Path Integral Subtleties

Mid-point Rule

Table of Contents:

The derivation of the path integral for a Hamiltonian with a velocity-dependent potential, such as the

(p-a)

magnetic interaction-^-—, may be more subtle where A is the vector potential. The discretized ver-

sion requires a mid-point rule for the Hamiltonian. Evaluation along some space-time path q{t) starting from the point (l t,, t, + e) then

where, the path satisfies the requirement

The paths are continuous and the velocity q(f), associated to them is infinite as

This subtle nature of the paths may be observed for a particle in a magnetic field described by the Lagrangian:

Here, e, c, A and v=q are, respectively, the electronic charge, speed of light, vector potential and velocity of the particle. For the correct quantization of the Lagrangian 4.53, the finite N discretized form of the vector potential contribution can be written:

where A is evaluated at the mid-point of the interval. The principal objective is the selection of an appropriate point where the variables are evaluated.

Lagrangian Formulation of Path Integration

We represent path integration 4.47 in an alternative form convenient in various applications and physically instructive and motivated by the observation that 4.47 imitates the Hamiltonian formulation of classical mechanics. Since, classically, the Hamiltonian and Lagrangian mechanics can be equally employed to describe dynamical evolution, then it is instructive to find a Lagrangian analogue of 4.47. Focusing on Hamiltonians where the dynamics is free then the kinetic energy dependence is quadratic in p and so, the Lagrangian formulation of path integration can be inferred from 4.47 by Gaussian integration.

The development of the transition amplitude via path integration as a new space-time approach to non-relativistic quantum mechanics was developed by Richard Feynman in his 1942 PhD Thesis [5] in the mid-1940s, following a hint from an earlier paper by Dirac [7,13]. Apparently, Dirac’s motivation was to formulate quantum mechanics from the Lagrangian rather than the Hamiltonian formulation of classical mechanics. So, making an attempt to understand the transition amplitude in a purely quantum mechanical sense then for the replacement

the classical Hamiltonian function in 4.47 has the following form in the generic case:

Here, U(q) is the potential energy associated with the particle.

Our goal now is to integrate over the momenta in 4.47, withholding only integrations over coordinates with the aid of expression 4.43. Transforming the finite multiple integral over the momenta with the i^,h momentum integral takes the form where exponent of the integral is quadratic in the momentum variable p (integral is Gaussian in p):

Complex Gaussian Integral

The integral on the right-hand side of 4.57 may be easily evaluated via the standard Gaussian integral:

This integral in equation 4.57 is an (extended) Fresnel integral: or

It should be noted that the real parameters a can take any sign.

The root of a complex variable z is understood as the principal value. This implies the branch of the two-valued square root function that has a positive real part:

So, the integrand in 4.61, however, is imaginary and hence the necessity to introduce a supplementary procedure for its evaluation which should be the analytic continuation of the integral 4.61 by a rotation

of the integration variable through the angle -y in the complex plane. This rotation transforms the integral into Gaussian form in 4.59 that may be easily evaluated. The rotation of the integration variable through у in the complex plane yields the same result. From this analytic continuation, we arrive at the result from integrating out the momenta only: