Gravitational operator algebras and the role of noncommutative conditional probability in quantum geometry

A central goal of quantum gravity is to fit geometry into the formalism of quantum mechanics. In the last several years, it has been appreciated that the role of geometry in quantum theory is in fact very multifaceted. This reflects and expands upon the dual role played by geometry in classical physics; not only is geometry a classical dynamical feature in its own right, it is also a fundamental structure on top of which every other dynamical feature is built. Likewise, the quantization of geometry does not only entail a promotion of classical geometric variables to quantum operators, but also a rather intimate realignment of the fundamental structure of the full algebra of observables of any system in which dynamical, quantum gravity is present. In this thesis, we undertake a detailed study of the multifaceted role of geometry by observing how it is encoded in classical, semiclassical, and fully quantum analyses of gravity. In Chapter 2, we introduce the extended phase space as a symplectic geometric approach to accounting for the full set of dynamical degrees of freedom in subregions for general gauge theories and gravity. The extended phase space underscores the important role played by large gauge transformations, supported on codimension two submanifolds of spacetime called corners, which are genuine symmetries of a gauge theory rather than degeneracies of its symplectic form. The inclusion of Noether charges generating large gauge transformations as Hamiltonian functions is a defining feature of the extended phase space. In Chapter 3, we propose a quantization of the extended phase space in terms of a von Neumann algebraic construction called the crossed product. The crossed product quantizes both the ordinary phase space and the aforementioned Noether charges to operators acting on an extended Hilbert space. In the gravitational context, the inclusion of these extended degrees of freedom has profound implications for the entanglement structure of the resulting theory. Familiar divergences which are encountered in ordinary quantum field theories are lifted and rigorous notions of density operators and von Neumann entropies can be assigned to states in the resulting subregion algebras. In Chapter 4, we formalize this result by proving a theorem which establishes when the crossed product of a type III von Neumann algebra $M$ with a locally compact group $G$ is semifinite. We show that this will be the case provided $G$ admits as a subgroup the modular automorphism of a faithful, semifinite, normal weight ω on $M$, and ω is (quasi)-invariant with respect to the action of $G$ on $M$. Gauge invariant measurements are intrinsically conditional -- depending upon the specification of a dynamical quantum reference frame which plays the role of a measuring apparatus. In Chapter 5, we identify the new degrees of freedom central to the extended phase space and the crossed product as quantum reference frames. These degrees of freedom allow for fields and operators in the non-extended theory to be dressed to satisfy constraints associated with genuine gauge symmetries. Crucially, these reference frames are formed from degrees of freedom from within the theory; quantum gauge theories come equipped with the ability to measure themselves. This observation resounds the important role of large gauge transformations in facilitating the gluing of subregions in a manifestly gauge invariant fashion. In Chapter 6, we contextualize the preceding results by observing that the extended phase space and the crossed product can be understood as subcases of a more general object called a quantum orbifold. Finally, in Chapter7, we analyze the quantum information theoretic properties of gravitational subregion algebras in light of the extensions addressed above. We demonstrate that the von Neumann entropy of a generic state in the gravitational crossed product algebra can be interpreted directly as a generalized entropy in the holographic sense. Moreover, we illustrate that this generalized entropy can be regarded as a factorized entropy under the presence of a (noncommutative) conditional expectation implementing a form of non-exact quantum error correction. This observation motivates a definition for the area operator in fully non-perturbative quantum gravity as the log of a relative conditional density operator whose expectation value computes the conditional entropy of a chosen state.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Klinger, Marc S.

Contributors dc:contributor

• Leigh, Robert
• Stone, Michael
• Faulkner, Thomas
• Berman, David

• Quantum Gravity
• Von Neumann Algebras
• Quantum Error Correction
• Bulk Reconstruction
• Emergent Geometry
• Quantum Information Theory
• Gauge Theory
• Quantum Reference Frames
• Crossed Products
• Entanglement
• Entropy
• Generalized Entropy
• Extended Phase Space
• Mathematical Phyiscs

Statement dc:rights

• 2025 by Marc S. Klinger. All rights reserved.

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129826

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129826