Title: Introduction to 𝒫𝒯-symmetric quantum theory
Abstract: Abstract 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 (𝒫𝒯 symmetry) without losing any of the essential physical features of quantum mechanics. Theories defined by non-Hermitian 𝒫𝒯-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 𝒫𝒯-symmetric quantum theories. Acknowledgments I am grateful to D. Hook for his careful reading of the manuscript. I thank the Theoretical Physics Group at Imperial College for its hospitality and the UK Engineering and Physical Sciences Research Council, the John Simon Guggenheim Foundation and the US Department of Energy for financial support. Notes *Permanent address: Department of Physics, Washington University, St. Louis, MO 63130, USA †Other examples of complex Hamiltonians having 𝒫𝒯 symmetry are H = [pcirc]2 + ◯4(i◯)δ, H = [pcirc]2 + ◯6(i◯)δ, and so on (see [Citation7]). These classes of Hamiltonians are all different. For example, the Hamiltonian obtained by continuing H in (5) along the path δ:0→8 has a different spectrum from the Hamiltonian that is obtained by continuing H = [pcirc]2 + ◯6(i◯)δ along the path δ:0→4. This is because the boundary conditions on the eigenfunctions are different. ‡An important technical issue concerns the definition of the operator (i◯) N when N is noninteger. This operator is defined in coordinate space and is used in the Schrödinger equation Hφ = Eφ, which reads −φ(x) + (ix)Nφ (x) = Eφ(x). The term (ix)N ≡ exp [N log (ix)] uses the complex logarithm function log (ix), which is defined with a branch cut that runs up the imaginary axis in the complex-x plane. This is explained more fully in section 3. §If a system is defined by an equation that possesses a discrete symmetry, the solution to this equation need not exhibit that symmetry. For example, the differential equation ÿ(t) = y(t) is symmetric under time reversal t→ – t. The solutions y(t) = exp(t) and y(t) = exp( – t) do not exhibit time-reversal symmetry while the solution y(t) = cosh (t) is time-reversal symmetric. The same is true of a system whose Hamiltonian is 𝒫𝒯-symmetric. Even if the Schrödinger equation and corresponding boundary conditions are 𝒫𝒯 symmetric, the solution to the Schrödinger equation boundary value problem may not be symmetric under space – time reflection. When the solution exhibits 𝒫𝒯 symmetry, we say that the 𝒫𝒯 symmetry is unbroken. Conversely, if the solution does not possess 𝒫𝒯 symmetry, we say that the 𝒫𝒯 symmetry is broken. †If a function satisfies a linear ordinary differential equation, then the function is analytic wherever the coefficient functions of the differential equation are analytic. The Schrödinger Equationequation (10) is linear and its coefficients are analytic except for a branch cut at the origin; this branch cut can be taken to run up the imaginary axis. We choose the integration contour for the inner product (18) so that it does not cross the positive imaginary axis. Path independence occurs because the integrand of the inner product (18) is a product of analytic functions.