Ward–Takahashi identity

A relation among quantum-field-theory Green’s functions that expresses the consequences of a continuous symmetry, especially gauge invariance. It generalizes current conservation from classical fields to quantum correlation functions.

Noether’s theorem: A theorem connecting continuous symmetries of an action with conserved currents. The identity is the quantum-field-theory expression of the conservation law associated with a continuous symmetry.
Continuous symmetry: A transformation symmetry whose parameters vary continuously and which leaves a system’s laws or action invariant. The identity follows from the invariance of the quantum theory under such transformations.
Renormalization: A procedure for absorbing ultraviolet divergences into parameters and fields while defining finite predictions. Ward–Takahashi relations constrain how fields and interaction vertices renormalize in gauge theories.
Slavnov–Taylor identities: Relations among Green’s functions that encode gauge symmetry in non-Abelian quantum field theories after gauge fixing. They extend the gauge-symmetry constraints of the Ward–Takahashi identity to non-Abelian theories.
Green’s function: A function that encodes a system’s response or correlations and often serves as an inverse of a differential operator. Ward–Takahashi identities constrain the relations among these quantum correlation functions.
Gauge symmetry: A local redundancy in the mathematical description of a physical system, represented by transformations depending on spacetime position. Gauge invariance is the principal symmetry whose quantum consequences the identity captures.
Gauge fixing: A choice of representative from the redundant field configurations related by gauge transformations. Calculations require gauge fixing, while the identity tracks constraints inherited from gauge invariance.
BRST symmetry: A fermionic symmetry of gauge-fixed theories that combines gauge transformations with ghost fields. Its invariance produces Slavnov–Taylor identities, the non-Abelian counterpart of Ward–Takahashi constraints.
Noether current: A conserved current associated with a continuous symmetry of a physical system. Its conservation supplies the current relation that becomes an identity among quantum correlators.
Correlation function: A function measuring statistical or quantum correlations between observables at specified points or momenta. The identity is a constraint connecting correlation functions of different orders.

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