[FOM] An alternative to the real numbers as a basis for physics

Gabriel Stolzenberg gstolzen at math.bu.edu
Sat Mar 4 11:43:05 EST 2006

   In his message of Friday March 3, Joe Shipman wrote,

> There is a growing literature on the use of alternative completions
> of Q (that is, the p-adic numbers) as a basis for physics. Since
> experimental measurements are always expressed as rational numbers,
> this is less wacky than it may seem.

   I thiink you mean "rational intervals."  But this doesn't effect
the point you ae making.

> The following paper is a good introduction:
> http://front.math.ucdavis.edu/hep-th/0602044
> Many more references can be found here:
> http://www.maths.ex.ac.uk/~mwatkins/zeta/physics7.htm

> Here is a quote from Yu. I. Manin:
> "On the fundamental level our world is neither real nor p-adic; it is
> adelic.

   Really?  What is the evidence for this claim?  Does the physics
community accept it?  Are there interesting arguments against it?

>                                                             We can
> equally well spiritually project it upon its non-Archimediean side
> and calculate most important things arithmetically.

   I'd like to see how this goes, whatever it means to "spiritually"
project.  (Is this in the references?)

> The relations between 'real' and 'arithmetical' pictures of the world
> is that of complementarity, like the relation between conjugate
> observables in quantum mechanics."

   Very likely I don't understand, but I don't see any more content
here than in, "The relation between pictures of the world in different
coordinates systems is like the relation between conjugate observables
in quantum mechanics."  (Note. I'm not suggesting that this is a good
analogy, only that it's similar to the one that Manin makes.)

> I've never seen any discussion of this on the FOM. Is there anyone
> who knows both enough math and enough physics to comment?

   I don't know enough math and physics but I may know enough math
and Manin.  Manin is a very great mathematician but I would not
trust him on a subject of this kind.  (Which doesn't mean that what
he says is wrong.)

   Joe, if you have the references you mention above, what more do
you want?  Why not ask some theoretical physicists you respect/trust
about this stuff?

   Speaking only for myself, I wouldn't take any of this seriously
until either a physicist I trust has signed off on it or I've seen
some convincing examples.  If there are examples in your references,
maybe you could show us one or two.

   Gabriel Stolzenberg

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