Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
arXiv:2412.15502 · doi:10.1007/JHEP07(2025)063
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 generalized entropy formula in a semiclassical limit. The observers thus used are also known as quantum reference frames (QRFs); as noted in our companion work [arXiv:2405.00114], 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 generalized 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.
103 pages, 4 figures, comments welcome. v3: minor typos corrected
References in corpus (68)
- Holographic Derivation of Entanglement Entropy from AdS/CFT
- Generalized gravitational entropy
- Reference frames, superselection rules, and quantum information
- Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime
- Quantum corrections to holographic entanglement entropy
- Black hole entropy in Loop Quantum Gravity
- Notes on Some Entanglement Properties of Quantum Field Theory
- Local subsystems in gauge theory and gravity
- Partial and Complete Observables for Hamiltonian Constrained Systems
- An Algebra of Observables for de Sitter Space
- Quantum mechanics and the covariance of physical laws in quantum reference frames
- Decomposition of entanglement entropy in lattice gauge theory
- Entanglement entropy of electromagnetic edge modes
- Gravity and the Crossed Product
- The Trinity of Relational Quantum Dynamics
- Edge modes of gravity -- I: Corner potentials and charges
- Diffeomorphism-invariant observables and their nonlocal algebra
- Quantum Time
- Large N algebras and generalized entropy
- Observables, gravitational dressing, and obstructions to locality and subsystems
- Local phase space and edge modes for diffeomorphism-invariant theories
- Conditional probabilities with Dirac observables and the problem of time in quantum gravity
- Time from quantum entanglement: an experimental illustration
- A change of perspective: switching quantum reference frames via a perspective-neutral framework
- Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems
- Edge modes of gravity -- II: Corner metric and Lorentz charges
- The Role of Type III Factors in Quantum Field Theory
- Generalized entropy for general subregions in quantum gravity
- Quantizing time: Interacting clocks and systems
- Quantum clocks observe classical and quantum time dilation
- Switching internal times and a new perspective on the 'wave function of the universe'
- Physics within a quantum reference frame
- Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy
- Equivalence of approaches to relational quantum dynamics in relativistic settings
- Quantum reference frames for general symmetry groups
- Notes on the type classification of von Neumann algebras
- Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics
- Group Averaging for de Sitter free fields
- Spacetime Quantum Reference Frames and superpositions of proper times
- Effective relational dynamics of a nonintegrable cosmological model
- How to switch between relational quantum clocks
- Quantum Relativity of Subsystems
- Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
- Edge modes as reference frames and boundary actions from post-selection
- Time in quantum physics: From an external parameter to an intrinsic observable
- Can chaos be observed in quantum gravity?
- Quantum reference frame transformations as symmetries and the paradox of the third particle
- von Neumann algebras in JT gravity
- Construction of quantum Dirac observables and the emergence of WKB time
- Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory
- Crossed product algebras and generalized entropy for subregions
- On leading order gravitational backreactions in de Sitter spacetime
- Edge modes as dynamical frames: charges from post-selection in generally covariant theories
- Generalized Black Hole Entropy is von Neumann Entropy
- Internal quantum reference frames for finite Abelian groups
- Relational observables, reference frames, and conditional probabilities
- Non-commutative Geometry from Perturbative Quantum Gravity
- Relative Quantum Time
- Quantum Reference Frames at the Boundary of Spacetime
- Operational Quantum Reference Frame Transformations
- Identification is Pointless: Quantum Coordinates, Localisation of Events, and the Quantum Hole Argument
- Solving the Problem of Time in Mini-superspace: Measurement of Dirac Observables
- A Type Approximation of the Crossed Product
- A covariant regulator for entanglement entropy: proofs of the Bekenstein bound and QNEC
- Semifinite von Neumann algebras in gauge theory and gravity
- Measurement events relative to temporal quantum reference frames
- Nonrelativistic spatiotemporal quantum reference frames
- Non-commutative coordinates from quantum gravity