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 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. Upper critical dimension in field theory Determining the upper critical dimension of a field theory is a matter of linear algebra. It 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. A Lagrangian may be written as a sum of terms, each consisting of an integral over a monomial of coordinates and fields . Examples are the standard -model and the isotropic Lifshitz tricritical point with Lagrangians see also the figure on the right. This simple structure may be compatible with a scale invariance under a rescaling of the coordinates and fields with a factor according to Time is not singled out here — it is just another coordinate: if the Lagrangian contains a time variable then this variable is to be rescaled as with some constant exponent . The goal is to determine the exponent set . One exponent, say , may be chosen arbitrarily, for example . In the language of dimensional analysis this means that the exponents count wave vector factors (a reciprocal length