Feynman Path Integral Formulation

132 4 Hamiltonian and Wheeler-DeWitt Equation
which, up to irrelevant constant contributions, can be written as combinations of
lattice differences in the time direction,
tr 1 a 0
[
U † (t + 1) − U † (t) ] 1
a 0
[
U(t + 1) − U(t)
]
. (4.148)
In the limit as a 0 → 0 these turn into a combination of time derivatives of the form
˙U † ˙U, and in this limit the exponent inside the path integral involves therefore the
quantity
L =
a
4g
∑ 2 tr ˙U † ˙U + ∑
links
✷
1
4ag 2 tr[ UUU † U † + h.c. ] . (4.149)
The next step is an elimination of the ˙U angular variables in favor of the local generators
of rotations (Creutz, 1977). First from the above action one can construct a
Hamiltonian in the usual way, by defining
H =
∑
links nm
(
which in this case gives
H =
˙U † nm
∂L
∂ ˙U nm
†
a
4g
∑ 2 tr ˙U † ˙U − ∑
links
✷
+ ˙U nm
∂L
∂ ˙U nm
)
− L( ˙U,U, ˙U † ,U † ) , (4.150)
1
4ag 2 tr[ UUU † U † + h.c. ] . (4.151)
The ˙U variables can now be eliminated by introducing generators of local rotations
Ei a (n), defined on the links (with spatial directions labeled by i, j = 1,2,3) and
satisfying the commutation relations
[E a i (n),U j (m)] = T a U i (n)δ ij δ nm , (4.152)
along with the SU(N) generator algebra commutation relation
[
]
Ei a (n),E b j (m) = if abc Ei c (n)δ ij δ nm . (4.153)
Since E a generates local rotations, it can be written explicitly in term of the U’s. An
infinitesimal local gauge rotation of the U link matrices is achieved by
U i (n) → (1 + iε a T a ) U i (n) . (4.154)
The generator for such a symmetry of the original Lagrangian L is by Noether’s
theorem
E a =
∂L (iT
∂ ˙U a U) ij + ∂L
ij ∂ ˙U † (iT a U) † ij
ij
a (
= i tr ˙U †
4g 2 T a U − h.c. ) . (4.155)