Journal of Physics B Atomic and Molecular Physics

Correspondence identities: I

A Norcliffe, Ian Colin Percival

Published Sep 1, 1968• 21 citations

Abstract

The complete quantum-dynamical solutions for a free particle moving in one dimension and for a free particle moving on a ring are obtained in terms of classical path sums. The propagators and Green operator kernels, and, in particular, the kernel of the spectral operator I E = δ( E - H ) are expressed explicitly in this way. Relations between quantum and classical mechanics which hold for low quantum numbers and relatively small values of the classical action are discussed. They are the correspondence identities.

DOI: 10.1088/0022-3700/1/5/303

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Journal of Physics B Atomic and Molecular Physics

Correspondence identities: I

A Norcliffe, Ian Colin Percival

Published Sep 1, 1968• 21 citations

Publisher

Abstract

DOI: 10.1088/0022-3700/1/5/303

Related papers

Correspondence identities. V. The zero-energy Coulomb problem

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For Pt. IV see abstr. A41769 of 1969. The classical and quantal problem of a charged particle moving in an attractive Coulomb potential with total energy zero is considered. A complete correspondence identity is obtained by expressing the kernel of the quantum-mechanical spectral operator IE= delta (E-H) exactly in terms of the classical action function in momentum representation. Use is made of the dynamical symmetry properties exhibited by both the classical and quantal problems, and a path-integral method of evaluating the spectral operator kernel is also presented.For Pt. IV see abstr. A41769 of 1969. The classical and quantal problem of a charged particle moving in an attractive Coulomb potential with total energy zero is considered. A complete correspondence identity is obtained by expressing the kernel of the quantum-mechanical spectral operator IE= delta (E-H) exactly in terms of the classical action function in momentum representation. Use is made of the dynamical symmetry properties exhibited by both the classical and quantal problems, and a path-integral method of evaluating the spectral operator kernel is also presented.

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Physical Review AFeb 1, 198863 citations

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Detlev Buchholz A, Klaus Fredenhagen

Abstract The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its invariance properties under variations of the action. These relations determine a dynamical algebra of bounded operators which encodes all properties of the corresponding quantum theory. This novel approach is applied to non-relativistic particles, where quantum mechanics emerges from it. The method works also in interacting quantum field theories and sheds new light on the foundations of quantum physics.Abstract The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its invariance properties under variations of the action. These relations determine a dynamical algebra of bounded operators which encodes all properties of the corresponding quantum theory. This novel approach is applied to non-relativistic particles, where quantum mechanics emerges from it. The method works also in interacting quantum field theories and sheds new light on the foundations of quantum physics.

arXiv:1905.04250Foundations of PhysicsMay 4, 20204 citations

{\mathit{q}}_{\mathit{f}}

{\mathit{p}}_{\mathit{f}}

)/\ensuremath{\partial}(

{\mathit{q}}_{\mathit{i}}

{\mathit{p}}_{\mathit{i}}

)], where (

{\mathit{q}}_{\mathit{i}}

{\mathit{p}}_{)}

and (

{q}_{f}

{p}_{f}

) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.