\({\mathcal P}(\phi)_ 2\) models with variable coefficients
For the equation (in one space dimension) \((*)\quad(\square +m^ 2)\phi +4(\rho_ 0+\rho(t,x))\phi^ 3=0, \rho_ 0>0\), the Euclidean approach (imaginary time correlation function) used successfully in the \(\rho =const\). case cannot be applied as one generally does not have a distinguished vacuum state, and the reconstruction to real time would be a great difficulty anyway. Using the Fock representation \((\phi^ 3\) understood as the Wick product \(:\phi^ 3:)\) it is shown for localized \(\rho\) (if \((diameter\quad(\sup p (\rho))+1)\cdot \| \rho \|_{\infty}\) is sufficiently small) that for any cut-off \(\ell\) time evolution operators exist. The obtained fields solve (*), are independent of (sufficiently large) \(\ell\), and fulfil locality and causality. As two-dimensional Lorentz manifolds are conformally flat (locally: \(g_{\mu \nu}=\Lambda \eta_{\mu \nu}),\) equation (*) on curved spacetime can be reduced to the Minkowski case. This procedure would be significant only if the solutions obtained are in some sense independent of the choice of \(\Lambda\). Such a conformal invariance is not settled, it is conjectured by which other renormalized products the Wick products have to be replaced.
scientific article; zbMATH DE number 3857615
scientific article