# Worldline formalism for a confined scalar field

## TL;DR

This paper extends the worldline formalism to handle scalar fields confined within boundaries, specifically a D-dimensional ball, by restricting path integrals to boundary-enclosed worldlines, enabling boundary condition analysis.

## Contribution

It introduces a method to implement boundary restrictions in the worldline formalism for confined scalar fields, verified through heat-kernel coefficient calculations.

## Key findings

- Successfully implemented boundary restrictions for scalar fields in the worldline formalism.
- Computed initial heat-kernel coefficients for confined scalar fields.
- Demonstrated potential for generalizing the approach to other boundary conditions.

## Abstract

The worldline formalism is a useful scheme in quantum field theory which has also become a powerful tool for numerical computations. The key ingredient in this formalism is the first quantization of an auxiliary point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field which is confined within some boundaries one needs to restrict the path integration domain of the auxiliary point-particle to a specific subset of worldlines enclosed by those boundaries. We show how to implement this restriction for the case of a scalar field confined to the $D$-dimensional ball under Dirichlet and Neumann boundary conditions, and compute the first few heat-kernel coefficients as a verification of our construction. We argue that this approach could admit different generalizations.