F.G. Scholtz

Quasi-Hermitian operators in quantum mechanics and the variational principle

We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. This involves the construction of a metric (if it exists) for the given set of non-Hermitian observables. We discuss uniqueness of this metric. We also show that it is not always necessary to construct the metric for the whole set of observables under consideration, but that it is sufficient for some calculational purposes to construct it for a subset only, even though this metric is, in general, not unique. The restricted metric turns out to be particularly useful in the implementation of a variational principle, which we also formulate.

Formulation, interpretation and application of non-commutative quantum mechanics

In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert?Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on positive operator valued measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non-commutativity are identified.

Twisted Galilean symmetry and the Pauli principle at low energies

We show the twisted Galilean invariance of the noncommutative parameter, even in the presence of spacetime noncommutativity. We then obtain the deformed algebra of the Schrodinger field in configuration and momentum space by studying the action of the twisted Galilean group on the non-relativistic limit of the Klein–Gordon field. Using this deformed algebra we compute the two-particle correlation function to study the possible extent to which the previously proposed violation of the Pauli principle may impact at low energies. It is concluded that any possible effect is probably well beyond detection at current energies.

Observation of exceptional points in electronic circuits

Two damped coupled oscillators have been used to demonstrate the occurrence of exceptional points in a purely classical system. The implementation was achieved with electronic circuits in the kHz-range. The experimental results perfectly match the mathematical predictions at the exceptional point. A discussion about the universal occurrence of exceptional points—connecting dissipation with spatial orientation—concludes this paper.

Path-integral action of a particle in the noncommutative plane

Noncommutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on noncommutative configuration space. Taking this as a departure point, we formulate a coherent state approach to the path-integral representation of the transition amplitude. From this we derive an action for a particle moving in the noncommutative plane and in the presence of an arbitrary potential. We find that this action is nonlocal in time. However, this nonlocality can be removed by introducing an auxilary field, which leads to a second class constrained system that yields the noncommutative Heisenberg algebra upon quantization. Using this action, the propagator of the free particle and harmonic oscillator are computed explicitly.

Spectrum of the non-commutative spherical well

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular, we discuss the infinite and finite non-commutative spherical wells in two dimensions. Using this, bound states and scattering can be discussed unambiguously. Here we focus on the infinite well and solution for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.

A unifying perspective on the Moyal and Voros products and their physical meanings

The Moyal and Voros formulations of non-commutative quantum field theory have been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular, we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective, the two formulations are therefore completely equivalent, but we also argue that only the Voros formulation admits a consistent physical interpretation. These considerations are elucidated by considering the free-particle transition amplitude in the two representations.

Strings from position-dependent noncommutativity

We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative space-time relations taken here to have position dependent structure constants. Some of the new variables are non-Hermitian in the most natural choice. We construct their Hermitian counterparts by means of a Dyson map, which also serves to introduce a new metric operator. We propose PTlike symmetries, i.e.antilinear involutory maps, respected by these deformations. We compute minimal lengths and momenta arising in this space from generalized versions of Heisenbergs uncertainty relations and find that any object in this two dimensional space is string like, i.e.having a fundamental length in one direction beyond which a resolution is impossible. Subsequently we formulate and partly solve some simple models in these new variables, the free particle, its PT-symmetric deformations and the harmonic oscillator.

Noncommutative quantum mechanics - a perspective on structure and spatial extent

We explore the notion of spatial extent and structure, already alluded to in earlier literature, within the formulation of quantum mechanics on the noncommutative plane. Introducing the notion of position and its measurement in the sense of a weak measurement (positive operator-valued measure), we find two equivalent pictures in a position representation: a constrained local description in position containing additional degrees of freedom and an unconstrained nonlocal description in terms of the position without any other degrees of freedom. Both these descriptions have a corresponding classical theory which shows that the concept of extended, structured objects emerges quite naturally and unavoidably there. It is explicitly demonstrated that the conserved energy and angular momentum contain corrections to those of a point particle. We argue that these notions also extend naturally to the quantum level. The local description is found to be the most convenient as it manifestly displays additional information about the structure of quantum states that is more subtly encoded in the nonlocal, unconstrained description. Subsequently, we use this picture to discuss the free particle and harmonic oscillator as examples.

Moyal products—a new perspective on quasi-Hermitian quantum mechanics

The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-Hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-Hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.