On the relation between perspective-neutral, algebraic, and effective quantum reference frames

TL;DR

This paper explores the relationships between three different approaches to quantum reference frames in gauge theories, demonstrating their equivalence under certain conditions and analyzing how physical quantities like uncertainties depend on the reference frame.

Contribution

It establishes the equivalence of effective, algebraic, and perspective-neutral approaches to quantum reference frames for ideal cases, and connects these formalisms with the Page-Wootters gauge-fixing method.

Findings

The three QRF approaches are equivalent for ideal QRFs with sharp orientations.

QRF transformations in the perspective-neutral framework agree semiclassically with effective approach transformations.

Uncertainties and fluctuations are frame-dependent, especially in effective and algebraic approaches.

Abstract

The framework of internal quantum reference frames (QRFs) constitutes a universal toolset for dealing with symmetries in quantum theory and has led to new revelations in quantum gravity, gauge theories and foundational physics. Multiple approaches have emerged, sometimes differing in scope and the way symmetries are implemented, raising the question as to their relation. Here, we investigate the relation between three approaches to QRFs for gauge symmetries, namely the effective semiclassical, algebraic, and perspective-neutral (PN) approaches. Rather than constructing Hilbert spaces, as the PN approach, the effective approach is based on a quantum phase space parametrized by expectation values and fluctuations, while the emphasis of the algebraic approach is on the state space of complex linear functionals on a kinematical algebra. Nevertheless, external frame information is treated as gauge in all three formalisms, manifested in constraints on states and algebra. We show that these three approaches are, in fact, equivalent for ideal QRFs, distinguished by sharp orientations, which is the previous setting of the first two approaches. Our demonstration pertains to single constraints, including relativistic ones, and encompasses QRF changes. In particular, the QRF transformations of the PN framework agree semiclassically with those of the older effective approach, by which it was inspired. As a physical application, we explore the QRF covariance of uncertainties and fluctuations, which turn out to be frame-dependent. This is particularly well-suited for the effective and algebraic approaches, for which these quantities form a natural basis. Finally, we pave the way towards extending these two approaches to non-ideal QRFs by studying the projection and gauge-fixing operations of the Page-Wootters formalism, built into the PN framework, on algebraic states.