From quantum time to manifestly covariant QFT: On the need for a quantum-action-based quantization

TL;DR

This paper proposes a quantum-action-based quantization approach to make Lorentz covariance manifest in quantum field theory, addressing limitations of standard formulations and connecting to recent concepts of spacetime states.

Contribution

It introduces a novel second-quantized formalism with quantum action that achieves manifest covariance in QFT, overcoming issues in naive many-body constructions.

Findings

A no-go theorem shows standard Dirac quantization collapses to traditional QFT.

The proposed formalism yields a spacetime version of quantum mechanics (SQM).

The approach links to recent ideas on states over time and emergent time in quantum gravity.

Abstract

In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.