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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?

Qmechanic
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shouldknowbetter
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1 Answers1

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The general diffeomorphism symmetry in the target space is not a symmetry of the world line theory or, analogously, the world sheet theory! A general spacetime diffeomorphism changes the metric tensor $g_{\mu\nu}(X^\alpha)$ which plays the role of the "coupling constants" (coefficients defining the action, e.g. your exponent) in the world line or world sheet theory! If a transformation changes the values of coupling constants, then it's clearly not a symmetry, not even classically.

The isometry, a diffeomorphism that actually preserves the metric at each point, is a symmetry of the world line theory or the world sheet theory both at the classical and quantum level.

I guess that the confusion that led to the question were the omnipresent misleading comments about "background independence". One may be tempted to say that the diff symmetry is there because we may also change the background metric. But if we do, we are changing the rules of the game. The full spacetime dynamics (at least in string theory) ultimately allows us to change the spacetime metric by creating condensates of gravitons in a state etc. But in the world line theory or the world sheet theory, this "emergent" process has a different interpretation: the spacetime background metric has to be considered as a fixed collection of coupling constants and it just happens that we may prove that the "full theory" with one metric field configuration is equivalent to another but that is something else than saying that any particular world line or world sheet theory has a diff symmetry! It doesn't.

Sorry I haven't mentioned the word "Unruh" because I believe that the core of the aforementioned paradox has nothing to do with the Unruh effect.

Luboš Motl
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