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  • American Institute of Physics (AIP)  (7)
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  • 1
    Electronic Resource
    Electronic Resource
    Ray representations of N(≤2)+1-dimensional Galilean group (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 3210-3215 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: For a quantum mechanical system, modeled via unitary or antiunitary representations of a symmetry group G with Lie algebra g, the Hamiltonian H is of special interest. If H is an element of g or of the universal enveloping algebra U(g), a generic time dependence for any element in U(g) is given through the Heisenberg picture. As an example we consider a system with Gal(N) as symmetry group with H as one of the generators. For N≥3 one gets from ray representations the free Schrödinger equation. For N=1 a peculiarity occurs: the "free'' equation has an interaction term, which results from the construction and parametrization of unitary ray representations. For N=2 there is another special feature: there exist ray representations of the universal covering group Gal(2)(large-closed-square), which induce no ray representations of Gal(2). Furthermore, for these representations it is not possible to construct a Schrödinger equation. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531026
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  • 2
    Electronic Resource
    Electronic Resource
    Quantum logic as a consequence of realistic measurements on deterministic systems (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 250-253 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown why, from the deterministic point of view, quantum logic and not classical logic is the appropriate language for describing realistic measurements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529127
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  • 3
    Electronic Resource
    Electronic Resource
    The stationary KdV hierarchy and so(2,1) as a spectrum generating algebra (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 4995-5003 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The family FLκ of all potentials V(x) for which the Hamiltonian H=−d2/dx2+V(x) in one space dimension possesses a high-order Lie symmetry is determined. A subfamily FSGA(2) of FLκ, which contains a class of potentials allowing a realization of so(2,1) as spectrum generating algebra of H through differential operators of finite order, is identified. Furthermore and surprisingly, the families FSGA(2) and FLκ are shown to be related to the stationary KdV hierarchy. Hence, the "harmless" Hamiltonian H connects different mathematical objects: high-order Lie symmetry, the realization of so(2,1)-spectrum generating algebra and families of nonlinear differential equations. We describe in a physical context the interplay between these objects. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533011
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  • 4
    Electronic Resource
    Electronic Resource
    Vector bundles over configuration spaces of nonidentical particles: Topological potentials and internal degrees of freedom (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 3943-3952 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We consider configuration spaces of nonidentical pointlike particles. The physically motivated assumption that any two particles cannot be located at the same point in space–time leads to nontrivial topological structure of the configuration space. For a quantum mechanical description of such a system, we classify complex vector bundles over the configuration space and obtain potentials of topological origin, similar to those that occur in the fiber bundle approach to Dirac's magnetic monopole or in Yang–Mills theory. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532101
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  • 5
    Electronic Resource
    Electronic Resource
    Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 49-63 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532786
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  • 6
    Electronic Resource
    Electronic Resource
    On topological effects in quantum mechanics: The harmonic oscillator in the pointed plane (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 1053-1059 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold M as its configuration space is based on a quantization method, which, in general, reflects global properties of M, i.e., some of the observables of S will "feel'' the topology: There are topological effects and inequivalent quantizations on M. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold M, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on M, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528375
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  • 7
    Electronic Resource
    Electronic Resource
    Boson operator realizations of su(2) and su(1,1) and unitarization (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 594-600 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Boson operator realizations of su(2) and su(1,1) are obtained. Scalar products are introduced on "Fock spaces'' (Verma modules) spanned by generators of the Heisenberg algebra H and by generators of su(2). These scalar products unitarize certain of the representations of H, or of su(1,1). It is shown that the Gel'fand–Dyson realization of su(1,1) implies a scalar product that unitarizes H, while the Primakoff–Holstein realizations imply a scalar product that unitarizes su(1,1). The relationship between the Gel'fand–Dyson boson operators a° and the Primakoff–Holstein boson operators b° is obtained making use of the two distinct scalar products. Generalized "vacuum states'' are defined that are formed by polynomials in the creation–annihilation operator pairs a°a. A representation ρ of H and su(1,1) on the states a°m and an is discussed. For this representation a1=a||0〉≠0, but rather (a°a)||0〉 =0. The states of this representation space consist of boson states and boson-hole states. All the familiar results of H and su(1,1) representation theory are preserved within the representation ρ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528424
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