Wick rotation

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Assuming the metric is \eta_{\mu\nu} = \operatorname{diag}(1,-1,-1,-1...)\,, then define the Euclidean vector \ell_E\,:

\ell^0_E = i\ell\,,
\vec{\ell}_E=\vec{\ell}.

This has the effect of changing the metric to g_{\mu\nu} = \operatorname{diag}(-1,-1,-1,-1...)\,. This over-all sign may normally be factored out of expressions, so that, effectively, the metric becomes \delta_{\mu\nu} = \operatorname{diag}(1,1,1,1...)\,.

[edit] Example

Consider the action for a free scalar field:

S = \int\!d^dx\, \frac{1}{2}\eta_{\mu\nu} \partial_\mu \phi \partial_\nu \phi - \frac{1}{2}m^2 \phi^2\,

Then d^dx = -i d^dx_E\,, so that

S_E\, = -i \int\!d^dx_E\, \frac{1}{2} (-\delta_{\mu\nu}) \partial_\mu \phi \partial_\nu \phi - \frac{1}{2}m^2 \phi^2\,,
= i \int\!d^dx_E\, \frac{1}{2} \delta_{\mu\nu} \partial_\mu \phi \partial_\nu \phi + \frac{1}{2}m^2 \phi^2\,,
= i \int\!d^dx_E\, \frac{1}{2} \partial_\mu \phi \partial_\mu \phi + \frac{1}{2}m^2 \phi^2\,.

In the last line, both indices are lowered to remind the reader that he or she is working in Euclidean space, where raising or lowering indices has no effect. This usage is, however, not universal, since contracting raised and lowered indices is a helpful book-keeping device.

The effect, and indeed the purpose of this Wick rotation is to make the path integral better defined:

Z = \int\!\mathcal{D}\phi\, e^{i S[\phi]} \to \int\!\mathcal{D}\phi\, e^{- S_E[\phi]}\,,

where now we define the Wick rotated action to be

S_E = \int\!d^dx_E\, \frac{1}{2} \partial_\mu \phi \partial_\mu \phi + \frac{1}{2}m^2 \phi^2\,.

Note also that the naïve propagator

\Delta(p) = \frac{i}{p^2 - m^2},

is replaced by

\Delta_E(p) = \frac{i}{p^2 + m^2},

which has no poles along the real axis, so that it's Fourier transform is uniquely defined.

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