Quantum mereology: Factorizing Hilbert space into subsystems with quasiclassical dynamics

We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any preexisting structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into “system” and “environment.” Such a decomposition can be defined by looking for subsystems that exhibit quasiclassical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) and remain localized around approximately classical trajectories. We present an in-principle algorithm for finding such a decomposition by minimizing a combination of entanglement growth and internal spreading of the system. Both of these properties are related to locality in different ways. This formalism is relevant to questions in the foundations of quantum mechanics and the emergence of spacetime from quantum entanglement

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 superpo- sitions 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.Comment: Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation. 6 page

Quantum Space, Quantum Time, and Relativistic Quantum Mechanics

We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being treated on an equal footing in Hilbert space, we show symmetry transformations to be implemented by unitary basis changes in Hilbert space, giving them a stronger quantum mechanical footing. Global symmetries, such as Lorentz transformations, modify the decomposition of Hilbert space; and local symmetries, such as U(1) gauge symmetry are diagonal in coordinate basis and do not alter the decomposition of Hilbert space. We briefly discuss extensions of this paradigm to quantum field theory and quantum gravity

Mad-Dog Everettianism: Quantum Mechanics at Its Most Minimal

To the best of our current understanding, quantum mechanics is part of the most fundamental picture of the universe. It is natural to ask how pure and minimal this fundamental quantum description can be. The simplest quantum ontology is that of the Everett or Many-Worlds interpretation, based on a vector in Hilbert space and a Hamiltonian. Typically one also relies on some classical structure, such as space and local configuration variables within it, which then gets promoted to an algebra of preferred observables. We argue that even such an algebra is unnecessary, and the most basic description of the world is given by the spectrum of the Hamiltonian (a list of energy eigenvalues) and the components of some particular vector in Hilbert space. Everything else—including space and fields propagating on it—is emergent from these minimal elements

Emergence of Gravitational Potential and Time Dilation from Non-interacting Systems Coupled to a Global Quantum Clock

We study gravitational back-reaction within relational time formulations of quantum mechanics by considering two versions of time: a time coordinate, modelled as a global quantum degree of freedom, and the proper time of a given physical system, modelled via an internal degree of freedom serving as a local quantum "clock". We show that interactions between coordinate time and mass-energy in a global Wheeler-DeWitt-like constraint lead to gravitational time dilation. In the presence of a massive object this agrees with time dilation in a Schwarzchild metric at leading order in $G$. Furthermore, if two particles couple independently to the time coordinate we show that Newtonian gravitational interaction between those particles emerges in the low energy limit. We also observe features of renormalization of high energy divergences.Comment: Essay written for the Gravity Research Foundation's 2023 Awards for Essays on Gravitatio