The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes

TL;DR

This paper characterizes homogeneous, isotropic, and quasifree states for Dirac fields on Robertson-Walker spacetimes, constructs adiabatic states, and establishes a microlocal Hadamard condition ensuring their physical and mathematical consistency.

Contribution

It introduces a microlocal Hadamard condition for spinor fields, constructs adiabatic vacuum states of arbitrary order, and proves their local quasi-equivalence and Hadamard property.

Findings

Adiabatic states of infinite order are Hadamard.

Two high-order adiabatic states are locally quasi-equivalent.

The microlocal Hadamard condition entails the usual short distance behaviour.

Abstract

We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on Robertson-Walker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order $n$ corresponding to some Cauchy surface. We then show that any two such states (of sufficiently high order) are locally quasi-equivalent. We propose a microlocal version of the Hadamard condition for spinor fields on arbitrary spacetimes, which is shown to entail the usual short distance behaviour of the twopoint function. The polarisation set of these twopoint functions is determined from the Dencker connection of the spinorial Klein-Gordon operator which we show to equal the (pull-back) of the spin connection. Finally it is demonstrated that adiabatic states of infinite order are Hadamard, and that those of order $n$ correspond, in some sense, to a truncated Hadamard series and will therefore allow for a point splitting renormalisation of the expected stress-energy tensor.