  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.
