The breakdown of Ehrenfest's theorem imposes serious limitations on quaternionic quantum mechanics (QQM). In order to determine the conditions in which the theorem is valid, we examined the conservation of the probability density, the expectation value and the classical limit for a non-antihermitian formulation of QQM. The results also indicated that the non-anti-hermitian quaternionic theory is related to non-hermitian quantum mechanics, and thus the physical problems described with both of the theories should be related.

A quaternionic analog of the Aharonov-Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics (QQM). A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.

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After a review on recent results on quaternic quantum mechanics (HQM), we present further consistency tests that reinforce its compatibility with the usual complex quantum mechanics (CQM). The novel results comprises the Virial theorem, the quantum quaternic Lorentz force, the existence of a quaternic magnetic monopole and the redefinition of the expectation value.

We consider pulsating strings in Lunin-Maldacena backgrounds, specifically in deformed Minkowski spacetime and deformed AdS 5 × S 5 . We find the relation between the energy and the oscillation number of the pulsating string when the deformation is small. Since the oscillation number is an adiabatic invariant it can be used to explore the regime of highly excited string states. We then quantize the string and look for such a sector. For the deformed Minkowski background we find a precise match with the classical results if the oscillation number is quantized as an even number. For the deformed AdS 5 × S 5 we find a contribution which depends on the deformation parameter.