A Quantum Theory of the Non-Linear Realization of a Group on Its Sub-Group

概要

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A quantum theory of the non-linear realization is proposed for the groups with a subgroup which can be realized non-linearly by a single self-conjugate preferred field. In the non-linear realization of a group on its sub-group, the displacement operators in Minkowski space can be written down at once as a sum of two terms. Not only the coefficients of the two terms but also the equal-time commutation relations between field operators and the equation of motion can be determined by the property of the displacement operators. Furthermore, all of them can be proved to be covariant under the internal group. Thus we can construct a quantum theory of the non-linear realization in a closed form without referring to the Lagrangian formalism or the variation method, where it is difficult to maintain the order of the operators correctly throughout all steps of manipulations. But it is argued that the Fock space cannot be spanned without violating the symmetry. Our method can be applied to any non-linear system once the energy-momentum tensor is specified. The difference between non-linear systems and linear ones and that between the Lagrangian formalism and ours are also discussed.
理論物理学刊行会の論文
1971-08-25