Crossing Equations and Positivity

Crossing says that different OPE pairings reconstruct one four-point function. Positivity is separate: for identical Hermitian operators in a reflection-positive theory, an orthonormal exchanged basis turns identical-pair OPE contractions into nonnegative squares. Degeneracies and mixed external operators replace scalar squares by positive-semidefinite matrices. Without the conjugation and Hilbert-space hypotheses, crossing remains valid but the positive cone can disappear.

Required background. OPE Convergence, Associativity, and Domain Control justifies equality of channel sums. Conformal Blocks and Casimir Equations fixes block normalization. Conjugation and Reflection Positivity supplies the positive radial form. Helpful background. Hilbert Space, Positivity, and Unitary Evolution supplies the general positivity framework.

Identical-scalar crossing

Normalize a Hermitian scalar by

$$\langle\phi(x)\phi(0)\rangle=\frac1{(x^2)^{\Delta_\phi}}$$

and write

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(u,v)} {(x_{12}^2x_{34}^2)^{\Delta_\phi}}.$$

In the $s$ channel,

$$\mathcal G(u,v) =1+\sum_{\mathcal O\neq\mathbf1} \lambda_{\phi\phi\mathcal O}^{\,2} g_{\Delta,\ell}(u,v).$$

The identity block is one. Locality of identical bosons selects even spin. Exchanging $x_1$ and $x_3$ sends $(u,v)\to(v,u)$ and gives

$$v^{\Delta_\phi}\mathcal G(u,v) =u^{\Delta_\phi}\mathcal G(v,u).$$

Define the crossing vector

$$F_{\Delta,\ell}(u,v) =v^{\Delta_\phi}g_{\Delta,\ell}(u,v) -u^{\Delta_\phi}g_{\Delta,\ell}(v,u).$$

Then the exact sum rule is

$$\boxed{ \sum_{\mathcal O\neq\mathbf1} \lambda_{\phi\phi\mathcal O}^{\,2} F_{\Delta,\ell}(u,v) =u^{\Delta_\phi}-v^{\Delta_\phi}. }$$

Every sign and power is tied to the displayed prefactor and block convention. Crossing equations with $F$ defined oppositely are equivalent only if the identity side changes sign as well.

Why the weights are nonnegative

Insert a sphere separating $(1,2)$ from $(3,4)$ in a reflection-positive Euclidean configuration. The inner pair creates a radial state. Expanding it in an orthonormal primary-descendant basis and pairing it with its reflected conjugate gives norms. For a nondegenerate Hermitian primary,

$$\lambda_{\phi\phi\mathcal O}\in\mathbb R, \qquad \lambda_{\phi\phi\mathcal O}^{\,2}\geq0.$$

If $m$ primaries share the same $(\Delta,\ell)$, an orthonormal change of degeneracy basis rotates their OPE vector. The invariant weight is

$$\sum_{a=1}^{m}\lambda_a^2\geq0.$$

In a nonorthonormal basis it is $\lambda^aG_{ab}\lambda^b$, positive only when $G$ is positive definite on the physical sector. Gauge-fixed fields, nonunitary models, and logarithmic modules need not satisfy this.

Positivity also depends on external ordering. A generic coefficient $\lambda_{12\mathcal O}\lambda_{34\mathcal O}$ can have either sign. For complex fields the positive configuration pairs an operator product with its reflected Hermitian conjugate, not an arbitrary four-point ordering.

From correlator data to a positive sum rule

The flow diagram shows where crossing and positivity enter. Inspect the two separate gates after the block construction.

Kinematic structures and block functions do not carry positivity by themselves. A convergent OPE sum reconstructs one channel; associativity supplies crossing; a positive radial inner product and Hermitian conjugate ordering supply nonnegative scalar weights. Degenerate or mixed systems use positive-semidefinite OPE matrices. The figure is schematic and does not turn a gap assumption or a truncated equation into an existence proof.

Gate	Required inputs	Result	Failure mode
Family construction	$\Delta,\ell$, external dimensions, block normalization	$g_{\Delta,\ell}$	Shadow or normalization mismatch
Channel reconstruction	Complete spectrum, OPE tensors, radial domain	$\mathcal G_s$	Truncation without a tail bound
Associativity	Two convergent representations on an overlap	$\mathcal G_s=\mathcal G_t$	Formal rearrangement outside convergence
Identity isolation	Unique normalized vacuum sector	Known right-hand side	Extra dimension-zero sectors omitted
Scalar positivity	Identical Hermitian external field, reflection positivity, orthonormal basis	$\lambda^2\geq0$	Nonunitary or nonconjugate ordering
Matrix positivity	Complete degenerate/mixed OPE vectors and positive $G$	Positive-semidefinite outer products	A component is incorrectly treated as a square
Gap assumption	Explicit sector and spin range	Restricted search space	A numerical exclusion is stated without the assumption

Linear functionals and the claim they support

A real linear functional $\alpha$ acting on crossing vectors gives

$$\sum_{\mathcal O\neq\mathbf1} \lambda_{\phi\phi\mathcal O}^{\,2} \alpha[F_{\Delta,\ell}] =\alpha[u^{\Delta_\phi}-v^{\Delta_\phi}].$$

If a hypothesized spectrum would require

$$\alpha[F_{\Delta,\ell}]\geq0$$

for every allowed nonidentity operator while the identity side has the opposite strict sign, the hypothesis is inconsistent. This is an exclusion conditional on the assumed CFT axioms, external data, gaps, and functional domain. It is not evidence that a theory exists at every point not excluded.

For matrix crossing, $\alpha$ acts entrywise and must produce a positive-semidefinite matrix in each allowed sector. Checking only diagonal entries is insufficient.

The original numerical-bootstrap use of a positive identical-scalar sum rule is developed in Rattazzi et al. 2008, §§ 2–3, with general conventions and limitations reviewed in Poland, Rychkov, and Vichi 2019, §§ IV–V.

Reproducible checks

A reproducible calculation should compare one fixed generalized-free one-dimensional correlator across exact blocks, channel conventions, later functionals, and later inversion stages. At this chapter’s stage, the bounded continuation is only the scalar correlator, Casimir, radial convergence, and direct crossing baseline; later controls belong to later chapters.

Mixed Correlators and Global-Symmetry Sectors turns scalar positivity into basis-covariant positive-semidefinite OPE matrices. Numerical approximation, precision, solver, and certificate validation remain with the numerical-bootstrap chapters.

Common pitfalls

Crossing implies positivity. Crossing is associativity. Positivity additionally uses Hermiticity, conjugate ordering, and a positive radial inner product.

Every OPE product is a square. Mixed external operators give products or matrices. Only suitable identical-pair contractions are nonnegative scalar squares.

Not excluded means realized. A finite family of functionals proves exclusions inside its tested assumptions. The surviving region is not an existence theorem.

Exercises

Derive the identity-isolated sum rule.

Solution

Insert $\mathcal G=1+\sum_{\mathcal O\neq\mathbf1}\lambda^2g_{\Delta,\ell}$ into

$$v^{\Delta_\phi}\mathcal G(u,v) -u^{\Delta_\phi}\mathcal G(v,u)=0.$$

The identity contributes $v^{\Delta_\phi}-u^{\Delta_\phi}$. Moving it to the other side gives the displayed equation with right-hand side $u^{\Delta_\phi}-v^{\Delta_\phi}$.

References

• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI; Open PDF
• Rattazzi, Riccardo, Vyacheslav S. Rychkov, Erik Tonni, and Alessandro Vichi. “Bounding Scalar Operator Dimensions in 4D CFT.” Journal of High Energy Physics 2008, no. 12 (2008): 031. DOI; Open PDF