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There are two natural ways in which quantum mechanics can be complexified such that both operators and state space are extended into complex domain. These two extensions can be characterised by the use of quaternion and their split-signature cousin coquaternion. Quaternionic extensions lead to a version of quaternionic quantum mechanics; while coquaternionic extensions are closely related to PT-symmetric quantum mechanics where depending on the parameters in the Hamiltonian the structure of the state space as well as dynamics exhibit radical transitions. In both cases, the mathematical structure of the theory demands that the ambient (Euclidean) space-time has dimension six. Upon further complexification we obtain an octavic representation for which ambient space-time has dimension ten. Here I present in detail the dynamical aspects of spin particles in these cases, and show how dimensional reduction can take place so that physics in three-space are restored. The formulation also suggests the possibility that the existence of extra dimensions postulated in string theories can be probed at low energies. The talk is based on joint work with Eva-Maria Graefe (Imperial College). |
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