# Irvine-Munich Workshop on the Foundations of Classical and Quantum Field Theories (December 14th 2014)

## Idea and Motivation

Since its introduction in the nineteenth century, the concept of a field as an independent physical entity has become central to modern physics. Many highly successful classical theories, most notably electromagnetism and gravitation, were reformulated along field-theoretic lines, leading to advances in the understanding of their respective subject matters as well as enriching knowledge of field theory itself. The great development of these classical field theories influenced the founders of quantum mechanics, with the subsequent formulation of *quantum* field theories providing molds from which the Standard Model of particle physics has been cast. Yet the profound difficulties of finding a quantum field theory that incorporates gravitation have forced physicists and philosophers to confront the foundations of field theory with more scrutiny. While the multifaceted nature of the difficulties with quantum field theory may have reopened inquiry into classical field theories in hopes of finding potent analogies, this study has developed a rich life of its own. This workshop brings together communities of researchers working from different viewpoints on the foundations of both classical and quantum field theory.

## Speakers

- Erik Curiel (MCMP/LMU)
- Benjamin Feintzeig (UC Irvine)
- Samuel Fletcher (MCMP/LMU)
- Brian Padden (MCMP/LMU)
- Sarita Rosenstock (UC Irvine)
- Michael Stoeltzner (MCMP/South Carolina)
- Karim Thébault (MCMP/LMU)
- James Weatherall (UC Irvine)

## Program

## Abstracts

### Erik Curiel: A Generalization of the Geometry of the Euler-Lagrange Equation

I isolate a simple algebraic structure on two families of vector fields on a manifold that is necessary for those families to be, respectively, the kinematically possible and the interaction vector fields of a classical system. This algebraic structure naturally yields a generalization of the structures that characterize the intrinsic geometry of a tangent bundle and allow the formulation of the Euler-Lagrange equation, the so-called almost-tangent structure. Because the algebraic structure is definable on infinite-dimensional manifolds, it may be possible to use it as a foundation for novel formulations of field theory in a Lagrangian-like framework.top

### Benjamin Feintzeig: Unitary Inequivalence in Classical Systems

I provide an algebraic formulation of classical field theories and use this to probe our interpretation of algebraic theories more generally. I show that the problem of unitarily inequivalent representations, as discussed in Ruetsche (2011), arises in classical theories just as in quantum theories, and I argue that this gives reason to not be a Hilbert Space Conservative when interpreting algebraic theories.top

### Samuel Fletcher: Classical Field Theory and Intertheoretic Reduction: A Prolegomenon

In 1986, Ehlers set out a program on how to understanding the approximative relationships between different physical theories. However, he essentially only investigated the case of classical and relativistic spacetime theories, which have a number of special features that distinguish them from broader classes of physical theories. To what extent, then, can the Ehlers program be successful? I outline some of the challenges facing the program's generalization and argue that they can largely be overcome for classical field theories.top

### Brian Padden: Relativistic Quantum Particles the Feynman Way

It is often believed, especially in light of theorems by Malament and others, that there is no relativistic theory of localizable quantum particles. However, an example of exactly such a theory seems to exist, and in fact occupies an important place in the storied history of quantum field theory: Feynman’s path integral approach to quantum electrodynamics. We introduce Feynman’s theory and show that, up to a few minor issues, it is satisfactory. Then, we turn to the theorems stating that such a theory is impossible and discuss which premises are violated by the Feynman theory.top

### Sarita Rosenstock: On Fiber Bundle and Holonomy Interpretations of Yang-Mills Theories

In the philosophy of Yang-Mills theories, there is an ongoing debate between rival interpretations that can be grouped into two rough categories: “holonomy interpretations” (supported by, e.g., Healey, Belot, and Lyre) and “fiber bundle interpretations” (supported by, e.g., Arntzenius, Maudlin, and Leeds). I present a theorem that I interpret as providing a precise sense in which these interpretations are equivalent.top

### Michael Stoeltzner: Possible Worlds and Random Walks: the Principle of Least Action as a Thought Experiment

Over the centuries, no other principle of classical physics has to a larger extent nourished exalted hopes of a universal theory, has constantly been plagued by mathematical counterexamples, and has ignited metaphysical controversies than has the principle of least action (PLA). The aim of this paper is first to survey a series of modern approaches, among them the structural realist readings of Planck and Hilbert, a neo-Kantian relativized a priori principle, and more recent discussions about modality within the context of analytic metaphysics. But these considerations seem outrun by the broad applicability of the PLA beyond classical physics. In the case of Feynman’s path integral, the PLA does no longer amount to the distinction of the actual dynamics among the possible ones, but to the definition of a stochastic process to which all possibilities contribute with a certain probability. To reach a unified philosophical picture of all the various applications of the PLA and its kin, I suggest to consider them as a thought experiment about the applicability of mathematics to a physical problems.top

### Karim Thébault: A New Prescription for the Quantization of Refoliation Invariant Field Theories

Imagine a loaf of bread that we can irregularly cut up into a sequence of slices. The loaf is spacetime and the slices are instantaneous spatial surfaces. A foliation is a parameterization of a spacetime by a time ordered sequence of spatial slices. In a field theory such a parametrization can be local in the sense that it is defined for every point on every spatial slice. Diffeomorphism invariance implies that spacetimes described by general relativity that are related by refoliations are physically equivalent. Classically the symmetry is therefore directly connectable to the idea that only the coordinate-free information contained in a spacetime geometry has a physical basis. The implications of this symmetry for quantization are notoriously problematic. Here we offer a new prescription for the canonical quantization of gravity that side-steps the issues with refoliations via the adoption of the 'shape dynamics’ reformulation. We then offer our thoughts as to whether this is a satisfactory resolution for the problem for understanding refoliation symmetry in the context of a quantum field theory of gravity.top

### James Weatherall: Fiber Bundles, Yang-Mills Theory, and General Relativity

I articulate and discuss a geometrical interpretation of Yang-Mills theory. Analogies and disanalogies between Yang-Mills theory and general relativity are also considered.top

## Location

## Contact & Registration

For information about practical matters and registration, please contact one of the organizers: Samuel Fletcher (Samuel.Fletcher@lrz.uni-muenchen.de) or Karim Thébault (Karim.Thebault@lrz.uni-muenchen.de).