Non-Hermitian Quantum Theory and its Holomorphic Representation: Introduction and Some Applications

Present Hermitian Quantum Theory, i.e. Quantum Mechanics and Quantum Field Theory, is revised and replaced by a consistent non-Hermitian formalism called non-Hermitian Quantum Theory (NHQT) or (Anti)Causal Quantum Theory ((A)CQT) after lining out some inherent inconsistencies and problems arising in the context of causality, which is observed to introduce an indefinite metric in canonical commutation relations. Choosing some (very selective) historical approach to introduce necessary terminology and explain complications when quantizing non-Hermitian systems in the presence of an indefinite metric we propose a way how to construct a causal, analytic, Poincare invariant, and local NHQT, the spacial representation of which is related to the so-called holomorphic representation used in complex analysis. Besides providing a revised antiparticle, spinor, and probability concept, a new neutrino Lagrangean, two distinct time-reversal operations, and generalized non-Hermitian Poincare transformations, we will apply NHQT to consider three important issues: PT-symmetry and non-Hermitian similarity transforms, non-Hermitian supersymmetry, and the construction of an asymptotically free theory of strong interactions without gluons.