Lagrangian quantum theory

Abstract

We present a consistent Lagrangian quantum theory for one degree of freedom based on the Fréchet derivative. We consider a quantum mechanical system with one independent coordinate q and Lagrangian L(q, q dot ) = a(q) q dot 2 + b(q) q dot + c(q) . In a consistent quantum theory, the canonical equal-time commutation relation (c.c.r.), [q, q dot ] = i/2a(q) must be preserved in time; that is to say [q, q ̈ ] = (i/2a) . Now the acceleration q ̈ is given in terms of q and q dot by the Euler-Lagrange equation; the precise form of this depends on which variations δq are assumed to leave the action stationary. We show that the constraint that q ̈ preserve the c.c.r. restricts the class of allowable variations of q for which the action integral is stationary to functions of the form δq(q,t) = g(t)(a(q)) −1 2 where g(t) is a c-number function which vanishes at the end-point of the time interval. This specification retains its form under a change of variable q → q( r). Thus unless the coefficient a( q) is a constant, c-number variations are not permissible. This means that the Euler-Lagrange equation of motion is not the naive form with c-number variations which is usually assumed. Noether's theorem, which relates the invariance of a Lagrangian under an infinitesimal coordinate transformation to an associated conserved quantity, only holds if the infinitesimal transformation is an allowable variation, that is to say one for which the action is stationary. It is proved that only variations of the form a − 1 2 can leave the Lagrangian invariant, so that the possibility of an invariance of L unaccompanied by a conserved quantity is removed. A Hamiltonian H(q, q dot ) is given which generates the time transformations given by the Euler-Lagrange equation. Finally, the allowability condition is shown to be the condition for which rearrangement of terms in the Lagrangian by means of the c.c.r. does not alter the ultimate Euler-Lagrange equation of motion.