Critical dimension

In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. Above the upper critical dimension the critical exponents of the theory become the same as that in mean field theory. An elegant criterion to obtain the critical dimension within mean field theory is due to V. Ginzburg.

Since the renormalisation group sets up a relation between a phase transition and a quantum field theory, this has implications for the latter and for our larger understanding of renormalization in general. Above the upper critical dimension, the quantum field theory which belongs to the model of the phase transition is a free field theory. Below the lower critical dimension, there is no field theory corresponding to the model.

In the context of string theory the meaning is more restricted: the critical dimension is the dimension at which string theory is consistent assuming a constant dilaton background without additional confounding permutations from background radiation effects. The precise number may be determined by the required cancellation of conformal anomaly on the worldsheet; it is 26 for the bosonic string theory and 10 for superstring theory.

Determining the upper critical dimension of a field theory is a matter of linear algebra. It nevertheless is worthwhile to formalize the procedure because it yields the lowest-order approximation for scaling and essential input for the renormalization group. It also reveals conditions to have a critical model in the first place.

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