Quaternion quantum mechanics: Second quantization and gauge fields

Abstract

Recent work on algebraic chromodynamics has indicated the importance of a systematic study of quaternion structures in quantum mechanics. A quaternionic Hilbert module, a closed linear vector space with many of the properties of a Hilbert space is studied. The propositional system formed by the subspaces of such a space satisfy the axioms of quantum theory. There is a hierarchy of scalar products and linear operators, defined in correspondence with the types of closed subspaces (with real, complex or quaternion linearity). Real, complex, and quaternion linear projection operators are constructed, and their application to the definition of quantum states is discussed. A quaternion linear momentum operator is defined as the generator of translations, and a complete description of the Euclidean symmetries is obtained. Tensor products of quaternion modules are constructed which preserve complex linearity. Annihilation-creation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The tensor product spaces provide representations for algebras with dimensionality increasing with particle number. The algebraic structure of the gauge fields associated with these algebras is precisely that of the semi-classical fields introduced by Adler.