Anomalous target space diffeomorphisms for one-loop world-line integrals

The Schwinger effect can be calculated in the world-line formalism by coupling the particle to the target space potential $A$.

My question relates to how this calculation might extend to computing particle creation in an accelerating frame of reference, i.e. the Unruh effect. Consider the one-loop world-line path integral:

$$Z_{S_1} ~=~ \int^\infty_0 \frac{dt}{t} \int d[X(\tau)] e^{-\int_0^t d\tau g^{\mu\nu}\partial_\tau X_\mu \partial_\tau X_\nu},$$

where $g_{\mu\nu}$ is the target space metric in a (temporarily?) accelerating reference frame in flat space and the path integral is over periodic fields on $[0,t]$, $t$ being the modulus of the circular world-line. If the vacuum is unstable to particle creation, then the imaginary part of this should correspond to particle creation.

Since diffeomorphism invariance is a symmetry of the classical 1-dimensional action here, but not of $Z_{S^1}$, since it depends on the reference frame, can I think of the Unruh effect as an anomaly in the one-dimensional theory, i.e. a symmetry that gets broken in the path integral measure when I quantize?

This question would also apply to string theory: is target space diffeomorphism invariance anomalous on the worldsheet?

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I'm a bit disappointed that this hasn't got any responses from people in the know. My first instinct was to question whether you could apply the worldline machinery to the Unruh/Hawking effect at all. The worldline calculations that I've looked at are deeply concerned with the vacuum of an interacting QFT - they make extensive use of the effective action. In contrast, the Unruh effect arises even in a free QFT, just from a Bogoliubov transformation of the vacuum. So is the worldline formalism applicable at all to Unruh style particle creation? – twistor59 May 11 at 9:49
I guess the initial thought was that the graviton interaction, even though it is only implementing a change of reference frame, is sourcing the particles. According to some notes by Bastianelli the measure should also change in a covariant way $DX(t) -> \sqrt{g(X(t))} DX(t)$ so perhaps this solves the conundrum: everything changes covariantly so $Z_{S_1}$ is also covariant. However Emparan uses worldline methods for a similar calculation, so I am still a bit confused. – shouldknowbetter May 11 at 15:53
Oh I see where you're coming from - the Bastianelli example with the metric minimally (or whatever) coupled to the matter field via the Ricci curvature clarifies the sort of action you had in mind. And you were wondering about whether breaking of diffeomorphism invariance has a role to play somewhere...At least I understand the question now. Thanks! – twistor59 May 11 at 16:27