PT-symmetry, invariance under the combined operations of spatial reflection and time reversal, arises naturally in non-Hermitian physics and has been attracting an increasing amount of attention in the literature. Here, we examine probabilistic structures within the unbroken and broken PT-symmetric regimes from a C*-algebraic perspective. We show that PT-unbroken probability is directly compatible with the operator-algebraic formulation of standard QM. However, though PT-broken Hamiltonians may generate well-defined non-unitary dynamics, basic structural properties required for quantum probability typically fail. We analyze this failure via relation to the GNS construction, ultimately demonstrating that an extension to a C*-algebraic probability theory on a Hermitian supersystem is required to define a meaningful probability calculus within the PT-broken system itself. Thereby, consistent probability assignment must be extrinsic to the PT-broken phase, and must be inherited from a dilation to a Hermitian system. We offer a novel interpretation of the broken PT-symmetric regime as a non-ordered, non-probabilistic subalgebra of a larger Hermitian, fully quantum system. Additionally, our work clarifies the structural importance of positivity in standard quantum mechanics.
PT-symmetry, invariance under the combined operations of spatial reflection and time reversal, arises naturally in non-Hermitian physics and has been attracting an increasing amount of attention in the literature. Here, we examine probabilistic structures within the unbroken and broken PT-symmetric regimes from a C*-algebraic perspective. We show that PT-unbroken probability is directly compatible with the operator-algebraic formulation of standard QM. However, though PT-broken Hamiltonians may generate well-defined non-unitary dynamics, basic structural properties required for quantum probability typically fail. We analyze this failure via relation to the GNS construction, ultimately demonstrating that an extension to a C*-algebraic probability theory on a Hermitian supersystem is required to define a meaningful probability calculus within the PT-broken system itself. Thereby, consistent probability assignment must be extrinsic to the PT-broken phase, and must be inherited from a dilation to a Hermitian system. We offer a novel interpretation of the broken PT-symmetric regime as a non-ordered, non-probabilistic subalgebra of a larger Hermitian, fully quantum system. Additionally, our work clarifies the structural importance of positivity in standard quantum mechanics.
Abstract PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.
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 A quantum mechanical system with unbroken super-and parity-time (PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (A PT) conjugation to redefine a new version of the inner product without any additional considerations. This PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.
We consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Poschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
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 (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-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. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C 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 PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.