Abstract
We now turn to four dimensions and consider the generalisation of the anomalous Ward identities and renormalisation group equations which underpinned the derivation of the c-theorem in two dimensions. We begin with the renormalisation of the energy-momentum tensor and the Weyl consistency conditions.
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Notes
- 1.
- 2.
More generally, we have the following metric variations of the curvature squared terms, which will be used in Chap. 7:
$$\displaystyle\begin{array}{rcl} \frac{1} {\sqrt{-g}} \frac{\delta } {\delta g^{\mu \nu }(x)}\,\left (R^{\kappa \lambda \rho \sigma }R_{\kappa \lambda \rho \sigma }\right )(y)& =& 2R_{\mu \lambda \rho \sigma }R_{\nu }^{\lambda \rho \sigma }\delta (x,y) + 4R_{\mu \rho \nu \sigma }D^{\rho }D^{\sigma }\delta (x,y) {}\\ \frac{1} {\sqrt{-g}} \frac{\delta } {\delta g^{\mu \nu }(x)}\,\left (R^{\rho \sigma }R_{\rho \sigma }\right )(y)& =& 2R_{\mu \rho }R_{\nu }^{\rho }\delta (x,y) + \left (R_{\mu \nu }D^{2} - 2R_{\mu \rho }D^{\rho }D_{\nu } + g_{\mu \nu }R^{\rho \sigma }D_{\rho }D_{\sigma }\right )\delta (x,y) {}\\ \frac{1} {\sqrt{-g}} \frac{\delta } {\delta g^{\mu \nu }(x)}\,R^{2}(y)& =& 2R\left (R_{\mu \nu } + \Delta _{\mu \nu }\right )\delta (x,y). {}\\ \end{array}$$ - 3.
Notice that there is no ambiguity in this section in taking all the derivatives with respect to x, since in the flat spacetime limit they are always acting on functions of (x − y). Later, when we consider curved backgrounds, we need to be careful to act with covariant derivatives appropriate to the points x or y.
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Shore, G. (2017). Local RGE and Weyl Consistency Conditions in Four Dimensions. In: The c and a-Theorems and the Local Renormalisation Group. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-54000-9_5
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