192 The very early universe
and
(∂iU)U−^1 =−U(
∂iU−^1)
, (4.175)
the variation of the first term in the integrand can be written as
δ(
(∂iU)U−^1)
=U
(
∂i(
U−^1 δU))
U−^1. (4.176)
The variation of the other two terms gives a similar contribution, and therefore
δν∝∫
tr(
εijk∂i(
U−^1 δU)(
∂jU−^1)
(∂kU))
d^3 x. (4.177)Upon integration by parts there arise terms of the formεijk∂i∂j...,which vanish
because of the antisymmetry of theεsymbol. Hence the winding number does not
change under continuous nonsingular transformations.
To prove the existence of homotopy classes with different winding numbers, we
construct them explicitly for the case of theSU( 2 )group. AnySU( 2 )matrix can
be written as
U(χ,e)=cosmχ 1 −i(e·σ)sinmχ, (4.178)wheree=(e 1 ,e 2 ,e 3 )is the unit vector andσ=(σ 1 ,σ 2 ,σ 3 )are the three Pauli
matrices combined as a vector.
Problem 4.27Verify this last statement. (HintAn arbitrarySU( 2 )matrix has the
same form as the matrixζin (4.47).)
Thus, the elements of theSU( 2 )group can be parameterized by the unit vector
in four-dimensional Euclidean space,
lα=(cosmχ,esinmχ),and they can be thought of topologically as the elements of the three-dimensional
sphere. Let us takeχandeto be functions of the spatial coordinatesxand identify the
points at spatial infinity (|x|→∞).ThenU(x)is an unambiguous function of the
spatial coordinates ifχ(|x|→∞)=πandmis an integer or zero. We can interpret
the functionU(x)as describing the mapping from the 3-sphereS^3 (Euclidean space
xwith infinity mapped to a point) to the 3-sphere of the elements of theSU( 2 )
group. For the mapping (4.178), the spatial coordinatesxwrap around this sphere
mtimes. The mappings with+mand−mcorrespond to different orientations and
therefore should be distinguished. Using the identity (4.175), (4.174) simplifies to
ν=−1
8 π^2∫
tr{
U−^1 (∂rU)[(
∂φU−^1)
(∂θU)−(
∂θU−^1)(
∂φU)]}
drdθdφ, (4.179)