Feynman Path Integral Formulation

6.15 Gauge Fields 213
6.15 Gauge Fields
In the continuum a locally gauge invariant action coupling an SU(N) gauge field to
gravity is
I gauge = − 1 ∫
4g 2 d 4 x √ gg μλ g νσ Fμν a Fλσ a , (6.171)
with Fμν a = ∇ μ A a ν − ∇ ν A a μ + gf abc A b μA c ν and a = 1...N 2 − 1.
On the lattice one can follow a procedure analogous to Wilson’s construction
on a hypercubic lattice, with the main difference that the lattice is now simplicial.
Given a link ij on the lattice one assigns group element U ij , with each U an N × N
unitary matrix with determinant equal to one, and such that U ji = Uij
−1 . Then with
each triangle (plaquettes) Δ labeled by the three vertices ijkone associates a product
of three U matrices,
U Δ ≡ U ijk = U ij U jk U ki . (6.172)
The discrete action is then given by (Christ Friedberg and Lee, 1982)
I gauge = − 1 g 2 ∑
Δ
c
V Δ
A 2 Δ
Re [tr(1 − U Δ )] , (6.173)
with 1 the unit matrix, V Δ the 4-volume associated with the plaquettes Δ, A Δ the
area of the triangle (plaquettes) Δ, and c a numerical constant of order one. If one
denotes by τ Δ = cV Δ /A Δ the d − 2-volume of the dual to the plaquette Δ, then the
quantity
τ Δ
A Δ
= c V Δ
A 2 Δ
, (6.174)
is simply the ratio of this dual volume to the plaquettes area. The edge lengths l ij
and therefore the metric enter the lattice gauge field action through these volumes
and areas.
One important property of the gauge lattice action of Eq. (6.173) is its local
invariance under gauge rotations g i defined at the lattice vertices, and for which U ij
on the link ijtransforms as
U ij → g i U ij g −1
j . (6.175)
These leave the product
[
]
tr (g i U ij g −1
j )(g j U jk g −1 )(g k U ki g −1
i
k
= tr [ U ij U jk U ki
]
, (6.176)
and therefore the action invariant. One can further show that the discrete action of
Eq. (6.173) goes over in the lattice continuum limit to the correct Yang-Mills action
for manifolds that are smooth and close to flat.