Geometric Phase in Quaternionic Quantum Mechanics

Quaternion quantum mechanics is examined at the level of unbroken SU(2) gauge symmetry. A general quaternionic phase expression is derived from formal properties of the quaternion algebra. pacs 03.65.Bz, 03.65.Ca,11.15.Kc, 02.30.+g 1 Quaternion Quantum Mechanics Quantum mechanics defined over general algebras have been conjectured since 1934 [1]. In 1936 Birkoff and von Neumann noted that the propositional calculus implies in a representation of pure states of a quantum system by rays on a Hilbert space defined over any associative division algebra [2]. This means that quantum theory would be limited to the real, complex and quaternion algebras. Standard textbooks explain the complex formulation of 1 quantum mechanics by means of the double slit experiment and the complex phase difference of the wave functions. It is possible to use a real quantum theory, but at the cost of introducing a special operator J satisfying J = 1 and J = −J , so that at the end the complex structure emerges again. The final argument for complex algebra as minimum requirement appears with the spinor structure. In fact, the spinor representations of the rotation group requires the existence of solutions of quadratic algebraic equations related to the invariant operators, which are guaranteed only over a complex algebra[3]. The development of quaternion quantum mechanics started with D. Finkelstein in 1959, its relativistic and particle aspects were studied by G. Emch and E. J. Schremp [4, 5, 6]. A comprehensive reference list can be found in [7]. In an attempt to interpret quaternion quantum mechanics, C. N. Yang suggested that the isospinor symmetry should be contained in the group of automorphisms of the quaternion algebra [4]. Indeed, supposing that the spin angular momentum ~ M associated with SO(3) and the isospin ~ I given by a representation of SU(2), are both present in a single state, their spinor representations are given by the Pauli matrices acting separately on the spinor space M and the isospinor space I respectively, generated by two independent complex bases (1, i) and (1, j). The direct sum M ⊕ I does not close as an algebra, except if third imaginary unit k = ij is introduced, producing a quaternion algebra. The automorphisms of this algebra carries the spin-isospin combined symmetry. According to this interpretation, quaternion quantum mechanics would be effective at the energy level in which the spin and isospin symmetries remain combined. When this combined symmetry breaks down, the isospin angular momentum may lead to an extra spin degree of freedom [8, 9, 10, 11, 12]. The existence and effectiveness of quaternion quantum mechanics at higher energies must be experimentally verified. In one of the experiments proposed by A. Peres, a neutron interferometer with thin plates made of materials with varying proportions of neutrons and protons is used, where the phase difference in one or another case is measured [13]. In principle this experiment could be adapted to a variable beam intensity, so that a phase difference between the complex and the quaternion could be detected for different energy levels. Quaternions keep a one-to-one correspondence with space-time vectors.