Wick rotation, scalar field and invariants

1. Nov 3, 2012

Melsophos

One point about Wick rotation is puzzling me and I can not find explanations in books. It concerns the invariants formed from scalar product and solutions to equation. So I will expose my way of reasoning to let you see if it is correct and at the end ask more specific questions.

$$ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = - dt^2 + d\vec x^2$$

Now if we go to euclidean time
$$t = -i\tau$$
(the sign being chosen to have an exponential decay: $e^{-iHt}\to e^{-H\tau}$) then the metric becomes euclidean
$$ds^2 = \delta_{\mu\nu} dx^\mu dx^\nu = dt^2 + d\vec x^2$$
and
$$S = \int d^d x \; L = -i \int d^d x_E \; L = i \int d^d x_E \; L_E = i S_E$$
so that
$$S = i S_E, \quad L = - L_E, \quad Z = \int d\phi\; e^{iS} = \int d\phi\; e^{-S_E}$$

Now take the scalar field lagrangian
$$L = -\frac{1}{2} \Big((\partial^\mu \phi)^2 + m^2 \phi^2 \Big) - V(\phi)$$
This gives the usual Klein-Gordon equation with potential:
$$(-\Delta + m^2) \phi = V'(\phi)$$
Plugging plane-waves into the free equation ($V(\phi) = 0$) gives the mass-shell condition
$$p^2 = -m^2$$
The Green function is (without care about epsilon factors)
$$G(p) = \frac{1}{p^2 + m^2}$$

Applying the Wick rotation gives
$$L_E = \frac{1}{2} \Big((\partial_E^\mu \phi)^2 + m^2 \phi^2 \Big) + V(\phi)$$
and the equation
$$(-\Delta_E + m^2) \phi = -V'(\phi)$$
Fourier transform gives the Green function
$$G_E(p) = \frac{1}{p_E^2 + m^2}$$
which have no poles since $p_E^2 \ge 0$. This just says that plane-waves are not eigenfunctions of the Laplacian (but instead exponentials; which seems to agree with the fact that euclidean has a better behavior – and so there is no waves propagating in euclidean space).

Now my problem is with the previous mass-shell condition: if we Wick rotate it, then it becomes
$$p_E^2 = -m^2$$
which seems to be in disagreement with the fact that scalar product in euclidean space is positive-definite. So does this mean that we should give up this relation and totally forget about it? And in this case, does this happen also to all invariant constructed from spacelike vectors? At a first glance I would have change the sign for timelike scalars, but then this gives the Helmholtz equation with plane-wave solutions and this reintroduces the pole in Green function, so it does not seem to be a good solution.
The fact that the mass-shell does not hold anymore seems in agreement that plane-waves are not eigenfunctions.