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+ dx, x; A + bA)] U[g(x)] (x) U[g(x + dX)] t(x)
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U[g(x
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(12-11)
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as desired. In infinitesimal form, this yields (12-12) which means that DI' (x) transforms as (x). Expanding (12-10) to first order in ba results in
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(12-13a)
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NONABELIAN GAUGE FIELDS
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where Oa is considered as transforming under the adjoint representation. More explicitly,
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oAl'a(x) = (\oaa(x)
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+ Cbcaoab(x)Al'ix)
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(12-13b)
For a finite gauge transformation, Eq. (12-10) gives
AI'(x) --+ 9 AI'(x)
= [Ol'g(X)]g-l(X) + g(x)AI'(X)g-l(X)
(12-14)
As compared to the abelian case, a new feature has emerged. For a constant g (or oa), AI' transforms as a charged field belonging to the adjoint representation, as indicated by the second term on the right-hand sides of Eqs. (12-13) and (12-14). In classical electrodynamics, two potentials that only differ, locally, by a nonsingular gauge transformation are physically equivalent, and are then characterized by the same field strength tensor Fl'v. We want to construct the analogous curvature tensor Fl'v in the nonabelian case. To this end, we consider the parallel displacement of some representative cp along an infinitesimal closed path C. After returning to the starting point, cp has rotated by
g(C; A)
Pexp
dx' A(X)]
If [ is the dimension of C, we expand this element up to order [2: g(C;A)= e+
L + LL
dX'A
X2>Xj
dX2'A(X2)dxl'A(Xl)
+ OW)
where, as before, the ordering implies that the curve has been parametrized with = x(O) = x(1). We also expand AI'[x(s)]:
AI'[x(s)]
= AI'(xo) + [XV(s) - xo]ovAI'(xo) + ...
and get
g(C; A)
fJ",
Fl'v
dxl' /\ dxV(ol'Av - ovAI' - [AI" Av])
+ OW)
(12-15)
where .s1 is the infinitesimal area bounded by C. We set
== ol'Av - ovAI' - [AI',Av]
= (ol'Ava - ovAl'a - CbcaAl'bAvc)ta
(12-16)
thereby generalizing the definition of electric and magnetic field. This tensor is often referred to as the strength tensor. A useful identity follows from the previous derivation: (12-17) [DI" Dv] = -Fl'v In a gauge transformation, g( C; A) transforms as
g( C ; A) --+ g(xo)g( C ; A)g - l(XO)
[cf. Eq. (12-10)]. Hence, F transforms as a charged field belonging to the adjoint
QUANTUM FIELD THEORY
representation
F(x) -4 g(X)F(X)g-l(X)
(12-18)
or in infinitesimal form
F -4F bF bFllva(x)
+ bF
(12-19)
= [blX(X), F] = CbcablXb(X)FIlVC(X)
In particular, the covariant derivative of F reads
DllabFvpbta
= [DIl' Fvp] = 0IlFvp - [AIl,Fvp]
In electromagnetism, the exterior derivative of the differential form A.(x)dx , viz iF .,dx closed form of degree two:
dx' is a
(12-20)
opF., + cyclic permutations
This property is equivalent to the homogeneous Maxwell equations div B = 0, curl E + oB/ot = O. Reciprocally, given such a closed form, Poincare's lemma entails the local existence of a potential A from which F is derived (Sec. 1-1-2). The situation is slightly different in the nonabelian case. There does exist an analogous identity (12-21) as a consequence of Jacobi's identity and Eq. (12-17). Notice, however, that Eq. (12-21) assumes the existence of A, as it appears in the covariant derivative. Moreover, one can show that if F,p and A. satisfy (12-21), F is not necessarily the strength tensor associated with A. Accordingly, in contrast with the abelian case, the field strength tensor F does not determine uniquely all gauge-invariant quantities.
If Fllv vanishes in the neighborhood of a point, All is a pure gauge
Fllv = 0 ;> 3g(x): AIl(x) = [Ollg(X)]g-l(X)
(12-22)
Indeed, if F = 0, the integral of All along a path C from the origin to x does not depend on the curve
g(X) = g(x, 0; A) = P exp
dx A(X)]
by definition of F. This element g(x) satisfies (12-22). Conversely, a pure gauge All has obviously a vanishing F. Owing to the gauge arbitrariness, we may sometimes demand that the potential locally satisfy a definite condition. This is referred to as a choice of gauge. For instance, let nil be a fixed four-vector. There exists a gauge transformation, A -4 A', such that
n A'(x)
(12-23)
This is the so-called axial gauge.
NONABELIAN GAUGE FIELDS
To show that this is possible, let us introduce a four-vector N" such that N nolO (for instance, N = n, if n2 # 0). Every point x may be written in a unique way:
x = .l.(x)n
+ X.L
X.L =
x~'N =
with
.l.(x)=N'n
x- n - N'n
Consider the segment C, x(s) = s.l.(x)n g(x)=Pexp Introducing
+ X.L(O :S s :s 1), and the integral
=Pexp
II dx } {Jo dSds'A[x(s)]
[I Jo
'(x)
d.l.n A(.l.n+x.L)
we observe that
From the definition of g(x), we have
-,-g(x) = n' og(x) = n' A(x)g(x)
d(x)
which implies n' A' = O.
Similarly, we may show that through a gauge transformation we can always fulfil locally the Lorentz gauge condition (12-24) or any condition obtained from (12-23) or (12-24) by replacing the right-hand side by a function taking its value in the Lie algebra.
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