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Register a new userMoyal products -- a new perspective on quasi-hermitian quantum mechanics
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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.
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Bell suggested that a new perspective on quantum mechanics was needed. We propose a solution of the measurement problem based on a reconsideration of the nature of particles. The solution is presented with an idealized model involving non-locality or non-separability, identified in 1927 by Einstein and implicit in the standard interpretation of single slit (or hole) diffraction. Considering particles as being localizable entities leads to an `induced collapse model, a parameter-free alternative to spontaneous collapse models, that affords a new perspective on, emph{inter alia}, nuclear decay.
In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dysons work, both technically and contextually. Here we recast Dysons work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly Hermitian. Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity.
Using the notions of frame transform and of square integrable projective representation of a locally compact group $G$, we introduce a class of isometries (tight frame transforms) from the space of Hilbert-Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group $Gtimes G$. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable star product which mimics, at the level of functions, the original product of operators. A phase space formulation of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.
Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be $eta$-pseudo-Hermitian with Hermitian, positive-definite and non-singular $eta$ operators. We show that despite the fact that an $eta$ operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is singular, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the help of the spectral decomposition.
$mathcal{PT}$-symmetric quantum mechanics has been considered an important theoretical framework for understanding physical phenomena in $mathcal{PT}$-symmetric systems, with a number of $mathcal{PT}$-symmetry related applications. This line of research was made possible by the introduction of a time-independent metric operator to redefine the inner product of a Hilbert space. To treat the dynamics of generic non-Hermitian systems under equal footing, we advocate in this work the use of a time-dependent metric operator for the inner-product between time-evolving states. This treatment makes it possible to always interpret the dynamics of arbitrary (finite-dimensional) non-Hermitian systems in the framework of time-dependent $mathcal{PT}$-symmetric quantum mechanics, with unitary time evolution, real eigenvalues of an energy observable, and quantum measurement postulate all restored. Our work sheds new lights on generic non-Hermitian systems and spontaneous $mathcal{PT}$-symmetry breaking in particular. We also illustrate possible applications of our formulation with well-known examples in quantum thermodynamics.
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