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

TL;DR

<3-5 sentence high-level summary>The paper shows that gravitational entropy, defined for subregions in perturbative quantum gravity, depends on the observer through quantum reference frames (QRFs). By treating clocks as dynamical quantum degrees of freedom and using perspective-neutral QRFs, the authors construct gauge-invariant observable algebras as crossed products, yielding well-defined entropies that reproduce generalized entropy in a semiclassical limit. They generalize to arbitrarily many clocks (including degenerate or periodic clocks), develop density-operator formalisms, and analyze semiclassical and antisemiclassical regimes, revealing linear corrections tied to entanglement between clocks and fields. Through explicit examples, they demonstrate how observer-dependence arises from subsystem relativity and clock properties, highlighting fundamental implications for the interpretation of gravitational entropy in quantum gravity.

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.

Figures (5)

• Figure 2.1: The Schrödinger evolution of $\ket*{\psi_{\vert C}(\tau)}$ stemming from $\left\lvert{\Psi'}\right)$ can be obtained in two ways. The tangential decoding is achieved by acting with $\mathcal{R}'_{\vert C}(\tau)$ on $\left\lvert{\Psi}\right)$ for different values of $\tau$. Alternatively, the transversal decoding first requires the trajectory $U_S(\tau) \left\lvert{\Psi'}\right)$ for different values of $\tau$ on which we then act with the same $\mathcal{R}'_{\vert C}(0)$.
• Figure 3.1: Depicted is the maximally extended Schwarzschild spacetime together with its time Killing flow lines. The Killing boost Hamiltonian $H_\xi$ simultaneously maps the Cauchy slice $\Sigma$ forwards in time in the right region and backwards in time in the left region. The role of the clocks can here be played by the ADM Hamiltonian defined the right and left asymptotic boundary $H_{\rm ADM}^{(2)}=H_R-H_L$ with the relative minus sign accounting for the opposite directions of the Killing vector $\xi$ in both regions.
• Figure 3.2: The static patch of de Sitter space depicted with its Killing flow lines. The killing flow generated by $H_\xi$ moves the cauchy slice $\Sigma$ forwards in time in the right patch but backwards in time in its complement, the left patch. The clocks measuring the boost time would be $H_1$ in the right patch and $H_2$ in the left patch respectively. One can think of them being localised along the $r=0$ worldlines. The backwards evolution in the complement leads to a relative minus sign between the two clock Hamiltonians.
• Figure 3.3: Commutative diagram showing two orders in which constraints may be imposed. CLPW Chandrasekaran:2022cip and JSS Jensen:2023yxy move counterclockwise from the top, imposing non-idealness (bounded energy) of the clocks only after the constraint is implemented. Our approach starts with the kinematical Hilbert space $\mathcal{H}_{\rm kin}$, where clocks already have the final restriction on their energy spectra in place, and imposes the constraint on this. This diagram shows a comparison at the gauge invariant level, culminating on the physical Hilbert space. An analogous diagram instead implements the gauge constraint via gauge fixing, for instance culminating in the reduced perspective of frame $C_2$, as we do in the main text below. The diagram on the right shows the similar two orders of imposing the constraints on the algebras.
• Figure 8.1: Clock $C_2$ starts at time reading 0 and is then split into two branches where one branch makes a flyby encounter with a massive object while the other is kept far away (or we accelerate them by different amounts). Due to time dilation, the flyby branch will lack behind in time by an amount $\Delta \tau$ w.r.t. the other branch. After recombination of both branches, we thus achieve a superposition of two clock states. Meanwhile, clock $C_1$ just evolves as is without splitting into multiple branches.

Theorems & Definitions (1)

• Example 1