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Hamiltonian Lattice Field Theory

Hamiltonian lattice QFT regulates space while retaining continuous time. It makes states, Gauss constraints, and unitary evolution explicit, while introducing independent limits in spacing, volume, local Hilbert dimension, and observation time. This chapter connects that formulation to Euclidean transfer matrices and physical observables.

Enter this chapter

Section titled “Enter this chapter”

Begin with Regulated Hamiltonian Field Theory to specify the local algebra, Hilbert space, Hamiltonian, and observable together.

Topics

Section titled “Topics”
  1. Regulated Hamiltonian Field Theory — Defines the finite-regulator quantum system.
  2. Lattice Gauge Hamiltonians and Gauss’s Law — Constructs electric and magnetic gauge dynamics.
  3. Transfer Matrices between Euclidean and Hamiltonian QFT — States when a Euclidean kernel defines a positive Hamiltonian theory.
  4. Local Hilbert-Space Regulators, Quantum Links, and Finite Gauge Groups — Separates local-state truncation from spatial discretization.
  5. Physical Gauge Hilbert Spaces and Constraint Enforcement — Builds or tests the physical Gauss-law sector.
  6. Real-Time Evolution, Scattering, and Observable Extraction — Connects finite-time signals to response and scattering observables.
  7. Hamiltonian Continuum Limits and Euclidean Cross-Validation — Compares independently tuned Hamiltonian and Euclidean limits.

How the ideas fit together

Section titled “How the ideas fit together”

Enforcing Gauss’s law selects the physical sector, but does not remove errors from a finite link representation or spatial lattice. Conversely, a numerically accurate time evolution may remain outside that sector if the encoding or Hamiltonian violates the constraint. Transfer matrices provide a comparison with Euclidean spectra when positivity and normalization conditions are met. Agreement is meaningful only after matching the state, observable, and renormalized inputs and separating the independent regulator limits. Kogut and Susskind 1975, pp. 395–408.

Where to go next

Section titled “Where to go next”

Continue to Light-Front QFT to identify the null coordinates and the generator of evolution.

References

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