Path integral formulation: Difference between revisions

Revision as of 12:36, 9 October 2005

Revision as of 17:35, 29 October 2005

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is the [[action (physics)|action]] of the classical problem in which one investigates the path starting at time t=0 and ending at time t=T, and Dx denotes integration over all paths. In the classical limit, <math>\hbar\to0</math>, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.

The connection with [[statistical mechanics]] follows. Perform the [[Wick rotation]] t&rarr;it, iei.e., make time imaginary. Then the path integral resembles the [[partition function (statistical mechanics)|partition function]] of statistical mechanics defined in a canonical ensemble with temperature <math>1/T\hbar</math>.

Clearly, such a deep analogy between [[quantum mechanics]] and [[statistical mechanics]] cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by