Feynman Path Integral Formulation

134 4 Hamiltonian and Wheeler-DeWitt Equation
which for example for SU(2) means all j = 0. In this limit the Hamiltonian has the
simple form
H = g2
2a ∑ Ei a Ei a . (4.161)
links
Then the vacuum is a state for which each link is in a color singlet state
E a i |0〉 > = 0 . (4.162)
The lowest order excitation of the vacuum is a “boxciton” state, with one unit of
chromo-electric field on each link of an elementary lattice square, with energy
E ✷ = 4 · g2 N 2 − 1
2a 2N
. (4.163)
The mass of the lowest excitation in the theory is usually referred to as the mass
gap, the energy gap between the vacuum and the first excited state. Note that if one
creates states out of the vacuum by having the Hamiltonian act on it, there is no
need to separately enforce the Gauss law constraint, as the states obtained in this
way automatically satisfy the constraint.
By the same kind of arguments, the static potential between a quark and an antiquark
pair, separated by a distance R, increases linearly with distance causing linear
confinement at strong coupling. This is due to the fact that, given two static quarks
separated form each other by this distance R [and described by fermion operators
ψ(n) and ψ † (n)], a number of link variables U proportional to the distance between
the two has to be laid down,
( )
ψ † (n) ∏ U ψ(n + R) , (4.164)
n →n+R
in order to construct the manifestly gauge invariant color singlet state. Then each
link variable which is not in the ground state costs a unit of energy proportional to
1/g 2 .
All of this applies to the strong coupling limit of the theory. Raleigh-Schrödinger
perturbation theory can then be used to compute corrections to such results, in principle
to arbitrarily high order in 1/g 2 . But ultimately one is interested in the limit
g 2 → 0, corresponding to the ultraviolet, asymptotic freedom fixed point of the nonabelian
gauge theory, and to the lattice continuum limit a → 0. Therefore in order to
recover the original theory’s continuum limit, one needs to examine a limit where
the mass gap in units of the lattice spacing goes to zero, am(g) → 0.
It is easy to see that this limit corresponds to an infinite correlation length in
lattice units, since the correlation length is given by ξ (g)=1/m. The last result can
most easily be seen from the Lehman representation of the connected vacuum field
correlations in the Euclidean time (τ) direction