Unitarity bounds and short representations

Normalize a scalar primary state to one. The norm of a derivative in one chosen Cartesian direction is the expectation of a special conformal generator followed by a translation. The primary is killed by the lowering generator, so only their commutator survives. Rotations act trivially on a scalar primary, leaving twice its dimension. A negative dimension would therefore create a negative-norm descendant immediately and is excluded under these assumptions.

This first-level test is necessary but not sufficient. For scalars, the contracted level-two descendant gives a stronger condition. Its norm contains a factor that vanishes at the free-scalar dimension. The separate dimension-zero solution includes the identity, whose translation descendants vanish; it must not be incorrectly ruled out by the positive-dimension scalar bound. In the usual irreducible theory with a unique vacuum, the physical dimension-zero scalar sector is the identity rather than an arbitrary new propagating field.

For a spinning primary, first-level descendants lie in the tensor product of a vector with the primary's rotation representation. The norm matrix is not a single scalar number: rotations mix the tensor components. Decomposing into irreducible rotation sectors identifies its least-positive eigenvalue. For a symmetric traceless tensor of positive integer spin, this gives a lower bound involving spin plus spacetime dimension minus two. Testing only a convenient component can miss the dangerous sector.

At equality, a particular descendant has zero norm. In a unitary Hilbert space it decouples after quotienting null states, leaving a shorter representation. For the symmetric traceless family, that null descendant is the divergence, yielding a conserved current equation at separated insertions. A spin-one conserved current has dimension d minus one, and the stress tensor has spin two and dimension d. Contact terms in Ward identities must still be handled separately from the vanishing separated-point divergence.

A scalar saturating its positive-dimension bound has a null Laplacian descendant. In the usual local unitary setting this is a free massless scalar equation, identifying a free scalar sector rather than proving that every other operator in the entire theory is free. An anomalous dimension above the bound restores a nonnull descendant. Additional algebraic symmetries can create other shortening conditions, but those follow from their enlarged representation algebra and cannot be inferred from the scalar calculation alone.

Passing all these unitarity tests is still not sufficient to construct a CFT. The proposed primaries must also fit consistent OPE coefficients, crossing, locality and the necessary stress-tensor structure. Conversely, a nonunitary conformal model may violate the positive-norm bounds without an algebraic contradiction because it drops their Hilbert-space assumption. The quadratic lab can illustrate the sign of a simple norm-like function, but it neither computes conformal descendant matrices nor checks a whole representation.