Linear Functionals and Positivity

A bootstrap functional is useful only on a specified domain: its basis, normalization, spin sectors, continuous dimension intervals, and matrix action are part of the mathematical object. A coefficient vector that is positive at a finite grid is not yet a separating functional for an infinite spectrum.

Required background. From Crossing Equations to Convex Optimization supplies the cone and exclusion logic. Helpful background. Forms, adjoints, and isometries clarifies the dual pairing and basis transformations.

Derivative and integral functionals

For scalar crossing, a finite derivative functional may be written

$$\alpha[f] =\sum_{m+n\leq\Lambda}a_{mn} \left.\partial_z^m\partial_{\bar z}^{n}f(z,\bar z)\right|_{z=\bar z=1/2},$$

after removing derivatives forbidden by exchange symmetry and choosing a real basis. Integral and analytic functionals replace derivatives by kernels, but the same questions remain: on which functions is the action defined, which endpoint or Regge conditions make integrations legitimate, and how is positivity established?

Rescaling the derivative basis changes the coefficients $a_{mn}$ and conditioning but not the exact functional. Store the transformation or the scaled basis itself; otherwise a saved coefficient vector cannot be evaluated independently.

Scalar and matrix positivity

For one identical correlator, the sector condition is

$$\alpha[\mathbf F_{\Delta,\ell}]\geq0 \quad \text{for every allowed }(\Delta,\ell).$$

In a mixed system, an exchanged primary has an OPE vector $\boldsymbol\lambda$ and contributes

$$\boldsymbol\lambda^{\mathsf T} \mathbf F_{\Delta,\ell} \boldsymbol\lambda.$$

The functional must therefore produce a positive-semidefinite matrix $\alpha[\mathbf F_{\Delta,\ell}]\succeq0$ on the entire sector. Checking diagonal entries is insufficient; all principal minors or an equivalent PSD representation are required. Antisymmetric or parity-odd sectors may carry convention-dependent crossing signs, so signs must be derived before a PSD block is assigned Poland, Rychkov, and Vichi 2019, §§VI.A–VI.C.

At an optimum, zeros of $\alpha[\mathbf F]$ can identify candidate operators. That observation supports later reconstruction, but it does not change the exclusion proof: positivity, not visual zero-finding, is the certified statement.

A rational sign fixture

For $v(x)=(1,x,x^2)$ at $x=0,1,2$, the functional with coefficients $(1,-2,1)$ evaluates to $(x-1)^2$. Exact rational arithmetic proves nonnegativity on the finite generator set, while changing the middle coefficient from $-2$ to $-3$ produces a negative value at $x=1$. Serialize the ordered basis $(1,x,x^2)$, the generator order, target, coefficients, and exact evaluations. This catches the common error of reading coefficients in a differently scaled basis.

For conformal blocks, replace the three generators by continuous families. A polynomial or rational positivity representation, interval arithmetic, or an analytic argument must close the intervals between points and the tail. Floating-point sampling can be a diagnostic, never the whole certificate Simmons-Duffin 2015, §§2–3.

Functional acceptance record

Item	What must be serialized	Rejection test
Crossing basis	Ordered components, tensor structures, sector labels	Permute two components without transforming coefficients
Functional basis	Derivative or kernel definitions and scaling	Re-evaluate in an unscaled basis
Normalization	Exact identity or objective condition and sign	Flip only the identity vector
Positivity domain	Spin set and dimension intervals, including tails	Insert a negative point between samples
Matrix sectors	Complete PSD blocks and OPE-vector order	Test a negative eigenvector with positive diagonals
Numerical data	Precision, rounding mode, residual tolerance	Truncate coefficients before verification

A reproducible benchmark should begin with an exact finite-cone fixture and then use a frozen one-dimensional generalized-free-boson correlator for the conformal-block stage. The fixture, derivative order, precision, and tolerances must be stated explicitly.

Failure tests

Grid test. Perturb the functional so it stays positive at stored nodes but becomes negative between them. A node-only verifier must reject its own scope.

PSD test. Use a $2\times2$ matrix with positive diagonal entries and negative determinant. Componentwise positivity is not matrix positivity.

Basis test. Rescale one derivative without transforming $a_{mn}$. The identity normalization and signs change.

Next, Block Approximations and Semidefinite Programs constructs continuum-positive representations, while Solver Certificates and Independent Verification evaluates their output.

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
• Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 06 (2015): 174. DOI. Open PDF