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Quaternion quantum mechanics: Second quantization and gauge fields. (English) Zbl 0558.46039
The properties of a quaternion Hilbert module (a closed vector space with positive definite norm, linear over the quaternions as a skew field for right multiplication) are studied. It supplements and extends the earlier investigations of [D. Finkelstein, J. M. Jauch, S. Schiminovich and D. Speiser, J. Math. Phys. 3, 207-220 (1962); 4, 788-796 (1963; Zbl 0124.226)]. Some of the methods used were developed for the treatment of a Hilbert module over the Clifford algebra of order seven by the authors, J. Math. Phys. 20, 269-298 (1979; Zbl 0416.22022). In [L. P. Horwitz and A. Soffer, J. Math. Phys. 24, 2780-2782 (1983)], it was found that the quaternion Hilbert module emerges as a GNS representation for a $$B^*$$ (and hence $$C^*)$$ algebra containing a subalgebra isomorphic to quaternions, and the Hahn-Banach theorem was proven for this structure [see also A. Soffer, Doctoral dissertation, Tel Aviv University (1984).].
Since the quaternion algebra contains complex and real subalgebras, a hierarchy of scalar products (quaternion, complex, real) exists, with which one associates corresponding vector spaces, all of which have the same norm and hence the same topology. Projection operators for the subspaces of these vector spaces are defined, and the Gleason theorem is applied for the development of the notion of quantum states. Pure states turn out not to be the simplest building blocks for the quaternion Hilbert module; the set of special pure states which are simplest are called primitive. Properties of symmetric operators are investigated in each vector space of the hierarchy. The translation group is used to define the (self-adjoint) momentum operator, and the momentum-coordinate quantum uncertainty relation is investigated. The minimum uncertainty state is primitive. A complete description of the Euclidean symmetries is obtained.
A tensor product is defined for the complex linear (”symplectic” representation, since no effective definition is available for the Kronecker product of quaternion linear spaces. Annihilation and certain operators are then defined, with which the Fock space can be constructed for a second-quantized (quantum field) theory. The gauge fields (sections on the fiber bundles for each sector of the Fock space) are shown to have a structure related to that found by S. L. Adler from a somewhat different approach [Phys. Rev D21, 550 (1980)], and references therein.

##### MSC:
 46N99 Miscellaneous applications of functional analysis 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 81T08 Constructive quantum field theory 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 22E70 Applications of Lie groups to the sciences; explicit representations 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 12E15 Skew fields, division rings
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