Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime

Abstract

We give an introduction to the techniques from microlocal analysis that have successfully been applied in the investigation of Hadamard states of free quantum field theories on curved spacetimes. The calculation of the wave front set of the two point function of the free Klein‐Gordon field in a Hadamard state is reviewed, and the polarization set of a Hadamard two point function of the free Dirac field on a curved spacetime is calculated.

A smooth introduction to the wavefront set

“…For a Dirac or photon field, two definitions are possible: (i) the (standard) wavefront set of the propagator, which is the union of the wavefront sets of all its components in a given basis; (ii) the polarization set, described in section 8.3.2. The polarization set of the Dirac propagator was investigated in detail [10,11]. The projection of the polarisation set on T M ˙* gives the (standard) wavefront set [66].…”

“…This idea was fully developed into a renormalized scalar field theory in curved spacetimes by Brunetti, Fredenhagen [8], Hollands and Wald [9]. This approach was rapidly extended to the case of Dirac fields [10][11][12][13][14][15][16], to gauge fields [17][18][19] and even to the quantization of gravitation [20]. This tremendous progress was made possible by a complete reformulation of quantum field theory, where the wavefront set of distributions plays a central role, for example to determine the algebra of microcausal functions and to define a spectral condition for timeordered products and quantum states [21][22][23][24].…”

Dirac fields and thermal effects in the de Sitter universe

“…A full check of the local Hadamard condition (see [23,25] for a modern point of view) would require also verifying that the subleading behaviour is of Hadamard form. However, here we are not using the local Hadamard condition to select the physical state.…”

Quantization of Dirac fields in static spacetime

“…A very general theory has been established by Dimock [5], for Dirac quantum fields on a hyperbolic Lorentzian manifold. For most up-todate research along this line, one can find recent papers by Hollands [6] and Kratzert [7], and references therein.…”