Complex numbers in physics

[Barry R. Smith]

Hi,

I'm a pure mathematician, working on the article "polynomial".  The article (for now) covers polynomials with real number coefficients, as those are the most common found in school and applications.  However, I mention that polynomials with other types of coefficients occur in applications, and I list complex numbers as one of my two examples.

I personally have never used a polynomial with non-real coefficients in an application, so I just wanted to get the stamp of approval from a real physicist that, yes indeed, you have at one point or another written down a polynomial with complex coefficients at some point during a quantum-mechanical calculation.  Or if you are pretty sure no physicist ever uses such "simple functions" (I'm guessing wave-functions and such always involve exponentials) except when the coefficients are real, then that would be good to know.

Also, the "complex number" page has a short section on complex numbers in physics.  In my opinion, it does not do a good job of explaining why they are essential to the foundations of quantum mechanics, rather than just a useful formalism when, say, pairs of real numbers would suffice just as well.  (I am given to understand that Penrose addresses this in "The road to reality", but it was too long for me to skim).  If it is possible to give a better discussion of this on a relatively elementary level, I think it would be a good improvement to the page.

Thanks for your help,
Barry Smith

[Paul Wormer]

I have used solid harmonics in complex form a lot. Solid harmonics are homogeneous polynomials in  x, y, and z. In complex form the coefficients are complex. And, of course, secular polynomials of complex Hermitian matrices are abundant in physics. They are nth degrees polynomials in one variable (the eigenvalue) with coefficients that are products of the matrix elements.

[Howard C. Berkowitz]

Just a note of admiration and appreciation: this is exactly the kind of interdisciplinary work we should be doing.

Another way to encourage this, besides the forums, is to make sure to use Related Articles subpages.  Several of us have been working on ways to increase the interconnection among articles, which is the inherent advantage of a wiki. I have some draft proposals nearly ready for formal submission.

[Barry R. Smith]

Thanks for the info.