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# Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy

## Details

**Authors** Julian De Vuyst, Stefan Eccles, Philipp A. Höhn, Josh Kirklin

**Year** 2025

**Publisher** Journal of High Energy Physics

**Kind of work** article

**Discipline** Physics

**Applied** false

[Read it at the publisher](https://doi.org/10.1007/JHEP07(2025)063) 
10.1007/JHEP07(2025)063

## In authors' words

### Abstract

A significant step towards a rigorous understanding of perturbative gravitational entropy was recently achieved by a series of works showing that a proper accounting of gauge invariance and observer degrees of freedom converts the Type III algebra of QFT observables in a gravitational subregion to a Type II crossed product, whose entropy reduces to the generalised entropy formula in a semiclassical limit. The observers thus used are also known as quantum reference frames (QRFs); as noted in our companion work, using different QRFs result in different algebras, and hence different entropies — so gravitational entropy is observer-dependent. Here, we provide an in-depth analysis of this phenomenon, with full derivations of many new results. Using the perspective-neutral QRF formalism, we extend previous constructions to allow for arbitrarily many observers, each carrying a clock with possibly degenerate energy spectra. We consider a semiclassical regime characterized by clocks whose energy fluctuations dominate over the fluctuations of the energy of the QFT. Unlike previous works, we allow the clocks and fields to be arbitrarily entangled. At leading order the von Neumann entropy still reduces to the generalised entropy, but linear corrections are typically non-vanishing and quantify the degree of entanglement between the clocks and fields. We also describe an 'antisemiclassical' regime as the opposite of the semiclassical one, with suppressed fluctuations of the clock energy; in this regime, we show how the clock may simply be 'partially traced' out when evaluating the entropy. Four explicit examples of observer-dependent entropy are then given, involving a gravitational interferometer, degenerate clock superselection, a semiclassical approximation applying to some clocks but not others, and differences between monotonic and periodic clocks.

### What they set out to do (purpose)

To extend the perspective-neutral quantum-reference-frame formalism to multiple observers with entangled clocks and analyze how gravitational entropy depends on the observer's reference frame.

### Who or what was studied (sample)

Not an empirical sample; a formal/theoretical derivation with four worked examples (a gravitational interferometer, degenerate clock superselection, a mixed semiclassical case, and monotonic versus periodic clocks).

### How they did it (methods)

Analytic derivation within the crossed-product/Type II algebra framework for gravitational subregions, extended to arbitrarily many observers with possibly entangled clocks, computing von Neumann entropy in semiclassical and 'antisemiclassical' regimes.

### What they found (results)

Extending the quantum-reference-frame formalism to multiple observers with entangled clocks, the study found that the gravitational entropy assigned to a system depends on which observer's quantum reference frame is used, with corrections beyond the leading-order generalized entropy formula that scale with the degree of clock-field entanglement.

## Commentary

### In short

Demonstrates that perspective is not incidental to but constitutive of the physical quantity measured, since the same gravitational system's entropy differs by observer or reference frame.

**Patterns it shows** P

**Added** 2026-09-28

**How to cite this** Julian De Vuyst, Stefan Eccles, Philipp A. Höhn, Josh Kirklin (2025). Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy. Journal of High Energy Physics.