Scattering and decay theory for quaternionic quantum mechanics, and the structure of inducedTnonconservation

We develop a scattering theory and decay theory for nonrelativistic quaternionic quantum mechanics. We show that the far-zone part of scattering states lies in the complex openC(1,i) subspace of quaternionic Hilbert space picked out by the kinetic part of the Hamiltonian; intrinsically quaternionic terms are present in the wave function but have exponential spatial decay. Hence, scattering phase shifts are necessarily complex in quaternion quantum mechanics, and the test for quaternionic effects suggested by Peres gives a null result. Integrating out the quaternionic components, the complex scattering problem can be expressed in terms of an optical potential $V_{\mathrm{opt}}$ which is Hermitian but time-reversal nonconserving. The corresponding decay problem for openC(1,i) initial states can be expressed in terms of the same optical potential. Solving the decay problem to order $V_{\mathrm{opt}{}^2}$, the phenomenological form of the induced T nonconservation is seen to be ‘‘milliweak,’’ with T nonconservation arising both from the mass and the decay matrices, and hence is compatible with the phenomenology of T nonconservation in the standard model. When the complex openC(1,i) part of the quaternionic Hamiltonian has bound states, the optical potential develops isolated pole singularities. These lead to resonances at scattering energies equal to the bound-state binding energies, with widths proportional to the square of the quaternionic part of the potential. Inclusion of a positive-rest-mass term in the Hamiltonian shifts the location of these resonances towards lower kinetic energy, and if large enough eliminates them altogether.