FOM: Complex Numbers
Mark Steiner
msteiner at dibinst.mit.edu
Thu Dec 18 17:03:23 EST 1997
Penrose is substantially correct on complex numbers:
In mathematics:
1. Complex analysis is not "two copies of real analysis." There are no
similar numbers you can make of "three copies of real analysis."
2. Using complex analysis you can explain theorems that you can prove, but
not explain, about the real numbers. (E.g. why 1/1+x^2 is not equal to its
expansion beyond abs(x) > 1, even though 1/1+x^2 is defined and
continuously differentially for all real x.)
3. Real analysis is only part of the picture, the tip of the iceberg.
Nineteenth century mathematicians likened the situation of just looking at
real analysis to being in Plato's cave, i.e. looking at shadows.
4. It is true that the coordinates of the complex number are real; but
mathematics should look for a coordinate free view of things. Modeling the
complex numbers as pairs of reals does not preserve, for example,
explanatory power, as above. (As I wrote this, I received Dana Scott's
ideas which I think are similar.)
In physics
1. There is a lot more in the world than what can be seen in the
laboratory; what we see in the laboratory is a random sampling of
information that happens to be available to our receptors. (This is a
general point.)
2. In quantum mechanics, groups usually act on complex vector spaces, not
directly on spacetime. Real vector spaces do not have enough degrees of
freedom to represent qm phenomena. (In a real vector space, for example,
an orthogonal transformation changes the coordinates of a unit vector with
respect to every orthonormal basis, but in the world, transferring the
laboratory in space changes position measurements but not momentum
measurements.)
3. Although complex numbers are used in physics in such a way that part of
the information of the complex number (its phase) is not physically
measurable (which is how you explain in the formalism why the unit vector
"doesn't move" with respect to the momentum basis while it "does move" with
the respect to the position basis when you transfer the labratory as
above), these very meaningless phase factors gain deep signifance in gauge
field theory.
In physics today, it is easy to start confusing the representation of the
world with the world. Steven Weinberg has spoken of the world as "being"
some kind of group representation, certainly a Pythagorean position. But
even short of that--if you have a mathematical formalism that tracks
reality (by this I mean that it continues to work when you play around with
it, make topological transformations on it, etc.) I see no reason to deny
that complex numbers are as "real" as real numbers.
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