The Bogoliubov Transformation in the Field Theory

Abstract

It is confirmed that the method of the Bogoliubov transformation is applicable also in the field theory. The original representation suited for a massless spinor field, ψ(0), can be transformed into another representation, in which the spinor field has a non-zero mass, for the Fermi interactions of any type, by the transformation. But the interaction responsible for the finite mass is the scalar part of the interaction itself or of the equivalent expression obtained by the Fierz transformation. As to the Yukawa interaction, the Bogoliubov transformation must be accompanied by a linear transformation for the Bose field, and the combined transformation is meaningful only for the scalar coupling of a scalar meson with mass. The situation about the φ4 interactions of spinless fields φ is completely the same as in the case of the Fermi interaction, but the method is not applicable to the φ3 interaction of a scalar field. Thus it is suggested that the quantities <<:ψ(0)ψ(0):>> for the Fermi interaction, <<φ(0)>> for the Yukawa interaction, and <<:φ(0)2:>> for the φ4 interactions correspond to the parameter of the long range order in the theory of superconductivity, respectively, where the notatin << >> means the expectation value in the transformed vacuum. It is also shown that Lehmann's method of the spectrum representation can be generalized to treat the canonical transformation, and that it gives a covariant and convenient method to derive the mass equation. In the last section, it is discussed what kinds of image of assumption must be introduced in order that the representation obtained by the Bogoliubov transformation can be used to the unified theories of elementary particles. Counter-examples to the too conventional method used in the theories of spontaneous breakdown of symmetries are also exhibited (§5 and App. I).