International Journal of Theoretical Physics, Vol. 42, No. 5, May 2003 ( C  2003) Holonomy in Quaternionic Quantum Mechanics Jerzy Dajka 1 and Marek Szopa 1,2 Received April 9, 2003 The generalization of geometric phase for the quantum systems described by quater- nionic quantum mechanics is given. The geometry of the quantum cyclic evolution is studied and the quaternionic Berry phase is shown to be given by the holonomy of the suitably defined fiber bundle. KEY WORDS: Berry phase; holonomy; quaternionic quantum mechanics. 1. INTRODUCTION The geometric ideas play an important role in physics. The quantum mechan- ical phenomena being geometric in its nature is the Berry phase (Anandan and Aharonov, 1990; Berry, 1984; Simon, 1983). After Berry’s discovery the phase has been formalized in terms of the connection on the suitably defined fiber bundle as a holonomy. The topological nature of holonomy leads to the observation that the geometric phase is a global feature of the quantum evolution. In this paper the results are extended in the framework of the quaternionic quantum mechanics (Adler, 1995). The topological considerations allow to recognize the difference between results concerning triviality of the holonomy obtained in the standard complex and quaternionic quantum mechanics. 2. GEOMETRIC PHASE IN QUATERNIONIC QUANTUM MECHANICS As a starting point, some concepts of quaternionic quantum mechanics will be introduced. Our approach is based on studies by Birkhoff and von Neumann, 1936; Finkelstein, Jauch, and Speiser, 1959; Adler, 1995. The space of states of the quantum system is a Hilbert space H(H) on the algebra of Hamilton’s quaternions H with H-valued scalar product h· | ·i : H × H → H. The time evolution of the state is governed by a group of unitary operators in H generated by the antihermitian 1 Institute of Physics, University of Silesia, Katowice, Poland. 2 To whom correspondence should be addressed at Institute of Physics, University of Silesia, 40-007 Katowice, Poland; e-mail: szopa@plktus11.bitnet. 1053 0020-7748/03/0500-1053/0 C  2003 Plenum Publishing Corporation