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The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes. (English) Zbl 0976.58023
Summary: We characterize the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on Robertson-Walker spacetimes. Using this characterisation, we construct adiabatic vacuum states of order $$n$$ corresponding to some Cauchy surface. It is demonstrated that any two such states (of sufficiently high order) are locally quasi-equivalent. We give a microlocal characterization of spinor Hadamard states and we show that this agrees with the usual characterization of such states in terms of the singular behaviour of their associated twopoint functions. The polarization set of these twopoint functions is determined and found to have a natural geometric form. We finally prove that our adiabatic states of infinite order are Hadamard, and that those of order $$n$$ correspond, in some sense, to a truncated Hadamard series and therefore allow for a point splitting renormalization of the expected stress-energy tensor.

##### MSC:
 58J90 Applications of PDEs on manifolds 83C99 General relativity 81T99 Quantum field theory; related classical field theories
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