Hadamard Renormalization of a 2-Dimensional Dirac Field

TL;DR

This paper applies Hadamard renormalization to a 2D Dirac field, deriving finite, state-independent results for quadratic operators like the stress-energy tensor, useful for numerical and analytical studies.

Contribution

It extends Hadamard renormalization to 2D Dirac fields, providing explicit divergent terms and finite operator values in various coordinate systems.

Findings

Derived divergent terms for the Dirac bispinor in 2D

Obtained finite expectation values for quadratic operators

Presented covariant expressions in different coordinate charts

Abstract

The Hadamard renormalization procedure is applied to a free, massive Dirac field $\psi$ on a 2 dimensional Lorentzian spacetime. This yields the state-independent divergent terms in the Hadamard bispinor $G^{(1)}(x, x') = \frac{1}{2} \left\langle \left[ \bar{\psi}(x'), \psi(x) \right] \right\rangle$ as $x$ and $x'$ are brought together along the unique geodesic connecting them. Subtracting these divergent terms within the limit assigns $G^{(1)}(x, x')$, and thus any operator expressed in terms of it, a finite value at the coincident point $x' = x$. In this limit, one obtains a quadratic operator instead of a bispinor. The procedure is thus used to assign finite values to various quadratic operators, including the stress-energy tensor. Results are presented covariantly, in a conformally-flat coordinate chart at purely spatial separations, and in the Minkowski metric. These terms can be directly subtracted from combinations of $G^{(1)}(x, x')$ - themselves obtained, for example, from a numerical simulation - to obtain finite expectation values defined in the continuum.