Covariant gradient expansion


Due to the relevance of the argument, after a nice discussion with a contribution of Carl Brannen, I decided to pursue this matter further. Indeed, the only way to have a covariant formulation of a gradient expansion is adding a time variable and taking the true time variable Wick rotated. In this way, for d=1+1 wave equation you will use d=2+1 wave equation and so on. In d=3+1 you will use d=4+1 wave equation. Let me explain with some equations what I mean. I consider again d=1+1 case as


but, instead to apply a gradient expansion to it, I apply this to the equation


being \Delta_2 = \partial_{xx}+\partial_{yy}. As usual, I rescale time variable as t\rightarrow\sqrt{\lambda}t and I take a solution series


Now I will get the set of equations




and so on. Let us note that, in this case, we can introduce two new spatial variables as z=x+iy and \bar z=x-iy. These are conjugate variables as you know. So, already at the leading order I have solved my equation. Indeed, I note that

\Delta_2=\partial_z\partial_{\bar z}

and so the Laplacian has the solution f(z)+g(\bar z) being f and g arbitrary functions. In this case the gradient expansion gives immediately the exact result making its application trivial as should be. Indeed, I take t=0 in the perturbation series and put iy=t and I get


that is the exact solution. Nice, it works! This means that a quantum field theory using gradient expansion exists and it is a strong coupling expansion. This result is surely less trivial than the one obtained above.

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