Casimir Effect and Global Theory of Boundary Conditions

TL;DR

This paper explores how boundary conditions affect the Casimir effect in quantum field theories, revealing topological and geometric constraints, singularities, and implications for renormalization and dualities.

Contribution

It provides a global geometric and topological analysis of boundary conditions in quantum fields, introducing a new Maslov index and examining their impact on physical phenomena.

Findings

Boundary conditions can cause singularities related to topology changes and edge states.

The space of boundary conditions has a non-trivial topology characterized by a Maslov index.

Global properties influence renormalization group flow and dualities in quantum field theories.

Abstract

The consistency of quantum field theories defined on domains with external borders imposes very restrictive constraints on the type of boundary conditions that the fields can satisfy. We analyse the global geometrical and topological properties of the space of all possible boundary conditions for scalar quantum field theories. The variation of the Casimir energy under the change of boundary conditions reveals the existence of singularities generically associated to boundary conditions which either involve topology changes of the underlying physical space or edge states with unbounded below classical energy. The effect can be understood in terms of a new type of Maslov index associated to the non-trivial topology of the space of boundary conditions. We also analyze the global aspects of the renormalization group flow, T-duality and the conformal invariance of the corresponding fixed points.