We develop relativistic non-Hermitian quantum theory and its application to neutrino physics in a strong magnetic field. It is well known, that one of the fundamental postulates of quantum theory is the requirement of Hermiticity of physical parameters. This condition not only guarantees the reality of the eigenvalues of Hamiltonian operators, but also implies the preservation of the probabilities of the considered quantum processes. However as it was shown relatively recently (Bender, Boettcher 1998), Hermiticity is a sufficient but it is not a necessary condition. It turned out that among non-Hermitian Hamiltonians it is possible to allocate a number of such which have real energy spectra and can ensure the development of systems over time with preserving unitarity. This type of Hamiltonians includes so-called parity-time (${\cal PT}$) symmetric models which is already used in various fields of modern physics. The most developed in this respect are models, which used in the field of ${\cal PT}$-symmetric optics, where for several years produced not only theoretical but experimental studies.
Non-Hermitian PT-symmetric relativistic quantum theory in an intensive magnetic field
We develop relativistic non-Hermitian quantum theory and its application to neutrino physics in a strong magnetic field. It is well known, that one of the fundamental postulates of quantum theory is the requirement of Hermiticity of physical parameters. This condition not only guarantees the reality of the eigenvalues of Hamiltonian operators, but also implies the preservation of the probabilities of the considered quantum processes. However as it was shown relatively recently (Bender, Boettcher 1998), Hermiticity is a sufficient but it is not a necessary condition. It turned out that among non-Hermitian Hamiltonians it is possible to allocate a number of such which have real energy spectra and can ensure the development of systems over time with preserving unitarity. This type of Hamiltonians includes so-called parity-time (PT) symmetric models which is already used in various fields of modern physics. The most developed in this respect are models, which used in the field of PT-symmetric optics, where for several years produced not only theoretical but experimental studies.
We review the methodology to theoretically treat parity-time- (PT-) symmetric, non-Hermitian quantum many-body systems. They are realized as open quantum systems withPTsymmetry and couplings to the environment which are compatible.PT-symmetric non-Hermitian quantum systems show a variety of fascinating properties which single them out among generic open systems. The study of the latter has a long history in quantum theory. These studies are based on the Hermiticity of the combined system-reservoir setup and were developed by the atomic, molecular, and optical physics as well as the condensed matter physics communities. The interest of the mathematical physics community inPT-symmetric, non-Hermitian systems led to a new perspective and the development of the elegant mathematical formalisms ofPT-symmetric and biorthogonal quantum mechanics, which do not make any reference to the environment. In the mathematical physics research, the focus is mainly on the remarkable spectral properties of the Hamiltonians and the characteristics of the corresponding single-particle eigenstates. Despite being non-Hermitian, the Hamiltonians can show parameter regimes, in which all eigenvalues are real. To investigate emergent quantum many-body phenomena in condensed matter physics and to make contact to experiments one, however, needs to study expectation values of observables and correlation functions. One furthermore, has to investigate statistical ensembles and not only eigenstates. The adoption of the concepts ofPT-symmetric and biorthogonal quantum mechanics by parts of the condensed matter community led to a controversial status of the methodology. There is no consensus on fundamental issues, such as, what a proper observable is, how expectation values are supposed to be computed, and what adequate equilibrium statistical ensembles and their corresponding density matrices are. With the technological progress in engineering and controlling open quantum many-body systems it is high time to reconcile the Hermitian with thePT-symmetric and biorthogonal perspectives. We comprehensively review the different approaches, including the overreaching idea of pseudo-Hermiticity. To motivate the Hermitian perspective, which we propagate here, we mainly focus on the ancilla approach. It allows to embed a non-Hermitian system into a larger, Hermitian one. In contrast to other techniques, e.g. master equations, it does not rely on any approximations. We discuss the peculiarities ofPT-symmetric and biorthogonal quantum mechanics. In these, what is considered to be an observable depends on the Hamiltonian or the selected (biorthonormal) basis. Crucially in addition, what is denoted as an 'expectation value' lacks a direct probabilistic interpretation, and what is viewed as the canonical density matrix is non-stationary and non-Hermitian. Furthermore, the non-unitarity of the time evolution is hidden within the formalism. We pick up several model Hamiltonians, which so far were either investigated from the Hermitian perspective or from thePT-symmetric and biorthogonal one, and study them within the respective alternative framework. This includes a simple two-level, single-particle problem but also a many-body lattice model showing quantum critical behavior. Comparing the outcome of the two types of computations shows that the Hermitian approach, which, admittedly, is in parts clumsy, always leads to results which are physically sensible. In the rare cases, in which a comparison to experimental data is possible, they furthermore agree to these. In contrast, the mathematically elegantPT-symmetric and biorthogonal approaches lead to results which, are partly difficult to interpret physically. We thus conclude that the Hermitian methodology should be employed. However, to fully appreciate the physics ofPT-symmetric, non-Hermitian quantum many-body systems, it is also important to be aware of the main concepts ofPT-symmetric and biorthogonal quantum mechanics. Our conclusion has far reaching consequences for the application of Green function methods, functional integrals, and generating functionals, which are at the heart of a large number of many-body methods. They cannot be transferred in their established forms to treatPT-symmetric, non-Hermitian quantum systems. It can be considered as an irony of fate that these methods are available only within the mathematical formalisms ofPT-symmetric and biorthogonal quantum mechanics.
The requirement of Hermiticity of a quantum mechanical Hamiltonian, for the description of physical processes with real eigenvalues which has been challenged notably by Carl Bender, is examined for the case of a Fock space Hamiltonian which is bilinear in two creation and annihilation operators. An interpretation of this model as a Schrödinger operator leads to an identification of the Hermitian form of the Hamiltonian as the Landau model of a charged particle in a plane, interacting with a constant magnetic field at right angles to the plane. When the parameters of the Hamiltonian are suitably adjusted to make it non-Hermitian, the model represents two harmonic oscillators at right angles interacting with a constant magnetic field in the third direction, but with a pure imaginary coupling, and real energy eigenvalues. It is now symmetric. Multiparticle states are investigated.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
In 1998 it was discovered that the requirement that a Hamiltonian be Dirac Hermitian (H = H†) can be weakened and generalized to the requirement that a Hamiltonian be PT symmetric ([H,PT] = 0); that is, invariant under combined space reflection and time reversal. Weakening the constraint of Hermiticity allows one to consider new kinds of physically acceptable Hamiltonians and, in effect, it amounts to extending quantum mechanics from the real (Hermitian) domain into the complex domain. Much work has been done on the analysis of various PT‐symmetric quantum‐mechanical models. However, only very little analysis has been done on PT‐symmetric quantum‐field‐theoretic models. Here, we describe some of what has been done in the context of PT‐symmetric quantum field theory and describe some possible fundamental applications.
Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.