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Energy is conserved in quantum field theory in flat spacetime. Does energy conservation still hold in quantum field theory in curved spacetime?

I already searched this site and Google about this, but didn’t find a clear answer .

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For simplicity, I'll focus on globally hyperbolic spacetimes, which are those in which we can write the spacetime manifold $\mathcal{M}$ as $\mathcal{M} = \mathbb{R} \times \Sigma$, where $\mathbb{R}$ represents a notion of a time coordinate and $\Sigma$ represents the spatial sections (snapshots of the entire spacetime at some value of a timelike coordinate). In this case, the spacetime is comprised of a foliation of spacelike surfaces $\Sigma_t$ at each time $t \in \mathbb{R}$.

Within General Relativity, energy conservation only holds in a global sense if there is some sort of time-translation symmetry (namely, a Killing vector field) is present. In that case, the spacetime is said to be stationary and it can be shown that (if the stress-energy tensor falls down sufficiently fast at spatial infinity or if $\Sigma$ has no boundary) the quantity $$E = \int_{\Sigma_t} T_{ab} n^a \xi^b \sqrt{h} \ \mathrm{d}^3x $$ does not depend on $t$, i.e., it is conserved. In the previous equation, $T_{ab}$ is the stress-energy-momentum tensor, $n^a$ is the unit normal vector orthogonal to $\Sigma_t$, $\xi^a$ is the timelike Killing field, and $h = \det h_{\mu\nu}$ is the determinant of the induced metric on $\Sigma_t$.

For an example of how energy conservation might fail in the absence of a timelike Killing vector field, see for example this answer.

In spite of that, a local form of conservation of energy still holds in arbitrary spacetimes. Namely, it follows from the Einstein Field Equations that $$\nabla_{a} T^{ab} = 0,$$ which is a covariant form of the continuity equation. This expression is indeed generalized to Quantum Field Theory in Curved Spacetimes. More specifically, when defining axiomatically the renormalized stress-energy-momentum tensor $\left\langle \hat{T}_{ab}\right\rangle_{\omega}$, where $\omega$ denotes the state being considered, one of the conditions imposed over it is precisely that $$\nabla_{a} \left\langle \hat{T}_{ab}\right\rangle_{\omega} = 0,$$ meaning local conservation of energy still holds. This is discussed in depth in Sec. 4.6 of Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics and, if I'm not mistaken, Wald's original papers dealing with the issues of defining the renormalized stress tensor axiomatically are doi: 10.1007/BF01609833 and doi: 10.1103/PhysRevD.17.1477.

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In the cases of interest, this is usually true. Instances where this is well defined is when there is a notion of a timelike Killing vector $\mathcal{K}^{\mu}$ on the spacetime in question.

This means that you can foliate the spacetime with a time parameter $t_{\mathcal{K}}$, so that for every time $t_{\mathcal{K}}$ there is a corresponding (3D) space-like hypersurface (usually called $\Sigma$: this is just a slice of constant $t_{\mathcal{K}}$).

If you have some quantum field coupled to the spacetime, it has a conserved stress-energy tensor $T^{\mu\nu}$ and it is possible to build a Hamiltonian corresponding to the energy conserved for observers moving along worldlines tangent to $\mathcal{K}$. It turns out this Hamiltonian is $$ H_{\mathcal{K}} = - \int_{\Sigma} d^3\Sigma\; K^{\mu}T_{\mu}^{\; \nu} \ . $$ With some work one can check that you recover the standard flat-space Hamiltonian if you look at the Killing vector associated with Minkowski time in the above. Similarly, if you consider the generator of boosts to be the Killing vector (which is timelike), you get the so-called Rindler Hamiltonian by the above construction.

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