Conformal field theory

A quantum field theory invariant under conformal transformations, which preserve angles while allowing local changes of scale. Its symmetry strongly constrains correlation functions and, in two dimensions, often makes the theory exactly solvable.

Conformal transformation: A transformation that preserves angles while locally rescaling distances. These transformations define the symmetry that distinguishes conformal field theories.
Conformal algebra: The Lie algebra generated by translations, rotations, dilations, and special conformal transformations. Its generators impose differential constraints on states, operators, and correlation functions.
Ising conformal field theory: The two-dimensional conformal field theory describing the critical point of the Ising model. Its exact operator spectrum makes it a standard example of a solvable critical theory.
Critical phenomena: Collective behavior near continuous phase transitions, where fluctuations occur across many length scales. Scale invariance at critical points often allows their long-distance physics to be described by conformal theories.
Quantum field theory: A framework combining quantum mechanics with fields distributed through space and time. Conformal field theory is a symmetry-restricted class within the broader family of quantum field theories.
Scale invariance: Invariance of a system's behavior under changes in length or energy scale. Global rescaling is part of conformal symmetry, though conformal invariance imposes further constraints.
Conformal bootstrap: A method that derives constraints on conformal theories from symmetry and consistency of correlation functions. Crossing symmetry and positivity can determine allowed operator dimensions and interaction strengths.
Minimal model: A two-dimensional conformal field theory with finitely many primary fields and rational central charge. Minimal models provide a classified family of exact, relatively simple theories.
Statistical mechanics: The framework connecting microscopic states to macroscopic properties of systems with many degrees of freedom. Two-dimensional lattice models at criticality often flow to conformal field theories.
Conformal anomaly: The failure of conformal symmetry in a quantum theory despite its presence in the classical description. Anomalies show that classical conformal invariance need not survive quantization.

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