We investigate the spinor solutions, the spectrum and the symmetry properties of a matrix-valued wave equation whose plane-wave solutions satisfy the superluminal (tachyonic) dispersion relation , where E is the energy, is the spatial momentum and m is the mass of the particle. The equation reads (iγ μ ∂ μ − γ 5 m )ψ = 0, where γ 5 is the fifth current. The tachyonic equation is shown to be invariant and invariant. The tachyonic Hamiltonian breaks parity and is non-Hermitian but fulfils the pseudo-Hermitian property , where P is the parity matrix and is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for ) and evanescent waves, which constitute resonances and anti-resonances with complex-conjugate pairs of resonance eigenvalues (for ). In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalue E implies that E *, − E and − E * also constitute resonance energies of H 5 . This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
We investigate the spinor solutions, the spectrum and the symmetry properties of a matrix-valued wave equation whose plane-wave solutions satisfy the superluminal (tachyonic) dispersion relation , where E is the energy, is the spatial momentum and m is the mass of the particle. The equation reads (iγ μ ∂ μ − γ 5 m )ψ = 0, where γ 5 is the fifth current. The tachyonic equation is shown to be invariant and invariant. The tachyonic Hamiltonian breaks parity and is non-Hermitian but fulfils the pseudo-Hermitian property , where P is the parity matrix and is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for ) and evanescent waves, which constitute resonances and anti-resonances with complex-conjugate pairs of resonance eigenvalues (for ). In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalue E implies that E *, − E and − E * also constitute resonance energies of H 5 . This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
The complex eigenvalues of some non-Hermitian Hamiltonians, e.g., parity-time-symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possessing one of four certain symmetries, the poles of the S-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.
In this paper, the relativistic Dirac equation in one dimension is investigated for a particle in an external electromagnetic field, with the property of position-dependent effective mass approximation (PDEM), in the absence of vector potential. By removing the lower spinor component and combining the pair of equations, a Schrödinger-like equation is obtained for the upper spinor component. Using canonical transformations and introducing two first-order Hermitian and anti-Hermitian differential operators, a formalism for pseudo-hermitic Hamiltonians with parity-time reversal symmetry (PT) has been obtained. Comparing the equation derived from pseudo-Hermitian Hamiltonian with the non-relativistic Schrödinger equation leads to a general formalism for one-dimensional solvable imaginary non-Hermitian potentials with real energy spectra. Also, using this process, the complex potentials of Pöschl-Teller and Scarf II with real energy spectra in Dirac equation with PDEM approximationand PT symmetry have been investigated and their application has been expressed. For some particular parameters we will see the phenomenon of energy-levels crossing. In fact, it means that energy levels disappear from the spectrum. Also, for the mentioned examples, potential figures are drawn.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.
This work presents a speculative theoretical model for a novel quantum gate inspired by the properties of tachyons, hypothetical particles associated with speeds greater than that of light. The research proposes the Tachyonic Phase Gate (TPG), a quantum gate based on non-Hermitian dynamics and PT symmetry, exploring how a tachyonic field could modify quantum states in ways inaccessible to conventional quantum gates. The article develops a description of the mathematical and conceptual architecture of the TPG, including its dynamic behavior, transitions between different physical phases, and effects on the states of individual qubits and entangled systems. A tachyonic entanglement gate for two qubits is also introduced, and potential connections with phenomena such as closed time-curves, relativistic causality, and advanced computational complexity are analyzed. Furthermore, the work compares TPG with standard quantum computing gates, showing that this approach could formally extend traditional models of quantum operations to non-unitary regimes inspired by imaginary mass physics and PT-symmetric systems. The study emphasizes that tachyons lack experimental evidence and that the proposed framework is explicitly speculative. However, some mathematical elements related to non-Hermitian dynamics and PT symmetry do have experimental support in modern optical systems, making the work a conceptual exploration of possible future extensions of quantum computing and theoretical physics.
We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.
This work is a revised version (Version 2) of a previous paper by the author, presented in a manner intended to emphasize more clearly the connection between its epistemological postulates and the subsequent development of its formalism. Several modifications have been introduced, and new sections have been added. As in the first version, the study concerns the physical interpretation of the imaginary quantities that arise in the formulation of superluminal relativity (including complex coordinates, velocities, angles, charges, and masses). In this framework, the kinematics of tachyons is analyzed, and the nature of the imaginary velocities appearing therein is examined. Furthermore, it is shown that a physical interpretation of the complex quantities (masses and charges) occurring both in quantum mechanics and in superluminal relativity is indeed possible. In this context, the complex nature of the electric charge is investigated, with particular attention to its divisibility. It is demonstrated that singular fractions of the electric charge exist whose squared values coincide with the observed quark charges. Within the same conceptual framework, the physical nature of a complex mass is also explored. It is shown that the imaginary component of such a mass implies that its gravitational field intensity behaves as a wave, propagating with the velocity of the wave associated with the particle itself. It is further demonstrated that the corresponding wavelength coincides with that given by the De Broglie relation, and that the resulting wave equation—second order in both spatial and temporal coordinates—is compatible with the Klein–Gordon wave equation. This constitutes a significant indication of the physical character of the associated wave and suggests an alternative interpretation to the orthodox probabilistic view of quantum mechanics.