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S1b. Projective geometry and quantum mechanics
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Projective geometry means that one works with rays instead of vectors
to designate points in a geometry.
Think of the 2-dimensional affine plane. The points are represented by
vectors in R^2. On the other hand, by moving an affine plane lying on
the floor a little upwards into the air (the same amount at every
point), one may think of each point as being represented by the ray
from an origin on the floor to the point on the plane.
(Actually, instead of the ray one should consider the whole line;
strictly speaking, a ray is only a half-line. But in quantum physics,
one custonmarily calls the 1-dimensional subspaces rays. Since the
coefficient field is complex, the rays are actually rotated complex
number planes.)
Similarly, lines are now 2-spaces through the origin. This gives
projective geometry (or homogeneous coordinates, which is the same in
more algebraic terms).
But now one also has some additional points, corresponding to rays
parallel to the affine plane. These points form the 'line at infinity'
= the 2-space through the origin parallel to the affine plane.
A slightly closer look reveals that the geometry has become more
complete: Now not only every two points have a unique connecting line
but also any two lines have a unique intersections - what were before
parallels are now lines intersecting 'at infinity'. Imagine two long,
straight rails of a railway track...
Thic can be extended to higher dimensions. n-dimensional affine geometry
can be respresented by rays through 0 in n+1 dimensional space, and can
be completed there to a projective geometry, in which the vector
subspaces are the geometrical objects. In Hilbert space one cannot
count anymore dimensions, but otherwise everything is similar.
Since, in quantum mechanics, state vectors are only defined up to a
phase (even when normalized), they correspond uniquely to rays
= 1-dimensional subspaces in Hilbert space. Hence quantum mechanics is
intrinsically projective.