Theory Of Fermions With Lorentz Violation

An extension of the standard model of particle physics incorporating the low-energy effects of spontaneous Lorentz and CPT violation in an underlying theory is considered. Within this framework, the quantum physics of free massive fermions is discussed and the conditions of causality and stability are investigated. Some constraints on the underlying theory are obtained. Among the topics investigated is the relativistic quantum mechanics of a modified Dirac equation. In particular, symmetries and roots of the associated dispersion relation, various properties of the eigenspinors, and the existence and construction of a hermitian hamiltonian are studied. Canonical quantization of the modified Dirac equation is performed and the requirements of causality and stability are discussed. It is found that these requirements are met below a certain energy scale. However, it is shown that above this scale, the model generically develops difficulty with microcausality, energy positivity, or both in some observer frame. Lorentz-violating dispersion relations with the desired low-energy behavior but consistent with causality and stability at all energy scales are shown to exist. It is argued that they must emerge from a nonlocal fundamental theory incorporating spontaneous Lorentz breaking. The similarity of such dispersion relations to that of a certain scalar within the field theory of the open bosonic string is demonstrated