The QCD Quark Propagator in Coulomb Gauge and - Institut für Physik

Appendix B
Gauge potential, field strength and
E- and B-fields
This subsections details on conventions used in chapter 3. Given a field φ(x) we define
the gauge transformation via
φ(x) → g(x)φ(x) ,
(B.1)
where g(x) = exp(ig 0 α a (x)T a ) and g 0 is the bare coupling constant. For the covariant
derivative D µ = ∂ µ − ig 0 A µ to transform correctly, i.e. D µ → gD µ , the gauge potential
A µ = A a µ T a has to transform like
A µ → gA µ g −1 − i (∂ µ g)g −1 .
g 0
Acting on a field operator φ = φ a T a the covariant derivative reads
(B.2)
[D µ φ] a = ∂ µ φ a + g 0 f abc A b µ φc . (B.3)
The field strength tensor can be defined by the commutator of the covariant derivative,
1
[[D µ , D ν ]φ] a = f abc Fµν b g φc ,
(B.4)
0
and hence
F b µν = ∂ µA b ν − ∂ νA b ν + g 0f bde A d µ Ae ν .
The electric and magnetic fields are defined in terms of the field strength tensor by
E a k = F a 0k , B a k = − 1 2 ε ijkF a
ij .
In terms of the gauge potential they are therefore
(B.5)
(B.6)
Ek a = −∂ k A a 0 + ∂ 0 A a k + g 0 f abc A b 0A c k ,
(B.7)
[
]
Bk a = −ε ifk ∂ i A a 1
j + g 0
2 fabc A b iA c j . (B.8)
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