It is easy to evaluate the green's function using path integral approach by evaluating classical action and using functional calculus method. Is it possible to evaluate path integral for harmonic oscillator directly by evaluating the integral for every time slice up to the last fixed time slice ? It is cumbersome yet I think it is possible. Is there any reference in which one solve the integral starting from the initial time slice to the final time slice?
It depends on how much you care about proving convergence and such nuisances. It is already done, sort of, at the very end of the original Rev. Mod. Phys. article by Feynman. But without any intermediate steps.
One problem of general nature occurs if you insist working directly with real time differences. It is simpler to do the integral with imaginary time, and do an analytic continuation to real time in the final answer. You will then discover that there is a branch cut along the real-time axis, which has to be resolved regardless of quantum mechanical approach being used.
Another problem is the overall normalisation of the integral. That more-or-less has to be put in by hand, to make the real time limit of the result consistent with unitarity of evolution.
A third, solvable, problem is to find the exact result of N iterated integrals, for arbitrary N and time slice Δt. I assume you want to work with some non-zero (arbitrary small) time-slice Δt^*. Then you must parametrise your variables cleverly in terms of (hyperbolic) trigonometric functions, so that you can make use of trigonometric addition formulas to build up your answer. The form of the result given by Feynman shows the appropriate parametrisation.
^* If you want to work directly with continuous time you will have to expand the paths into a trigonometric series, and integrate over the expansion coefficients)
thank you So much
//If you want to work directly with continuous time you will have to expand the paths into a trigonometric series, and integrate over the expansion coefficients//
Is there any reference for this method...?? Can you please suggest something for the above explanation??
Vigneshwaran> Is there any reference for this method...??
It is discussed at the link below; you will have to fill in some steps in the calculation yourself. The Fourier series mentioned in the link means a series of sine-functions.
https://en.wikipedia.org/wiki/Path_integral_formulation#Simple_harmonic_oscillator
Of course. It's in any textbook in quantum field theory and statistical mechanics-it's done in Drouffe and Itzykson, ``Statistical field theory'', for instance.
The most direct way is to work on the (one-dimensional) lattice, find the eigenvalues of the quadratic form, that defines the action, by Fourier transform, and compute the product.
(Another homework assignment.)
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