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 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 formula_1 and fields formula_2. Examples are the standard formula_3-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 formula_6 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 formula_8 with some constant exponent formula_9. The goal is to determine the exponent set formula_10. One exponent, say formula_11, may be chosen arbitrarily, for example formula_12. In the language of dimensional analysis this means that the exponents formula_13  count wave vector factors (a reciprocal length formula_14). Each monomial of the Lagrangian thus leads to a homogeneous linear equation formula_15 for the exponents formula_13. If there are formula_17 (inequivalent) coordinates and fields in the Lagrangian, then formula_17 such equations constitute a square matrix. If this matrix were invertible then there only would be the trivial solution formula_19. The condition formula_20 for a nontrivial solution gives an equation between the space dimensions, and this determines the upper critical dimension formula_21 (provided there is only one variable dimension formula_22 in the Lagrangian). A redefinition of the coordinates and fields now shows  that determining the scaling exponents formula_13 is equivalent to a dimensional analysis with respect to the wavevector formula_24, with all coupling constants occurring in the Lagrangian rendered dimensionless. Dimensionless coupling constants are the technical hallmark for the upper critical dimension. Naive scaling at the level of the Lagrangian does not directly correspond to physical scaling because a cutoff is required to give a meaning to the field theory and the path integral. Changing the length scale also changes the number of degrees of freedom. This complication is taken into account by the renormalization group. The main result at the upper  critical dimension is that scale invariance remains valid for large factors formula_6, but with additional formula_26 factors in the scaling of the coordinates and fields. What happens below or above formula_21 depends on whether one is interested in long distances (statistical field theory) or short distances (quantum field theory). Quantum field theories are trivial (convergent) below formula_21 and not renormalizable above formula_21. Statistical field theories are trivial (convergent) above formula_21 and renormalizable below formula_21. In the latter case there arise "anomalous" contributions to the naive scaling exponents formula_13. These anomalous contributions to the effective critical exponents vanish at the upper  critical dimension. It is instructive to see how the scale invariance at the upper critical dimension becomes a scale invariance below this dimension. For small external wave vectors the vertex functions formula_33 acquire additional exponents, for example formula_34. If these exponents are inserted into a matrix formula_35 (which only has values in the first column) the condition for scale invariance becomes formula_36. This equation only can be satisfied if the anomalous exponents of the vertex functions cooperate in some way. In fact, the vertex functions depend on each other hierarchically. One way to express this interdependence are the Dyson-Schwinger  equations. Naive scaling at formula_21 thus is important as zeroth order approximation. Naive scaling at the upper critical dimension also classifies terms of the Lagrangian as relevant, irrelevant or marginal. A Lagrangian is compatible with scaling if the formula_1- and formula_2 -exponents formula_40 lie on a hyperplane, for examples see the figure above. formula_13 is a normal vector of this hyperplane. Lower critical dimension. The lower critical dimension formula_42 of a phase transition of a given universality class is the last dimension for which this phase transition does not occur if the dimension is increased starting with formula_43. Thermodynamic stability  of an ordered phase depends on entropy and energy. Quantitatively this depends on the type of domain walls and their fluctuation modes. There appears to be no generic formal way for deriving the lower critical dimension of a field theory. Lower bounds may be derived with statistical mechanics arguments. Consider first a one-dimensional system with short range interactions. Creating a domain wall requires a fixed energy amount formula_44. Extracting this energy from other degrees of freedom decreases entropy by formula_45. This entropy change must be compared with the entropy of the domain wall itself. In a system of length  formula_46 there are formula_47 positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain formula_48. For nonzero temperature formula_49 and formula_46 large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at formula_51. Space dimension formula_52 thus is a lower bound for the lower critical dimension of such systems. A stronger lower bound formula_53 can be derived with the help of similar arguments for systems with short range interactions and an order parameter with a continuous symmetry. In this case the Mermin-Wagner -Theorem states that the order parameter expectation value vanishes in formula_54 at formula_51, and there thus is no phase transition of the usual type at formula_53 and below. For systems with quenched disorder a criterion given by Imry and Ma might be relevant. These authors used the criterion to determine the lower critical dimension of random field magnets.