Trace anomaly of a conformally invariant quantum field in curved spacetime

We analyze a point-separation prescription for renormalizing the stress-energy operator ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ of a quantum field in curved spacetime, based on the assumption that the expectation value $G(x, {x}^{\ensuremath{'}})=〈\ensuremath{\varphi}(x)\ensuremath{\varphi}({x}^{\ensuremath{'}})+\ensuremath{\varphi}({x}^{\ensuremath{'}})\ensuremath{\varphi}(x)〉$ has the form of a Hadamard elementary solution. An error is pointed out in the work of Adler, Lieberman, and Ng: The "locally determined" piece ${G}^{L}(x, {x}^{\ensuremath{'}})$ and "boundary-condition-dependent" piece ${G}^{B}(x, {x}^{\ensuremath{'}})$ of $G(x, {x}^{\ensuremath{'}})$ do not separately satisfy the wave equation in ${x}^{\ensuremath{'}}$, as required in their proof of the conservation of the boundary-condition-dependent contribution to ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$. This error affects the point-separation renormalization prescription given in my previous paper describing an axiomatic approach to stress-energy renormalization. It is now seen that this prescription yields a stress-energy tensor whose divergence is not zero but is the gradient of a local curvature term. However, this deficiency can be corrected by subtracting off this local curvature term times the metric tensor; as a direct consequence the trace of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ becomes nonvanishing. Given this result it is shown that any prescription for renormalizing ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ which is consistent with conservation (axiom 3), causality (axiom 4), and agreement with the formal expression for the matrix element between orthogonal states (axiom 1) must yield precisely this trace, modulo the trace of a conserved local curvature term. Hence, for consistency with the first four axioms and dimensional considerations, we find that the trace of the stress tensor of the conformally invariant scalar field must be ${T}_{\ensuremath{\mu}}^{\ensuremath{\mu}}={(2880{\ensuremath{\pi}}^{2})}^{\ensuremath{-}1}({C}^{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\delta}\ensuremath{\gamma}}{C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\delta}\ensuremath{\gamma}}+{R}^{\ensuremath{\alpha}\ensuremath{\beta}}{R}_{\ensuremath{\alpha}\ensuremath{\beta}}=\frac{1}{3}{R}^{2})$ plus an arbitrary constant times ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\nabla}}^{\ensuremath{\alpha}}R$. This confirms previous work of a number of authors on the existence of trace anomalies. For consistency with axiom 5 (no "local curvature terms containing third or higher derivatives of the matric"), the coefficient of the ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\nabla}}^{\ensuremath{\alpha}}R$ term must be zero. However, it is argued that if the expectation value $G(x, {x}^{\ensuremath{'}})$ is of the Hadamard form in the massless case, as assumed in defining the point-separation renormalization prescription, then axiom 5 cannot be satisfied and, indeed, a completely unambiguous prescription for ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ cannot be given without introducing a length scale.