Feynman Path Integral Formulation

4.13 Lattice Regularized Hamiltonian for Gauge Theories 133
In terms of the operators E a one then has for the first term in the Hamiltonian H
E a E a =
a
2g 4 tr ˙U † ˙U , (4.156)
after using the normalization condition on the SU(N) generators T a ,trT a T b = 1 2 δ ab .
This finally gives for the Hamiltonian of Wilson’s lattice gauge theory (Kogut and
Susskind, 1975)
H = g2
2a ∑ E a E a − ∑
links
✷
1
4ag 2 tr[ UUU † U † + h.c. ] . (4.157)
The first term in Eq. (4.157) is the lattice analog of the electric field term E 2 , while
the second term is a lattice discretized version, involving lattice finite differences,
of the magnetic field, (∇ × A) 2 term. In this picture the analog of Gauss’s law is a
constraint, which needs to be enforced on physical states at each spatial site n
6
∑
i=1
E a i (n)|Ψ〉 = 0 . (4.158)
In the special case of the group SU(2), the generators of group rotations in
Eq. (4.153) are just the usual angular momentum operators J a (n), a = 1,2,3. The
system can be regarded therefore as a collection of quantum rotators, with a kinetic
term defined on the links and proportional to J 2 [with eigenvalue j( j +1)], and a potential,
or link coupling, term. An appropriate basis in the extreme strong coupling
limit is then represented by a suitable product of angular momentum eigenstates
|Ψ〉 = ∏
n,i
| j,m〉 n,i . (4.159)
In this limit the B 2 term can be regarded as a perturbation, whose action on the
above state can then be determined from the commutation relation in Eq. (4.152).
Even simpler is the Abelian case U(1). Here only one angle variable θ μ (n) survives
on each link. In the position representation one writes for the electric field operator
E ≡ J θ = −i∂/∂θ, with integer eigenvalues m and eigenfunctions e imθ / √ 2π.
The remainder of the Hamiltonian then involves for each spatial plaquette the term
cosθ ✷ with
θ ✷ = θ μ (n)+θ ν (n + μ) − θ μ (n + ν) − θ ν (n) . (4.160)
It is characteristic of this lattice gauge theory model, compact electrodynamics, that
the gauge variables are angles. For weak enough coupling g = e one can consider
them as variables ranging over the whole real line, and ordinary QED is
recovered.
In general, and irrespective of the symmetry group chosen, the ground state in
the strong coupling g 2 → ∞ limit has all the SU(N) rotators in their ground state,