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Physical Review A

Generalizations of classical Poisson brackets to include spin

Kuo‐Ho Yang, Joseph O. Hirschfelder

Published Nov 1, 1980• 38 citations
Publisher

Abstract

The classical spin of a particle is essentially its angular momentum of rotation about an axis passing through its center of mass. The classical Poisson brackets are generalized to include spin for any system for which the rate of change of spin is given by $\frac{d\stackrel{\ensuremath{\rightarrow}}{s}}{\mathrm{dt}}=\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{s}\ifmmode\times\else\texttimes\fi{}(\frac{\ensuremath{\partial}H}{\ensuremath{\partial}\stackrel{\ensuremath{\rightarrow}}{s}})$ where $H$ is the Hamiltonian. In a previous paper, we showed that this equation is satisfied by the Breit-Pauli Hamiltonian for a molecular system of electrons and nuclei interacting with an electromagnetic field. This Hamiltonian contains all of the fine-structure terms with the exception of the Lamb shift. Thus, our generalized Poisson brackets should apply to almost all molecular problems. Our generalized Poisson brackets are the same as those which Sudarshan and Mukunda derived by group-theoretic arguments for the direct product of a pure spin group with the group of transformations of generalized coordinates and their conjugate momenta.

DOI
10.1103/physreva.22.1814

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The classical spin of a particle is essentially its angular momentum of rotation about an axis passing through its center of mass. The classical Poisson brackets are generalized to include spin for any system for which the rate of change of spin is given by ds\ensuremath→dt=\ensuremath−s\ensuremath→\ifmmode×\else\texttimes\fi(\ensuremath∂H\ensuremath∂s\ensuremath→)\frac{d\stackrel{\ensuremath{\rightarrow}}{s}}{\mathrm{dt}}=\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{s}\ifmmode\times\else\texttimes\fi{}(\frac{\ensuremath{\partial}H}{\ensuremath{\partial}\stackrel{\ensuremath{\rightarrow}}{s}})dtds\ensuremath→​=\ensuremath−s\ensuremath→\ifmmode×\else\texttimes\fi(\ensuremath∂s\ensuremath→\ensuremath∂H​) where HHH is the Hamiltonian. In a previous paper, we showed that this equation is satisfied by the Breit-Pauli Hamiltonian for a molecular system of electrons and nuclei interacting with an electromagnetic field. This Hamiltonian contains all of the fine-structure terms with the exception of the Lamb shift. Thus, our generalized Poisson brackets should apply to almost all molecular problems. Our generalized Poisson brackets are the same as those which Sudarshan and Mukunda derived by group-theoretic arguments for the direct product of a pure spin group with the group of transformations of generalized coordinates and their conjugate momenta.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
Geometric Quantum Mechanics is a formulation demonstrating how quantum theory may be cast in the language of Hamiltonian phase-space dynamics. Within this framework, the classical properties of spin 1/2, spin 1 and spin 3/2 particles have been studied. The correspondence between the Poisson bracket and commutator algebras for these systems was shown by explicitly computing the value of the commutator of spin operators and comparing it with the Poisson bracket of the corresponding classical observables. This study was extended by comparing the Casimir operator and its classical counterpart. The results showed that there exists a correspondence between classical and quantum Casimir operators at least for the case of spin 1/2. This research clearly shows the limit of classical notions to describe the purely quantum concept.
The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.
A covariant, symplectic, classical dynamical system is presented whose quantization, by replacement of the Poisson brackets with commutators, gives precisely the Dirac electron theory. For the classical system the velocity and momentum are independent dynamical variables, as in Dirac electron; it undergoes a real Zitterbewegung so that the spin is the orbital angular momentum of the Zitterbewegung. Thus the longstanding problem of representing the configuration space of quantum spin by a classical model is solved. Three different forms of the dynamical equations are given.
An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.