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NONABELIAN GAUGE FIELDS
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12-1-1 The Gauge Field AI' and the Tensor Fl'v
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The model considered by Yang and Mills was based on the isotopic symmetry, with the group SU(2) of global in variance. Was it possible to transform this invariance into a local one, so that the reference frame used to define the isospins could vary from point to point If so, the information that a particle produced at a given space-time point was a proton, say, was meaningless for a different observer, unless there existed a way to compare their two frames. This is the role of the gauge field, very much as in electrodynamics relative phases of charged fields at different points make sense only when compared via the electromagnetic potential. To be more explicit, let us consider N fields (Lorentz scalars for simplicity) transforming according to an irreducible representation U of some compact Lie group G: (12-1) 4>(X) ---+ (g4>)(x) = U(g)4>(x) where U(g) is an N x N unitary (or orthogonal) matrix. We assume the lagrangian invariant under the transformation (12-1). We want to build a more complex theory, still invariant when g depends on the space-time point x. Let g(x) be such a G-valued function. The derivatives 81'4> have no longer any intrinsic meaning as we compare fields 4>(x + bx) and 4>(x) that transform independently under (12-1). It is necessary to introduce a new object to allow this comparison. Consider an infinitesimal transformation (12-2) where e is the identity of G and the t a are elements of the Lie algebra satisfying the commutation relations (12-3)
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In the representation associated to the fields 4>, antihermitian matrices T a represent the elements ta of the Lie algebra, so that under the transformation g of Eq. (12-2),
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4>(x) b4>(x)
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(g4>)(x) = 4>(x) + b4>(x) b!Xa T a4>(x) == b!X4>
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(12-4a)
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If the infinitesimal parameters b!Xa(x) depend on x, these transformations will be called (local) gauge transformations, and the transformation group~the gauge group~wi11 formally be the infinite product Ilx G x :
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b4>(X)
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b!X(x)4>(x)
b!Xix) T a4>(x)
(12-4b)
We introduce the gauge fields x ---+ Al'a(x). They are vector fields and carry an index of the adjoint representation of G. We denote AI' the corresponding element of the Lie algebra : (12-5) and we will use the same notation for any of its representations, Al'a(x)Ta. What is understood should be clear from the context.
QUANTUM FIELD THEORY
To an infinitesimal path (x, x the group
+ dx),
the gauge field associates an element in
e + dxl' AI'(x)
+ dx, x; A) =
(12-6)
providing the means to compare two neighboring frames. This generalizes to a finite path C going from Xl to X2. If S --+ x(s), 0::;; S ::;; 1 is a parametrization of this path: s(O) = Xl, s(1) = X2, the element of G associated to C is
g( C ; A)
= P exp
~~ . A )
=2: o
dsn- l '"
dx dx dSld-'A[x(sn)]"'-d 'A[X(Sl)] Sn Sl
(12-7)
The path-ordering operation P is the analog of the familiar time ordering T. In the language of differential geometry, the AI' form a connection and define a parallel displacement of geometrical objects belonging to representation spaces of the group. A parallel displacement of from X to X + dx is defined through
t(x)
9(x+dx,x) (x)
(x)
+ dx' A(x) (x)
dxl'[ol' - AI'(x)] (x)
(12-8)
The definition of the covariant derivative follows naturally:
dxI'DI' (x) == (x
+ dx) -
t(x)
DI' == 01' - AI'(x)
(12-9a)
or, in components,
(DI')A B
= Ol'bA B - (TC)A B Al'c(x)
(12-9b)
In particular, in the adjoint representation (Tc)a b = CCba = -C bca, b (DI')a b = Ol'b a + CbcaAl'c(x)
(12-9c)
If G is the group U(1), this reduces to the familiar concept in electrodynamics. The gauge field is endowed with a transformation law under gauge transformations such that t(x) transforms as (x + dx). To this end, it is sufficient that
+ dx, x; A + bA) = g(x + dx)g(x + dx, x; A)g-l(X)
t(x)
(12-10)