Trace anomaly on a quantum spacetime manifold

In this paper we investigate the trace anomaly in a space-time where single events are delocalized as a consequence of short distance quantum coordinate fluctuations. We obtain a modified form of heat kernel asymptotic expansion which does not suffer from short distance divergences. Calculation of the trace anomaly is performed using an IR regulator in order to circumvent the absence of UV infinities. The explicit form of the trace anomaly is presented and the corresponding $2D$ Polyakov effective action and energy-momentum tensor are obtained. The vacuum expectation value of the energy-momentum tensor in the Boulware, Hartle-Hawking and Unruh vacua is explicitly calculated in a $rt$ section of a recently found, noncommutative inspired, Schwarzschild-like solution of the Einstein equations. The standard short distance divergences in the vacuum expectation values are regularized in agreement with the absence of UV infinities removed by quantum coordinate fluctuations.