The conditional probabilities and the empirical laws in a free scalar QFT in curved spacetime
Hideyasu Yamashita
TL;DR
The paper introduces prior conditional probabilities as a state-free mechanism to express empirically verifiable laws in quantum field theory on curved spacetime, focusing on free scalar (Klein–Gordon) fields and their canonical commutation relations. It develops both a general explicit formula for prior probabilities of CCR-based measurements and a covariant reformulation, then validates the approach through concrete prototypes: the harmonic oscillator and finite-dimensional CCR. By reducing KG-field observables to finite-dimensional subspaces, the authors derive explicit empirical-law expressions that relate sequential measurement outcomes, highlighting how CCR structure translates into calculable conditional probabilities. This framework aims to clarify the empirical content of QFTCS beyond the absence of a distinguished vacuum, with potential implications for how predictive content is extracted in curved backgrounds.
Abstract
Unlike QFT in Minkowski spacetime (QFTM), QFT in curved spacetime (QFTCS) suffers from a conceptual obscurity on the empirical (experimentally verifiable/falsifiable) laws. We propose to employ the notion of prior conditional probabilities to describe a part of the empirical laws of QFTCS. This is interpreted as a quantum conditional probability without no information on the initial state. Hence this notion is expected to be free from the inevitable vagueness of the empirical meaning of quantum states in QFTCS. More generally in quantum physics, this notion seems free from the conceptual problems on state reductions. We confine ourselves to the probabilistic laws of the free scalar fields (Klein-Gordon fields) in curved spacetime, which require some reconsideration on the empirical meaning of the canonical commutation relation (CCR). We give some examples of empirical laws in terms of prior conditional probabilities, concerning the CCR and the free scalar QFTCS.