Foundations of Physics Letters

, Volume 2, Issue 4, pp 371–382 | Cite as

An approach to quanitzation of constrained systems

  • M. S. Marinov
Article

Abstract

The dynamics of a classical system, specified by a HamiltonianH0(q,p) and a number of constraint functionsϕ k (q,p) in Euclidean phase space (q,p), is described by means of a new HamiltonianH(q,p), which is an invariant of the (closed) Poisson-bracket Lie algebra (H0,ϕ k ). Fixed values ofϕ k (not necessarily zero) are given by initial conditions, and they are conserved along the trajectories determined by the Hamilton equations. The quantization is performed by the standard Heisenberg commutation relations in the embedding phase space, while the constraint functions are put in correspondence with constraint operators\(\hat \phi _k\) which generate the Lie algebra of quantum commutators. A subset of commuting constraint operators may be chosen to have certain values in the initial state; and as soon as the Hamiltonian operatorĤ is an invariant of the Lie algebra, these conditions are maintained permanently. Simple examples are presented. Systems with both Bose and Fermi degrees of freedom (and constraints) can be treated universally.

Key words

quantum theory constraints classical mechanics Lie algebras 

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Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • M. S. Marinov
    • 1
    • 2
  1. 1.The Racah Institute of PhysicsHebrew UniversityJerusalemIsrael
  2. 2.NorditaCopenhagenDenmark

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