Editing Path integral formulation (section)

=== Path integral and the partition function ===

The path integral is just the generalization of the integral above to all quantum mechanical problems—
: <math>Z = \int  e^\frac{i\mathcal{S}[\mathbf{x}]}{\hbar}\, \mathcal{D}\mathbf{x} \quad\text{where }\mathcal{S}[\mathbf{x}]=\int_0^{t_f} L[\mathbf{x}(t),\dot\mathbf{x}(t)]\, dt</math>
is the [[action (physics)|action]] of the classical problem in which one investigates the path starting at time {{math|''t'' {{=}} 0}} and ending at time {{math|''t'' {{=}} t<sub>f</sub>}}, and <math>\mathcal{D}\mathbf{x}</math> denotes the integration measure over all paths. In the classical limit, <math>\mathcal{S}[\mathbf{x}]\gg\hbar</math>, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.<ref name="Feynman-Hibbs">{{harvnb|Feynman|Hibbs|Styer|2010|pp=29–31}}</ref>

The connection with [[statistical mechanics]] follows. Considering only paths that begin and end in the same configuration, perform the [[Wick rotation]] {{math|''it'' {{=}} ''ħβ''}}, i.e., make time imaginary, and integrate over all possible beginning-ending configurations. The Wick-rotated path integral—described in the previous subsection, with the ordinary action replaced by its "Euclidean" counterpart—now resembles the [[partition function (statistical mechanics)|partition function]] of statistical mechanics defined in a [[canonical ensemble]] with inverse temperature proportional to imaginary time, {{math|{{sfrac|1|''T''}} {{=}} {{sfrac|i''k''<sub>B</sub>''t''|''ħ''}}}}. Strictly speaking, though, this is the partition function for a [[statistical field theory]].

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
: <math>|\alpha;t\rangle=e^{-\frac{iHt}{\hbar}}|\alpha;0\rangle</math>
where the state {{mvar|α}} is evolved from time {{math|''t'' {{=}} 0}}. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time {{mvar|iβ}} is given by
: <math>Z = \operatorname{Tr} \left[e^{-H\beta}\right]</math>
which is precisely the partition function of statistical mechanics for the same system at the temperature quoted earlier. One aspect of this equivalence was also known to [[Erwin Schrödinger]] who remarked that the equation named after him looked like the [[diffusion equation]] after Wick rotation. Note, however, that the Euclidean path integral is actually in the form of a ''classical'' statistical mechanics model.