Abstract: Gibbs explained probability in terms of frequentism using the notion of ensemble equipped with a ‘density-in-phase’ function (a non-negative real-valued function on phase-space). The same procedure is applied to a decoherent history space, with parameter space M replacing phase space, equipped with a quantum state, replacing the density-in-phase by the modulus square of the associated wave-function (as a non-negative real-valued function on M). So long as M includes at least one continuous variable, I show that probability is similarly explained in terms of frequentism. The ensembles consist of finite numbers of equi-amplitude decohering microstates, whose superposition is the quantum state.
There is a natural alternative, following Boltzmann, defining microstates as equi-volume microstates of non-zero amplitude (for given volume measure on M (for Boltzmann, Liouville measure). It has long been known that probabilities similarly defined (but in terms of macrostates) are diachronically inconsistent; I show that on the Boltzmann variant, they are synchronically inconsistent as well.
The Gibbs method, as applied to the Everett interpretation, is then further a form of actual frequentism, as contrasted with hypothetical frequentism, as all the microstates exist in a superposition. No limiting procedure is needed. This contrasts with my earlier work on this topic in https://arxiv.org/abs/2201.06087, which involved diachronic probabilities. The present talk is based on http://arxiv.org/abs/2404.12954, just posted, and is purely synchronic. As such it can be extended to contexts independent of decoherence altogether; if the basis is allowed to vary, the constraint on M can be dropped as well.
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Philosophy of Physics Seminar Convenor for TT24: Christopher Timpson | Philosophy of Physics Group Website
