# upper critical dimension in field theory

Is there field theory which describe a second order phase transition without upper critical dimension ? Mermin-Wagner says something about lower critical dimension but nothing about upper dimension.

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You need a way to declare that two field theories in different dimensions are "the same". There are natural ways to do this for simple examples, but it is easy to make up a model where there is no upper critical dimension because you extrapolate wrong. – Ron Maimon Apr 28 '12 at 5:32

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– Manishearth♦ May 7 '12 at 6:09
Thanks @Manishearth for pointing that out to PanAkry, I was just about to leave the same 'citizenship patrol' comment ;) – NewAlexandria Dec 29 '12 at 23:54

## 1 Answer

The upper critical dimension is the dimension where the statistical field theory is well described by a mean field theory. It is also the dimension where the fluctuation theory turns into a free field theory. You can avoid having an upper critical dimension by tuning the kinetic terms properly:

Consider the Euclidean action:

$$S= \int |q|^{2n} |\phi|^2 + \lambda \phi^4 d^n x$$

This field theory never has an upper critical dimension. But this is because the dimensional extrapolation is wrong. For any fixed power of q, there is an upper critical dimension.

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Does this action correspond to some physical model? It is not clear : "This field theory never has an upper critical dimension... there is an upper critical dimension"! – PanAkry Apr 28 '12 at 12:25
@PanAkry: Not really physical. The point of this example is that I continued the theory wrong into higher dimensions. The right continuation holds the power of q fixed (the exponent doesn't change with n), and this continuation does have an upper critical d. – Ron Maimon Apr 28 '12 at 14:26
Why the downvote? It answers the question--- this is a silly mathematical continuation without upper critical dimension. – Ron Maimon May 9 '12 at 22:12
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