Thus, as long as we refrain from turning the particle propagator into an expression that has multiple components like a vector, there are exactly two ways in which influences on the behavior of a particle can be introduced into the mathematical formalism of quantum mechanics. In either case the behavior of a particle is represented by a differential geometry. In the first case, the particle acted on follows a geodesic of a Finsler geometry (which can differ between particle species). In the second case, it follows a geodesic of a pseudo-Riemannian geometry (which is the same for all particle species).
So what exerts these influences? What causes these effects? What acts on classical particles by bending the geodesics they follow? The causes could include minds, spirits, goblins, and what have you. The fact is that we don’t know. Physicists, however, pretend that they do. Particle physics is based on the assumption that it is only particles (including their aggregates) that can affect particles (including their aggregates). (There is nothing wrong with pretended knowledge as long as it is acknowledged to be hypothetical and not treated as dogma and/or upheld in the face of contrary evidence.) So let’s pretend and see where it leads us.
Thus far we have assumed that the potentials V(x,y,z,t) and A(x,y,z,t) in the action increment (2.17.1) have precise values. Yet particles are fuzzy: neither their positions nor their momenta are in possession of exact values. If these potentials represent effects that particles have on particles, their values cannot be exact. Fuzzy causes have fuzzy effects.
How do we make room for fuzzy potentials? We do so in the same way that we made room for fuzzy particles. In the case of a single particle, this meant calculating the propagator <B,tB|A,tA> by summing contributions from spacetime paths leading from (A,tA) to (B,tB). If we are dealing with a physical system that has N degrees of freedom, A and B are points in the system’s N-dimensional configuration space, and we again sum contributions from all paths that lead from (A,tA) to (B,tB). And although the potentials V and A are not physical systems but mathematical tools representing the effects that particles have on particles, if we want them to be fuzzy, we sum contributions from paths that lead from an initial configuration A to a final configuration B in their infinite-dimensional configuration spacetime. (Configuration spacetime is configuration space plus one additional dimension — time.)
The electromagnetic effects of the distribution and motion of classical particles on the motion of a classical particle are usually calculated in two steps:
- given the distribution and motion of all particles but one, one determines the geodesic along which V and A “evolve” from a given initial configuration to a given final configuration in their configuration spacetime,
- given the values V(x,y,z,t) and A(x,y,z,t), one determines the geodesic along which that one particle moves from a given position A at a given time tA to a given position B at a given time tB in its configuration spacetime.
To find the spacetime path followed by a particle subject to the influences that are encapsulated by V and A, we looked for that path G in the particle’s configuration spacetime which minimizes or maximizes the action ∫C dS relative to all neighboring paths, using the action increment (2.17.1).
To find the path followed by the potentials when the distribution and motion of particles is given, we must likewise look for that path in the configuration spacetime of V and A which minimizes or maximizes the action ∫C dS relative to all neighboring paths. Because now the distribution and motion of the particles is fixed, the term −mc2ds of (2.17.1), which does not contain the potentials, is of no consequence and thus may be dropped. Because previously the values of V and A were fixed, a term containing only the potentials was of no consequence and therefore had been omitted. This term, whose actual form leaves nothing to the physicist’s discretion, must now be included.
What we find is that the “evolution” of the so-called electric field E and the so-called magnetic field B from any initial configuration to any final configuration is characterized by the second pair of Maxwell’s equations. (The first pair is implied by how E and B are defined in terms of the potentials V and A.) Together with the Lorentz force law, Maxwell’s equations form the complete set of fundamental equations of the classical electromagnetic theory.
Thus far we have also assumed that the sixteen components of the metric tensor g in the action increment (2.17.2) have precise values. Yet, once again, particles are fuzzy. If these components represent effects that particles have on particles, their values cannot be exact. Fuzzy causes have fuzzy effects.
But now we run into a problem. If the components of the metric had exact values, we could think of them as functions of spacetime points. But if the metric is fuzzy, so are the distances between spacetime points. And if the distances between spacetime points are fuzzy, then it is physically meaningless to speak of spacetime points, inasmuch as physically meaningful positions and times are defined by the distances or intervals between them. Hence if the metric is fuzzy, it is inconsistent to treat its components as functions of spacetime points. Physicists haven’t yet learned how to circumnavigate this conundrum, which may be the reason — or at least one of the reasons — why as yet there is no such thing as a testable quantum theory of gravity. This, however, does not deter us from using the principle of least action, whose origin is quantum-mechanical to the core, to obtain the classical theory of gravity.
We already know that gravitational effects are described by the geodesics of a pseudo-Riemannian spacetime geometry. What remains to be found is an equation or set of equations that tells us how the geodesics of this geometry are determined. To this end we must add to the action another term, one that depends only on the components of the metric. The actual form of this term again leaves little to the physicist’s discretion. The result is Einstein’s equation, which is at the heart of the general theory of relativity.
This Quantum World 