Matter relative to quantum hypersurfaces

TL;DR

This paper extends the Page-Wootters formalism to quantum field theory by describing fields relative to quantum hypersurfaces, demonstrating formal equivalence with standard QFT and revealing frame-dependent particle creation effects.

Contribution

It introduces a relational framework for quantum fields on hypersurfaces, unifying the Page-Wootters approach with the Tomonaga-Schwinger equation and quantum reference frame transformations.

Findings

Conditional wave functionals satisfy the Tomonaga-Schwinger equation.

Relational Dirac observables and quantum deparameterization are unitarily equivalent to the Page-Wootters formalism.

Frame transformations lead to a frame-dependent particle creation effect.

Abstract

We explore the canonical description of a scalar field as a parameterized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces $\mathsf{X}$ relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals $|\psi_\phi[\mathsf{X}]\rangle$ interpreted as the state of the field relative to the hypersurface $\mathsf{X}$, thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparameterization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect.