Hadamard renormalization of a two-dimensional Dirac field

The Hadamard renormalization procedure is applied to a free, massive Dirac field $\ensuremath{\psi}$ on a two-dimensional Lorentzian spacetime. This yields the state-independent divergent terms in the Hadamard bispinor ${G}^{(1)}(x,{x}^{\ensuremath{'}})=\frac{1}{2}⟨[\overline{\ensuremath{\psi}}({x}^{\ensuremath{'}}),\ensuremath{\psi}(x)]⟩$ as $x$ and ${x}^{\ensuremath{'}}$ are brought together along the unique geodesic connecting them. Subtracting these divergent terms within the limit assigns ${G}^{(1)}(x,{x}^{\ensuremath{'}})$, and thus any operator expressed in terms of it, a finite value at the coincident point ${x}^{\ensuremath{'}}=x$. 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 ${G}^{(1)}(x,{x}^{\ensuremath{'}})$---themselves obtained, for example, from a numerical simulation---to obtain finite expectation values defined in the continuum.

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