Spacetime states and covariant quantum theory

TL;DR

This paper proposes a covariant formulation of quantum mechanics that treats time on equal footing with other variables by using spacetime smeared states, addressing issues in quantum gravity.

Contribution

It introduces a covariant quantum theory framework using spacetime smeared states, removing the need for a preferred time variable in quantum mechanics.

Findings

States can be described without a special time variable.

Quantum dynamics can be expressed via a covariant propagator.

The approach aligns quantum mechanics with relativistic covariance.

Abstract

In it's usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turns, gives again time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics? We observe that the preferred role of time in quantum theory is the consequence of an idealization: that measurements are instantaneous. Canonical quantum theory can be given a covariant form by dropping this idealization. States prepared by non-instantaneous measurements are described by "spacetime smeared states". The theory can be formulated in terms of these states, without making any reference to a special time variable. The quantum dynamics is expressed in terms of the propagator, an object covariantly defined on the extended configuration space.