Dynamical locality of the free scalar field

TL;DR

This paper investigates the dynamical locality property of the free scalar field, showing that massive theories satisfy it while massless ones violate it due to gauge symmetry, with implications for classical and quantum field theories across dimensions.

Contribution

It provides a detailed analysis of dynamical locality for the free Klein-Gordon field, demonstrating conditions under which it holds or fails in classical and quantum contexts.

Findings

Massive Klein-Gordon field obeys dynamical locality in all spacetime dimensions.

Massless Klein-Gordon field violates dynamical locality due to gauge symmetry.

Restoring dynamical locality by working with the massless current in certain dimensions.

Abstract

Dynamical locality is a condition on a locally covariant physical theory, asserting that kinematic and dynamical notions of local physics agree. This condition was introduced in [arXiv:1106.4785], where it was shown to be closely related to the question of what it means for a theory to describe the same physics on different spacetimes. In this paper, we consider in detail the example of the free minimally coupled Klein--Gordon field, both as a classical and quantum theory (using both the Weyl algebra and a smeared field approach). It is shown that the massive theory obeys dynamical locality, both classically and in quantum field theory, in all spacetime dimensions $n\ge 2$ and allowing for spacetimes with finitely many connected components. In contrast, the massless theory is shown to violate dynamical locality in any spacetime dimension, in both classical and quantum theory, owing to a rigid gauge symmetry. Taking this into account (equivalently, working with the massless current) dynamical locality is restored in all dimensions $n\ge 2$ on connected spacetimes, and in all dimensions $n\ge 3$ if disconnected spacetimes are permitted. The results on the quantized theories are obtained using general results giving conditions under which dynamically local classical symplectic theories have dynamically local quantizations.