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I am currently working through the Fujikawa paper "Comments on Chiral and Conformal Anomalies". I have, however, had some issues getting around some notation, and perhaps a little of the logic, he uses.

The theory the paper concerns is: $$ S = \frac{1}{2} \int d^4x \sqrt{-g} \Big[g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi + \frac{1}{6} R \phi^2 \Big]. $$

In the paper Fujikawa introduces the half-weight fields to be able to write a covariant path integral measure: $$ \tilde{\phi}(x) \ \hat{=} \ g^\frac{1}{4}(x)\phi(x) \\ \tilde{S}[g, \tilde{\phi}] \equiv S[g, \phi] \ . $$ He later writes $$ \frac{\delta_g \tilde{S}}{\delta\alpha(x)} + \frac{\delta_\tilde{\phi} \tilde{S}}{\delta\alpha(x)} = 0 \\ \frac{\delta_g Z}{\delta \alpha(x)} \Big\vert_{\alpha = 0} \ \ \hat = \ \ \langle T_{\mu \nu}(x) \rangle g^{\mu \nu}(x) \sqrt{g} \ = \ \Big\langle \frac{\delta_g \tilde{S}}{\delta \alpha(x)} \Big\rangle \Big\vert_{\alpha = 0} \ . $$ Here $\alpha(x)$ is the parameter of the Weyl transformation, $T_{\mu\nu}(x)$ is the energy-momentum tensor

My questions relate to the choice of syntax he uses, and to the definition of the energy-momentum tensor in the particular expression above.

Firstly, is the interpretation that $$ \frac{\delta_g}{\delta\alpha(x)} $$ means to transform just the metric by the rules of the transformation we are interested in, and then to functionally differentiate with respect to the parameter of that transformation, correct?

The second question concerns that we classically define the symmetric energy-momentum tensor as the source of the gravitational field: $$ \Theta^{\mu \nu} = -\frac{2}{\sqrt{g}} \frac{\delta S}{\delta g_{\mu \nu}} \ . $$

We can show that this should be traceless for a Weyl-invariant theory, but I am not sure why we should expect $T_{\mu\nu}$ to be similarly traceless as we obtain it using a new definition. In particular, that the new definition depends on the covariant integral measure $\prod_x\mathcal{D}\tilde{\phi}(x)$ rather than the naive $\prod_x\mathcal{D}\phi(x)$ measure. In previous questions I found the answers solely pertain to the classical tensor object, my confusion lies, I suspect, in the construction of the expectation object.

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