Editing Path integral formulation (section)

== Euclidean path integrals ==
It is very common in path integrals to perform a [[Wick rotation]] from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals.

=== Wick rotation and the Feynman–Kac formula ===
If we replace <math>t</math> by <math>-it</math>, the time-evolution operator <math>e^{-it\hat{H}/\hbar}</math> is replaced by <math>e^{-t\hat{H}/\hbar}</math>. (This change is known as a [[Wick rotation]].) If we repeat the derivation of the path-integral formula in this setting, we obtain<ref>{{harvnb|Hall|2013|loc=Section 20.3.}}</ref>
: <math>\psi(x,t)=\frac{1}{Z}\int_{\mathbf{x}(0)=x} e^{-S_{\mathrm{Euclidean}}(\mathbf{x},\dot{\mathbf{x}})/\hbar}\psi_0(\mathbf{x}(t))\, \mathcal{D}\mathbf{x}\,</math>,
where <math>S_{\mathrm{Euclidean}}</math> is the Euclidean action, given by
: <math>S_{\mathrm{Euclidean}}(\mathbf{x},\dot{\mathbf{x}})=\int\left[ \frac{m}{2}|\dot\mathbf{x}(t)|^2+V(\mathbf{x}(t))\right] \,dt</math>.
Note the sign change between this and the normal action, where the potential energy term is negative. (The term ''Euclidean'' is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.)

Now, the contribution of the kinetic energy to the path integral is as follows:
: <math>\frac{1}{Z}\int_{\mathbf{x}(0)=x} f(\mathbf{x})e^{-\frac{m}{2}\int |\dot\mathbf{x}|^2dt}\, \mathcal{D}\mathbf{x}\,</math>
where <math>f(\mathbf{x})</math> includes all the remaining dependence of the integrand on the path. This integral has a rigorous mathematical interpretation as integration against the [[Wiener process|Wiener measure]], denoted <math>\mu_{x}</math>. The Wiener measure, constructed by [[Norbert Wiener]] gives a rigorous foundation to [[Brownian motion#Einstein.27s theory|Einstein's mathematical model of Brownian motion]]. The subscript <math>x</math> indicates that the measure <math>\mu_x</math> is supported on paths <math>\mathbf{x}</math> with <math>\mathbf{x}(0)=x</math>.

We then have a rigorous version of the Feynman path integral, known as the [[Feynman–Kac formula]]:<ref>{{harvnb|Hall|2013|loc=Theorem 20.3.}}</ref>
: <math>\psi(x,t)=\int e^{-\int V(\mathbf{x}(t))\,dt/\hbar}\,\psi_0(\mathbf{x}(t)) \,d\mu_x(\mathbf{x})</math>,
where now <math>\psi(x,t)</math> satisfies the Wick-rotated version of the Schrödinger equation,
: <math>\hbar \frac{\partial}{\partial t}\psi(x,t) = -\hat H \psi(x,t)</math>.
Although the Wick-rotated Schrödinger equation does not have a direct physical meaning, interesting properties of the Schrödinger operator <math>\hat{H}</math> can be extracted by studying it.<ref>{{harvnb|Simon|1979}}</ref>

Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory.<ref>{{harvnb|Glimm|Jaffe|1981|loc=Chapter 19.}}</ref> On the other hand, it is much more difficult to give a meaning to path integrals (even Euclidean path integrals) in quantum field theory than in quantum mechanics.<ref group=nb>For a brief account of the origins of these difficulties, see {{harvnb|Hall|2013|loc=Section 20.6.}}</ref>

=== 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.