---
title: "Free fields solve the crossing equation exactly"
description: "Generalized free fields solve crossing exactly, with every OPE coefficient positive. They give the bootstrap a test case and mark where bounds can end."
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last_modified: "2026-10-29"
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[Conformal bootstrap](https://sandboxphysics.com/stories/conformal-bootstrap/)

L2 · Chapter 4

# Free fields solve the crossing equation exactly

Sandbox Physics · 2026-10-29

Before asking which spectra are ruled out, it helps to have spectra that certainly work. Generalized free fields are the simplest. Their four-point function is built from two-point functions alone, it obeys crossing by construction, and its operator list can be read off line by line. They will reappear in the next chapter as the theory that sits exactly on the one-dimensional bound.

## Before you begin

The crossing equation and the block expansion from the previous chapter, and Taylor series. You do not need to know how free fields arise in a Lagrangian.

- Write the closed-form four-point functions of the generalized free boson and fermion, and their exchanged spectra.
- Read OPE coefficients off a Taylor series, and see why truncating the sum spoils crossing away from z equal to one half.

## Correlators built from pairs

A generalized free field has a four-point function that is the sum of the three ways to pair four points, each pair contributing a two-point function. For a fermion one pairing enters with a minus sign. On a line this gives the closed forms below.

Swapping $z$ and $1-z$ only permutes the three pairings, so crossing holds exactly. Nothing about the exchanged operators was put in by hand; they follow from expanding the closed form.

$$
A_\psi(z)=1+\Bigl(\frac{z}{1-z}\Bigr)^{2\Delta_\psi}-z^{2\Delta_\psi},\qquad A_\phi(z)=1+\Bigl(\frac{z}{1-z}\Bigr)^{2\Delta_\phi}+z^{2\Delta_\phi}
$$

These hold for any external dimension. The worked examples use $\Delta=1/2$.

Source: D. Gaiotto · D. Mazáč

## The exchanged operators

Expanding each closed form in blocks gives a ladder of two-particle operators. The fermion exchanges dimensions $2\Delta_\psi+1$, $2\Delta_\psi+3$ and so on. The boson starts one step lower, at $2\Delta_\phi$.

Every coefficient $c_n^2$ is positive and falls quickly with $n$. For $\Delta=1/2$ the fermion values are $1$, $1/10$, $1/126$, $1/1716$, and the boson values are $2$, $1/3$, $1/35$, $1/462$.

$$
\Delta_n^{\psi}=2\Delta_\psi+2n+1,\qquad \Delta_n^{\phi}=2\Delta_\phi+2n,\qquad n=0,1,2,\dots
$$

The fermion has no operator below its first rung. That empty interval is what the bound in the next chapter is about.

Source: D. Gaiotto

## Worked example: reading coefficients off a Taylor series

Take the boson with $\Delta_\phi=1/2$. Its closed form minus the identity is $2z+z^2+z^3+\dots$. The lightest block is $k_1=-\log(1-z)$. Matching the $z$ term fixes $c_1^2=2$.

Then $2k_1$ gives $z^2$ with coefficient $1$, which already matches, so no operator of dimension $2$ is needed. At $z^3$ it gives $2/3$ against the required $1$. The gap is filled by a block $k_3=z^3+\dots$ with $c_3^2=1/3$.

The same bookkeeping on the fermion gives $c_2^2=1$ from the $z^2$ term and $c_4^2=1-9/10=1/10$ from the $z^4$ term. The numbers agree with the general formula to every digit printed, which is an independent check of both.

With these coefficients, two boson operators already give $A(0.3)=1.728406$ against the exact $1.728571$. The full sum matches to six digits.

$$
A_\phi(z)-1=2z+z^2+z^3+\dots,\qquad k_1(z)=-\log(1-z)=z+\tfrac12z^2+\tfrac13z^3+\dots
$$

Matching powers of $z$ term by term fixes one coefficient at a time and shows which dimensions are absent.

Source: D. Gaiotto · D. Mazáč

## What truncation does

Keep only the first three fermion operators. At $z=1/2$ the crossing residual is exactly zero, because both sides are evaluated at the same point. At $z=0.8$ it is $-4.46\times10^{-3}$, and at $z=0.9$ it reaches $-3.36\times10^{-2}$. Ten operators push the residual at $z=0.9$ down to $-5.87\times10^{-6}$.

This is the honest version of a numerical statement: a finite spectrum satisfies crossing only approximately, and the error depends on where you look. At $z=1/2$ both channels converge equally fast, so numerical studies take derivatives of the crossing equation there.

Source: D. Mazáč · D. Simmons-Duffin

## Try it in the Lab

### [Compare the fermion and the boson](https://sandboxphysics.com/labs/crossing-two-expansions/)

1. Open the free-boson preset and note the first exchanged operator. Compare it with the free-fermion preset.
2. Predict how the mismatch curve will look if the first coefficient is wrong, then pick that preset.
3. With the too-few-operators preset, drag the point toward the right end and watch the residual grow.

## Exercises with explained answers

1. For a fermion with $\Delta_\psi=0.8$, what is the dimension of the first exchanged operator?

   Show the answer

   $2\Delta_\psi+1=2.6$. No operator lies between the identity and this value.

2. Using $A_\psi-1=z^2+z^3+z^4+\dots$ for $\Delta_\psi=1/2$ and $k_2=z^2(1+z+\tfrac{9}{10}z^2+\dots)$, show that no operator of dimension $3$ appears.

   Show the answer

   With $c_2^2=1$, the block $k_2$ already supplies $z^3$ with coefficient $1$, exactly what $A_\psi$ needs. A dimension-3 operator would add an extra $z^3$, so its coefficient must vanish.

3. Flip the sign of the fermion coefficient $c_4^2$ and keep everything else. Would crossing still hold?

   Show the answer

   No. The crossing residual at $z=0.3$ becomes $0.114$. Unitarity would also be violated, because it requires every $c^2\ge 0$.

[Previous: One four-point function expanded two ways gives the crossing equation](https://sandboxphysics.com/stories/conformal-bootstrap/four-point-function-two-expansions/) [Next: How one linear functional rules out a whole spectrum](https://sandboxphysics.com/stories/conformal-bootstrap/linear-functionals-and-exclusion/)

## Sources and what we recomputed

1. [D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982v1 (2016)](https://arxiv.org/abs/1602.07982v1)

   Version 1. Eqs. (157)-(158): the four-point function as a sum over conformal multiplets, one conformal block each. Eqs. (164)-(168): the Casimir equation for blocks and their closed form in two dimensions, built from the one-variable function k. Eqs. (194)-(199), Sections 10.5-10.7: the crossing equation for identical scalars as a sum of vectors, the derivative functionals and the assumptions behind the scalar-gap bound. Fig. 26: 2D crossing vectors. Fig. 27 and Table 1: 2D bounds for derivative cutoffs 6 to 28 with the kink at the Ising point. Fig. 28: the 3D bound with its kink.

2. [D. Mazáč, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, arXiv:1611.10060v1; JHEP 04 (2017) 146](https://arxiv.org/abs/1611.10060v1)

   Version 1. Eqs. (2.12)-(2.17): the four-point function of identical operators on a line, the cross-ratio, the one-variable block and the crossing equation. Eq. (2.18) and Fig. 2: the numerical gap bound, which the text says converges to twice the external dimension plus one as derivatives are added. Eqs. (2.19)-(2.23): the generalized free fermion that saturates it and the zeros of the extremal functional. Fig. 1: the action of a typical extremal functional.

3. [D. Gaiotto, D. Mazáč and M. F. Paulos, Bootstrapping the 3d Ising twist defect, arXiv:1310.5078v2; JHEP 03 (2014) 100](https://arxiv.org/abs/1310.5078v2)

   Version 2. Section 4.1: one-dimensional generalized free fields. Eqs. (4.17) and (4.20) give the exchanged dimensions and OPE coefficients of the fermionic and bosonic solutions; the course resums them against the closed-form correlators.