research35 min studyLesson 736 in Physics

Crossing symmetry and reflection positivity

Combine alternative OPE channels with the precise positivity needed for bootstrap inequalities.

Before you begin

Conformal blocks and Casimir equations

On this page

The question for today

What you’ll understand

Derive the identical-scalar crossing prefactors.
Distinguish reflection positivity from pointwise positivity.
Check a free-field four-point function and diagnose an identity-only truncation.

The same Euclidean four-point correlator can be expanded by fusing different pairs of operators. Crossing symmetry requires those descriptions to agree where they represent the same analytically defined correlator. Reflection positivity supplies additional inequalities by interpreting special reflected configurations as Hilbert-space norms. These principles work together but make different statements: a crossing-symmetric formula is not automatically reflection positive, and a few nonnegative coefficients do not establish a complete local conformal field theory.

Keep the distance prefactor when exchanging channels

Use the identical real scalar normalization and cross-ratios defined earlier, initially in a reflection-positive Euclidean CFT with dimension d at least three. Exchanging points one and three interchanges u and v but also changes the distance prefactor that was factored out. Comparing the complete correlators gives a crossing equation with external-dimension powers. Setting the two reduced functions equal without those powers would impose the wrong relation except in special cases.

Substituting the conformal block expansion turns crossing into an infinite linear relation among family contributions, with coefficients given by squared real OPE coefficients in this identical-operator setting. The identity contribution is fixed. Different exchanged dimensions and spins remain unknown CFT data. For mixed correlators, global-symmetry representations, multiple tensor structures or complex operators, positivity can instead involve matrices of coefficients and conjugations, so the simplest one-correlator sum cannot be copied universally.

v^{\Delta_\phi}\mathcal G(u,v)=u^{\Delta_\phi}\mathcal G(v,u),\qquad \sum_{\mathcal O}\lambda_{\phi\phi\mathcal O}^2\!\left[v^{\Delta_\phi}g_{\Delta,\ell}(u,v)-u^{\Delta_\phi}g_{\Delta,\ell}(v,u)\right]=0

This is crossing under 1 ↔ 3 with the fixed prefactor r12^(−2Δφ)r34^(−2Δφ). The identity block and its unit coefficient are included in the sum.↔ Scroll wide equations horizontally.

Translate reflected insertions into a norm

Euclidean reflection positivity pairs insertions on one side of a reflection surface with conjugate insertions on the other. For a linear combination of prepared states, the reflected correlator must give a nonnegative quadratic form. In radial quantization the reflection is inversion together with the appropriate operator weights and tensor action. A single correlator taking a positive value at one arbitrary configuration is a much weaker statement than positivity for every allowed state combination.

An orthonormal intermediate-state expansion of a suitable reflected four-point function explains why identical real-scalar family coefficients can be organized as nonnegative squares. It does not imply that every individual OPE coefficient is positive, nor that every crossing combination of blocks has positive sign at every point. The subtraction between channels is essential. Nonunitary conformal theories may satisfy crossing while failing these positivity assumptions, so a unitary bootstrap exclusion does not automatically apply to them.

|\Psi\rangle=\sum_i a_i|\Psi_i\rangle,\qquad \langle\Psi|\Psi\rangle=\sum_{i,j}a_i^*G_{ij}a_j\ge0,\qquad G\succeq0

G is the Gram matrix of radially or time-reflected state preparations, with conjugation and transformation factors included. Positive semidefiniteness is a condition on all coefficients ai, not just on diagonal entries.↔ Scroll wide equations horizontally.

Check an exact example before interpreting numerical bounds

For a normalized free real scalar in four dimensions, Wick contractions give a reduced four-point function made from three pairings. Its external dimension is one, and the resulting simple expression satisfies the crossing prefactor relation exactly. The identity term alone does not satisfy the relation at generic unequal cross-ratios. This illustrates why additional families are needed even in a free theory and why checking only one symmetric point can miss a failed channel relation.

The free example is a genuine specified model; an arbitrary crossing-symmetric trial function is not thereby a construction of another local theory. To test positivity one also needs the appropriate expansion, norms and operator data. The probability lab can illustrate normalization and nonnegative weights in an exact binomial distribution. Those outcome probabilities have no conformal blocks or channel exchange, so their positivity neither solves crossing nor certifies a candidate CFT.

\mathcal G_{\rm free}(u,v)=1+u+\frac{u}{v}\qquad(d=4,\ \Delta_\phi=1)

The expression follows from the three Wick pairings for a unit-normalized free real scalar. It is an exact model benchmark, not a universal interacting four-point function.↔ Scroll wide equations horizontally.

Work it through

Verify both sides of crossing

For the displayed free scalar take $u=1/4$ , $v=1/2$ . Find the reduced function in both channel orders and check $v\mathcal G(u,v)=u\mathcal G(v,u)$ .

In the first order, $\mathcal G(1/4,1/2)=1+1/4+(1/4)/(1/2)=7/4$ . The identity alone would have supplied only the first one.
After exchanging the ratios, $\mathcal G(1/2,1/4)=1+1/2+2=7/2$ . The reduced functions are unequal, as expected because their prefactors differ.
The weighted values are $(1/2)(7/4)=7/8$ and $(1/4)(7/2)=7/8$ . Equality holds for the complete crossing relation without requiring the reduced function itself to be symmetric.

The result

The reduced values are 7/4 and 7/2, while both correctly weighted crossing sides equal 7/8.

The living notebook

Explore the model

Interactive lab

Interactive lab · probability

Nonnegative weights are only one ingredient

Reset and set BOTH Success probability p = 0.25 and Trials n = 1; predict weights 0.75 and 0.25. Reset and set p = 0.75 and n = 1; predict exchanged weights, still totaling one. This is normalized binomial bookkeeping, not reflected CFT correlators, OPE coefficients or a crossing test.

Exact probabilityHighlighted outcomeMean np

Mean · np: 5
Variance · np(1−p): 2.5
Total probability: 1

Success probability p0.5Trials n10

t = 0 s

These are exact binomial probabilities for independent trials with one fixed success probability. Playback tours the distribution; it does not resample outcomes. 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 forPositivity must be imposed on the mathematical objects that the physical argument actually uses.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Reflection positivity concerns norms of all prepared state combinations with the appropriate reflection and conjugation. It is stronger and more structured than a pointwise sign check.

Your turn

Try it yourself

Keep $\Delta_\phi=1$ but approximate the reduced correlator by its identity contribution $\mathcal G=1$ . At $u=1/4$ , $v=1/2$ , compute the crossing residual $v\mathcal G(u,v)-u\mathcal G(v,u)$ . Would evaluating only at u = v detect this failure?

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

Give me a hint

The identity is a valid family contribution but generally not the complete four-point function.

Compare with the solution

The residual is $1/2-1/4=1/4$ , so this truncation violates crossing at the supplied point. At u = v the same antisymmetric residual vanishes identically, regardless of this missing physics, and would not expose the failure. Derivatives near a symmetric point can carry nontrivial crossing constraints, but the value of an automatically vanishing combination alone cannot certify the full equation or the adequacy of an identity-only approximation.

Keep the idea

Take it with you

Crossing compares complete correlators with channel-dependent prefactors.
Reflection positivity constrains reflected-state Gram matrices.
An exact model check and a numerical consistency condition are different evidence.

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