Elements of a non-Hermitian quantum theory without Hermitian conjugation - scalar product and scattering
Frieder Kleefeld
备注：
6 pages, no figures, contribution to the Proceedings of the workshop on "Unquenched Hadron Spectroscopy: Non-Perturbative Models and Methods of QCD vs. Experiment (EEF70)" (1-5 September, 2014, Univ. of Coimbra, Coimbra, Portugal)
The descripition of in a Hermitian setting seemingly nonlocal and nonperturbative phenomena like confinement or superconductivity is most conveniently performed by generalizing quantum theory to a non-Hermitian regime where these phenomena appear perturbative and local. The short presentation provides a clue how this can be done on the basis of Lorentz covariance while preserving the analyticity of the theory. After deriving with the help of Lorentz covariance a quantum scalar product without making any use of metric or complex conjugation we sketch how the formalism of scattering theory can be extended analytically to a non-Hermitian regime.
Quantumly controlled measurement, Hermitian conjugation and normalization in matrix-manipulation algorithms
Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS（俄罗斯科学院化学物理与药物化学联邦问题研究中心）；Institute of Quantum Computing and Computer Theory, School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机科学与工程学院量子计算与计算机理论研究所）；School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
Non-Hermitian random matrices have been utilized in such diverse fields as dissipative and stochastic processes, mesoscopic physics, nuclear physics, and neural networks. However, the only known universal level-spacing statistics is that of the Ginibre ensemble characterized by complex-conjugation symmetry. Here we report our discovery of two other distinct universality classes characterized by transposition symmetry. We find that transposition symmetry alters repulsive interactions between two neighboring eigenvalues and deforms their spacing distribution. Such alteration is not possible with…
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We present a large class of non-Hermitian non-PT-symmetric two-component Dirac Hamiltoninas with real energy spectra. These Hamiltonians are invariant under the combined action of "charge" conjugation (two-component transpose) and space-parity. Examples are given from the two subclasses of these systems having localized and/or continuum states with real energies.
We demonstrate the extension to PT-symmetric field theories of the Goldstone theorem, confirming that the spontaneous appearance of a field vacuum expectation value via minimisation of the effective potential in a non-Hermitian model is accompanied by a massless scalar boson. Laying a basis for our analysis, we first show how the conventional quantisation of the path-integral formulation of quantum field theory can be extended consistently to a non-Hermitian model by considering PT conjugation instead of Hermitian conjugation. The extension of the Goldstone theorem to a PT-symmetric field theo…
We develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity a…
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Symmetries associated with complex conjugation and Hermitian conjugation, such as time-reversal symmetry and pseudo-Hermiticity, have great impact on eigenvalue spectra of non-Hermitian random matrices. Here, we show that time-reversal symmetry and pseudo-Hermiticity lead to universal level statistics of non-Hermitian random matrices on and around the real axis. From the extensive numerical calculations of large random matrices, we obtain the five universal level-spacing and level-spacing-ratio distributions of real eigenvalues, each of which is unique to the symmetry class. Furthermore, we an…
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This is an introduction to antilinear operators. In following E.P.Wigner the terminus "antilinear" is used as it is standard in Physics. Mathematicians prefer to say "conjugate linear". By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case. Basics of antilinearity is explained in sections 2, 3,4, 7 and in subsection 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators, the Hermitian adjoint, classification of antilinear normal oper…
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The recently introduced two- and three-parameter ($p,q$)- and ($p,q,μ$)-deformed extensions of the Heisenberg algebra were explored under the condition of their connectedness with the respective nonstandard (other than known ones) deformed quantum oscillator algebras. In this paper we show that such connection dictates certain new $η(N)$-pseudo-Hermitian conjugation rule between the creation and annihilation operators, with $η(N)$ depending on the particle number operator $N$. In turn, that leads to the related $η(N)$-pseudo-Hermiticity of the position/momentum operators, though the involved H…
A class of non-Hermitian quadratic su(2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realisations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a s…
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians $H=p^2+x^2(ix)^ν$ with $ν\in(-2,\infty)$ that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of $H$ for the cases that $ν$ is an even integer, so that $H=p^2\pm x^{2+ν}$, and give a proof of the discreteness of the spectrum of $H$ for all $ν\in(-2,\infty)$. Furthermore, we study the consequences of defining a square-well Hamilton…
An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric …
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We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creati…
A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric $θ$ to another, more involved but correct positive-definite physical metric $Θ$. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerge…
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The work contains a detailed investigation of free neutral (Hermitian) or charged (non-Hermitian) scalar fields and the describing them (system of) Klein-Gordon equation(s) in momentum picture of motion. A form of the field equation(s) in terms of creation and annihilation operators is derived. An analysis of the (anti-)commutation relations on its base is presented. The concept of the vacuum and the evolution of state vectors are discussed.
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We describe recent progress in understanding the continuous symmetry properties of non-Hermitian, PT-symmetric quantum field theories. Focussing on a simple non-Hermitian theory composed of one complex scalar and one complex pseudoscalar, we revisit the derivation of Noether's theorem to show that the conserved currents of non-Hermitian theories correspond to transformations that do not leave the Lagrangian invariant. We illustrate the impact that this has on the consistent formulation of (Abelian) gauge theories by studying a non-Hermitian extension of scalar quantum electrodynamics. We consi…
Recent years have seen a fascinating pollination of ideas from quantum theories to elastodynamics---a theory that phenomenologically describes the time-dependent macroscopic response of materials. Here, we open route to transfer additional tools from non-Hermitian quantum mechanics. We begin by identifying the differences and similarities between the one-dimensional elastodynamics equation and the time-independent Schrodinger equation, and finding the condition under which the two are equivalent. Subsequently, we demonstrate the application of the non-Hermitian perturbation theory to determine…
We present an intuitive and scalable algorithm for the diagonalization of complex symmetric matrices, which arise from the projection of pseudo--Hermitian and complex scaled Hamiltonians onto a suitable basis set of "trial" states. The algorithm diagonalizes complex and symmetric (non--Hermitian) matrices and is easily implemented in modern computer languages. It is based on generalized Householder transformations and relies on iterative similarity transformations T -> T' = Q^T T Q, where Q is a complex and orthogonal, but not unitary, matrix, i.e, Q^T equals Q^(-1) but Q^+ is different from Q…
We investigate Non-Hermitian quantum field theoretic model with $ιgϕ^3$ interaction in 6 dimension. Such a model is PT-symmetric for the pseudo scalar field $ϕ$. We analytically calculate the 2-loop $β$ function and analyse the system using renormalization group technique. Behavior of the system is studied near the different fixed points. Unlike $gϕ^3$ theory in 6 dimension $ιgϕ^3$ theory develops a new non trivial fixed point which is energetically stable.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-compon…
Covariant formulation of the Berry connection in non-Hermitian systems
Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University（帕拉茨基大学光学联合实验室与捷克科学院物理研究所，理学院，帕拉茨基大学）
We study $(1+1)$ dimensional Dirac equation with non Hermitian interactions, but real energies. In particular, we analyze the pseudoscalar and scalar interactions in detail, illustrating our observations with some examples. We also show that the relevant hidden symmetry of the Dirac equation with such an interaction is pseudo supersymmetry.
A series of recent papers ``Faster than Hermitian Quantum Mechanics'' and related articles made a point of the possibility of a non-Hermitian, but PT-symmetric, operator to play the role of a Hamiltonian. In particular, they show that with an appropriate choice of an inner product, the evolution generated by such an operator will conserve the norm and scalar product. Here we observe that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian, and the whole exercise is to a certain degree redundant.
Motivated by the existence of bi-Hamiltonian classical systems and the correspondence principle, in this paper we analyze the problem of finding Hermitian scalar products which turn a given flow on a Hilbert space into a unitary one. We show how different invariant Hermitian scalar products give rise to different descriptions of a quantum system in the Ehrenfest and Heisenberg picture.
