Editing Path integral formulation (section)

=== 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>