Green’s functions in the presence of a bubble wall

Field theoretical tools are developed so that one can analyze quantum phenomena such as transition radiation that must have occurred during the Higgs condensate bubble expansion through plasma in the early universe. Integral representations of Bosonic and Fermionic propagators are presented for the case that particle masses are varied continuously during the passage through the bubble wall interface between symmetry-restored and symmetry-broken regions. The construction of propagators is based on the so-called eigenfunction expansion method associated with self-adjoint differential operators, developed by Weyl, Stone, Titchmarsh, Kodaira and several others. A novel method of field quantization in the presence of the bubble wall is proposed by using the spectral functions introduced in constructing the two-point Green’s functions.