# Poster presented by Norbert Ligterink

**Poster Session**
**Topic:** Theoretical Developments

**Authors:** N. E. Ligterink, N. R. Walet, and R. F. Bishop

### A Hamiltonian Many-Body Approach to SU(N) Lattice Gauge Theory

**Abstract:** The study of the non-perturbative aspects of QCD is one of the most difficult challenges posed by this theory. One way to access these properties is through the study of the vacuum wave functional, which can only be determined in a Hamiltonian framework. Unfortunately, a Hamiltonian can only be formulated using a gauge fixing procedure. We construct a lattice Hamiltonian using a maximal-tree gauge inspired by the approach of M\"uller and R\"uhl, and Bronzan$^1$, which is in its turn embedded within the standard temporal gauge. In this approach we define a maximal tree as a spanning tree that connects all lattice points, passing through each lattice point once, apart from the origin, which is taken to be the root of the tree. On the links of the tree the gauge field is fixed to be unity, in the chargeless sector. The resulting Hamiltonian, which still depends on the choice of tree, is complicated and contains long-range interactions originating from the gauge-fixing procedure. It is hoped that these interactions have some bearing on confinement and other non-perturbative aspects of the theory.As a first step towards application of more powerful many-body techniques to this Hamiltonian, we apply the variational principle with two simple choices of wave functional to this Hamiltonian. We find that the Schr\"odinger equation reduces to a one-plaquette problem.The one-plaquette Hamiltonian is diagonalised completely, in a basis of eigenstates of the electric operator$^2$ incorporating the coupling induced by the magnetic potential. We shall exhibit the spectrum of the one-plaquette problem for U(1), SU(2), SU(3), SU(4), and SU(5), and shall relate this to the old work by Robson and Webber.$^3$. We shall disucss how we can use these results as a starting point for an investigation into the leading spatial correlations and the construction of many-body wave functionals for the vacuum state and low-lying excitations, using techniques similar to those discussed in Ref. 4. $^1$V.F. M\"uller and W. R\"uhl, Nucl. Phys. B 230 (1984) 49; J.B. Bronzan, Phys. Rev. D 31 (1985) 2020. $^2$ H. Weyl, The Classical Groups (Princeton University Press, 1946); J.E. Hetrick, Int. J. Mod. Phys. {\bf A9}, 3153 (1994); J. Hallin, Class. Quant. Grav. {\bf 11}, 1615 (1994). $^3$ D. Robson and D.M. Webber, Z.Phys. C 7 (1980) 53. $^4$ N.E. Ligterink, N.R. Walet and R.F. Bishop, Ann. Phys. (NY) 267 (1998) 97, and in preparation.

Massimo Campostrini,
`campo@mailbox.difi.unipi.it`