Constructing States and Effective Hamiltonians in Lattice QCD

We formulate SU(3) Hamiltonian lattice QCD in terms of the plaquette variables and determine the relevant subspaces of the Hilbert space for the vacuum wave functional and its approximations. We analyze the oneand two-plaquette problems. In an earlier paper [1] we formulated the fully gauge-fixed SU(N) lattice gauge Hamiltonian and studied the one-plaquette problem for several values of N . This paper is restricted to the case of SU(3) lattice gauge theory. For this case I discuss ways to go beyond the one-plaquette approximation. The Hamiltonian approach to lattice QCD, although more complicated, has advantages over the Euclidean formulation. [2, 3] In general, one can say that the Hamiltonian approach allows one to split the problem in parts and tackle each part separately, thereby generating more understanding and restricting the eventual computational effort. This should be formulated consistently as restrictions on the full physical Hilbert space to a proper subspace, or likewise, either by formal arguments, or from calculations. One important step in this process is the gauge fixing, such that all the degrees of freedom are physical, and an excitation spectrum, determined from an ab-initio many-body approach, such as RPA, is actually physical, and does not contain oscillations in the gauge parameter. The first and foremost part of the full QCD problem is the determination of the vacuum wave functional. ∗present address: Physics Department, University of Pittsburgh (PA), U.S.A. 1 The dimensionless QCD lattice Hamiltonian can be split into three parts: H = 2 ∑