Imprimitivity theorem and quaternionic quantum mechanics

Conference · Wed Apr 01 00:00:00 EST 1981 · Hadronic J.; (United States)

OSTI ID:6645337

Truini, P ^[1]; Biedenharn, L C; Cassinelli, G

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1. Duke Univ., Durham, NC

The study of quantum mechanics in quaternionic Hilbert space is not new in physics; indeed, it goes back to the first attempts in formulating quantum mechanics from an axiomatic point of view, and later, from a more physical viewpoint, to the survey of Finkelstein et al. But recently a new physical motivation for such investigations has been given by the work of Adler in algebraic chromostatics, since it has been pointed out that quaternionic quantum mechanics, as a model of an intrinsically non-Abelian gauge theory, fits the Adler algebraic version of chromostatics. The present paper aims at a critical understanding of quaternionic quantal structure through the examination of the elementary case of a single free particle in non-relativistic space-time. We follow the suggestion by Mackey to exploit the imprimitivity theorem, together with the notion of induced representation, in deriving the theory for the free particle system, starting from the axiomatic standpoint. We show that there exists a unique unitary skew adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic unit.

• OSTI ID: 6645337
• Report Number(s): CONF-8008162-
• Journal Information: Hadronic J.; (United States), Vol. 4:3; Conference: 3. workshop on Lie-admissible formulations, Boston, MA, USA, 4 Aug 1980
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
QUANTUM MECHANICS
HILBERT SPACE
ALGEBRAIC FIELD THEORY
ELEMENTARY PARTICLES
QUANTUM OPERATORS
AXIOMATIC FIELD THEORY
BANACH SPACE
FIELD THEORIES
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
QUANTUM FIELD THEORY
SPACE
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics