Nuno Costa Dias, Joao Nuno Prata
We discuss a recent method proposed by Kryukov and Walton to address boundary-value problems in the context of deformation quantization. We compare their method with our own approach and establish a connection between the two formalisms.
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In the paper quant-ph/0503212, Barone and Halayel-Neto (BH) claim that charge quantization in quantum mechanics can be proven without the need for the existence of magnetic monopoles. In this paper it is argued that their claim is untrue.
In Phys. Rev. A 70, 032104 (2004), M. Montesinos and G. F. Torres del Castillo consider various symplectic structures on the classical phase space of the two-dimensional isotropic harmonic oscillator. Using Dirac's quantization condition, the authors investigate how these alternative symplectic forms affect this system's quantization. They claim that these symplectic structures result in mutually inequivalent quantum theories. In fact, we show here that there exists a unitary map between the two representation spaces so that the various quantizations are equivalent.
The structure of Pais-Uhlenbeck oscillator in the equal-frequency limit has been recently studied by Mannheim and Davidson [Phys.Rev. A71 (2005), 042110]. It appears that taking this limit, as presented in the above paper, is quite subtle and the resulting structure of space of states - involved. In order to clarify the situation we present here the proper way of taking the equal-frequency limit, first under the assumption that the scalar product in the space of states is positive defined. We discuss also the case of indefinite metric space of states. We show that, irrespective of the way the …
We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called $\*$-genvalue (``stargenvalue'') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the…
In the comment to the article by J.Baez and K.Krasnov (hep-th/9703112) are discussed some topics related with application of certain constructions to non-trivial principal bundles.
In deformation quantization (a.k.a. the Wigner-Weyl-Moyal formulation of quantum mechanics), we consider a single quantum particle moving freely in one dimension, except for the presence of one infinite potential wall. Dias and Prata pointed out that, surprisingly, its stationary-state Wigner function does not obey the naive equation of motion, i.e. the naive stargenvalue (*-genvalue) equation. We review our recent work on this problem, that treats the infinite wall as the limit of a Liouville potential. Also included are some new results: (i) we show explicitly that the Wigner-Weyl transform …
From Newton's equations to the Schrödinger equation: domain walls as the quantization condition
本文提出一种伴生粒子介质，其牛顿动力学在粗粒化极限下涌现出薛定谔方程，通过二元极性提供量子化条件，并重现盒子、谐振子及双缝干涉现象。
There is no conclusive theoretical argument that would require the quantization of the gravitational field. A paradox presented by Mari et al. [Sci. Rep. 6, 22777 (2016)] and its resolution by Belenchia et al. [Phys. Rev. D 98, 126009 (2018)] did not change anything about this assertion. Despite being interesting in other respects, efforts of a recent work by Ryding, Aurell, and Pikovski [arXiv:2107.07514] to further establish the inconclusiveness of the question of necessity of quantization are futile.
Bragg diffraction is comparable to a hard-wall reflection if the Bragg condition is exactly fulfilled. However, in a neutron interferometer in the gravitational field (COW experiment) this is not the case and the momentum transfers should not be described by hard wall reflection, as it has been done in the commented article. Instead, and quite similar to the atom case, each mirror or beam splitter creates a constant momentum transfer given by the reciprocal lattice vector of the Bragg crystal. To lowest order there are no differences between the atom and the neutron case.
Normally, the half-harmonic oscillator is active when $x>0$ and absent when $x<0$. From a canonical quantization perspective, this leads to odd eigenfunctions being present while even eigenfunctions are absent. In that case, only the usual odd eigenfunctions will appear if the wall slides to negative infinity. However, if an affine quantization is used, sliding the wall away shows that all the odd and even eigenfunctions are encountered, exactly like any full-harmonic oscillator. We provide numerical support for this.
A recent paper [J. Math. Phys. {\bf 59}, 082105 (2018)] constructs a Hamiltonian for the (dissipative) damped harmonic oscillator. We point out that non-Hermiticity of this Hamiltonian has been ignored to find real discrete eigenvalues which are actually non-real. We emphasize that non-Hermiticity in Hamiltonian is crucial and it is a quantal signature of dissipation.
The nonlinear quantization of the domain wall (relativistic membrane of codimension 1) is considered. The membrane dust equation is considered as an analogue of the Hamilton-Jacobi equation, which allows us to construct its quantum analogue. The resulting equation has the form of a nonlinear Klein-Fock-Gordon equation. It can be interpreted as the mean field approximation for a quantum domain wall. Dispersion relations are obtained for small perturbations (in a linear approximation). The group speed of perturbations does not exceed the speed of light. For perturbations propagating along the do…
This is a response to a recently reported comment [1] on paper [J. Math. Phys.59, 082105 (2018)] regarding the quantization of damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of [2] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be firstly achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can …
In (quant-ph/0701141) Rajeev studied quantization of the damped simple harmonic oscillator and introduced a complex-valued Hamiltonian (which is normal). In this note we point out that the quantization is interpreted as a quantum mechanics with {\bf complex time}. We also present a problem on quantization of classical control systems.
We reconsider the problem of quantising a particle on the $D$-dimensional sphere. Adopting a Lagrangian method of reducing the degrees of freedom, the quantum Hamiltonian is found to be the usual Schrödinger operator without any boundary term. The equivalence with the Dirac Hamiltonian approach is demonstrated, either in the cartesian or in the curvilinear basis. We also briefly comment on the path integral approach.
This note is a somewhat-lighthearted comment on a recent paper by David Wallace, arXiv:0906.2718[quant-ph] entitled "A formal proof of the Born rule from decision-theoretic assumptions".
We discuss the representations of the algebra of quantization, the canonical commutation relations, in a scalar quantum field theory with spontaneously broken U(1) internal symmetry, when a topological defect of the vortex type is formed via the condensation of Nambu-Goldstone particles. We find that the usual thermodynamic limit is not necessary in order to have the inequivalent representations needed for the existence of physically disjoint phases of the system. This is a new type of inequivalence, due to the nontrivial topological structure of the phase space, that appears at finite volume.…
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