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1. What the question is asking (in plain language)

We have a scalar field that lives on a curved space‑time with metric (g_{\mu\nu}(x)). The quadratic part of its (Euclidean) action can be written with the differential operator

[ \boxed{ \;G^{-1}(x,y;g)=\delta^{(4)}(x-y)\,\sqrt{-g(y)}\, \bigl(g^{\mu\nu}(y)\nabla_\mu^{(y)}\nabla_\nu^{(y)}+m^{2}\bigr) \;} ]

which is the inverse propagator (the “kernel’’ of the kinetic operator). The one‑loop effective action contains the functional determinant of this operator, (\log\det G^{-1}).

The problem is to differentiate that determinant with respect to the metric, i.e. to compute

[ \frac{\delta}{\delta g^{\mu\nu}(x)}\, \log\det G^{-1}\; . ]

In flat space a similar derivative with respect to a scalar background field (\varphi) gives the familiar result

[ \frac{\delta}{\delta\varphi(x)}\log\det G^{-1}= V’’’(\varphi(x))\,G(x,x)\; . ]

Here we have to do the same thing, but the operator depends on the metric in a non‑trivial way (through the factor (\sqrt{-g}), the inverse metric (g^{\mu\nu}) and the covariant derivatives).
The goal is to obtain an expression that is exact (no steps skipped) and to explain why it looks divergent and how the divergence is usually handled.

2. Full derivation, step by step

Below we work in Euclidean signature (the Wick‑rotated version of the original Lorentzian theory). All functional traces are taken over the space of square‑integrable functions on the manifold.

2.1 Basic functional‑determinant identity

For any (formally) invertible operator (\mathcal{O})

[ \boxed{\;\frac{\delta}{\delta\lambda}\,\log\det \mathcal{O} =\operatorname{Tr}!\bigl(\mathcal{O}^{-1}\,\delta_{\lambda}\mathcal{O}\bigr) =\operatorname{Tr}!\bigl(G\,\delta_{\lambda}\mathcal{O}\bigr)\;} \tag{2} ]

where (G\equiv\mathcal{O}^{-1}) is the Green function (the propagator) defined by

[ \mathcal{O}_x\,G(x,y)=\frac{\delta^{(4)}(x-y)}{\sqrt{-g(x)}} . ]

Equation (2) is the curved‑space analogue of the flat‑space identity used in the question.

2.2 Write the operator in a convenient form

Define the covariant Laplacian (the d’Alembertian)

[ \Box\equiv g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\;, \qquad\text{so that}\qquad \mathcal{O}\equiv G^{-1}= \sqrt{-g}\,(\,-\Box+m^{2}\,). ]

(With the metric signature ((+,-,-,-)) the kinetic operator is (-\Box+m^{2}); the overall sign is irrelevant for the functional derivative.)

2.3 Variation of the operator

The metric appears in three places:

  1. the overall factor (\sqrt{-g});
  2. the inverse metric inside (\Box);
  3. the connection hidden inside the covariant derivatives.

We vary each piece.

Variation of the determinant factor

[ \delta\sqrt{-g}= -\frac12\sqrt{-g}\;g_{\alpha\beta}\,\delta g^{\alpha\beta}. \tag{3} ]

Variation of the Laplacian

[ \delta\Box =\delta(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}) =\underbrace{\delta g^{\mu\nu}\,\nabla_{\mu}\nabla_{\nu}}{\text{explicit metric}} \;-\;g^{\mu\nu}\,\delta\Gamma^{\lambda}{\mu\nu}\,\nabla_{\lambda}, \tag{4} ]

where the variation of the Christoffel symbols is

[ \boxed{\; \delta\Gamma^{\lambda}{\mu\nu} =\frac12 g^{\lambda\rho} \bigl(\nabla{\mu}\delta g_{\rho\nu} +\nabla_{\nu}\delta g_{\rho\mu} -\nabla_{\rho}\delta g_{\mu\nu}\bigr) \;} \tag{5} ]

and (\delta g_{\mu\nu}= -g_{\mu\alpha}g_{\nu\beta}\,\delta g^{\alpha\beta}).

Putting (3)–(5) together we obtain

[ \boxed{ \delta\mathcal{O} =\sqrt{-g}\Bigl[ -\frac12 g_{\alpha\beta}\,\delta g^{\alpha\beta}\;(!-\Box+m^{2}!) +\delta g^{\mu\nu}\nabla_{\mu}\nabla_{\nu} -g^{\mu\nu}\,\delta\Gamma^{\lambda}{\mu\nu}\nabla{\lambda} \Bigr]. } \tag{6} ]

2.4 Insert the variation into the trace formula

Using (2) we write

[ \delta\log\det G^{-1} =\operatorname{Tr}!\bigl(G\,\delta\mathcal{O}\bigr) =\int!d^{4}x\sqrt{-g(x)}\, \bigl\langle x\big|\,G\,\delta\mathcal{O}\,\big|x\bigr\rangle . \tag{7} ]

Because (G) is the inverse of (\mathcal{O}), the operator product (G\,\delta\mathcal{O}) can be evaluated by acting the derivatives on the coincident Green function (G(x,y)) and then setting (y\to x). Denoting

[ G(x,x)\equiv \lim_{y\to x}G(x,y), \qquad \bigl(\nabla_{\mu

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