Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 193

wherer,θ,φare the spherical coordinates inxspace. Let us assume for simplicity
thatχ(x)=χ(r)and

e(x)=(sinθcosφ,sinθsinφ,cosθ).

Then, taking into account thatU−^1 (χ)=U(−χ)and using the Pauli matrix property

σiσj=δij+iεijkσk,

from which follows(eσ)^2 = 1 ,we obtain

U−^1 (∂rU)=−im(eσ)

dχ

dr

, (4.180)

(

∂φU−^1

)

(∂θU)−

(

∂θU−^1

)(

∂φU

)

=− 2 isin^2 (mχ)sinθ(eσ). (4.181)

Substituting these expressions into (4.179) and integrating over the angles, we
derive the desired result:ν=m.Thus the homotopy class characterized by the
winding numbermcorresponds to the mapping which wrapsmtimes around the
SU( 2 )sphere.
To what extent are these results particular to theSU( 2 )group? First of all note
that every mapping ofS^3 into theU( 1 )Abelian group is continuously deformable
to the trivial mapping. Hence the vacuum has a trivial structure in this case and
there is no analog of winding number. As for a non-AbelianSU(N)group, one can
show by methods beyond the scope of this book that any continuous mapping ofS^3
into an arbitrarySU(N)group can be continuously deformed to a mapping into an
SU( 2 )subgroup ofSU(N).Therefore all results derived for theSU( 2 )group are
valid for an arbitrarySU(N)group; in particular, (4.174) for the winding number
requires no alteration.

Barrier heightTwo vacua, with different winding numbers, are separated by a
barrier. To demonstrate this we will use the identity

tr

(

FF ̃

)

=∂α

[

trεαβγ δ

(

FβγAδ−

2

3

igAβAγAδ

)]

, (4.182)

whereF ̃αβ≡^12 εαβγ δFγδis the tensor dual toF.

Problem 4.28Verify (4.182).

Let us consider two vacuum configurations (4.173) with winding numbersν 0 and
ν 1 specified on two different space-like hypersurfaces. Then, integrating (4.182)
and using the Gauss theorem, we obtain
∫
tr

(

FF ̃

)

d^4 x=

16 π^2

g^2

(ν 1 −ν 0 ). (4.183)