Euclidean Quantum Fields with Spin on an Indefinite Inner Product State Space

Abstract

It is shown that for a Wightman quantum field theory with space inversion symmetry, the Wightman functions in Euclidean points can be obtained as certain linear combinations of vacuum expectation values of Euclidean fields, on a state space with an indefinite inner product. The coefficients of these linear combinations only depend on the transformation properties of the fields, not on their interactions. The use of an indefinite inner product enables us to keep the number of Euclidean fields the same as that of Wightman fields, except that in the case of hermitian fermion fields, the corresponding Euclidean fields are not hermitian. For free Dirac fields, the construction is exhibited explicitly and a Euclidean invariant Hilbert space metric on the state space, making the indefinite inner product continuous, is introduced.