Lagrangian method and Feynman path integrals

Philip R. Wallace

G. E. Hahne

A calculation is presented that shows that Feynman's path integral implies Ostrogradsky's Hamiltonian for nonsingular Lagrangians with second derivatives. The procedure employs the stationary phase approximation to obtain the limiting change of the wave function per unit time. By way of introduction, the method is applied anew to the case of nonsingular Lagrangians with only first derivatives, but not necessarily quadratic in the velocities. A byproduct of the calculation is an alternate derivation of the Legendre transformation of taking general classical Lagrangians into Hamiltonians. In both the first and second derivative cases, the outcome contains precisely the classical Hamiltonian, which represents the so-called "symbol" of a (not necessarily Hermitean) pseudodifferential operator acting on the wave function at an instant of time. The derivation herein argues for a claim that Feynman's method starts with a classical Lagrangian and ends with a classical Hamiltonian---nonclassical operator-ordering prescriptions in the passage from classical to quantum Hamiltonians require external input, and are generally not inherent in Feynman's path integral formalism.

Peter Woit

Donald H. Kobe

Gauge invariance in both quantum and classical mechanics is investigated using the Feynman path integral formulation of quantum mechanics. The Feynman path integral expresses the wave function in terms of the Lagrangian. A gauge transformation in classical mechanics is the addition of a total derivative of a gauge function to the Lagrangian. A gauge transformation of the second and first kind in quantum mechanics is obtained by adding and subtracting, respectively, a total derivative to the Lagrangian in the Feynman path integral.

Branko G Dragovich, Lj. Nesic

The Feynman path integral method plays even more important role in p-adic and adelic quantum mechanics than in ordinary quantum theory. p-Adic path integral is defined as a natural generalization of the standard one. Here 1 , we give a brief review of our recent investigations on this subject. In particular, the path integrals for two-dimensional models with quadratic Lagrangians are evaluated.