19 From quantum to classical (causes)

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:

  1. 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,
  2. 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).

The World According to Quantum MechanicsTo 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.

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