The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional

We argue in a model-independent way that the Hilbert space of quantum gravity is locally finite-dimensional. In other words, the density operator describing the state corresponding to a small region of space, when such a notion makes sense, is defined on a finite-dimensional factor of a larger Hilbert space. Because quantum gravity potentially describes superpositions of different geometries, it is crucial that we associate Hilbert-space factors with spatial regions only on individual decohered branches of the universal wave function. We discuss some implications of this claim, including the fact that quantum field theory cannot be a fundamental description of Nature. Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation. e-mail: [email protected],[email protected],[email protected] ar X iv :1 70 4. 00 06 6v 1 [ he pth ] 3 1 M ar 2 01 7 In quantum field theory, the von Neumann entropy of a compact region of space R is infinite, because an infinite number of degrees of freedom in the region are entangled with an infinite number outside. In a theory with gravity, however, if we try to excite these degrees of freedom, many states collapse to black holes with finite entropy S = A/4G, where A is the horizon area [1, 2]. It is conceivable that there are degrees of freedom within a black hole that do not contribute to the entropy. However, if such states were low-energy, the entropy of the black hole could increase via entanglement, violating the Bekenstein bound. If they are sufficiently high-energy that they don’t become entangled, exciting them would increase the size of the black hole, taking it out of the “local region” with which it was associated. The finiteness of black hole entropy therefore upper bounds the number of degrees of freedom that can be excited within R, and therefore on the dimensionality of HR, the factor of Hilbert space associated with R (as the dimensionality of Hilbert space is roughly the exponential of the number of degrees of freedom). Similarly, a patch of de Sitter space, which arguably represents an equilibrium configuration of spacetime, has a finite entropy proportional to its horizon area, indicative of a finite-dimensional Hilbert space [3–7]. The most straightforward interpretation of this situation is that in the true theory of nature, which includes gravity, any local region is characterized by a finite-dimensional factor of Hilbert space. Some take this statement as well-established, while others find it obviously wrong. Here we argue that the straightforward interpretation is most likely correct, even if the current state of the art prevents us from drawing definitive conclusions. This discussion begs an important question: what is “the Hilbert space associated with a region”? Quantum theories describe states in Hilbert space, and notions like “space” and “locality” should emerge from that fundamental level [8–10]. Our burden is therefore to understand what might be meant by the Hilbert space of a local region, and whether that notion is well-defined in quantum gravity. We imagine that the fundamental quantum theory of nature describes a density operator ρ acting on a Hilbert space HQG. The entanglement structure of near-vacuum states in spacetime is very specific, so generic states in HQG won’t look like spacetime at all [9, 10]. Rather, in phenomenologically relevant, far-from-equilibrium states ρ, there will be macroscopic pointer states representing semiclassical geometries. To that end, we imagine a decomposition HQG = Hsys ⊗Henv, (1) where the system factor Hsys will describe a region of space and its associated long-wavelength fields, and the environment factor Henv is traced over to obtain our system density matrix, ρsys = Trenv ρ. The environment might include microscopic degrees of freedom that are either irrelevant or spatially distant. Then decoherence approximately diagonalizes the system density matrix in the pointer basis, |Ψa〉 ∈ Hsys, ρsys = ∑