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
material.
.
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