Hadamard Renormalization of a 2-Dimensional Dirac Field

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.