UC Berkeley

This dissertation addresses a number of related questions concerning perturbative "path" integrals. Perturbative methods are one of the few successful ways physicists have worked with (or even defined) these infinite-dimensional integrals, and it is important as mathematicians to check that they are correct.

Chapter 0 provides a detailed introduction.

We take a classical approach to path integrals in Chapter 1. Following standard arguments, we posit a Feynman-diagrammatic description of the asymptotics of the time-evolution operator for the quantum mechanics of a charged particle moving nonrelativistically through a curved manifold under the influence of an external electromagnetic field. We check that our sum of Feynman diagrams has all desired properties: it is coordinate-independent and well-defined without ultraviolet divergences, it satisfies the correct composition law, and it satisfies Schrödinger's equation thought of as a boundary-value problem in PDE.

Path integrals in quantum mechanics and elsewhere in quantum field theory are almost always of the shape ∫ f e^s for some functions f (the "observable") and s (the "action"). In Chapter 2 we step back to analyze integrals of this type more generally. Integration by parts provides algebraic relations between the values of ∫ (-) e^s for different inputs, which can be packaged into a Batalin–Vilkovisky-type chain complex. Using some simple homological perturbation theory, we study the version of this complex that arises when f and s are taken to be polynomial functions, and power series are banished. We find that in such cases, the entire scheme-theoretic critical locus (complex points included) of s plays an important role, and that one can uniformly (but noncanonically) integrate out in a purely algebraic way the contributions to the integral from all "higher modes," reducing ∫ f e^s to an integral over the critical locus. This may help explain the presence of analytic continuation in questions like the Volume Conjecture.

We end with Chapter 3, in which the role of integration is somewhat obscured, but perturbation theory is prominent. The Batalin–Vilkovisky homological approach to integration illustrates that there are generalizations of the notion of "integral" analogous to the generalization from cotangent bundles to Poisson manifolds.

The AKSZ construction of topological quantum field theories fits into this approach; in what is usually called "AKSZ theory," everything is still required to be symplectic.

Using factorization algebras as a framework for (topological) quantum field theory, we construct a one-dimensional Poisson AKSZ field theory for any formal Poisson manifold M. Quantizations of our field theory correspond to formal star-products on M. By using a ``universal'' formal Poisson manifold and abandoning configuration-space integrals in favor of other homological-perturbation techniques, we construct a universal formal star-product all of whose coefficients are manifestly rational numbers.