Path integral formulation: Difference between revisions

Revision as of 09:44, 11 November 2021

Revision as of 16:00, 30 June 2022

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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{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''t_f}}, 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 which begin and end in the same configuration, perform the [[Wick rotation]] {{math|''it'' {{=}} ''τħβ''}}, 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''_B''τt''|''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

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:<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|iTiβ}} is given by

: <math>Z = \operatorname{Tr} \left[e^\frac{-HT}{H\hbarbeta}\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.

== Quantum field theory ==