Abstract
We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary operator which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution only for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schrödinger equation is the only possible choice, up to a shift of the global phase.
Highlights
INTRODUCTIONOur understanding of quantum theory has significantly improved by investigating alternatives to quantum theory and analyzing how these alternatives would or would not be at variance with observations or with our expectations on the structure of a physical theory
In quantum mechanics there is an intimate relation between the Hamiltonian H as the energy observable and the generator of time shifts − i H as it occurs in the Schrödinger equation
We studied the structure of universal dynamics in quantum theory using three main axioms (DC1)–(DC3)
Summary
Our understanding of quantum theory has significantly improved by investigating alternatives to quantum theory and analyzing how these alternatives would or would not be at variance with observations or with our expectations on the structure of a physical theory. A consistent dynamics in quaternionic quantum theory has been formulated, at the price of a superselection rule, where only a subspace of all self-adjoint operators can be used as the Hamiltonian of a system[17]. In contrast to previous work[17,26], we are interested in the universal case where the set of Hamiltonians is unrestricted, that is, every self-adjoint operator must induce some dynamics. We find that this is only possible for one-level or two-level systems and that the corresponding Schrödinger-type equation is necessarily of the form ψ(t) = [AH + HA − tr(H)A]ψ(t),.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
