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Hadronic J.; (United States)

Remarks on the Problematic Aspects of Heisenberg-lie Symplectic Formulations

Ruggero María Santilli

Published Feb 1, 1980• 3 citations
OpenAlex

Abstract

A number of problematic aspects of conventional quantum mechanical formulations have been recently focused. A few rudimentary remarks are presented in the hope of contributing toward a more adequate identification of the open problems, as a prerequisite for their future resolution.

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Abstract This paper contains the skeleton of one lecture, and of two raporteur's interventions, made at the Sanibel Symposium in February–March, 1979. The task asked of this lecturer was to present an outline of the program of nonequilibrium statistical mechanics, and to indicate some of the recent developments in our understanding of this program. Section I lists a few of the principal physical ingredients of the theory. The possible implementation of certain aspects of the program is demonstrated in Sec. II with the help of three exactly soluble models. Finally in Sec. III we extrapolate from these particular examples in two directions: the stochastic aspect of statistical mechanics, and some remarks on the theory of the quantum measurement process. The presence, at the Symposium, of several distinguished lecturers who presented their own original, and most interesting, contributions hopefully made appear less outrageous—at least at the time—the severe omissions due to the personal choices this particular speaker had to make to keep within the time limitations inherent to such a gathering. It is now hoped that, even when taken out of the full context of the Symposium, the rearranged text which follows might nevertheless help focusing attention on some of the important problems remaining to be solved. It is a pleasure to acknowledge here the stimulating hospitality of the Symposium Scientific Director, Professor P.‐O. Löwdin.

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On the Symplectic Propagation of the Spin-MInt Algorithm for Non-Adiabatic Quantum Dynamics

James R. Rampton, Lauren E. Cook, Timothy J. H. Hele

Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general $K$ electronic states. A review of the symplectic nature of coadjoint orbits of the $\mathfrak{su}(K)$ Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition $\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}$ exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.

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Errata: Lie-Group Dynamical Formalism and the Relation between Quantum Mechanics and Classical Mechanics

Thomas F. Jordan, E. C. G. Sudarshan

Reviews of Modern Physics•Apr 1, 1962•2 citations
Abstract This paper contains the skeleton of one lecture, and of two raporteur's interventions, made at the Sanibel Symposium in February–March, 1979. The task asked of this lecturer was to present an outline of the program of nonequilibrium statistical mechanics, and to indicate some of the recent developments in our understanding of this program. Section I lists a few of the principal physical ingredients of the theory. The possible implementation of certain aspects of the program is demonstrated in Sec. II with the help of three exactly soluble models. Finally in Sec. III we extrapolate from these particular examples in two directions: the stochastic aspect of statistical mechanics, and some remarks on the theory of the quantum measurement process. The presence, at the Symposium, of several distinguished lecturers who presented their own original, and most interesting, contributions hopefully made appear less outrageous—at least at the time—the severe omissions due to the personal choices this particular speaker had to make to keep within the time limitations inherent to such a gathering. It is now hoped that, even when taken out of the full context of the Symposium, the rearranged text which follows might nevertheless help focusing attention on some of the important problems remaining to be solved. It is a pleasure to acknowledge here the stimulating hospitality of the Symposium Scientific Director, Professor P.‐O. Löwdin.
At the 1927 Solvay conference Born and Heisenberg presented a joint report on quantum mechanics. I suggest that the significance of this report lies in that it contains a ‘final’ formulation of the statistical interpretation of quantum mechanics that goes beyond Born’s origi-nal proposal. In particular, this formulation imports elements from Heisenberg’s work as well as from the transformation theory of Dirac and Jordan. I suggest further a reading of Born and Heisenberg’s position in which the wave function is an effective notion. This can make sense of a remarkable aspect of their presentation, namely the fact that the ‘quantum mechanics’ of Born and Heisenberg apparently lacks wave function collapse.
Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general KKK electronic states. A review of the symplectic nature of coadjoint orbits of the su(K)\mathfrak{su}(K)su(K) Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition MJMT=J\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}MJMT=J exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.