3Skyrmions
III Classical and Quantum Solitons
3.5 Fermionic quantization of the B = 1 hedgehog
We begin by quantizing the
B
= 1 Skyrmion. The naive way to do this is to
view the Skyrmion as a rigid body, and quantize it. However, if we do this, then
we will find that the result must have integer spin. However, this is not what
we want, since we want Skyrmions to model protons and nucleons, which have
half-integer spin. In other words, the naive quantization makes the Skyrmion
bosonic.
In general, if we want to take the Skyrme model as a low energy effective
field theory of QCD with an odd number of colours, then we must require the
B = 1 Skyrmion to be in a fermionic quantum state, with half-integer spin.
As a field theory, the configuration space for a baryon number
B
Skyrme field
is
Maps
B
(
R
3
→ SU
(2)), with appropriate boundary conditions. These are all
topologically the same as
Maps
0
(
R
3
→ SU
(2)), because if we fix a single element
U
0
∈ Maps
B
(
R
3
→ SU
(2)), then multiplication by
U
0
gives us a homeomorphism
between the two spaces. Since we imposed the vacuum boundary condition, this
space is also the same as Maps
0
(S
3
→ S
3
).
This space is not simply connected. In fact, it has a first homotopy group of
π
1
(Maps
0
(S
3
→ S
3
)) = π
1
(Ω
3
S
3
) = π
4
(S
3
) = Z
2
.
Thus,
Maps
0
(
S
3
→ S
3
) has a universal double cover. In our theory, the wave-
functions on
Maps
(
S
3
→ S
3
) are not single-valued, but are well-defined functions
on the double cover. The wavefunction Ψ changes sign after going around a
non-contractible loop in the configuration space. It can be shown that this is
not just a choice, but required in a low-energy version of QCD.
This has some basic consequences:
(i) If we rotate a 1-Skyrmion by 2π, then Ψ changes sign.
(ii)
Ψ also changes sign when one exchanges two 1-Skyrmions (without rotating
them in the process). This was shown by Finkelstein and Rubinstein.
This links spin with statistics. If we quantized Skyrmions as bosons, then
both (i) and (ii) do not happen. Thus, in this theory, we obtain the
spin-statistics theorem from topology.
(iii)
In general, if
B
is odd, then rotation by 2
π
is a non-contractible loop,
while if
B
is even, then it is contractible. Thus, spin is half-integer if
B
is
odd, and integer if B is even.
(iv)
There is another feature of the Skyrme model. So far, our rotations are
spatial rotations. We can also rotate the value of the pion field, i.e. rotate
the target 3-sphere. This is isospin rotation. This behaves similarly to
above. Thus, isospin is half-integer if B is odd, and integer if B is even.