Algebraic quantum field theory, the Hadamard condition and Ward identities
Markus B. Fr\"ob
In (perturbative) algebraic quantum field theory, the construction of interacting fields, including renormalisation, is done independently of the quantum state of the system. A crucial ingredient in the construction is a Green's function of Hadamard form. For gauge theories, so far one has been restricted to a special gauge (Feynman gauge in Yang-Mills theories), where this Green's function was explicitly known in a general curved background.
We show how to construct Green's functions in a general linear covariant gauge, both for Yang-Mills theories and linearised gravity. These functions fulfil certain divergence and trace identities, which can be interpreted as Ward identities in the free theory, and are a prerequisite for the gauge independence of the interacting quantum theory.