research35 min studyLesson 731 in Physics

Conformal algebra and primary operators

Specify Euclidean conformal representations before interpreting dimensions, spin and descendants.

Before you begin

On this page

The question for today

What you’ll understand

Fix signature and generator conventions.
Distinguish a primary from its derivative descendants.
Count conformal generators and organize levels without claiming a full spectrum.

At a conformal fixed point, local observables organize into representations that link their position dependence to their scaling dimensions and spin. This is stronger structure than dimensional analysis alone, but it does not determine every operator dimension or interaction coefficient. Begin in flat Euclidean space with integer dimension d at least three. Whenever positivity is invoked, assume a reflection-positive local theory with an appropriate radial Hilbert space, rather than assuming every scale-invariant statistical model is unitary.

Specify the algebra and what it acts on

Translations, rotations, dilatations and special conformal transformations preserve angles locally. A finite special conformal transformation can be constructed from inversion, translation and another inversion, on a patch avoiding its singular locus. For dimensions above two, the connected conformal algebra is finite dimensional. Euclidean rotations use the positive metric delta; Lorentzian conformal symmetry has a different real form. Analytic continuation and operator adjoints therefore require stated conventions rather than replacing a time coordinate casually.

Choose generators without explicit factors of i in the radial commutators. Dilatation raises a translation descendant's dimension by one, while a special conformal generator lowers it. Rotations organize tensor indices. The algebra is a set of consistency rules for representations, not a list of numerical scaling dimensions. Also, scale invariance by itself need not imply conformal invariance without additional assumptions about locality, stress tensors and the spectrum; conformality is an explicit premise here.

[D,P_\mu]=P_\mu,\qquad [D,K_\mu]=-K_\mu,\qquad [K_\mu,P_\nu]=2\delta_{\mu\nu}D-2M_{\mu\nu}

These are dimensionless radial-generator conventions with Euclidean indices. In the reflection-positive radial inner product, D† = D, Pμ† = Kμ and Mμν† = −Mμν.↔ Scroll wide equations horizontally.

Identify lowest-weight operators before taking derivatives

A primary at the origin is annihilated by all special conformal generators and has a definite dilatation eigenvalue. It may be a scalar or carry an irreducible rotation representation. Calling an operator primary is therefore stronger than saying it has a scaling dimension. Acting with translations produces derivatives, called descendants, whose dimensions increase by integer levels. A derivative can carry spin even when its parent is scalar; spin alone does not identify a new primary.

At a given level, tensor products can split into several rotation representations, and some combinations may be null. The surviving states form one conformal family. Different primaries can share the same dimension and spin, in which case additional labels and a basis in the degenerate space are necessary. Gauge-dependent elementary fields are not automatically members of the physical local-operator Hilbert space; the discussion concerns properly defined local observables and their physical representation content.

K_\mu|\mathcal O\rangle=0,\qquad D|\mathcal O\rangle=\Delta|\mathcal O\rangle,\qquad D(P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle)=(\Delta+n)P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle

Δ and the descendant level n are dimensionless. The equation states eigenvalues, not a claim that every formal descendant is nonzero or linearly independent.↔ Scroll wide equations horizontally.

Separate kinematics from dynamical data

A useful description of a CFT specifies its primary dimensions, rotation representations and operator-product coefficients. Symmetry then fixes many descendant coefficients and coordinate structures. It still leaves a difficult consistency problem: those data must produce correlators compatible with crossing, locality and positivity. An arbitrary table of plausible dimensions is not a constructed theory. Additional conserved currents or supersymmetry can enlarge the algebra and impose further representation constraints beyond the conformal generators counted here.

Counting the generators is a simple check on which symmetry is being used. In two dimensions the same count describes only the global subalgebra; local holomorphic transformations can enlarge it to Virasoro symmetry. In higher dimensions there is no corresponding generic infinite set. The spectral lab can illustrate finite versus extended sums of modes, but those sine modes are neither conformal descendants nor independent primary operators and cannot calculate a CFT spectrum.

N_{\rm generators}=d+\frac{d(d-1)}2+1+d=\frac{(d+1)(d+2)}2

The terms count translations, rotations, dilatation and special conformal transformations. This finite count applies to the global conformal algebra; it does not count operators in the theory.↔ Scroll wide equations horizontally.

Work it through

Count generators and descendant energies

In a Euclidean d = 3 CFT, count each type of global conformal generator. A scalar primary has $\Delta=1.20$ . What dimensions do its formal first- and second-level descendants have?

There are three translations, three independent rotations, one dilatation and three special conformal generators. Their sum is ten, matching $(3+1)(3+2)/2$ .
The commutator of D with one translation adds one to the eigenvalue. A nonzero first-level descendant therefore has dimension $2.20$ , independent of the direction of its derivative.
Two translations add two, giving dimension $3.20$ . These eigenvalues do not assert that all index combinations are independent, positive-norm or nonnull; those questions require the representation and its inner product.

The result

The global algebra has ten generators, and the formal descendant dimensions are 2.20 and 3.20.

The living notebook

Explore the model

Interactive lab

Interactive lab · spectral

A mode count is not an operator count

Reset and set BOTH Mode count N = 2 and Coefficient decay p = 1. Predict less fine spatial structure than a comparison after Reset and setting N = 6 and p = 1 at equal time or equal Step counts. These are terms sin(nx−t)/nᵖ, not conformal generators, primaries or descendants.

Finite sumHighest included mode

Sum at x = π/2: 0.867
Smallest wavelength: 2π / 5
Highest-mode coefficient: 0.2

Mode count N5Coefficient decay p1

t = 0 s

A finite sum of sin(nx − t)/nᵖ. Playback changes t; increasing the cutoff includes finer spatial scales. This is not a proof about a continuum limit or renormalization. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe organization of a mathematical sum needs its own physical interpretation.

Pause & reason

Check your understanding

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Read the checkpoint explanation

The primary is a lowest-weight state for the conformal algebra. Spinning and noninteger-dimension primaries are allowed; translations generate higher descendants.

Your turn

Try it yourself

For d = 4, count the global conformal generators and determine the dimension of a nonzero level-three descendant of a primary with $\Delta=2.50$ . Does this count fix the number of primaries?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Count antisymmetric rotation pairs separately from the two groups of vector generators.

Compare with the solution

There are four translations, six rotations, one dilatation and four special conformal generators, giving 15. The descendant dimension is $2.50+3=5.50$ . Neither result fixes the number of primaries: a finite-dimensional symmetry algebra can have infinitely many inequivalent representations in a field theory. Dimensions and multiplicities require dynamical CFT data and consistency conditions, while null relations can reduce the number of descendants within a particular representation.

Keep the idea

Take it with you

Conformal representations require signature and generator conventions.
Primaries can carry spin; derivatives organize their descendants.
Finite generator counting does not determine a CFT operator spectrum.

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