This Quantum World

Wave functions versus propagators

Pseudo-problems

At the heart of the quantum formalism is the amplitude (a.k.a. propagator)

< x_f , t_f | x_i , t_i > .

In the case of a single particle, x_i and x_f are points in "ordinary" 3-space. In general, they are points in the system's configuration space, which has as many dimensions as the system has degrees of freedom. (The absolute square of this amplitude, integrated with respect to x_f over a region R of the configuration space, gives the conditional probability of finding the system in R if the appropriate measurement is made at the time t_f, the condition being that the particle was last "seen" at x_i at the time t_i.)

It is customary to introduce the so-called "wave function" Ψ(x,t) by requiring that it satisfy the equation

Ψ(x_f,t_f) = ∫ dx_i < x_f , t_f | x_i , t_i > Ψ(x_i,t_i) .

Wave functions and propagators provide the same information. Knowing either one can calculate the other. Yet some proposed quantum ontologies (most notably, Everett's "many-worlds interpretation") transmogrify the wave function into a representation of "what ultimately exists" and treat the propagator as nothing more than a computational tool.

Whence this partiality? I should have thought that if the propagator is nothing more than a computational tool, then so is the wave function.

One obvious reason for the preference of wave functions over propagators is the historical precedence of Schrödinger's "wave mechanics" over Feynman's propagator-based formulation of quantum mechanics.

In Feynman's formulation, the propagator < x_f , t_f | x_i , t_i > is defined by applying Rule B to alternatives ("paths") that are defined as the continuum limit of sequences of outcomes of exact position measurements. (Since under the conditions stipulated by Rule B, alternatives are defined in terms of the outcomes of unperformed measurements, we need not worry how exact position measurement can possibly be made.)

In relativistic quantum mechanics, where the sum-over-paths formalism proves itself vastly superior, additional summations occur. There the symbol x_i stands for a set of incoming particles observed at the time t_i (usually set to -∞), while x_f stands for a set of outgoing particles observed at the time t_f (usually set to +∞). One additional summation extends over all possible ways in which particles can be created or annihilated in the meantime. (The summation over particle paths now applies to the propagators connecting the "vertices" at which particles are created and/or annihilated.) Apart from that there is a spacetime integral for each vertex, due to the complete indefiniteness of the vertex positions.

Another reason for the preference of wave functions over propagators is the primacy of absolute over conditional probabilities in Kolmogorov's mathematical foundation of probability theory.

This, too, is a fortuitous case of historical precedence. There now exists an alternative to Kolmogorov's axiomatic formulation of probability theory, due to A. Rényi. Every result of Kolmogorov's theory can be translated into Rényi's theory, and this is based entirely on conditional probabilities. If the wave function is taken as representing "what ultimately exists," then its dependence on measurement outcomes is forgotten or ignored, and the probabilities it assigns to possible measurement outcomes are regarded as absolute, which is to say determined by "what ultimately exists." In reality, quantum-mechanical probabilities are conditional, which is to say determined by measurement outcomes.

But the main reason why the wave function is widely regarded as the primary concept is the evolutionary paradigm...

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