How to renormalize quantum field theory in curved spacetime? (or in Schwarzschild spacetime?)

 I want to calculate n-point functions $$<0|Tφ(x_1)...φ(x_n)|0>$$ in massless $φ^4$ theory in Schwarzschild metric. The spacetime coordinates $x_i$ are outside the blackhole horizon. (For simplicity, I choose the vacuum state $|0>$ as Hartle-Hawking state, which has a good singularity behavior in the overall region of Kruskal coordinates.)

 Instead, it is OK if n-point functions of $φ^4$ theory in Euclidian Schwarzschild spacetime can be calculated. Spacetime coordinates are Wick rotated$(t=-iτ)$, and the theory is defined by Euclidian path integral, $$<φ(x_1)...φ(x_n)>=∫ \mathcal Dφφ(x_1)...φ(x_n)\exp{\left(-∫^β_0dτ∫d^3x\mathcal L_E\right)} $$ where β is inverse Hawking temperature, and $$\mathcal L_E=\frac12 φ(-\square)φ+\frac1{4!}λφ^4.$$ The periodicity of Euclidian action arises from that I chose Hartle Hawking vacuum. the Euclidian n-point functions seem to be analytic continuations of the above Lorentzian ones.

 This theory is similar to thermal field theory in flat spacetime. A difference is that since the $\square$ is covariant d'Alembertian, you should use corresponding Green function as a propagator when you calculate n-point functions diagrammatically.

However there may be a probrem. Unlike flat spacetime case, $\exp(-ipx)$ is not mode functions(the eigenfunctions of $\square$), so we cannot perform fourier transformation of the euclidian action $∫^β_0dτ∫d^3x\mathcal L_E$. Therefore we cannot translate "position-space Feynmann rule" into "momentum space Feynman rule". (Therefore there may be no counterpert to momentum conservation law at each vertex in flat spacetime.)

This seems to be problematic because we cannot regularizing theory imposing momentum cut off of loop momentum integrals. Dimensional regularization also make little sense for me because 4-ε dimension Schwarzschild metric seems to be pathological.(and if you consider more general metric, there is no natunal generalization to D-dimension)

Is there any method which can renormalize the n-point functions?

 I read the famous paper written by P. candelas, in which the renormalized 2-point function of free field,$$\lim_{x'→x}<0|φ(x)φ(x')|0>_{ren}$$ was calculated by subtracting some divergent term. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.21.2185

this may give the answer to the value of 1-loop 2-point diagram, enter image description here

bacause the loop in the diagram is equal to $\lim_{x'→x}<0|φ(x)φ(x')|0>_{ren}$. How to generalize this argument to 2-loop diagram or 1-loop diagram of $φ^3$ interaction?

  • $\begingroup$ before even going to curved backgrounds, have you done the two loop 2-pt amplitude computation in flat spacetime? $\endgroup$ – Wakabaloola Apr 17 at 19:01
  • $\begingroup$ I have studied resummation and calculated "semi 2-loop" 2-point amplitude which only contains the correction from daisy diagrams, but I have not done full 2-loop calculation (correction from lollipop diagram). Are there some difficulties for doing that? $\endgroup$ – Takumi Hayashi Apr 18 at 2:31

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