---
title: "Crossing Symmetry: One Four-Point Function, Two Expansions Interactive Lab | Sandbox Physics"
description: "Does the same operator data give the same correlator when you expand around either pair of points? Yes, when the operator data are consistent."
canonical_url: "https://sandboxphysics.com/labs/crossing-two-expansions/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/crossing-two-expansions/"
last_modified: "2026-10-30"
---

FR37 · Conformal bootstrap

# Crossing Symmetry: One Four-Point Function, Two Expansions

**Does the same operator data give the same correlator when you expand around either pair of points?** Yes, when the operator data are consistent. A four-point function can be built by fusing either pair of operators first, and the two sums must match at every point. The exact data of a free fermion or free boson pass this test everywhere; moving one operator, changing one coefficient or keeping too few terms makes the two sums disagree.

Edit the operators of a 1D free field and compare the s- and t-channel sums of its four-point function. Consistent data agree at every cross-ratio.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Four identical operators sit on a line: the first at the origin, the second at a movable point, the third at one and the fourth at infinity. The s-channel sums conformal blocks for the operators exchanged between the first pair; the t-channel sums the same data for the other pairing. The default data are the exact operator dimensions and OPE coefficients of a generalised free fermion or boson (Gaiotto, Mazáč and Paulos 2013, Eqs. 4.17 and 4.20); the dashed curve is the closed-form correlator. Editing the leading operator or cutting the sum short gives data that no longer solve crossing. Everything is computed; no measured data are shown.

Two expansions of one correlator

Crossing residual, in powers of ten

More numbers and how they are computed

- **Free-field closed form at the same point**: $0$
- **Where the mismatch is largest**: $0$
- **Second exchanged operator**: $0$
- **Operators kept in each sum**: $0$
- **External operator dimension**: $0$

The exported file lists every operator with its edited and free-field dimension and coefficient, plus the two sums, the closed form and the residual on 161 cross-ratios.

Physics tutorial

## One correlator, two expansions

**Background** At a critical point the theory is described by its operators: their scaling dimensions and how strongly each appears when two operators approach. For four operators on a line, everything depends on one number, the cross-ratio.

**Why it matters** You can fuse the first two operators, or the second and third. Both routes must give the same answer. That single requirement is strong enough to test any proposed list of operators.

Start with the essentials

- **Focus question**: Does the same operator data give the same correlator when you expand around either pair of points?
- **One-sentence intuition**: The free fermion and free boson data make the two sums agree at every cross-ratio. Moving one operator, changing one coefficient or keeping too few terms breaks the agreement, and the mismatch is largest where one expansion converges slowly.

### Core mathematical model

#### Conformal block on a line

$$
k_\Delta(z)=z^{\Delta}\,{}_2F_1(\Delta,\Delta;2\Delta;z)
$$

Each exchanged operator of dimension $\Delta$ contributes its block, weighted by the squared OPE coefficient $c^2$.

#### Crossing equation

$$
(1-z)^{2\Delta_{\mathcal O}}\Bigl[1+\sum_{\Delta}c_\Delta^2\,k_\Delta(z)\Bigr]=z^{2\Delta_{\mathcal O}}\Bigl[1+\sum_{\Delta}c_\Delta^2\,k_\Delta(1-z)\Bigr]
$$

The left side expands around $z=0$, the right side around $z=1$. The Lab plots their difference as the crossing residual.

#### Free fermion data

$$
\Delta_j=2\Delta_{\mathcal O}+2j+1,\qquad c_j^2=\frac{2\,(2\Delta_{\mathcal O})_{2j+1}^2}{(2j+1)!\,(4\Delta_{\mathcal O}+2j)_{2j+1}}
$$

These closed forms resum to $A(z)=1+\bigl(\tfrac{z}{1-z}\bigr)^{2\Delta_{\mathcal O}}-z^{2\Delta_{\mathcal O}}$.

### Common difficulties

#### Agreement at one point is not crossing

**Typical misconception** If the two channels agree at the symmetric point, the data are consistent.

**Better mental model** At $z=\tfrac12$ the residual vanishes for any data by symmetry. Crossing is a statement about every $z$; its odd derivatives at the symmetric point carry the content.

### Run the experiment

1. 01

   #### Watch both channels

   Let the second point sweep along the line.

   **What to observe:** The cyan bars shrink to a few operators near the first point, the amber bars near the third. Both totals stay equal.

2. 02

   #### Move one operator

   Drag the flashing operator sideways, or choose Move one operator.

   **What to observe:** The residual grows away from the symmetric point and the badge reports a mismatch.

3. 03

   #### Cut the sum short

   Keep only two operators.

   **What to observe:** The s-channel still works near the first point, but fails as the second point approaches the third.

Physical conclusion

Crossing turns a list of operators into a checkable equation: consistent data agree everywhere, and any wrong entry shows up as a residual.

## Related experiments and articles

Conformal bootstrap · Reading path

### From a critical chain to the kink in the 2D bound

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **One four-point function expanded two ways gives the crossing equation**](https://sandboxphysics.com/stories/conformal-bootstrap/four-point-function-two-expansions/)
2. [Article **Free fields solve the crossing equation exactly**](https://sandboxphysics.com/stories/conformal-bootstrap/free-fields-as-exact-solutions/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)