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New Horizons in Mathematical Physics

C in Complex PT-symmetry

Biswanath Rath

Published Nov 16, 2018• 0 citations
Publisher

Abstract

We propose a model PT -symmetry operator(H) in the form of (2x2) matrix and find corresponding charge operator (C) such that it stisfies all the necessary appropriate commutation relations [H,PT]=0;[H,C]=0,[C,PT]=0 and [H,CPT]=0 in order to justify a model in complex quantum system like that to preserve the necessary conditions in Hilbert space.

DOI
10.22606/nhmp.2018.24003

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We propose a model PT -symmetry operator(H) in the form of (2x2) matrix and find corresponding charge operator (C) such that it stisfies all the necessary appropriate commutation relations [H,PT]=0;[H,C]=0,[C,PT]=0 and [H,CPT]=0 in order to justify a model in complex quantum system like that to preserve the necessary conditions in Hilbert space.
If a HamiltonianH has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT . The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT -symmetric Hamiltonian H = 12p + 1 2x 2 + i x using perturbative techniques. It is also shown how to construct the operator C for H = 12p 2 + 12x 2 − x using nonperturbative methods [1].
Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT-symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT-symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.
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.
Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that ${\mathcal{T}}^{2}=\ensuremath{-}1$ for fermionic systems. In PT\mathcal{PT}PT-symmetric quantum mechanics an operator has both PT\mathcal{PT}PT and CPT\mathcal{CPT}CPT adjoints. Fermionic operators $\ensuremath{\eta}$, which are quadratically nilpotent (${\ensuremath{\eta}}^{2}=0$), and algebras with PT\mathcal{PT}PT and CPT\mathcal{CPT}CPT adjoints can be constructed. These algebras obey different anticommutation relations: $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{PT}}+{\ensuremath{\eta}}^{\mathcal{PT}}\ensuremath{\eta}=\ensuremath{-}\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{PT}}$ is the PT\mathcal{PT}PT adjoint of $\ensuremath{\eta}$, and $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{CPT}}+{\ensuremath{\eta}}^{\mathcal{CPT}}\ensuremath{\eta}=\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{CPT}}$ is the CPT\mathcal{CPT}CPT adjoint of $\ensuremath{\eta}$. This paper presents matrix representations for the operator $\ensuremath{\eta}$ and its PT\mathcal{PT}PT and CPT\mathcal{CPT}CPT adjoints in two and four dimensions. A PT\mathcal{PT}PT-symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.