The action principle in quantum mechanics

Abstract

The Euler-Lagrange equation derived from Schwinger's action principle (1951) has been shown by Kianget al. (1969) and Linet al. (1970) to lead to inconsistencies for quadratic lagrangians of the form $$\bar L(\dot q,q) = \tfrac{1}{2}\dot q^j g_{jk} (q)\dot q^k - V(q)$$ except in the Euclidean caseg jk =δ jk . This inadequacy is linked to Schwinger's specification that the variations of operators bec-numbers. We reformulate the action principle by introducing the concept of ‘proper’ Gauteaux variation of operators to find the most general class of admissible variation consistent with the postulated quantisation rules. This new action principle, applied to the LagrangianL, yields a quantum Euler equation consistent with the Hamilton-Heisenberg equations.