QCD on an Infinite Lattice

Abstract

We construct a mathematically well–defined framework for the kinematics of Hamiltonian QCD on an infinite lattice in \({\mathbb{R}^3}\) , and it is done in a C*-algebraic context. This is based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph e.a.. To extend this model to an infinite lattice, we need to take an infinite tensor product of nonunital C*-algebras, which is a nonstandard situation. We use a recent construction for such situations, developed by Grundling and Neeb. Once the field C*-algebra is constructed for the fermions and gauge bosons, we define local and global gauge transformations, and identify the Gauss law constraint. The full field algebra is the crossed product of the previous one with the local gauge transformations. The rest of the paper is concerned with enforcing the Gauss law constraint to obtain the C*-algebra of quantum observables. For this, we use the method of enforcing quantum constraints developed by Grundling and Hurst. In particular, the natural inductive limit structure of the field algebra is a central component of the analysis, and the constraint system defined by the Gauss law constraint is a system of local constraints in the sense of Grundling and Lledo. Using the techniques developed in that area, we solve the full constraint system by first solving the finite (local) systems and then combining the results appropriately. We do not consider dynamics.

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Correspondence to Hendrik Grundling.

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Communicated by Y. Kawahigashi

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Grundling, H., Rudolph, G. QCD on an Infinite Lattice. Commun. Math. Phys. 318, 717–766 (2013). https://doi.org/10.1007/s00220-013-1674-5

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Keywords

  • Gauge Transformation
  • Wilson Loop
  • Constraint System
  • Inductive Limit
  • Approximate Identity