Hi! I noticed this question about exercise 15.5 seemed to be
outstanding for a while. Let me repost the post from sci.math here (by
Knarfian) and give some thoughts, and then mention some references:

--begin quote

Hello,

I have been working my way through Roger Penrose's fantastic book "The
Road to Reality", and I am nearing the end of the Math section. There
is a statement and a question in section 15.4 that has myself and
another friend stymied. (Unfortunately, the solutions are not
available yet.)

On page 336, he is describing the Clifford bundle / Hopf fibration:
"In fact, in turns out that each point of our sphere S^3 can be
interpreted as a unit-length 'spinorial' tangent vector to S^2 at one
of its points". Then, he goes on to propose a problem [15.5] in a
footnote:

[15.5] Show this. Hint: Take the tangent vector to be u(d/dv) -
v(d/du) + x(d/dy) - y(d/dx).

His definition of the the sphere in S^3: abs(w)^2+abs(z)^2=1, which is
u^2+v^2+x^2+y^2=1, where w=u+iv and z=x+iy. S^2 is defined as the
Riemann Sphere, where each point is associated with a Complex 1-D
subspace of the C^2, Aw+Bz=0, where the ratio of A/B is unique for
every point on S^2.

I understand the relation between unit quaternions, rotation, and S^3,
thanks to David Lyons' paper "An Elementary Introduction to the Hopf
Fibration". What I am missing is this notion of a spinorial tangent
vector to S^2.

After this point, Penrose also talks about 2 to 1 mappings between
antipodal points on S^3 and "ordinary tangent vectors" to S^2. I'd
really like to understand both of these concepts, and I feel I'm pretty
close - but not quite there.

Thanks in advance for any insight, or references to relevant background

--end quote



Here's my take. The Hopf bundle (or Clifford bundle, as Penrose calls
it here) is certainly a fascinating topic and I'm unlikely to do it
full justice in a short post.

Recall that the fibre bundle in question is the "Clifford bundle"
S^1->S^3->CP^1 (here we identify the complex projective line with the
Riemann 2-sphere). This is the sphere bundle for the "tautological
line bundle" C^1->C^2->CP^1, where it's "tautological" because CP^1
*is* the set of complex lines through the origin in C^2, and so if you
take C^2 as the total space, and you project each complex line to the
corresponding point in CP^1, then that projection map is a line bundle
(since the fibre over a point is that complex line that *is* the point).

So, the special unitary group SU(2) acts on the vector space C^2, and
preserves the unit sphere S^3. SU(2) is identified with the group of
unit quaternions (e.g., Pauli spin matrices) and is also Sp(1). Since
SU(2) acts on C^2 complex-linearly, it also takes complex lines to
complex lines. Thus it acts on CP^1. You can write the action out on
ratios A/B.

The tangent bundle of C^2 is of course a trivial bundle, and thus we
can talk about the basis vectors d/du, d/dv, d/dx, and d/dy in each
tangent space over each point. They are the differential operators
that "point in the directions of the coordinate axes", i.e., at a
given point, they take functions to numbers (the basis vector d/dx in
the tangent space at p is an operator taking f to the number "df/dx
evaluated at p").

So what is this beast: u(d/dv) - v(d/du) + x(d/dy) - y(d/dx)?

As a simpler example, take S^1 embedded in a R^2 with coordinates (u,
v). Then a point on the circle is (u, v) with u^2 + v^2 = 1. What is
an example of a tangent vector to S^1 at that point? Note that the
tangent bundle of S^1 sits as a subset of the tangent bundle to R^2
because of the embedding so we can talk meaningfully of which tangent
vectors over a point (u, v) are tangent to S^1. If you draw a picture,
you'll see that you want, for a point (u, v) on the circle, one
tangent vector points in the direction (v, -u). Since it is in the
tangent space, when you write it in terms of the basis of the tangent
space, you v d/du - u d/dv.

So, the same kind of computation serves to demonstrate that at a point
p = (u, v, x, y) = (u + vi, x + yi) in S^3 (inside C^2), we can find
that the vector u(d/dv) - v(d/du) + x(d/dy) - y(d/dx), which sits in
the tangent space T_p (C^2), is in fact a tangent vector for S^3, i.e.
it sits in the subspace T_p(S^3).

I had written up to this point, but I have been rather tied up with
other things and have not had the chance to finish constructing a
really clear explanation of what's going on. So I will refer to two
sources:

1. Penrose and Rindler's "Spinors and Space-Time" has a discussion of
spinorial tangent vectors in chapter 1 with more details.

2. Gilkey's Invariance Theory, the Heat Equation, and the
Atiyah-Singer Index Theorem has a very explicit construction of the
spin structure on the tangent bundle of CP^1 on pages 167-170. This
book is available free at http://www.emis.de/monographs/gilkey/ .
The basic idea is that if you can take the transition functions for
the tangent bundle of CP^1, which are in SO(2) (which is a circle S^1)
and replace them all with elements from the 2-fold cover (which I
think some people write as U(1) -> SO(2)) given by squaring, i.e.,
e^{it} -> e^{2it}, then that is a "spin structure" on the tangent
bundle. It's kind of like being able to take the "square root" of the
bundle.

Hope this helps!

Best wishes,
Francis Fung