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The Vandermonde determinant in Physics

+ 4 like - 0 dislike
2369 views

The Vandermonde polynomial / determinant |V| popped up in some operator algebra I was exploring along with the $n$-simplices, and as I was trying to come to a better understanding of these little fellows, I stumbled across some relations to Physics (as usual).

I) Coulomb interaction energy:

In "Notes on Fekete points on complex manifolds, " Julien Meyer states that minimizing the interaction energy for a system of equally charged particles in a plane is equivalent to maximizing the magnitude of |V| constructed from their mutual positions. 

On page 10 of  Zabrocki's "Matrix models and and growth processes: from viscous flow to the quantum hall effect,"  the log of |V| appears again in the Coloumb interaction energy of a Dyson gas.

II) The Fadeev-Popov determinant in quantum field theory:

Ryan Thorngren in the MathOverflow question "Wonderful applications of the Vandermonde determinant" describes the connection to |V|. 

III) Wave functions:

Sabine Jansen in "Fermionic and bosonic Laughlin state on thick cylinders" presents a many-body wave function for particles on a cylinder that is the power of |V| times a Gaussian as a model in the fractional Hall effect. The use of the |V| in wave function expressions (and random matrix theory) seems to be closely related to |V|'s basic invariance under substitution of orthogonal polynomials in the standard definition of the Vandermonde matrix.

IV) Partition functions:

Powers of |V| occur in the Mehta integral, the partition function for a gas of point charges moving on a line and attracted to the origin. (Cf. also "Beta integrals" by Warnaar.)

(Edit 9/2015) See also The importance of the Selberg integral by Forrester and Warnaar and the Wronskian determinant associated with the Euler beta function integral and, therefore, the Veneziano amplitude in Dual N-point functions in PGL(N-2,C)-invariant formalism by Hanson (page 5). 

Where else does the Vandermonde determinant appear in Physics?

asked May 16, 2014 in Theoretical Physics by Tom Copeland (290 points) [ revision history ]
edited Sep 21, 2015 by Tom Copeland

1 Answer

+ 4 like - 0 dislike

Powers of the Vandermonde ($\beta=1,2,4$)  determinant appear in the context of Random Matrix theory. This is a vast and active field of research that I don't feel competent enough to comment on. However, a wonderful starting point would be the table of contents of a special issue of the Journal of Physics edited by P J Forrester, N C Snaith and J J M Verbaarschot. Looking through various articles in that issue has been on my wish list for a long time. :-(

answered May 17, 2014 by suresh (1,535 points) [ no revision ]

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