Crossing and Positivity in One Dimension

In a reflection-positive one-dimensional CFT, an ordered identical-scalar correlator is a positive sum of $SL(2,\mathbb R)$ blocks. Crossing then becomes a convex feasibility problem: a putative spectrum is impossible if a linear functional has one sign on every allowed block, including the identity. The force of the conclusion comes from the complete list of hypotheses, not from crossing symmetry alone.

Required background. One-Dimensional Conformal Blocks and Casimir Equations fixes the physical block and normalization. Crossing and Positivity supplies reflection-positive OPE coefficients. Helpful background. Bounded and Compact Operators clarifies continuity and interchange of limits.

The positive crossing sum rule

For an identical Hermitian bosonic primary $\phi$ of dimension $\Delta_\phi$, define

$$F_\Delta(z)=z^{-2\Delta_\phi}g_\Delta(z) -(1-z)^{-2\Delta_\phi}g_\Delta(1-z).$$

Crossing on $0<z<1$ is

$$\sum_{\mathcal O\in\phi\times\phi} a_{\mathcal O}F_{\Delta_{\mathcal O}}(z)=0, \qquad a_{\mathcal O}=\lambda_{\phi\phi\mathcal O}^2\ge0, \qquad a_{\mathbf1}=1.$$

Each $F_\Delta$ is odd under $z\leftrightarrow1-z$. At $z=1/2$, only odd derivatives provide independent scalar equations. Degenerate primaries contribute through their summed weight $a_\Delta$.

The nonnegative coefficients require all of the following: a positive Hilbert-space inner product, Hermitian identical external operators, a reflection-compatible ordering, unit two-point normalization, and a convergent OPE in the domain where a functional is applied. For nonidentical operators the coefficients form positive-semidefinite matrices in a mixed system; for nonunitary theories even scalar coefficients need not be positive.

Separating a forbidden cone

Let $\alpha$ be a real linear functional on crossing vectors. Suppose a proposed gap $\Delta_{\rm gap}$ is such that

$$\alpha(F_0)>0, \qquad \alpha(F_\Delta)\ge0 \quad\text{for every }\Delta\ge\Delta_{\rm gap}.$$

If the identity is the only operator below the gap, applying $\alpha$ to crossing gives a sum of nonnegative terms with a strictly positive identity contribution. It cannot vanish. The assumed gap is excluded.

This is a separation theorem, not a construction theorem. Failure to find $\alpha$ in a finite search space does not produce a CFT, and finding an allowed point in a truncated system does not prove that an untruncated spectrum exists.

The finite-dimensional numerical version of this convex separation argument, including derivative functionals and positivity domains, is reviewed in Poland, Rychkov, and Vichi 2019, §§4–7. Analytic one-dimensional functionals require the stronger endpoint, swapping, and Regge controls constructed by Mazáč and Paulos 2019, §§3–5.

A transparent derivative certificate

Take $\Delta_\phi=1$ and define

$$\alpha[f]=f'''\!\left(\frac12\right)-49f'\!\left(\frac12\right).$$

With the block normalization of the preceding page,

$$\alpha(F_0)=32, \qquad \alpha(F_\Delta)>0\quad\left(\Delta\ge\frac92\right).$$

Here is a short proof of the second sign. Put $C_\Delta=\Delta(\Delta-1)$ and $g=g_\Delta(1/2)$. The Casimir equation reduces the functional to

$$\alpha(F_\Delta) =2g\left[ (32C_\Delta+28)\frac{g_\Delta'(1/2)}{g_\Delta(1/2)} -(384C_\Delta-16) \right].$$

The Euler representation defines a positive probability measure

$$d\mu_\Delta(t)\propto t^{\Delta-1}(1-t)^{\Delta-1} \left(1-\frac t2\right)^{-\Delta}dt, \qquad 0<t<1,$$

and gives

$$\frac{g_\Delta'(1/2)}{g_\Delta(1/2)} =2\Delta\left[1+ \mathbb E_{\mu_\Delta}\!\left(\frac{t}{2-t}\right) \right].$$

The beta weight before the last factor is symmetric about $1/2$, while $(1-t/2)^{-\Delta}$ is increasing. Therefore $\mathbb E_{\mu_\Delta}(t)\ge1/2$. Since $t/(2-t)$ is increasing and convex, Jensen’s inequality yields

$$\mathbb E_{\mu_\Delta}\!\left(\frac{t}{2-t}\right)\ge\frac13, \qquad \frac{g_\Delta'(1/2)}{g_\Delta(1/2)}\ge\frac{8\Delta}{3}.$$

For $\Delta\ge9/2$ this ratio is at least $12$, and substitution leaves a strictly positive bracket. Hence any unitary solution with these hypotheses contains a nonidentity primary with $\Delta<9/2$. This deliberately low-order certificate is not optimal: analytic extremal functionals prove the sharper $\Delta_{\rm gap}\le2\Delta_\phi+1$ for the integer and half-integer cases covered by the theorem, with generalized-free fermion data saturating the bound Mazáč 2017, §§3–5.

The hierarchy is pedagogically useful. A short derivative functional shows exactly why convex separation yields a rigorous exclusion. A nonlocal analytic functional can be optimal, but then its kernel, endpoint behavior, swapping with the OPE sum, and Regge assumptions must all be proved.

From derivative searches to certified bounds

A finite derivative ansatz has the form

$$\alpha[f]=\sum_{m=0}^{N}c_m \,\partial_z^{2m+1}f(z)\big|_{z=1/2}.$$

To turn a numerical candidate into a certificate:

1. fix $\Delta_\phi$, all symmetry or parity sectors, and the proposed gap;
2. compute block derivatives with controlled truncation and precision;
3. normalize $\alpha(F_0)=1$ or another declared nonzero value;
4. prove positivity on each continuous allowed interval, not merely on a grid;
5. bound the large-$\Delta$ tail analytically;
6. rerun at higher precision and with an independent block representation.

For polynomial or rational approximations, interval arithmetic can cover compact dimension ranges while asymptotic estimates cover the tail. A grid scan is a diagnostic, never the final proof: a narrow negative region between grid points invalidates the exclusion.

Functional sum rules and their domain

If $\omega$ is represented by an integral kernel, schematically

$$\omega[f]=\int_{\mathcal C}dz\,h(z)f(z),$$

then the formal step

$$\omega\!\left[\sum_\Delta a_\Delta F_\Delta\right] =\sum_\Delta a_\Delta\omega[F_\Delta]$$

requires uniform bounds that control OPE endpoints and any contour at infinity. Different Regge behavior can change which kernels are admissible. Complete functional bases and Polyakov-block reorganizations belong to Analytic Functionals and Polyakov Blocks; the present chapter uses only the crossing-separation logic.

The figure makes the dependency order explicit. Read it from left to right: an ordering fixes the real cross-ratio interval and the crossing vector, reflection positivity supplies the cone, and a functional separates a proposed spectrum only after its domain and sign conditions are justified.

One-dimensional crossing and functional separation. The ordering sector and statistics determine the crossing equation; positivity is an additional reflection assumption, and a functional certificate applies only on its proved spectral domain. The generalized-free branch is an exact normalization test. The diagram is schematic and not to scale.

The same dependencies in structured form are:

Stage	Mathematical object	Required condition	Valid conclusion
Ordering	$x_1<x_2<x_3<x_4$ and $0<z<1$	Statistics and prefactor fixed	One real boundary value of the correlator
Blocks	$g_\Delta(z)=z^\Delta{}_2F_1(\Delta,\Delta;2\Delta;z)$	Physical Casimir solution and OPE normalization	One conformal-family contribution
Crossing	$\sum_\Delta a_\Delta F_\Delta(z)=0$	Compatible continuation and complete channel sum	Equality of two ordered channel expansions
Positive cone	$a_\Delta\ge0$	Hermitian external operator and reflection-positive sector	Convex spectral feasibility problem
Functional	$\alpha(F_0)>0$ and $\alpha(F_\Delta)\ge0$ on the assumed spectrum	Swapping, endpoint, and continuous-interval signs proved	Exclusion of that spectrum, not construction of a CFT

What a bound does and does not say

Result	Justified conclusion	Not justified
A positive functional exists	Every CFT satisfying the stated assumptions violates the proposed gap or coefficient range	A particular model realizes the boundary
No functional is found at order $N$	The chosen finite search did not exclude the point	The point is allowed in the exact problem
Successive derivative orders converge numerically	Evidence for a limiting bound, with a stated extrapolation	A theorem without error control
An extremal spectrum solves truncated equations	A candidate explanation of boundary data	Existence, uniqueness, or locality of a full CFT

This distinction prevents the two most common overclaims: turning exclusion into existence, and turning a sharp boundary feature into identification with a named model.

Common pitfalls

Forgetting the identity sign. The normalization of $F_\Delta$ and $\alpha$ can be reversed, but the identity must lie strictly on the forbidden side while all assumed nonidentity blocks lie on the other.

Checking positivity only at sample dimensions. The spectrum is not confined to the sampling grid. Certify every continuous interval and the asymptotic tail.

Using scalar positivity in a mixed system. Mixed correlators carry matrices of OPE products. Positivity is matrix-valued and depends on the chosen reflection pairing.

Exercises

Using $F_\Delta(1-z)=-F_\Delta(z)$, show that all even derivatives vanish at $z=1/2$.

Solution

Set $z=1/2+y$. Then $F_\Delta(1/2-y)=-F_\Delta(1/2+y)$, so its Taylor series contains only odd powers of $y$. Every even derivative at $y=0$ therefore vanishes.

Explain why the derivative certificate above excludes a gap $\Delta_{\rm gap}\ge9/2$.

Solution

Under that assumption the identity contributes $a_0\alpha(F_0)=32>0$, and every nonidentity primary contributes $a_\Delta\alpha(F_\Delta)\ge0$. Their sum is strictly positive, contradicting the functional applied to crossing, which must give zero.

References

• Mazáč, D. “Analytic Bounds and Emergence of $\mathrm{AdS}_2$ Physics from the Conformal Bootstrap.” Journal of High Energy Physics 2017, 146 (2017). arXiv. DOI.
• Mazáč, D., and Paulos, M. F. “The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices.” Journal of High Energy Physics 2019, 162 (2019). arXiv. DOI.
• Poland, D., Rychkov, S., and Vichi, A. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91, 015002 (2019), §§4–7. arXiv. DOI.