# The Vandermonde determinant in Physics

+ 4 like - 0 dislike
2488 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?

edited Sep 21, 2015

+ 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 (1,545 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.