# Euclidean QFT definition

I have a question on Euclidean field theories and their relationship with QFT defined on a Minkowski spacetime.

In order to compute the generating function $Z$, one has to compute the integral

$$Z = \int \mathcal D \phi \exp{\left(i\int \mathrm {d}^4x \mathcal L\left[\phi \right]\right)}$$

where for the sake of simplicity I assumed only a scalar field.

Now, quoting my teachers, the convergence of this integral is bad because we have an imaginary exponent, thus an oscillating integral (BAD!).

A possible solution to this is to analytically continue the lagrangian and then rotate the axis of the $\mathrm{dx^0}$ integration in the exponent.

$$\left( x^0, \vec{x}\right) \rightarrow \left( -i\tilde{x}, \vec{\tilde{x}}\right)$$

where this means that i just changed the axis of integration by a $\pi/2$ clockwise rotation.

(This works because we assume that when we rotate the axis of integration, we do not cross any pole ; otherwise the integral, computed via residue theorem, would give us a different result I suppose)

So, after this step, we have to redefine the integral variable to the euclidean variable.

$$\left(\tilde{x}^0, \vec{\tilde{x}}\right) = \left(-ix^4_E, \vec{x}_E\right)$$

Next step is to introduce euclidean fields, defined such that

$$iS\left[ \phi \right] = -S_E\left[ \phi_E \right]$$