FR04-A · A charge needs its flux

Gauss Law on a Lattice

Can every charge and electric link be chosen independently? They cannot be chosen independently. At each site, the outgoing electric flux minus the incoming flux must equal the matter charge plus any fixed external charge. Moving a particle therefore requires changing the adjoining flux.

Move electric flux between neighboring sites and find where Gauss law fails. Add a particle and its antiparticle, then find the links that make the configuration physical.

Violating sites $2$

Largest Gauss residual $1$

Total matter charge $0$

Electric energy $2$

Repair the broken link

Selected site

n

2

Flux to its right

E_n

0

Click a matter sphere to change its occupation. Drag a link up or down to change its flux.

Reading the scene and units

Cyan links have positive electric flux; violet links have negative flux. Gold and coral matter spheres show positive and negative charge; stronger color and glow mean a larger charge expectation. Their sizes mark sites. White arcs show occupation. Double gold rings mark fixed external sources, with signs identifying the charges. Site numbers begin at zero.

$a=\hbar=1$ sets lattice units. Spheres and links show basis values in the builder and quantum expectations in the evolving Labs.

Use left and right arrows to select a site, up and down to change its outgoing flux, and Space to toggle occupation. Rebuilding flux follows the left boundary; incompatible total charge remains visible at the right boundary.

The builder uses a six-site open chain and a unit electric cutoff. Rebuilding an edited state can require a larger cutoff; the notice then explains that the configuration exceeds the retained range.

Model assumptions

The open chain has six staggered-fermion sites, integer electric flux and two fixed external charges. The builder displays individual basis configurations. It tests the local Gauss constraint; it does not evolve a quantum state. This one-dimensional Abelian model is distinct from three-dimensional QCD.

Deterministic model calculation

Check each site

Two neighboring sites detect the broken link.

Selected site equation

G_2=-1

The record saves occupations, internal flux, fixed charges, boundary flux and every local residual.

Continue the course: Gauss law, strings and gauge protection

Start with the essentials

Focus question: Can every charge and electric link be chosen independently?
One-sentence intuition: A matter hop and the electric-flux change on its link are one allowed operation.

Core mathematical model

Charge relative to the staggered background

b_n=\frac{1-(-1)^n}{2},\qquad Q_n=N_n-b_n

Sites are numbered from zero. The reference product state fills odd sites and leaves even sites empty. An occupied even site carries positive charge; an empty odd site carries negative charge.

The local Gauss generator

G_n=E_n-E_{n-1}-Q_n-q_n^{\rm ext},\qquad G_n=0

Electric flux points to the right. Both boundary fluxes vanish. A physical configuration satisfies this relation at every site, including the external charges at the ends.

Common difficulties

A vanishing total is not enough

Typical misconceptionPositive and negative Gauss residuals cancel, so the state is physical.

Better mental modelEvery local residual must vanish separately. The sum only checks the total charge against the two boundary fluxes.

Run the experiment

01
Find the two affected sites
Start with A broken link. Drag the dark central link upward, or select its left site and increase Flux to its right.
What to observe: One changed link affects the outgoing flux of one site and the incoming flux of its neighbor. Restoring the link removes both violations.
02
Make a screening pair
Choose A screening pair. Compare the two gold and coral matter charges with An allowed string.
What to observe: The matter charges reduce the flux between them. All local Gauss residuals still vanish.
03
Change an occupation alone
Select a site and press Toggle occupation, or click the matter sphere in the scene. Then use Rebuild flux.
What to observe: An isolated occupation change usually violates the constraint and can change the total charge. Rebuilding from the left boundary reveals any incompatible charge at the right boundary.

Gauss Law on a Lattice

Check each site