Hamiltonian Lattice Field Theory

Hamiltonian lattice field theory regulates space while retaining continuous real time, as in the canonical lattice-gauge construction of Kogut and Susskind 1975, pp. 395–408. It makes states, constraints, and unitary evolution explicit, but it also introduces several independent approximations: spatial spacing and volume, local Hilbert dimension, basis truncation, constraint enforcement, time evolution, and finite observation windows. This chapter develops the formulation and shows how its spectra and observables are cross-validated against Euclidean lattice results.

Choose the missing Hamiltonian step

Question	Page	Decisive evidence
What exactly defines the regulated theory?	Regulated Hamiltonian field theory	Algebra, representation, domain, state, constraints, observables, target limits
How are gauge dynamics and Gauss’s law built?	Lattice gauge Hamiltonians and Gauss’s law	$[H,G_x]=0$ including matter and boundary charge
When does a Euclidean action define $H$ ?	Transfer matrices between Euclidean and Hamiltonian QFT	Reflection positivity and a positive transfer operator
How may each site or link be made finite?	Local Hilbert-space regulators	Exact algebraic relations plus convergence in local dimension
How is the physical gauge sector represented?	Physical gauge Hilbert spaces	Independent state count and quantified constraint leakage
How are response or scattering data extracted?	Real-time evolution and observable extraction	Preparation, time-step, window, boundary, and asymptotic controls
Do Hamiltonian and Euclidean regulators share a target?	Continuum limits and Euclidean cross-validation	Matched dimensionless observables and a discrepancy decomposition

The order is important. A unitary finite matrix can be simulated exactly and still be the wrong QFT because its physical sector, counterterms, or regulator-removal path is wrong.

The regulated specification

A Hamiltonian calculation should state

\mathcal C_H= \{\mathcal A_a,\mathcal H_{a,d},\mathcal D(H),H_{a,d}(\mathbf g), G_x,\rho_0,\{O_i\},L,\text{boundaries},\text{limit order}\}.

$\mathcal A_a$ is the regulated local algebra, $\mathcal H_{a,d}$ its representation with local dimension $d$ when finite, $\mathcal D(H)$ the operator domain, $G_x$ any constraints, and $\rho_0$ the prepared state. The bare parameters $\mathbf g$ are tuned by renormalized conditions rather than held fixed mechanically as $a$ or $d$ changes.

For gauge theories, the physical space is not the full link-by-link tensor product:

\mathcal H_{\mathrm{phys}}= \{\lvert\psi\rangle:G_x\lvert\psi\rangle=q_x\lvert\psi\rangle \ \text{for all }x\}.

The prescribed charges $q_x$ include dynamical matter and boundary flux. Exact projection, gauge-invariant bases, energy penalties, and postselection have different guarantees and must not share one label.

Two validation branches

A positive Euclidean transfer matrix $T$ can yield $H=-a_t^{-1}\log T$ , so Euclidean exponential decay and Hamiltonian energies can be compared at the same finite regulator. Direct real-time evolution gives unequal-time and retarded observables unavailable from noisy analytic continuation. The two branches meet only after their conventions, states, operators, and renormalized parameters are matched.

The chapter’s shared physical-sector diagram shows where exact statements end. In particular, a penalty Hamiltonian does not turn a leaky state into a physical state, and a Euclidean action without reflection positivity does not furnish a positive-metric Hamiltonian by this route.

Independent limits

A schematic target observable has the dependence

O=O(a,L,d,\Lambda_B,\delta t,T,\epsilon_G;\mathbf g),

where $\Lambda_B$ is a basis cutoff, $\delta t$ an evolution step, $T$ a measurement window, and $\epsilon_G$ a constraint-violation diagnostic. A credible calculation varies each relevant axis while the others are controlled. Typical ordering is

\delta t\to0,\quad T\to\infty,\quad d,\Lambda_B\to\infty, \quad L\to\infty,\quad a\to0,

but nonuniform convergence can require joint fits. The order is part of the scientific claim, especially when a local cutoff changes symmetry, when long-time errors grow, or when finite-volume scattering precedes an infinite-volume limit.

Chapter standard

For every spectrum or time-dependent observable, report the finite regulated Hamiltonian, physical-sector definition, state preparation, operator matching, time and boundary conditions, all cutoff axes, and at least one independent benchmark. Label a result “regulator level” until the required axes are actually extrapolated.

Chapter outcomes

Given a proposed Hamiltonian calculation, identify which of the seven chapter stages is missing and name an observable test that would expose the omission.
Write a complete regulated calculation specification and separate its spatial, local-Hilbert, basis, constraint, time-evolution, observation-window, and continuum limits.

References

Kogut, J. B., and Susskind, L. (1975). Hamiltonian formulation of Wilson’s lattice gauge theories. Physical Review D, 11, 395–408. DOI.