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N. Geometrical Aspects of Gauge Fields
FIGURE N.5
2
Closed loop ABCD in q
1
− q space.
N.2 Geometrical curvature and the gauge field strength
tensor
Consider a small closed loop in our (possibly curved) two-dimensional space
1
1
– see figure N.5 – whose four sides are the coordinate lines q = a, q =
a + δa, q
2 = b, q
2 = b + δb. We want to calculate the net change (if any) in
δV
α as we parallel transport V → around the loop. The change along A → B is
∫ 2
q =b,q
1 =a+δa
(δV
α ) AB = −
Γ
α
γ1 V
γ dq
1
q 2 =b,q 1 =a
≈ −δaΓ
α
γ1 (a, b)V
γ (a, b)
(N.20)
to first order in δa, while that along C → D is
∫ 2
∫
q =b+δb,q
1 =a
(δV
α ) CD = −
Γ
α
γ1 V
γ dq
1
q 2 =b+δb,q 1 =a+δa
2
q =b+δb,q
1 =a+δa
= +
Γ
α
γ1 V
γ dq
1 .
q =b+δb,q 1 =a
2
≈ δaΓ
α
γ1 (a, b + δb)V
γ (a, b + δb).
(N.21)
Now
∂Γ
α
γ1
Γ
α
γ1 (a, b + δb) ≈ Γ
α
γ1 (a, b) + δb
(N.22)
∂q 2
and, remembering that we are parallel-transporting V → ,
γ
V
γ (a, b + δb) ≈ V
γ (a, b) − Γ V
δ δb.
(N.23)
δ2
Combining (N.20) and (N.21) to lowest order, we find
[
]
∂Γ
α
γ1
γ
(δV
α ) AB + (δV
α ) CD ≈ δaδb
V
γ
− Γ
α
γ1 Γ V
δ
(N.24)
δ2
∂q 2