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
Other articles related to "dimension, critical dimension":
... is an electrode that has at least one dimension (the critical dimension) smaller than 25 μm ... of 5 μm are commercially available and electrodes with critical dimension of 0.1 μm have been made ... Electrodes with even smaller critical dimension have been reported in the literature, but exist mostly as proofs of concept ...
... There appears to be no generic formal way for deriving the lower critical dimension of a field theory ... Space dimension d=1 thus is a lower bound for the lower critical dimension of such systems ... These authors used the criterion to determine the lower critical dimension of random field magnets ...
Famous quotes containing the words dimension and/or critical:
“Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.”
—Shoshana Zuboff (b. 1951)
“To take pride in a library kills it. Then, its motive power shifts over to the critical if admiring visitor, and apologies are necessary and acceptable and the fat is in the fire.”
—Carolyn Wells (1862–1942)