# Quantum Field Theory with (2,2) metric

Does someone know some reference which treats QFT in a space with the following non-Lorentzian signature: $g_{\mu\nu}=\text{diag}(-1,-1,1,1)$. I'm interested in basic stuff like the shape of the scalar propagator and Feynman rules in such a space.

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This doesn't seem like a question in its current form. What specifically do you want to ask about 2+2D QFT? – David Z May 27 '14 at 16:50
– diffeomorphism May 27 '14 at 16:56
Related: physics.stackexchange.com/q/43322/2451 and links therein. – Qmechanic May 27 '14 at 17:47

Meddling with the spacetime dimensions and/or signature is usually much more troubesome than what appears. At wikipedia you can find a rough discussion, but the first thing that comes to mind is that the Huygens principle is only valid in spatial dimensions greater or equal than 3 and odd.

Now, regarding your 2+2 spacetime, the situation is much worse. Let's try the easiest possible field theory, namely the massless scalar field. The equation should be then

$(-\partial_t^2-\partial_u^2+\partial_x^2+\partial_y^2)\phi=0$,

where I named the extra timelike dimension $u$. In the usual field theory in 3+1 we start with the classical solutions of the equation (in this case it would be the just the wave equation), write the general solution as a sum of Fourier terms $\sum_k a_k e^{i(kx-\omega t)}$ and "promote" the coefficients to ladder operators $\hat{a}_k$ and its adjoints.

What about in 2+2? We should try the same. Now, as explained in this answer in our math counterpart, there is a theorem stating that ultrahyperbolic PDEs, like the one that concerns you, are not Hadamard well-posed, meaning that either the solution does not exist, or it's not unique, or it is not stable. Violating the first two conditions invalidate the Fourier expansion, and violating stability means that no meaningful perturbation theory exists, even at classical level. Stability is also a problem regarding initial conditions. Failure of Hadamard well-posedness implies that if you change the boundary conditions by a small value then the full solutions is completely different from the initial one. This invalidades everything about physics, since it makes impossible to deal with the finite accuracy of experiments.

Existence for the free 2+2 is easy from separation of variables, although I cannot say the same for uniqueness since I have not proven, nor seen it proven, completeness of the basis thus obtained. Nevertheless, even if uniqueness could be established we would then certainly violate stability, and then we would not be able to perform perturbation theory as usual.

This are the reasons why it makes no sense to look at 2+2 spacetimes, even if only from a mathematical point of view.

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