Modular Theory and Spacetime Structure in QFT

Abstract

In quantum field theory (QFT), physical quantities are represented by a net of operator algebras indexed by spacetime region. Each local algebra comes equipped with a state-dependent pair of modular operators encoding vital information about the algebra's internal structure, the local state, and the relational structure of the net. These operators and the associated structure theory are the subject of Tomita-Takesaki modular theory. Despite its importance, the physical significance of modular theory remains murky, and philosophers of QFT have largely ignored it. This dissertation aims to help close the gap. It explores a series of deep connections between modular theory and spacetime geometry established by the Bisognano-Wichmann theorem (which identifies the modular operators attached to spacelike wedge regions in the vacuum sector with certain generators of the Poincare group).

Chapter 1 orients my project within the current philosophical literature on the foundations of QFT. I advocate a cosmopolitan stance towards the subject, incorporating tools from Lagrangian, constructive, and algebraic field theory alike. Adopting this stance, chapter 2 introduces the core mathematical ideas behind modular theory and discusses several important physical applications centered around the Bisognano-Wichmann theorem.

Chapter 3 examines how modular theory can be used to give new geometric insight into the parity-charge-time (PCT) theorem. I argue that the key to understanding the mysterious connection between spatiotemporal orientation and charge structure entailed by the theorem lies in recognizing PCT symmetry as a single global reflection of quantum statespace.

Chapter 4 turns to the vexed problem of localization in QFT. Modular theory tells us that at a fundamental level the world is thoroughly entangled. In spite of this, via the split/nuclearity conditions, modular theory also supplies critical tools needed to help explain approximately-localized emergent entities like particles and chairs.

The final chapter contains a critique of the Connes-Rovelli thermal time hypothesis, a proposal to solve the problem of time in quantum gravity using modular theory. Although I conclude that the current proposal cannot provide a coherent gauge-free description of physical change in generally covariant settings, it does highlight an intriguing connection between modular structure and local dynamics in standard QFT.