Classical and Quantum Field Theories in the Lagrangian Formalism

Phys. Rev. 89, 4 – Published 1 January 1953
Peter G. Bergmann and Ralph Schiller

Abstract

In this paper, we have first examined the relationship between the transformation properties of a (nonquantum) covariant field theory and its constraints, generating functionals, conservation laws, and "superpotentials" purely within the Lagrangian formalism and indicated the relevance of these quantities for the problem of motion of particles (singularities) in the field. This discussion includes a presentation of actual methods of computation of these important quantities suitable for a very wide class of theories. In the second part of the paper, we have discussed the probable structure of a quantum covariant field theory, both in the Hamiltonian and in the Lagrangian formalism. In the Hamiltonian formalism, it is suggested that those field variables canonically conjugate to constraints are not observables in the physical sense nor operators in Hilbert space, and that the states of a system which alone can be regarded as Hilbert vectors are those consistent with all the constraints inherent in the theory and its transformation properties. This approach permits the characterization of legitimate observables even if the isolation of the "constraint variables" is not feasible, as in the general theory of relativity; observables must commute with all the constraints. They are, thus, invariants under the group of invariant transformations. It is asserted that this selection of observables, which is mathematically self-consistent, does not lead to the discard of quantities of physical interest. On the other hand, all the so-called paradoxes between constraints (subsidiary conditions) and commutation relations are thereby avoided. In the development of the Lagrangian quantization, we are proposing a new set of field equations which are different from the usual ones but which can be shown to permit the transition to the canonical formulation if desired. Our proposal is to assert the stationary character of the Feynman-Schwinger action operator not with respect to all, but only with respect to those variations that correspond to invariant transformations. As a result, the number of operator equations at each point of space-time is finite, though it is different from the number of equations in the nonquantum theory. These equations, though considerably weaker than what would be obtained if the action integral were to be made stationary with respect to all conceivable variations, are sufficient to yield all the usual conservation laws and also to permit the transition to the Hamiltonian form of the theory if desired. Commutation relations can be obtained for the field variables and their time derivations on the same hypersurface, simply by requiring that the field variables and their derivatives be algebraically independent of each other. The procedure employed breaks down if applied to a variable that is canonically conjugate to a constraint, an indication that in the Lagrangian formulation, too, the set of observables must be selected if paradoxes are to be avoided. Altogether it appears that the Lagrangian and the Hamiltonian quantizations, if set up properly, are largely equivalent; but this does not preclude the possibility that one may be more useful heuristically than the other.

DOI: http://dx.doi.org/10.1103/PhysRev.89.4

  • Received 12 August 1952
  • Published in the issue dated January 1953

© 1953 The American Physical Society

Authors & Affiliations

Peter G. Bergmann and Ralph Schiller*

  • Department of Physics, Syracuse University, Syracuse, New York

  • *Now at Universidade de Sao Paulo, Brasil.

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