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 meaning is more restricted: the critical dimension is 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; it is 26 for the bosonic string theory and 10 for superstring theory.
Read more about Critical Dimension: Upper Critical Dimension in Field Theory, Lower Critical Dimension
Famous quotes containing the words critical and/or dimension:
“Probably more than youngsters at any age, early adolescents expect the adults they care about to demonstrate the virtues they want demonstrated. They also tend to expect adults they admire to be absolutely perfect. When adults disappoint them, they can be critical and intolerant.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.4 (1985)
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)