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Pramana

Collective field in Hamiltonian and Lagrangian (path-integral) approach: An application to O(N) Heisenberg spin operator

Arunabha Guha

Published Feb 1, 1982• 0 citations
Publisher
DOI
10.1007/bf02847985

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The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.
Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ eff which depends on additional noncommutativity.