The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

https://iopscience.iop.org/article/10.1088/0034-4885/70/6/R03/pdf

Making sense of non-Hermitian Hamiltonians

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

The homotopy perturbation method for nonlinear oscillators with discontinuities

Comparison of homotopy perturbation method and homotopy analysis method

In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving) To express the Hermiticity of a Hamiltonian, one writes H = H †, where the symbol † denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space – time reflection symmetry (&#x1d4ab;&#x1d4af; symmetry) without losing any of the essential physical features of quantum mechanics Theories defined by non-Hermitian &#x1d4ab;&#x1d4af;-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian &#x1d4ab;&#x1d4af;-symmetric quantum theories

Introduction to PT-symmetric quantum theory

A recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the Lagrangian such as φ4 by (φ2)1+δ and then treating δ as a small parameter. It is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the Lane–Emden, Thomas–Fermi, Blasius, and Duffing equations.

A new perturbative approach to nonlinear problems

A novel perturbative technique for solving quantum field theory is proposed. In this paper we explore this scheme in the context of self-interacting scalar field theory. For a phi/sup 2//sup p/ theory the method consists of expanding a phi/sup 2(1+//sup delta//sup )/ theory in powers of delta. A diagrammatic procedure for computing the terms in this series is given. We believe that for any Green's function the radius of convergence of this series is finite and is, in fact, 1. Moreover, while the terms in the unrenormalized series are individually divergent, they are considerably less so than in the standard weak-coupling perturbation series. In simple, low-dimensional quantum-field-theory models, the delta expansion gives excellent numerical results. We hope this new technique will ultimately shed some light on the question of whether a (phi/sup 4/)/sub 4/ theory is free.

Novel Perturbative Scheme in Quantum Field Theory

We study two continuum methods of regulating the formal strong-coupling expansion of the Green's functions, obtained by expanding the path integral in powers of the kinetic energy (inverse free propagator). Our continuum regulations amount to introducing either a hard ($\ensuremath{\theta}$ function) or soft (Gaussian) cutoff $\ensuremath{\Lambda}$ in momentum space. The cutoff takes the place of the usual spatial cutoff, the lattice spacing, which arises when the path integral is defined as the continuum limit of ordinary integrals on a Euclidean space-time lattice. We find, by investigating free field theory and $g{\ensuremath{\varphi}}^{4}$ field theory in one dimension, that the $\ensuremath{\theta}$-function regulation is more accurate than the Gaussian and, unlike the Gaussian, preserves certain continuum Green's-function identities. The extension to field theories with fermions is trivial and we give the strong-coupling graphical rules for an arbitrary field theory with fermions and bosons in $d$ dimensions.

Continuum regulation of the strong-coupling expansion for quantum field theory

Using the method of finite elements we investigate the quantum behavior of a particle starting in an unstable equilibrium at the top of a potential hill and rolling down. In order to study the numerical accuracy of the method for large times we consider the exactly solvable model with V(q) = -1/2q/sup 2/ and an initial Gaussian wave function at t = 0, psi(q)proportionalexp(-1/2q/sup 2/) so that the initial state Vertical Bar0> has = = 1/2. We study the accuracy of the large-time approximations to based upon single finite elements of degree n. The Taylor series is exact up to t/sup 2n/ and even the coefficient of t/sup 2n/+2 is a very accurate representation of the exact coefficient. We then consider the convergence properties of the (N,n) approximations consisting of N iterations of the finite element of degree n. For these approximations the corrections to the Taylor-series coefficients in higher orders vanish as N/sup -2n/.

Quantum roll: A study of the long-time behavior of the finite-element method.

The operator difference equations that arise in the finite-element treatment of a quantum theory are implicit and therefore difficult to solve. By introducing a matrix formulation it is possible to circumvent the implicit character of these equations and obtain explicit closed-form solutions for arbitrary matrix elements of any operator.

L. M. Simmons

Papers

A new perturbative approach to nonlinear problems

Novel Perturbative Scheme in Quantum Field Theory

Continuum regulation of the strong-coupling expansion for quantum field theory

Quantum roll: A study of the long-time behavior of the finite-element method.

Matrix methods in discrete-time quantum mechanics.