The Bogoliubov Transformation in the Field Theory : The Mass Equation

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概要

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• 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 tranformation. 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 φ^4 interactions of spinless fields φ is completely the same as in the case of the Fermi interation, but the method is not applicable to the φ^3 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 φ^4 interactions correspond to the parameter of the long range order in the theory of superconductivity, respectively. where the notation ≪ ≫ 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 or assumption must be introduced in order that the representation obtained by the Bogoliubov transformation can be used to the unified theories of elementaryparticles. Counter-examples to the too conventional method used in the theories of spontaneous breakdown of symmetries are also exhibited (§5 and App.I).

• 理論物理学刊行会の論文
• 1964-10-25

著者

• Yamada Masao

  Department Of Cardiology Tokyo Medical University

• Tanabe Kosai

  Department Of Physics College Of Liberal Arts Saitama University

• Okabayashi Takao

  Department Of Physics University Of Tokyo

• Tanabe Kosai

  Department Of Physics University Of Tokyo

• YAMADA Masao

  Department of Physics, University of Tokyo

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