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On local representations of rotations on discrete configuration spaces

Marcel Reginatto, Michael J. W. Hall

A quantum mechanical spin-half system may be characterized abstractly as having a set of two-valued observables which generate infinitesimal rotations in three dimensions. We consider a concrete realization of such a two-level system within the formalism of ensembles on configuration space, an approach which is not only capable of describing quantum mechanical systems but allows also for theories that are generalizations of quantum theory. Such a spin-half system may be called a rotational bit or robit, to distinguish it from the standard quantum qubit. After reviewing ensembles on configuration space and examining the example of constructing representations of the Galilean Lie algebra for the free particle, we construct probabilistic models for ensembles that consist of one and two spin-half systems. In the case of a single spin-half system, there are two main requirements: the configuration space must be a discrete set, labeling the outcomes of two-valued spin observables, and these observables must provide an algebraic representation of so (3). The case of a pair of spin-half systems is more complicated, in that additional physical requirements concerning locality and subsystem independence must also be taken into account and now the observables must provide an algebraic representation of so (3) ⊕ so (3). We compare the resulting theories to the corresponding quantum mechanical systems.A quantum mechanical spin-half system may be characterized abstractly as having a set of two-valued observables which generate infinitesimal rotations in three dimensions. We consider a concrete realization of such a two-level system within the formalism of ensembles on configuration space, an approach which is not only capable of describing quantum mechanical systems but allows also for theories that are generalizations of quantum theory. Such a spin-half system may be called a rotational bit or robit, to distinguish it from the standard quantum qubit. After reviewing ensembles on configuration space and examining the example of constructing representations of the Galilean Lie algebra for the free particle, we construct probabilistic models for ensembles that consist of one and two spin-half systems. In the case of a single spin-half system, there are two main requirements: the configuration space must be a discrete set, labeling the outcomes of two-valued spin observables, and these observables must provide an algebraic representation of so (3). The case of a pair of spin-half systems is more complicated, in that additional physical requirements concerning locality and subsystem independence must also be taken into account and now the observables must provide an algebraic representation of so (3) ⊕ so (3). We compare the resulting theories to the corresponding quantum mechanical systems.

Journal of Physics Conference SeriesAug 1, 20180 citations

Representations of the Space Groups

Nikolai Borisovich Melnikov, Boris Il'ich Reser

Springer eBooksJan 1, 20220 citations

Quantum Chemistry Density Matrix Renormalization Group in the Discrete Variable Representation

Bing Gu, Jiajun Ren, Junzhe Zhang

We present a numerical implementation of the quantum chemistry density matrix renormalization group (DMRG) using the hybrid discrete variable representation (DVR)/Gaussian basis set. The z -axis of real space is discretized by a DVR basis set and each transversal plane is described by the eigenstates of the transversal core Hamiltonian, represented in a set of primitive Gaussian basis functions. Such a hybrid basis can reduce the computation of two-electron repulsion integrals. One main advantage of using the local DVR basis sets over finite-difference methods for real space discretization is that the kinetic energy operator matrix elements can be computed exactly. This method is first applied to a one-dimensional pseudo-hydrogen chain under screened Coulomb potential and then to a realistic hydrogen chain, whereby the DMRG/DVR results are comparable in accuracy to the full configuration interaction results.We present a numerical implementation of the quantum chemistry density matrix renormalization group (DMRG) using the hybrid discrete variable representation (DVR)/Gaussian basis set. The z -axis of real space is discretized by a DVR basis set and each transversal plane is described by the eigenstates of the transversal core Hamiltonian, represented in a set of primitive Gaussian basis functions. Such a hybrid basis can reduce the computation of two-electron repulsion integrals. One main advantage of using the local DVR basis sets over finite-difference methods for real space discretization is that the kinetic energy operator matrix elements can be computed exactly. This method is first applied to a one-dimensional pseudo-hydrogen chain under screened Coulomb potential and then to a realistic hydrogen chain, whereby the DMRG/DVR results are comparable in accuracy to the full configuration interaction results.

Journal of Chemical Theory and ComputationJul 2, 20250 citations

On the theory of collective rotations

Shinzo Nakai, Muneo Sasaki

Nuclear PhysicsMay 1, 19612 citations

Symmetries in Discrete-Time Mechanics

Mohammad Khorrami

Annals of PhysicsDec 1, 19960 citations

Quantum chaos in the Wigner representation

G. P. Berman, A. R. Kolovskii, Felix M. Izrailev +1 more

The dynamics of a familiar model of stochastic behavior-the quantum kicked rotator-is analyzed in the Wigner representation. Exact nonlocal maps defined on a discrete phase space are derived. The basic dynamics of a quantum kicked rotator can be described satisfactorily by means of a simplified map that incorporates only the discrete nature of the phase space.

Chaos An Interdisciplinary Journal of Nonlinear ScienceAug 1, 19915 citations

Local Representations of Rotations on Discrete Configuration Spaces

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