path.

For a massless particle the quantum phase frequency is zero, and a completely free photon (if such a thing existed) would just be represented by an entire null-cone. On the other hand, real photons are necessarily emitted and absorbed, so they corresponds to bounded null intervals. Consistent with quantum electrodynamics, the quantum phase of photon does not advance while in transit between its emission and absorption (unlike massive particles). According to this view, the oscillatory nature of macroscopic electromagnetic waves arises from the advancing phase of the source, rather than from any phase activity of an actual photon.

The spatial volume swept out by a mediating surface is a maximum when evaluated with respect to it's rest frame. When evaluated relative to any other frame of reference, the spatial contraction causes the swept volume to be reduced. This is consistent with the idea that the effective mass of a particle is inversely proportional to the swept volume of the propagating surface, and it's also consistent with the effective range of mediating particles being inversely proportional to their mass, since the electromagnetic force mediated by massless photons has infinite range, whereas the strong nuclear force has a very limited range because it is mediated by massive particles. Schematics of a stationary and a moving particle are shown below.

This is the same illustration that appeared in the discussion of Lorentz's "corresponding states" in Section 1.5, although in that context the shells were understood to be just electromagnetic waves, and Lorentz simply conjectured that all physical phenomena conform to this same structure and transform similarly. In a sense, the relativistic Schrödinger wave equation and Dirac's general argument for light-like propagation of all physical entities based on the combination of relativity and quantum mechanics (as discussed in Section 9.10) provide the modern justification for Lorentz's conjecture. Looking back even further, we see that by conceiving of a particle as a sequence of surfaces of finite extent, it is finally possible to answer Zeno's question about how a moving particle differs from a stationary particle in "a single instant". The difference is that the mediating surfaces of a moving particle are skewed in spacetime relative to those of a stationary particle, corresponding to their respective planes of simultaneity.

Some quantum interactions involve more than two particles. For example, if two coupled particles separate at point A and interact with particles at points B and C respectively, the interaction (viewed straight from the side) looks like this:

The mediating surface for the pair AB intersects with the mediating surface for AC at the two points of intersection of the dotted circles, but in full 4D spacetime the intersection of the two mediating spheres is a closed circle. (It's worth noting that these two surfaces intersect if and only if B and C are spacelike separated. This circle enforces a particular kind of consistency on any coherent waves that are generated on the two mediating surfaces, and are responsible for "EPR" type correlation effects.)

The locus of null-separated points for two lightlike-separated events is a degenerate quadratic surface, namely, a straight line as represented by the segment AB below:

The "surface area" of this locus (the intersection of the two cones) is necessarily zero, so these interactions represent the transits of massless particles. For two spacelike-separated events the mediating locus is a two-part hyperboloid surface, represented by the hyperbola shown at the intersection of two null cones below

This hyperboloid surface has infinite area, which suggests that any interaction between spacelike separated events would correspond to the transit of an infinitely massive particle. On this basis it seems that these interactions can be ruled out. There is, however, a limited sense in which such interactions might be considered. Recall that a pseudosphere can be represented as a sphere with purely imaginary radius. It's conceivable that observed interactions involving virtual (conjugate) pairs of particles over spacelike intervals (within the limits imposed by the uncertainty relations) may correspond to hyperboloid mediating surfaces.

(It's also been suggested that in a closed universe the "open" hyperboloid surfaces might need to be regarded as finite, albeit extremely huge. For example, they might be 35 orders of magnitude larger than the mediating surfaces for timelike interactions. This is related to vague notions that "h" is in some sense the "inverse" of the size of a finite universe. In a much smaller closed universe (as existed immediately following the big bang) there may be have been an era in which the "hyperboloid" surfaces had areas comparable to the ellipsoid surfaces, in which case the distinction between spacelike and time-like interactions would have been less significant.)

An interesting feature of this interpretation is that, in addition to the usual 3+1 dimensions, spacetime requires two more "curled up" dimensions of angular orientation to represent the possible directions in space. The need to treat these as dimensions in their own right arises from the non-transitive topology of the pseudo-Riemannian manifold. Each point [t,x,y,z] actually consists of a two-dimensional orientation space, which can be parameterized (for any fixed frame) in terms of ordinary angular coordinates q and f. Then each point in the six-dimensional space with coordinates [x,y,z,t,q,f] is a terminus for a unique pair of spacetime rays, one forward and one backward in time. A simple mechanistic visualization of this situation is to imagine a tiny computer at each of these points, reading its input from the two rays and sending (matched conservative) outputs on the two rays. This is illustrated below in the xyt space:

The point at the origin of these two views is on the mediating surface of events A and B. Each point in this space acts purely locally on the basis of purely local information. Specifying a preferred polarity for the two null rays terminating at each point in the 6D space, we automatically preclude causal loops and restrict information flow to the future null cone, while still preserving the symmetry of wave propagation. (Note that an essential feature of spacetime mediation is that both components of a wave-pair are "advanced", in the sense that they originate on a spherical surface, one emanating forward and one backward in time, but both converge inward on the particles involved in the interaction.

According to this view, the "unoccupied points" of spacetime are elements of the 6D space, whereas an event or particle is an element of the 4D space (t,x,y,z). In effect an event is the union of all the pairs of rays terminating at each point (x,y,z). We saw in Chapter 3.5 that the transformations of q and f under Lorentzian boosts are beautifully handled by linear fractional functions applied to their stereometric mappings on the complex plane.

One common objection to the idea that quantum interactions occur locally between null-separated points is based on the observation that, although every point on the mediating surface is null-separated from each of the interacting events, they are spacelike-separated from each other, and hence unable to communicate or coordinate the generation of two equal and opposite outgoing quantum waves (one forward in time and one backward in time). The answer to this objection is that no communication is required, because the "coordination" arises naturally from the context. The points on the mediating locus are not communicating with each other, but each of them is in receipt of identical bits of information from the two interaction events A and B. Each point responds independently based on its local input, but the combined effect of the entire locus responding to the same information is a coherent pair of waves.

Another objection to the "spacetime mediation" view of quantum mechanics is that it relies on temporally symmetric propagation of quantum waves. Of course, this objection can't be made on strictly mathematical grounds, because both Maxwell's equations and the (relativistic) Schrödinger equation actually are temporally symmetric. The objection seems to be motivated by the idea that the admittance of temporally symmetric waves automatically implies that every event is causally implicated in every other event, if not directly by individual interactions then by a chain of interactions, resulting in a non-sensical mess. However, as we've seen, the spacetime mediation view leads naturally to the conclusion that interactions between spacelike-separated events are either impossible or else of a very different (virtual) character than interactions along time-like intervals. Moreover, the stipulation of a preferred polarity for the ray pairs terminating at each point is sufficient to preclude causal loops.

Incidentally, two of the best known anecdotes about Paul Dirac can both be seen as expressions of the requirement for the initiation of any action to be contingent on its conclusion, just as the light from Tetrode's star can only be emitted if there is a reception, though it may be an eye located in the far distant future. One of these stories was told by Heisenberg regarding a journey that he and Dirac made together in 1929, and appears as the epigram of this chapter. The second story relates to the time when the young Dirac was visiting Bohr in Copenhagen. (Dirac later recalled that during his stay he and Bohr had many long talks, during which Bohr did practically all the talking). At one point while Bohr was dictating some remarks, he stopped in mid-sentence and told Dirac that he was having trouble finishing the sentence. Dirac said

At school I was taught never to start a sentence without knowing the end of it.

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