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.Comment: 14 pages. Submitted to J Phys A special issue on The Physics of Non-Hermitian Operator

Constraints to the sustainability of a ‘systematised’ approach to livestock marketing amongst smallholder cattle producers in South Africa

Commercialization of smallholder agriculture in South Africa is underpinned by reforms to improve livestock off-take in communal areas and engage smallholder farmers with formal markets. To achieve this, Custom Feeding Programmes (CFPs) were established to improve the condition of communal cattle prior to their sale into formal markets and to ‘systematise’ the informal marketing of cattle in communal areas by enabling participants to achieve higher informal market prices. We evaluate the sustainability of eight CFPs located in Eastern Cape Province in terms of their ability to add value to smallholder cattle production and encourage market participation. Communities with CFPs achieved a 16.6% mean cattle off-take rate, substantially higher than in most communal systems. Furthermore, cattle sold through CFPs attained a 17% higher mean selling price than those sold through other marketing channels. However, these benefits were mainly realized by better-off farmers with larger cattle herds and greater ability to transport animals to and from CFPs. More marginalized farmers, particularly women, had low participation. CFPs also face challenges to their sustainability, including inconsistent feed and water supplies, poor infrastructure and high staff turnover. Key to enhancing participation in CFPs, will be improving the way they are supported and embedded within communities

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 Heisenberg's 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.Comment: 11 pages, Late

Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem

In a previous paper (arXiv:math-ph/0604055) we introduced a very simple PT-symmetric non-Hermitian Hamiltonian with real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this note we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction -- based on a backward use of the spectral theorem for self-adjoint operators -- provides new insights into the nature of the model.Comment: LaTeX, 6 page

Exact PT-Symmetry Is Equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite-dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field theories.Comment: Few typos have been corrected and a reference update

Coherent States on Hilbert Modules

We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over $C^*$-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert $C^*$-modules which have a natural left action from another $C^*$-algebra say, $\mathcal A$. The coherent states are well defined in this case and they behave well with respect to the left action by $\mathcal A$. Certain classical objects like the Cuntz algebra are related to specific examples of coherent states. Finally we show that coherent states on modules give rise to a completely positive kernel between two $C^*$-algebras, in complete analogy to the Hilbert space situation. Related to this there is a dilation result for positive operator valued measures, in the sense of Naimark. A number of examples are worked out to illustrate the theory

Testing earthquake links in Mexico from 1978 to the 2017 M = 8.1 Chiapas and M = 7.1 Puebla Shocks

The M = 8.1 Chiapas and the M = 7.1 Puebla earthquakes occurred in the bending part of the subducting Cocos plate 11 days and ~600 km apart, a range that puts them well outside the typical aftershock zone. We find this to be a relatively common occurrence in Mexico, with 14% of M > 7.0 earthquakes since 1900 striking more than 300 km apart and within a 2 week interval, not different from a randomized catalog. We calculate the triggering potential caused by crustal stress redistribution from large subduction earthquakes over the last 40 years. There is no evidence that static stress transfer or dynamic triggering from the 8 September Chiapas earthquake promoted the 19 September earthquake. Both recent earthquakes were promoted by past thrust events instead, including delayed afterslip from the 2012 M = 7.5 Oaxaca earthquake. A repeated pattern of shallow thrust events promoting deep intraslab earthquakes is observed over the past 40 years

Crypto-Harmonic Oscillator in Higher Dimensions: Classical and Quantum Aspects

We study complexified Harmonic Oscillator models in two and three dimensions. Our work is a generalization of the work of Smilga \cite{sm} who initiated the study of these Crypto-gauge invariant models that can be related to $PT$-symmetric models. We show that rotational symmetry in higher spatial dimensions naturally introduces more constraints, (in contrast to \cite{sm} where one deals with a single constraint), with a much richer constraint structure. Some common as well as distinct features in the study of the same Crypto-oscillator in different dimensions are revealed. We also quantize the two dimensional Crypto-oscillator.Comment: 17 pages, Latex, enlarges version, added ref.s., accepted in J.Phys.A, slight alteration in reference section and text, matches journal versio

Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations

We use the theory of pseudo-Hermitian operators to address the problem of the construction and classification of positive-definite invariant inner-products on the space of solutions of a Klein-Gordon type evolution equation. This involves dealing with the peculiarities of formulating a unitary quantum dynamics in a Hilbert space with a time-dependent inner product. We apply our general results to obtain possible Hilbert space structures on the solution space of the equation of motion for a classical simple harmonic oscillator, a free Klein-Gordon equation, and the Wheeler-DeWitt equation for the FRW-massive-real-scalar-field models.Comment: 29 pages, slightly revised version, accepted for publication in Class. Quantum Gra

Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the $P$-unitary evolution of a physical system are also given.Comment: 20 page