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I am reading Sec. 1.12 of the Cosmology book by Weinberg.

In this section he explains the very simple model of quintessence which attempts to provide a dynamical explanation of the smallness of the cosmological constant today.

He considers an example of a time-dependent but space-independent scalar field in a space with Robertson-Walker metric

$$ \mathcal{L} = \int d^4 x \sqrt{-det(g)} \left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + V(\phi)\right], \tag{1} $$

where, in the simplest case, the following potential seems to make the job (naively)

$$ V(\phi) = \frac{M^{\alpha+4}}{\phi^\alpha} \tag{2} $$

for some positive constant $\alpha>0$ and mass scale $M$.

I am used to consider field theories where the potentials are analytic functions of the fields, especially in the point $\phi=0$, e.g. $\phi^4$-theory and standard effective field theories. In particular, all the effective field theories used by Beyond Standard Model phenomenologists involve positive power of fields and derivatives, e.g.

$$ V(\phi) = \sum_{n,k}\partial^n \phi^k $$

Up to now I thought that fields in denominators were allowed as long as we expand around some constant VEV $v$, for example $\phi \rightarrow v + \delta\phi$, in such a way that any potential of the form $V(\phi) = \phi^{-\alpha}$ is an expansion of analytic interaction terms involving the perturbation $\delta \phi$. Once we do this expansion, we quantize the theory around the minimum $\phi=v$.

Here instead the minimum of the potential is at $\phi=+\infty$ and I have never seen quantization around $\phi=+\infty$.

I have two very related questions.

Question

  • What is the physical meaning of the potential (2) which blows up for very small values of the field $\phi$?
  • Can we interpret this as a theory of particles? If yes, how do you quantize this theory in RW metric with zero curvature? (so, definition of one-particle states...)
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  • $\begingroup$ I'm not sure what you actually want to know here - why do you think the denominator is dangerous in this case of "infinitely large VEV", but are apparently totally fine with this in the case of a finite VEV? $\endgroup$ – ACuriousMind Nov 17 '18 at 15:22
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    $\begingroup$ The questions are actually three (as written in the OP). 1) What is a physical meaning of a potential like Eq.(2) which blows up for very small values of the field $\phi$? 2) Is this kind of potential completely fine? 3) Why do physicists (say, phenomenologists) usually work with interactions with positive power of fields? $\endgroup$ – newUser Nov 17 '18 at 15:34
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    $\begingroup$ @ACuriousMind Let me add another interesting question.Take the limit of flat space time. Is this lagrangian renormalizable? $\endgroup$ – newUser Nov 17 '18 at 15:34
  • $\begingroup$ @ACuriousMind Let me add another question yet. How do you quantize this theory? $\endgroup$ – newUser Nov 17 '18 at 15:36
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    $\begingroup$ @knzhou probably because that redefinition would mess up the kinetic term $\endgroup$ – Prof. Legolasov Nov 19 '18 at 10:17

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