9.10 Locality and Temporal Asymmetry

We've seen that the concept of locality plays an important role in the EPR thesis and the interpretation of Bell's inequalities, but what precisely is the meaning of locality, especially in a quasi-metric spacetime in which the triangle inequality doesn't hold? The general idea of locality in physics is based on some concept of nearness or proximity, and the assertion that physical effects are transmitted only between suitably "nearby" events. From a relativistic standpoint, locality is often defined as the proposition that all causal effects of a particular event are restricted to the interior (or surface) of the future null cone of that event, which effectively prohibits communication between spacelike-separated events (i.e., no faster-than-light communication). However, this restriction clearly goes beyond a limitation based on proximity, because it specifies the future null cone, thereby asserting a profound temporal asymmetry in the fundamental processes of nature.

What is the basis of this asymmetry? It certainly is not apparent in the form of the Minkowski metric, nor in Maxwell's equations. In fact, as far as we know, all the fundamental processes of nature are perfectly time-symmetric, with the single exception of certain processes involving neutral kaons. However, the original experimental evidence in 1964 for violation of temporal symmetry in the decay of neutral kaons was actually a demonstration of asymmetry in parity and charge conjugacy, from which temporal asymmetry was indirectly inferred on the basis of the CPT Theorem. As recently as 1999 there were still active experimental efforts to demonstrate temporal asymmetry directly. In any case, aside from the single rather subtle peculiarity in the behavior of neutral kaons, no one have ever found any evidence at all of temporal asymmetry in any fundamental interaction. How, then, do we justify the explicit temporal asymmetry in our definition of locality for all physical interactions?

As an example, consider electromagnetic interactions. Recall that the only invariant measure of proximity (nearness) in Minkowski spacetime is the absolute interval

(dt)² - (dx)² - (dy)² - (dz)²

which is zero between the emission and absorption of a photon. Clearly, any claim that influence can flow from the emission event to the absorption event but not vice versa cannot be based on an absolute concept of physical nearness. Such a claim amounts to nothing more or less than an explicit assertion of temporal asymmetry for the most fundamental interactions, despite the complete lack of justification or evidence for such asymmetry in photon interactions. It would be more consistent to regard a photon interaction as an indivisible whole, including the null-separated emission and absorption events on a symmetrical footing. This view is supported by the fact that once a photon is emitted, its phase does not advance while "in flight", because quantum phase is proportional to the absolute spacetime interval (which, as discussed in Section 2.1, is what gives the absolute interval its physical significance). Thus, if we take seriously the spacetime interval as the absolute measure of proximity, it seems that the transmission of a photon is, in some sense, a single event coordinated mutually and symmetrically between the points of emission and absorption.

Now, this image of a photon as a single unified event with a coordinated emission and absorption seems unsatisfactory to many people, partly because it doesn't allow for the concept of a "free photon", i.e., a photon that has been emitted but not absorbed. However, it's worth remembering that we have no direct experience of "free photons", nor of any "free particles", because ultimately all our experience is comprised of completed interactions.

Another possible objection is that this view doesn't allow for a photon to have wave properties, i.e., to have an evolving state while "in flight". However, this objection is based on a misperception. From the standpoint of quantum electrodynamics, the wave properties of electromagnetic radiation are actually wave properties of the emitter. All the potential sources of a photon have a certain (complex) amplitude for photon emission, and this amplitude evolves in time as we progress along the emitter's worldline. However, as noted above, once a photon is emitted, its phase does not advance. In a sense, the ancients who conceived of sight as something like a blind man's incompressible cane, feeling distant objects, were correct, because our retinas actually are in "direct" contact, via null intervals, with the sources of light. The null interval plays the role of the incompressible cane, and the wavelike properties we "feel" are really the advancing quantum phases of the source. (We might say there is less to photons than meets the eye.)

One might think that the reception amplitude for an individual photon must evolve as a function of its position, because if we had (contra-factually) encountered a particular photon one meter further away from its source than we did, we would surely have found it with a different phase. However, this again is based on a misconception, because the photon we would have received one meter further away (on the same timeslice) would necessarily have been emitted one light-meter earlier, carrying the corresponding phase of the emitter at that point on its worldline. When we consider different spatial locations relative to the emitter, we have to keep clearly in mind which points they correspond to along the worldline of the emitter.

Taking another approach, it might seem that we could "look at" a single photon at different distances from the emitter (trying to show that its phase evolves in flight) by receding fast enough from the emitter so that the relevant emission event remains constant, but of course the only way to do this would be to recede at the speed of light (i.e., along a null interval), which isn't possible. This is just a variation of the young Einstein's thought experiment about how a "standing wave" of light would appear to someone riding along side it. The answer is "it wouldn't, because you can't", i.e., because light exists only as completed interactions on null intervals.

Of course, if we attempted such an experiment, we would notice that as our speed of recession from the source gets closer and closer to c, the difference between the phases of the photons we receive becomes smaller and smaller (i.e., the "frequency" of the light gets red-shifted), and approaches zero, which is just what we should expect based on the fact that each photon is simply the lightlike null projection of the emitter's phase at a point on the emitter's worldline. Hence, if we stay on the same projection ray (null interval), we are necessarily looking at the same phase of the emitter, and this is true everywhere on that null ray. This leads to the view that the concept of a "free photon" is meaningless, and a photon is nothing but the communication of an emitter event's phase to some null-separated absorber event, and vice versa.

More generally, since the Schrodinger wave function propagates at c, it follows that every fundamental quantum interaction can be regarded as propagating on null surfaces. Dirac gave an interesting general argument for this strong version of Huygens' Principle in the context of quantum mechanics. In his "Principles of Quantum Mechanics" he noted that a measurement of a component of the instantaneous velocity of a free electron must give the value � c, which implies that electrons (and massive particles in general) always propagate along null intervals, i.e., on the local light cone. At first this may seem to contradict the fact that we observe massive objects to move at speeds much less than the speed of light, but Dirac points out that observed velocities are always average velocities over appreciable time intervals, whereas the equations of motion of the particle show that its velocity oscillates between +c and -c in such a way that the mean value agrees with the average value. He argues that this must be the case in any relativistic theory that incorporates the uncertainty principle, because in order to measure the velocity of a particle we must measure its position at two different times, and then divide the change in position by the elapsed time. To approximate as closely as possible to the instantaneous velocity, the time interval must go to zero, which implies that the position measurements must approach infinite precision. However, according to the uncertainty principle, the extreme precision of the position measurement implies an approach to infinite indeterminancy in the momentum, which means that almost all values of momentum - from zero to infinity - become equally probable. Hence the momentum is almost certainly infinite, which corresponds to a speed of � c. This is obviously a very general argument, and applies to all massive particles (not just fermions). This oscillatory propagation on null cones is discussed further in Section 9.11.

Another argument that seems to favor a temporally symmetric view of fundamental interactions comes from consideration of the exchange of virtual photons. (Whether virtual particles deserve to be called "real" particles is debatable; many people prefer to regard them only as useful mathematical devices in the machinery of quantum field theory, with no ontological status. On the other hand, it's possible to regard all fundamental particles that way, so in this respect virtual particles are not unique.) The emission and absorption points of virtual particles may be space-like separated, and we therefore can't say unambiguously that one happened "before" the other. The temporal order is dependent on the reference frame. Surely in these circumstances, when it's not even possible to say absolutely which side of the interaction was the emission and which was the absorption, those who maintain that fundamental interactions possess an inherent temporal asymmetry have a very difficult case to make. Over limited ranges, a similar argument applies to massive particles, since there is a non-negligible probability of a particle traversing a spacelike interval if it's absolute magnitude is less than about h²/(2pm)², where h is Planck's constant and m is the mass of the particle. So, if virtual particle interactions are time-symmetric, why not all fundamental particle interactions? (Needless to say, time-symmetry of fundamental quantum interactions does not preclude asymmetry for macroscopic processes involving huge numbers of individual quantum interactions evolving from some, possibly very special, boundary conditions.)

Experimentally, those who argue that the emission of a photon is conditioned by its absorption are in a somewhat awkward position, because if tests of Bell's inequalities had not already been performed, they are precisely the kinds of experiments one would propose to decide the matter. If in fact the emission of a particle is conditioned by its absorption, then one would predict violations of Bell's inequalities exactly as have been found by experiments. However, the crucial experiments have already been performed, so modern theorists are reduced to the business of making post-predictions, which can never be very impressive.

In any case, it's interesting to consider why temporal asymmetry seems to be such an important element in many peoples' intuitive conceptions of locality, in spite of the fact that there is very little (if any) direct evidence of temporal asymmetry in any fundamental laws or interactions. One reason may be the fear that without some such restriction, the Minkowskian spacetime manifold would be incapable of supporting any notion of locality at all, because the fact that triangle inequality fails in this manifold implies that there are null paths connecting every two points. This applies even to spacelike separated points if we allow the free flow of information in either direction along null surfaces. However, this need not be the case. The failure of the triangle inequality (actually, the reversal of it) does not necessarily imply that the manifold is unable to support non-trivial structure. There are absolute distinctions between the sets of null paths connecting spacelike separated events and the sets of null paths connecting timelike separated events, and these differences might be exploited to yield a structure that conforms with the results of observation. There is no reason this cannot be a "locally realistic" theory, provided we understand that locality in a quasi-metric manifold is non-transitive.

Is it possible to adopt non-transitive locality without abandoning realism? I would say yes, because realism is simply the premise that the results of our measurements and observations are determined by an objective world, and it's perfectly possible that the objective world might possess a non-transitive locality, commensurate with the non-transitive metrical aspects of Minkowski spacetime. Indeed, even before the advent of quantum mechanics and the tests of Bell's inequality, we should have learned from special relativity that locality is not transitive, and this should have led us to expect non-Euclidean connections and correlations between events, not just metrically, but topologically as well. From this point of view, many of the seeming paradoxes associated with quantum mechanics and locality are really just manifestations of the non-intuitive fact that the manifold we inhabit does not obey the triangle inequality, which is one of our most basic spatio-intuitions.

On the other hand, we should acknowledge that the Bell correlations can't be explained in a locally realistic way simply by invoking the quasi-metric structure of Minkowski spacetime, because if the timelike processes of nature were ontologically continuous it would not be possible to regard them as propagating on null surfaces. We also need our fundamental physical processes to consist of irreducible discrete interactions, as discussed in Section 9.11.