Passivity and microlocal spectrum condition

TL;DR

This paper proves that passive quantum states in vector-valued quantum fields on stationary spacetimes satisfy the microlocal spectrum condition, ensuring their two-point functions are of Hadamard form, which is crucial for quantum field theory in curved spacetime.

Contribution

It establishes that passive states of vector-valued quantum fields inherently satisfy the microlocal spectrum condition, extending previous results to include fermionic fields and an abstract algebraic framework.

Findings

Passive states fulfill the microlocal spectrum condition.

Two-point functions of passive states are of Hadamard form.

Results apply to both bosonic and fermionic fields.

Abstract

In the setting of vector-valued quantum fields obeying a linear wave-equation in a globally hyperbolic, stationary spacetime, it is shown that the two-point functions of passive quantum states (mixtures of ground- or KMS-states) fulfill the microlocal spectrum condition (which in the case of the canonically quantized scalar field is equivalent to saying that the two-point function is of Hadamard form). The fields can be of bosonic or fermionic character. We also give an abstract version of this result by showing that passive states of a topological *-dynamical system have an asymptotic pair correlation spectrum of a specific type.