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Quantum Reference Frames from Top-Down Crossed Products

Shadi Ali Ahmad, Wissam Chemissany, Marc S. Klinger, Robert G. Leigh

TL;DR

This work develops an algebraic framework for quantum reference frames by relating QRFs to crossed product algebras and introducing the G-framed algebra, an atlas of local crossed-product charts that encodes potentially inequivalent frames. Using Landstad duality, it shows that for a fixed covariant system $(M,G,\alpha)$ all covariant realizations yield isomorphic crossed products, while inequivalent frames arise when stitching together multiple covariant systems into a global, frame-dependent structure. The de Sitter static patch serves as a physically relevant example where multiple observers generate relational crossed products forming ${\mathfrak A}_{\mathrm dS}$, illustrating frame-dependent observables and relational densities. The G-framed construction generalizes the crossed product to capture Gribov ambiguities, obstructions to a single global QRF, and the global quotient viewpoint akin to an orbifold; it also provides a route to describe frame-independent data embedded in overlaps and to quantify relational entropy. Overall, the paper proposes a robust algebraic toolkit for constrained systems, gauge theories, and gravity that clarifies when frame dependence emerges and how to coherently sew together multiple quantum reference frames.

Abstract

All physical observations are made relative to a reference frame, which is a system in its own right. If the system of interest admits a group symmetry, the reference frame observing it must transform commensurately under the group to ensure the covariance of the combined system. We point out that the crossed product is a way to realize quantum reference frames from the bottom-up; adjoining a quantum reference frame and imposing constraints generates a crossed product algebra. We provide a top-down specification of crossed product algebras and show that one cannot obtain inequivalent quantum reference frames using this approach. As a remedy, we define an abstract algebra associated to the system and symmetry group built out of relational crossed product algebras associated with different choices of quantum reference frames. We term this object the G-framed algebra, and show how potentially inequivalent frames are realized within this object. We comment on this algebra's analog of the classical Gribov problem in gauge theory, its importance in gravity where we show that it is relevant for semiclassical de Sitter and potentially beyond the semiclassical limit, and its utility for understanding the frame-dependence of physical notions like observables, density states, and entropies.

Quantum Reference Frames from Top-Down Crossed Products

TL;DR

This work develops an algebraic framework for quantum reference frames by relating QRFs to crossed product algebras and introducing the G-framed algebra, an atlas of local crossed-product charts that encodes potentially inequivalent frames. Using Landstad duality, it shows that for a fixed covariant system all covariant realizations yield isomorphic crossed products, while inequivalent frames arise when stitching together multiple covariant systems into a global, frame-dependent structure. The de Sitter static patch serves as a physically relevant example where multiple observers generate relational crossed products forming , illustrating frame-dependent observables and relational densities. The G-framed construction generalizes the crossed product to capture Gribov ambiguities, obstructions to a single global QRF, and the global quotient viewpoint akin to an orbifold; it also provides a route to describe frame-independent data embedded in overlaps and to quantify relational entropy. Overall, the paper proposes a robust algebraic toolkit for constrained systems, gauge theories, and gravity that clarifies when frame dependence emerges and how to coherently sew together multiple quantum reference frames.

Abstract

All physical observations are made relative to a reference frame, which is a system in its own right. If the system of interest admits a group symmetry, the reference frame observing it must transform commensurately under the group to ensure the covariance of the combined system. We point out that the crossed product is a way to realize quantum reference frames from the bottom-up; adjoining a quantum reference frame and imposing constraints generates a crossed product algebra. We provide a top-down specification of crossed product algebras and show that one cannot obtain inequivalent quantum reference frames using this approach. As a remedy, we define an abstract algebra associated to the system and symmetry group built out of relational crossed product algebras associated with different choices of quantum reference frames. We term this object the G-framed algebra, and show how potentially inequivalent frames are realized within this object. We comment on this algebra's analog of the classical Gribov problem in gauge theory, its importance in gravity where we show that it is relevant for semiclassical de Sitter and potentially beyond the semiclassical limit, and its utility for understanding the frame-dependence of physical notions like observables, density states, and entropies.
Paper Structure (23 sections, 2 theorems, 119 equations, 1 table)

This paper contains 23 sections, 2 theorems, 119 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Note
  4. Overview of results
  5. Crossed Product Algebras: Two Approaches
  6. Bottom-up approach to crossed product algebras
  7. Top-down approach to crossed product algebras
  8. Algebraic Quantum Reference Frames
  9. Crossed products and quantum reference frames
  10. Systems with multiple reference frames
  11. The $G$-Framed algebra
  12. A $G$-framed algebra for de Sitter space
  13. Relational density states and entropies
  14. Discussion
  15. Frame-independent objects
  16. ...and 8 more sections

Key Result

Theorem 1

Let $A$ be a von Neumann algebra and $G$ a locally compact group. The algebra $A$ is isomorphic to a (regular) crossed product $M \rtimes_{\alpha} G$ for some covariant system $(M,G,\alpha)$ if and only if there exists a continuous homomorphism $\lambda: G \rightarrow A$ and a coaction $\delta: A \r What's more, the covariant system $(M,G,\alpha)$ is determined uniquely by $\delta$ and $\lambda$ a

Theorems & Definitions (4)

  • Theorem 1: Landstad's Classification Theorem
  • Definition 1: Internal Reference Frame
  • Theorem 2: Covariant Representation Induced by an Internal Frame
  • Definition 2: G-framed algebra