159
16.2. Discretization
Under the gauge transformation (16.29), O(x, y) transforms by
∫
{ie
Adx +ie[θ(x)−θ(y)]}
y
O(x, y) → φ
† (x)e
−ieθ(x) e
x
′
e
ieθ(y) φ(y) = O(x, y),
(16.32)
and it is therefore gauge invariant. The familiar ‘covariant derivative’ rule can
be recovered by letting y = x + dx for infinitesimal dx, and by considering the
gauge-invariant quantity
[
]
O(x, x + dx) − O(x, x)
lim
.
(16.33)
dx→0
dx
Evaluating (16.33) one finds (problem 16.4) the result
(
)
φ
† (x)
d − ieA φ(x)
(16.34)
dx
≡ φ
† (x)D x φ(x)
(16.35)
with the usual definition of the covariant derivative. In the discrete case, we
merely keep the finite version of (16.31), and replace φ
† (n 1 )φ(n 1 +1) in (16.28)
by the gauge invariant quantity
φ
† (n 1 )U (n 1 , n 1 + 1)φ(n 1 + 1),
(16.36)
where the link variable U is defined by
[
]
∫ n1a
′
U (n 1 , n 1 + 1) = exp ie
Adx .
(16.37)
(n1+1)a
Note that
U (n 1 , n 1 + 1) → exp[−ieA(n 1 )a]
(16.38)
in the small a limit.
¯
Similarly, the free Dirac term ψ ¯ (n 1 )γ
E ψ(n 1 + 1) − ψ(n 1 + 1)γ
E ψ(n 1 ) in
1
1
(16.18) is replaced by the gauge-invariant term
¯
¯
ψ(n 1 )γ 1
E U (n 1 , n 1 + 1)ψ(n 1 + 1) − ψ(n 1 + 1)γ 1
E U (n 1 + 1, n 1 )ψ(n 1 ). (16.39)
The generalization to more dimensions is straightforward. In the nonAbelian SU (2) or SU (3) case, ‘eA’ in (16.38) is replaced by gt
a A
a (n 1 ) where
the t’s are the appropriate matrices, as in the continuum form of the covariant
derivative. A link variable U (n 2 , n 1 ) may be drawn as in figure 16.1. Note
that the order of the arguments is significant: U (n 2 , n 1 ) = U
−1 (n 1 , n 2 ) =
U
† (n 1 , n 2 ) from (16.38), which is why the link carries an arrow.
Thus gauge invariant discretized derivatives of charged fields can be constructed. What about the Maxwell action for the U(1) gauge field? This does
not exist in only one dimension (∂ μ A ν − ∂ ν A μ cannot be formed), so let us
move into two. Again, our discussion of the geometrical significance of F μν as
16.2. Discretization
Under the gauge transformation (16.29), O(x, y) transforms by
∫
{ie
Adx +ie[θ(x)−θ(y)]}
y
O(x, y) → φ
† (x)e
−ieθ(x) e
x
′
e
ieθ(y) φ(y) = O(x, y),
(16.32)
and it is therefore gauge invariant. The familiar ‘covariant derivative’ rule can
be recovered by letting y = x + dx for infinitesimal dx, and by considering the
gauge-invariant quantity
[
]
O(x, x + dx) − O(x, x)
lim
.
(16.33)
dx→0
dx
Evaluating (16.33) one finds (problem 16.4) the result
(
)
φ
† (x)
d − ieA φ(x)
(16.34)
dx
≡ φ
† (x)D x φ(x)
(16.35)
with the usual definition of the covariant derivative. In the discrete case, we
merely keep the finite version of (16.31), and replace φ
† (n 1 )φ(n 1 +1) in (16.28)
by the gauge invariant quantity
φ
† (n 1 )U (n 1 , n 1 + 1)φ(n 1 + 1),
(16.36)
where the link variable U is defined by
[
]
∫ n1a
′
U (n 1 , n 1 + 1) = exp ie
Adx .
(16.37)
(n1+1)a
Note that
U (n 1 , n 1 + 1) → exp[−ieA(n 1 )a]
(16.38)
in the small a limit.
¯
Similarly, the free Dirac term ψ ¯ (n 1 )γ
E ψ(n 1 + 1) − ψ(n 1 + 1)γ
E ψ(n 1 ) in
1
1
(16.18) is replaced by the gauge-invariant term
¯
¯
ψ(n 1 )γ 1
E U (n 1 , n 1 + 1)ψ(n 1 + 1) − ψ(n 1 + 1)γ 1
E U (n 1 + 1, n 1 )ψ(n 1 ). (16.39)
The generalization to more dimensions is straightforward. In the nonAbelian SU (2) or SU (3) case, ‘eA’ in (16.38) is replaced by gt
a A
a (n 1 ) where
the t’s are the appropriate matrices, as in the continuum form of the covariant
derivative. A link variable U (n 2 , n 1 ) may be drawn as in figure 16.1. Note
that the order of the arguments is significant: U (n 2 , n 1 ) = U
−1 (n 1 , n 2 ) =
U
† (n 1 , n 2 ) from (16.38), which is why the link carries an arrow.
Thus gauge invariant discretized derivatives of charged fields can be constructed. What about the Maxwell action for the U(1) gauge field? This does
not exist in only one dimension (∂ μ A ν − ∂ ν A μ cannot be formed), so let us
move into two. Again, our discussion of the geometrical significance of F μν as