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1.
A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrödinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found. 相似文献
2.
The dynamical algebra of theq-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a welldefined algebraSU q(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra. 相似文献
3.
Jeeva Anandan 《Foundations of Physics Letters》2002,15(5):415-438
It is argued that there is no evidence for causality as a metaphysical relation in quantum phenomena. The assumptions that
there are no causal laws, but only probabilities for physical processes constrained by symmetries, leads naturally to quantum
mechanics. In particular, an argument is made for why there are probability amplitudes that are complex numbers. This argument
generalizes the Feynman path integral formulation of quantum mechanics to include all possible terms in the action that are
allowed by the symmetries, but only the lowest order terms are observable at the presently accessible energy scales, which
is consistent with observation. The notion of relational reality is introduced in order to give physical meaning to probabilities.
This appears to give rise to a new interpretation of quantum mechanics. 相似文献
4.
We present a unified approach to representations of quantum mechanics on non-commutative spaces with general constant commutators
of the phase-space variables. We find two phases and duality relations among them in arbitrary dimensions. Conditions for
the physical equivalence of different representations of a given system are analyzed. Symmetries and classification of phase
spaces are discussed. Especially, the dynamical symmetry of a physical system is investigated. Finally, we apply our analyses
to the two-dimensional harmonic oscillator and the Landau problem.
Received: 17 December 2002, Published online: 11 June 2003 相似文献
5.
V K B KOTA 《Pramana》2014,82(4):743-755
In this paper, an overview of some aspects of quantum phase transitions (QPT) in nuclei is given and they are: (i) QPT in interacting boson model (sdIBM), (ii) QPT in two-level models, (iii) critical point E(5) and X(5) symmetries, (iv) QPT in a simple solvable model with three-body forces. In addition, some open problems are also given. 相似文献
6.
The two-dimensional Dirac Hamiltonian with equal scalar and vector potentials has been proved commuting with the deformed orbital angular momentum L. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and similarly the harmonic oscillator potential possesses an SU(2) symmetry. The generators of the symmetric groups are derived for these two systems separately. The corresponding energy spectra are yielded naturally from the Casimir operators. Their non-relativistic limits are also discussed. 相似文献
7.
Phase Space is the framework best suited for quantizing superintegrable systems—systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear σ-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the use of Nambu Brackets (linked to Dirac Brackets) involving the extra invariants of superintegrable models. The quantization of Nambu Brackets is then successfully compared to that of Moyal, validating Nambu’s original proposal, while invalidating other proposals. 相似文献
8.
Kishin Mariwalla 《Physics Reports》1975,20(5):287-362
The object of this review is to discuss methods that enable one to trace the origin of symmetries and conservation laws in mechanics to geometrical symmetries of space-time. Starting with the basic Newtonian assumptions on absolute space and time classical mechanics is developed in configuration space and phase space independently together with the related structures such as force-less mechanics. Heuristic considerations on geometric symmetries in configuration space reveal their intimate relation to conservation laws. Using the methods of differential geometry this relationship is put on a formal footing and symmetry groups of all spherically symmetric single term potentials are classified. The method of infinitesimal canonical transformations is presented as an alternative method of deducing dynamical symmetries of an arbitrary system in phase space. These methods also apply to non-relativistic quantum theory. Possible extension to special and general relatively is also discussed. 相似文献
9.
Rupak Chatterjee 《Letters in Mathematical Physics》1996,36(2):117-126
It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique. 相似文献
10.
We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics. 相似文献
11.
Hans Tilgner 《International Journal of Theoretical Physics》1973,7(5):379-388
The Koecher construction of simple symmetric Lie algebras is used to realize colineation and conformai Lie algebras of non-linear transformations of a pseudo-orthogonal vector space in the canonical Weyl algebras, which are used in the Schrödinger representation. The realization maps the linear sub-algebras onto symmetrized polynomials of second degree, whereas the non-linear parts are mapped onto polynomials of first and third degree. For the two examples the Meyberg Jordan algebras are explicitly given. 相似文献
12.
The discrete heat equation is worked out to illustrate the search of symmetries of difference equations. Special attention
it is paid to the Lie structure of these symmetries, as well as to their dependence on the derivative’s discretization. The
case ofq-symmetries for discrete equations in aq-lattice is briefly considered at the end.
Talk delivered by J. Negro at the DI-CRM Workshop held in Prague, 18–21 June 2000.
This work has been partially supported by DGES of the Ministerio de Educación y Cultura of Spain under Projects PB98-0360
and the Junta de Castilla y León (Spain). 相似文献
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15.
Pengming Zhang P. A. Horváthy 《The European Physical Journal C - Particles and Fields》2010,65(3-4):607-614
The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schrödinger group, which also involves, in addition, Schrödinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally invariant relativistic theory, the recently discussed conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit. 相似文献
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New theorems about the existence of solution for a system of infinite linear equations with a Vandermonde type matrix of coefficients are proved. Some examples and applications of these results are shown. In particular, a kind of these systems is solved and applied in the field of the General Relativity Theory of Gravitation. The solution of the system is used to construct a relevant physical representation of certain static and axisymmetric solution of the Einstein vacuum equations. In addition, a newtonian representation of these relativistic solutions is recovered. It is shown as well that there exists a relation between this application and the classical Haussdorff moment problem. 相似文献
19.
In the Hilbert space formulation of classical mechanics, pioneered by Koopman and von Neumann, there are potentially more observables than in the standard approach to classical mechanics. In this Letter, we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the Koopman and von Neumann formulation is extended to include the evolution of differential forms. Because of their noninvariance, those extra observables have to be removed. This removal makes the superposition of states in the Koopman and von Neumann formulation, and as a consequence also in classical mechanics, impossible. 相似文献
20.
Michael Tsamparlis Andronikos Paliathanasis 《General Relativity and Gravitation》2011,43(6):1861-1881
It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion
in a Riemannian space are of the form [(x)\ddot]i+Gjki[(x)\dot]j[(x)\dot] k+f(xi)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x
i
) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the
homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical
system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted
that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection
of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions
involving the vectors and the potential function defining the dynamical system. The general results are applied to a number
of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b)
The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi
type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation
of the Bianchi I metric when the scalar field has an exponential potential. 相似文献