Symmetry, Actions, and Redundancy

The same field transformation can represent a physical symmetry, a redundant description, or a duality between theories. This chapter identifies the acted-on objects and invariant predictions before classifying the transformation or applying a selection rule.

Enter this chapter

Begin with What Is a Symmetry of a QFT? to specify the action on physical data.

Helpful background. Groups, Actions, Quotients, and Covers supply the language of actions, kernels, and quotients.

Topics

1. What Is a Symmetry of a QFT? — Defines an ordinary symmetry through an invertible action on physical data.
2. Internal, Spacetime, Discrete, and Antiunitary Symmetries — Separates independent classifications of transformations and their implementations.
3. Symmetry, Gauge Redundancy, and Duality — Distinguishes physical actions, equivalences of representatives, and dual descriptions.
4. Quantum Implementations, Projective Actions, and Central Extensions — Explains phase ambiguities in Hilbert-space implementations.
5. Multiplets, Invariants, and Selection Rules — Derives invariant-tensor restrictions on couplings and matrix elements.

How the ideas fit together

A symmetry can act nontrivially on charged operators or sectors while leaving selected neutral observables unchanged. Its faithful quotient therefore depends on the full declared set of physical objects, not one convenient probe. The comparison of transformation types makes this distinction explicit.

Projective implementation is another issue: operators on state vectors may compose only up to phase while acting consistently on physical rays. This phase ambiguity does not by itself establish an obstruction to coupling the theory to background fields. See Weinberg 1995, §§ 2.2 and 2.7.

Review the chapter

An invariant vacuum has a nonzero correlator of charged operators. What must their total charge satisfy for an ordinary $U(1)$ symmetry?

Solution

The correlator acquires a phase equal to the exponential of the sum of charges times the symmetry parameter. Invariance for every parameter requires total charge zero. This is a necessary selection rule; it does not guarantee a nonzero value.

Where to go next

Continue to Currents, Charges, and Quantum Ward Identities for the local constraints associated with continuous symmetries.

References

• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995. DOI.