General covariance for quantum states over time

Ali Mostafazadeh

N. L. Diaz

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.

Jan Naudts

Ernst Schmützer

Marc-Thierry Jaekel

Covariance of stochastic mechanics under changes of time and simultaneous time-dependent changes of coordinates is made explicit in the context of diffusion processes, and extended to the operator formulation. Covariant time derivatives are defined in both representations and used to express the equations of motion in a manifestly covariant way. They also provide general constraints on the correspondence between quantum states and stochastic processes.

Donald Salisbury