--- title: "Turning allowed configurations into a quantum model" description: "Build the finite Hamiltonian, derive why matter hopping must change link flux, and enumerate the physical basis before evolving a state." canonical_url: "https://sandboxphysics.com/stories/lattice-gauge/hamiltonian-basis/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/lattice-gauge/hamiltonian-basis/" last_modified: "2026-10-27" --- [Lattice gauge fields and string breaking](https://sandboxphysics.com/stories/lattice-gauge/) L2 · Chapter 2 # Turning allowed configurations into a quantum model Sandbox Physics · 2026-10-27 A charged particle cannot hop to its neighbor and leave the connecting electric field untouched. A gauge-preserving hop changes both. Enumerating the allowed configurations and connecting them by these hops produces a finite quantum model we can solve. ## Before you begin Complete the site-by-site charge addition in Chapter 1. A complex amplitude has a magnitude and a phase; its squared magnitude gives a probability. This chapter explains how a matrix connects those amplitudes. [Review the preceding chapter](https://sandboxphysics.com/stories/lattice-gauge/local-gauss/) - Explain the three terms in the Hamiltonian. - Check Gauss’s law at both ends of a hop. - Construct a finite matrix from allowed configurations. ## Define the finite world first We use an open one-dimensional $\mathrm{U}(1)$ model with sites numbered from zero. Matter occupations are zero or one. Internal link fluxes are integers. The browser retains four, six or eight sites and flux values $-E_{\max},\ldots,E_{\max}$, with a cutoff of one or two. Both boundary fluxes vanish and the external sources remain fixed. Set the lattice spacing and reduced Planck constant to one. Quote energies in units of a reference hopping strength and times in its inverse. The parameter $w$ sets hopping, $m$ sets staggered mass and $g^2$ sets the electric-energy cost. Converting these lattice parameters into laboratory units requires an apparatus mapping. Reading: David Tong · Sandbox Physics FR04 finite Hamiltonian ## Three contributions to one energy operator The Hamiltonian is a matrix. Its diagonal entries assign energies to configurations. Its off-diagonal entries transfer amplitude between them. The mass and electric terms are diagonal in our configuration basis. The hopping term moves one occupation between neighbors. The operator $\hat f_n$ removes a fermion and $\hat f_n^\dagger$ inserts one. Their product $\hat n_n=\hat f_n^\dagger\hat f_n$ measures occupation. The link operator $\hat U_n$ raises electric flux by one. The Hermitian conjugate supplies the reverse hop, making the matrix Hermitian and its energies real. $$ \hat H=-w\sum_{n=0}^{N-2}(\hat f_n^\dagger\hat U_n\hat f_{n+1}+\mathrm{h.c.})+m\sum_{n=0}^{N-1}(-1)^n\hat n_n+\frac{g^2}{2}\sum_{n=0}^{N-2}\hat E_n^2 $$ Hopping connects configurations. Positive mass favors occupied odd sites. Each internal link pays an energy proportional to flux squared. The external sources enter through the Gauss constraint that selects configurations. ## A leftward hop raises the link flux Move a particle from the right site to the left. Left occupation increases by one; right occupation decreases by one. Raising the connecting flux by one increases the left site’s flux difference and decreases the right site’s flux difference. Each changes by exactly the charge change at that site. The reverse process moves the particle right and lowers the flux. Omitting the link operator would create a pair of local violations. Preservation of the local law is a property of the matrix element that connects the two configurations. At an electric cutoff, a raising or lowering operation that would leave the retained range is removed. This hard truncation can preserve Gauss’s law while changing the link operator’s algebra at the edges. It therefore needs its own convergence check. $$ [\hat H,\hat G_n]=0\quad\text{for every site }n $$ Every allowed hop in the defined finite model preserves the local Gauss sector. Chapter 4 follows its time evolution. ## Build the electric fields from the matter Four half-filled sites have six occupation patterns. With endpoint sources, reconstruct the links for each pattern. A cutoff of one retains only $(0,1,0,1)$ and $(0,0,1,1)$. Each other pattern requires flux two somewhere. Raising the cutoff to two retains all six. Each retained configuration becomes a basis vector. A quantum state assigns a complex amplitude to every vector. The squared magnitudes add to one. Since electric flux is diagonal in this basis, its expectation is the probability-weighted average over configurations. This construction excludes forbidden configurations before building the Hamiltonian. The remaining space is the physical subspace. It saves memory and enforces the constraint exactly. Chapter 6 compares it with keeping extra configurations and assigning violations an energy penalty. $$ |\psi\rangle=\sum_{a=1}^{D}c_a|a\rangle,\qquad \sum_a|c_a|^2=1,\qquad \langle E_n\rangle=\sum_a|c_a|^2E_n(a) $$ $D$ is the number of retained physical configurations. The last expression applies because electric flux is diagonal in this basis. ## What this Hamiltonian describes The finite model describes a one-dimensional Abelian gauge field coupled to staggered fermions. It supports questions about local constraints, screening and string release. Three-dimensional QCD has a non-Abelian color field and a different spatial structure. The calculation here concerns string breaking in the stated one-dimensional model. A finite chain omits the short-distance modes of a continuous field. An electric cutoff also removes potentially accessible configurations. Understanding the finite Hamiltonian first, then enlarging its electric range and size, gives a controlled route toward harder calculations. ## Try the prediction ### [Check the same hopping rule on six sites](https://sandboxphysics.com/labs/gauss-law-builder/) 1. Choose “An allowed string” and reconstruct every link on the six-site chain. 2. Choose “A screening pair” and identify the two changed occupations and the changed field link. 3. Compare “Charges without flux” with the allowed screened configuration and explain its local violations. ## Exercises and explanations 1. Why does moving a particle from an odd site to an even site cost $2m$ in mass energy? Read the explanation The occupied odd site contributes $-m$. Removing that particle raises energy by $m$. Occupying the even site adds another $+m$, so the total increase is $2m$. 2. Does a superposition of physical basis configurations still satisfy the local constraint? Read the explanation Yes. Each Gauss generator is linear. It annihilates every retained basis vector and therefore annihilates any superposition of them. The full state remains inside the physical subspace. [Previous: Building an electric-flux chain with Gauss’s law](https://sandboxphysics.com/stories/lattice-gauge/local-gauss/) [Next: The energy cost of stretching an electric-flux string](https://sandboxphysics.com/stories/lattice-gauge/flux-energy/) ## Sources and calculation scope 1. [Sandbox Physics FR04 finite Hamiltonian, conventions and calculation scope](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr04/model-specification.md) Defines the course’s zero-based staggered charge, open boundaries, integer-flux hard cutoff, source geometry and deliberate gauge-breaking error. The accompanying paper-reproduction and numerical-validation notes specify which finite-model calculations were checked independently. This is our model definition and calculation record. 2. [David Tong, Gauge Theory (2018 lecture notes)](https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf) Sections 2.5.1 and 2.7 discuss static probes and dynamical matter; sections 4.2.1 and 4.3.1 introduce lattice links and two-dimensional staggered fermions; section 7.1 gives the electrostatic energy in one spatial dimension. Our finite Hamiltonian, indexing and hard cutoff are defined explicitly in this course.