In a quantum field theory, one may calculate an effective or running coupling constant that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the electric charge. In approximate calculations in several quantum field theories, notably quantum electrodynamics and theories of the Higgs particle, the running coupling appears to become infinite at a finite momentum scale. This is sometimes called the Landau pole problem. It is not known whether the appearance of these inconsistencies is an artifact of the approximation, or a real fundamental problem in the theory. However, the problem can be avoided if an ultraviolet or UV fixed point appears in the theory. A quantum field theory has a UV fixed point if its renormalization group flow approaches a fixed point in the ultraviolet (i.e. short length scale/large energy) limit. This is related to zeroes of the beta-function in the Callan-Symanzik equation.

The large length scale/small energy limit counterpart is the infrared fixed point.

4 References

Specific cases and details

Among other things, it means that a theory possessing a UV fixed point may not be an effective field theory, because it is well-defined at arbitrarily small distance scales. At the UV fixed point itself, the theory can behave as a conformal field theory.

The converse statement, that any QFT which is valid at all distance scales (i.e. isn't an effective field theory) has a UV fixed point is false. See, for example, cascading gauge theory.

Noncommutative quantum field theories have a UV cutoff even though they are not effective field theories.

If the UV fixed point is trivial (aka Gaussian), we say that we have asymptotic freedom.

If the UV fixed point is nontrivial, we say that we have "asymptotic safety". Theories with asymptotic safety may be well defined at all scales despite being nonrenormalizable in perturbative sense (according to the classical scaling dimensions).

Asymptotic safety scenario in quantum gravity

Steven Weinberg has proposed[1] that gravity may satisfy asymptotic safety. http://arxiv.org/abs/gr-qc/0610018, http://relativity.livingreviews.org/Articles/lrr-2006-5/, http://arxiv.org/abs/0911.2727,

See also

Ultraviolet divergence

Landau pole

Quantum triviality

Asymptotic Safety gravity

References

^ S. Weinberg. Ultraviolet Divergences In Quantum Theories Of Gravitation. In General Relativity: An Einstein centenary survey. Eds. S. W. Hawking and W. Israel, Cambridge University Press (1979), p.790

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