Derivation of Schwinger variational principles

Alexander L. Zubarev

C. L. Dolph, Robert K. Ritt

W. Lucht

G�nter Sauermann, H. Turschner

T. W. B. Kibble, John Charlton Polkinghorne

Abstract The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a number of points in relation to Schwinger’s quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger’s formalism. It is also found to be possible to obtain Green’s generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.

M. Z. Shaharir

The c-number variation in the quantum-mechanical action principle of Schwinger is extended to a q-number variation which uses an action integral analogous to the classical modified Hamilton action integral. An explicit realization of the admissible q-number variation for the hamiltonian operator, H, given by H(q,p,t)= 1 / 2 p j g jk (q)p k + 1 / 2 (A j (q),p j )+W(q,t) is discussed in terms of Gauteaux variation. The action principle yields the correct Hamilton-Heisenberg equations and a relationship between the lagrangian L and the hamiltonian H. It is also shown that while the fundamental commutation relation is successfully derived by Schwinger in his c-number variational principle, the same argument cannot be used in the present q-number variation. A new method of quantization is suggested.