Critical dimension

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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 renormalization group sets up a relation between a phase transition and a quantum field theory, this also has implications for the latter. 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 phrase critical dimension has a more restricted meaning. It indicates the dimension at which string theory is consistent assuming a constant dilaton background. The precise number may be determined by the required cancellation of conformal anomaly on the worldsheet and it equals 26 for the bosonic string theory and 10 for superstring theory.