Re: why do complex numbers work in physics?

From: Igor Khavkine (igor.kh_at_gmail.com)
Date: 12/04/04 

Date: Sat, 4 Dec 2004 21:15:25 +0000 (UTC)

alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0412021058.4771a6ad@posting.google.com>...
> The square root of -1 is used a lot in physics.
> But how does it relate to what,I suspect,most people would regard
> as the real world i.e real numbers (for example we speak of real
> probabilities
> and not imaginary probabilities - real probabilities are the "real"
> world).

The real numbers are not the real world. The world cares very little
about what mathematics we use to describe it. Complex numbers are
used in physics for the same reason that real numbers are: because
it's
possible. If you want an analogy, think about trying to solve a
differential equation. The only way general way to do it is to guess
a solution and then check whether it works. If a certain mathematical
structure (e.g. complex numbers) allows you to the right guess, by all
means use it. This reason is deep enough for physics. If you want to
know deep mathematical reasons why certain mathematical structures
come up in solutions of certain mathematical problems, that's a
different
question.

BTW, probability theory is not magic. It is based on common sense. If
you
can make sense of complex probabilities, feel free to construct a
probability theory based on them. But do not expect the converse, that
throwing in complex numbers into existing probability theory will
allow you to make sense of complex probabilities.

> Complex numbers can be represented by two orthogonal axes on a sheet
> of paper and so can real numbers.Since such representations are both
> geometrical
> entities,do complex numbers only relate to real numbers (and hence the
> "real" world) in the context of geometry? And since general relativity
> is a theory based on ideas of geometry, do complex numbers only relate
> to the real
> world in the context of general relativity i.e would an imaginary
> probability seem reasonable in the theory of general relativity?

Wow! Now that's a non sequitur if I've ever seen one. See my comments
above about probability and complex numbers. In order to avoid making
vaccuous speculative statements, you might want to familiarize
yourself with logic and rules of inference. It would also help to
realize that even though physics can be described by some mathematics,
not all mathematics describes some physics.

Igor

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