Path integral on matrix model

Question

I was looking at a 0-dimensional matrix model, where the variables are $N\cdot N$ Hermitean matrices. It had a gauge symmetry, e.g. $U(N)$. And in the path integral, the Faddeev-Popov trick was used. So instead of integrating over the set of matrices, one was reduced to integrating over the $N$ (real) eigenvalues. Or, I tried counting the number of degrees of freedom and I was puzzled:

1. $N$ by $N$ Hermitean matrices have $N^2$ real independent variables ($N(N-1)$ values for the triangular part (not counting the diagonal ones), plus $N$ real variables for the diagonal ones, giving us a total of $N^2$

2. $U(N)$ has real dimension $N^2$, classic result from group theory

3. At the end, I arrive at $N^2 - N^2 = 0$ variables instead of $N$ real eigenvalues.

I tried seeing it from another point but I can't derive what I'd like to. What I am thinking/doing wrong?

Answer 1

Your $N^2-N^2$ calculation is naive, well, it is incorrect because not all $N^2$ generators of $U(N)$ are changing the Hermitian matrix $M$. If a generic Hermitian matrix $M$ is transformed to $$ M\to U M U^{-1} ,\quad U U^\dagger = 1,$$ then $N$ directions in $U$ i.e. in $U(N)$ don't change $M$ at all. This is easily seen in the basis in which $M$ is diagonal: the matrices $$ U = {\rm diag} (e^{i\alpha_1},\dots , e^{i\alpha_N})$$ don't change $M$ at all. So the right difference is $N^2 - (N^2-N) = N$.

Answer 2

In this answer we will basically expand on Lubos Motl's correct answer using some other words and introducing some terminology.

In the Hermitian one-matrix model, the action

$$\begin{align}S~=~& {\rm Tr} L(H), \cr L(H)~=~&\sum_{n\in\mathbb{N}_0}c_n H^n,\cr c_n~\in~&\mathbb{R},\end{align}\tag{1}$$

is invariant under adjoint conjugation

$$ H\to UHU^{-1}\tag{2}$$

with unitary matrices $U$. Eq.(2) here plays the role of the gauge transformations. One therefore has $N^2$ real gauge parameters.

At the infinitesimal level $U=e^A$, the gauge transformation is

$$ \delta H ~=~[A, H],\tag{3}$$

where $A$ is an infinitesimal anti-Hermitian matrix.

On the other hand, the $N$ real eigenvalues $\lambda_1$, $\ldots$, $\lambda_N$ of an Hermitian matrix are gauge invariants, which cannot be changes by gauge transformations of adjoint type (2). Hence there are actually only $N(N−1)$ independent gauge parameters. Such a gauge algebra is called reducible.

For a diagonal Hermitian matrix $H$, the $N$ redundant gauge parameters may be identified with the diagonal matrix entries $A^1{}_1$ , $A^2{}_2$, $\ldots$, $A^N{}_N$, at the infinitesimal level.

The Faddeev-Popov trick (in its original formulation) applies to irreducible gauge symmetries, but one can make it work in this reducible case (i.e. the Hermitian one-matrix model) by properly identifying the independent gauge parameters, cf. above.

The Faddeev-Popov determinant becomes the square of the Vandermonde determinant of the eigenvalues $\lambda_1$, $\ldots$, $\lambda_N$.

Finally, it seems natural to mention that the presence of this Vandermonde determinant is a typical feature of (random) matrix models, and it leads to eigenvalue repulsion, cf. e.g. this Phys.SE post.