  We present a general procedure to renormalize the stress tensor two-point
correlation function on a Minkowski background in position space. The method is
shown in detail for the case of a free massive scalar field in the standard
Minkowski vacuum state, and explicit expressions are given for the counterterms
and finite parts, which are in full accordance with earlier results for the
massless case. For the general case in position space, only regularized --- but
not renormalized --- results have been obtained previously. After a Fourier
transformation to momentum space, we also check agreement with a previous
calculation there. We generalize our results to general Hadamard states.
Furthermore, the proposed procedure can presumably be generalized to the
important case of an inflationary spacetime background, where the transition to
momentum space is in general not possible.
