Кванттық өріс теориясында конформдық симметрия

Quantum field theory enjoying conformal symmetry
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions For this reason, the terms are often used interchangeably in the context of quantum field theory.

The number of independent conformal transformations is infinite in two dimensions, and finite in higher dimensions. This makes conformal symmetry much more constraining in two dimensions. All conformal field theories share the ideas and techniques of the conformal bootstrap. But the resulting equations are more powerful in two dimensions, where they are sometimes exactly solvable (for example in the case of minimal models), in contrast to higher dimensions, where numerical approaches dominate. The development of conformal field theory has been earlier and deeper in the two dimensional case, in particular after the 1983 article by Belavin, Polyakov and Zamolodchikov. The term conformal field theory has sometimes been used with the meaning of two dimensional conformal field theory, as in the title of a 1997 textbook. Higher dimensional conformal field theories have become more popular with the AdS/CFT correspondence in the late 1990s, and the development of numerical conformal bootstrap techniques in the 2000s.

Some conformal field theories in three and four dimensions admit a Lagrangian description in the form of a gauge theory, either abelian or non abelian. Examples of such CFTs are conformal QED with sufficiently many charged fields in or the Banks Zaks fixed point in .

Continuous phase transitions (critical points) of classical statistical physics systems with D spatial dimensions are often described by Euclidean conformal field theories. A necessary condition for this to happen is that the critical point should be invariant under spatial rotations and translations. However this condition is not sufficient: some exceptional critical points are described by scale invariant but not conformally invariant theories. If the classical statistical physics system is reflection positive, the corresponding Euclidean CFT describing its critical point will be unitary. Continuous quantum phase transitions in condensed matter systems with D spatial dimensions may be described by Lorentzian D+1 dimensional conformal field theories (related by Wick rotation to Euclidean CFTs in D+1 dimensions). Apart from translation and rotation invariance, an additional necessary condition for this to happen is that the dynamical critical exponent z should be equal to 1. CFTs describing such quantum phase transitions (in absence of quenched disorder) are always unitary.

World sheet description of string theory involves a two dimensional CFT coupled to dynamical two dimensional quantum gravity (or supergravity, in case of superstring theory). Consistency of string theory models imposes constraints on the central charge of this CFT, which should be c=26 in bosonic string theory and c=10 in superstring theory. Coordinates of the spacetime in which string theory lives correspond to bosonic fields of this CFT.

Conformal field theories play a prominent role in the AdS/CFT correspondence, in which a gravitational theory in anti de Sitter space (AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples are d = 4, N = 4 supersymmetric Yang–Mills theory, which is dual to Type IIB string theory on AdS5 × S5, and d = 3, N = 6 super Chern–Simons theory, which is dual to M theory on AdS4 × S7. (The prefix "super" denotes supersymmetry, N denotes the degree of extended supersymmetry possessed by the theory, and d the number of space time dimensions on the boundary.)