Conformal Symmetry and Representations
Conformal symmetry relates local changes of scale to the organization of quantum operators. This chapter follows that connection from spacetime transformations to operator families, positivity bounds, conserved currents, and counting. The focus is ordinary conformal symmetry in fixed dimension; scale invariance alone does not establish it.
Enter this chapter
Section titled “Enter this chapter”Begin with Conformal Geometry, Maps, and Compactification to distinguish a local transformation from a global symmetry.
Helpful background. Lie Groups, Lie Algebras, the Exponential Map, and the Adjoint Action supplies the group–algebra distinction. Local Composite-Operator Insertions supplies renormalized local operators. Lorentz-Field and Poincaré-Particle Representations supplies Lorentz and Spin representations. Multiplets, Invariants, and Selection Rules supplies the general language of symmetry multiplets.
Topics
Section titled “Topics”- Conformal Geometry, Maps, and Compactification — Determines where conformal coordinate maps exist and how compactification extends their domains.
- The Conformal Algebra and Its Generators — Derives the generators and fixes their commutators in Euclidean and Lorentzian signature.
- Primaries, Descendants, and Conformal Multiplets — Organizes a primary operator and its derivatives into a conformal family.
- Spin and Tensor Representations — Identifies the rotation or Lorentz representations needed for spinning operators.
- Unitarity Bounds and Null States — Uses positive descendant norms to constrain dimensions and identify null states.
- Conserved Currents and the Stress Tensor — Relates conserved multiplets to Ward identities and two-point normalizations.
- Characters and Conformal Multiplet Counting — Counts descendants while removing complete null submodules.
- Scale versus Conformal Invariance: Hypotheses and Counterexamples — Tests the hypotheses needed to improve a scale-invariant stress tensor to a conformal one.
How the ideas fit together
Section titled “How the ideas fit together”A primary is an operator from which translations generate a family of descendants. In a unitary theory, the inner product supplied by radial quantization constrains the norms of those descendants. A saturated bound can make a descendant null; removing its entire submodule then changes both the local equations obeyed by the operator and the character that counts its states. Conservation and multiplet counting are therefore two consequences of the same shortening mechanism. Simmons-Duffin 2017, §§ 4.3 and 7.3.
The algebra alone does not supply positivity. Likewise, a coordinate formula valid on one patch does not establish a global group action: its singularities, compactification, and spin cover still matter.
Review the chapter
Section titled “Review the chapter”Why does finding an operator at a familiar conservation-bound dimension not, by itself, prove conservation?
Solution
The dimension must belong to a representation with the appropriate positive inner product. Only then does saturation force the relevant descendant to be null; the state–operator correspondence turns that null relation into a local equation. A nonunitary representation can share the dimension without that conclusion.
Where to go next
Section titled “Where to go next”Continue to Radial Quantization and State–Operator Correspondence to construct the inner product and connect dimensions to energies.