Boundary Constraints as Explanations

In classical electromagnetism, the surface of a metallic conductor acts as a boundary constraint on the electric field. Normally these fields can point in any direction, but at the surface of a metal those fields are constrained: they must be aligned perpendicular to the surface.

In this case, at least, we can easily see the explanation. The metal acts as a boundary constraint, which always trumps randomness. In general, physicists only use random­ness when we have no other information to go on—but in the case of a boundary condition, we have much better information—making random-logic incorrect and obsolete.

True, one can also explain this alignment of the electric field in terms of dynamical rules, electrons moving around in the metal, etc. In other words, by extending the boundary into the time dimension, an alternate dynamical explanation is possible. But the crucial point is this: even if the dynamical explanation were not available, the boundary explanation would still go through, and it would explain a scenario that would otherwise seem inexplicably organized.

This same essential argument also applies to the Big Bang; all one needs is a boundary constraint on the universe, and this boundary can naturally explain the smooth character of the early universe. The only essential difference is that the nec­essary cosmological boundary is one dimension higher than the surface-boundary of a metallic conductor (3D spatial volumes have 2D boundaries; 4D spacetime-volumes such as our universe have 3D boundaries). The added dimension here means that the alternate dynamical explanation that worked for conductors is no longer available.

This is far from a novel idea; after all, the initial state of the universe is often referred to as an “initial boundary condition”. The only problem is that many physi­cists want to then explain this boundary condition, via dynamics or randomness. And as we’ve already seen, neither of those are going to work. Instead, the problem goes away if we simply treat boundary-explanations as fundamental in their own right, framing our physical theories such that the boundaries are just as central as the dynamics.

We use boundaries and boundary constraints all over physics, they’re just typi­cally viewed as stand-ins for other explanations rather than being fundamental. We imagine infinite thermal reservoirs, compute the normal modes of laser cavities, and pay special attention to the initial conditions of mechanical systems. Even in our most fundamental physical theories, using some basic Lagrangian density, physi­cists mathematically fix an external (3D) boundary on every spacetime region of interest.

In most of these cases one could make a case that the boundary condition isn’t really fundamental, instead due to dynamics or an earlier state. Even in the Lagrangian case, one could argue that there was a bigger boundary that subsumed the smaller one. But this ignores the clear truth that boundaries can be used to explain systems, in general. And as one expands the size of the system, one approaches the biggest 3D boundary of all—the cosmological boundary of the universe, where the “larger boundary” argument fails. Since we need an ultimate boundary to explain the success of our physical theories, the cosmological boundary must be contributing an essential part of the explanation.

One complaint here might be that the required boundary is unlikely, as viewed from a statistical perspective. But this gets the logical priority of explanation exactly backwards. Consider the case of the metallic conductor, where the same argument could be made. Someone who used only random statistics to analyze the bound­ary would conclude that metallic conductors were themselves highly improbable! Someone else who knew the boundary condition would have more information, and realize where the random-explainer had gone astray: they used the wrong probability distribution, based on a lack of information. The same is true for the Big Bang; it is simply incorrect to assume that all microstates on the cosmological boundary are equally likely, for that denies the very role of boundary conditions in limiting the possibilities.

A more sophisticated complaint would be that boundary constraints apply to microstates, not macrostates—and perfectly smooth microstates are very boring. If the early universe had no perturbations whatsoever, one might guess that the rest of the universe would have no interesting structure. One conventional solution here would be to add “quantum fluctuations” to the initial boundary, but such an approach would violate the very concept of a strict boundary constraint. A better solution, which also works for classical systems, is to note that typical boundaries used in physics only tend to smoothly constrain half the parameters on any surface. Even in the example of the metallic conductor, if you consider both electric and magnetic fields, exactly 3 out of the 6 components are constrained at the boundary, with the other 3 components unconstrained. Similarly, when one imposes boundaries in Lagrangian field theory, one imposes a boundary constraint on exactly half the relevant parameters (the field value, but not its normal derivative). This half-constrained information provides a well-known connection between classical states and quantum uncertainty [17, 18], connecting to the “quantum fluctuation” solution mentioned above.

One last complaint might be from those who just didn’t accept that boundary constraints were ultimately fundamental, and should in turn have some deeper expla­nation. And to that, I would have no objection—so long as the deeper explanation was neither dynamical nor random nor anthropic. Dropping back to one of these modes of explanation is the mistake made far too often. Such thinking might encourage one to view something like Roger Penrose’s “Weyl Curvature Hypothesis” in a more fundamental light [19]. But whether one treats the boundary itself as fundamental, or finds something deeper explaining the boundary in turn, we have finally made it to the point where we can draw a few basic conclusions.

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