Hermitian matrix model ($Z=\int dM{e}^{-S(M)}$)

A Hermitian matrix model replaces a spacetime field by a finite-dimensional Hermitian matrix. Its normalized integral supplies correlation functions and a zero-dimensional perturbative quantum field theory expansion. For an action built from traces of matrix powers, unitary matrix conjugation is a symmetry and the eigenvalues describe the invariant degrees of freedom. A quartic real integral needs a nonnegative highest-order coupling for convergence; a formal power series can be studied independently of this convergence issue.

Imposing vanishing off-diagonal entries with auxiliary integrations and an off-diagonal Grassmann Gaussian integral gives a determinant of the commutator map on the off-diagonal matrix units. Its eigenvalues are ${\lambda }_i-{\lambda }_j$, $i\ne j$. Up to a constant sign, their product is the squared Vandermonde determinant. It contributes $-2{\sum }_{i<j}\log |{\lambda }_i-{\lambda }_j|$ to the eigenvalue action, encoding repulsion between distinct eigenvalues. Diagonal zero modes and constant normalization factors must be treated separately.

For $S=a\mathrm{tr}(M^2)/2+b\mathrm{tr}(M^4)/4$ with $a>0$, the Wick theorem givesThere are eight adjacent external-attachment contractions with one free matrix-index sum and four opposite attachments with no free sum. The second tensor structure is needed at finite $N$; omitting it is an additional large-$N$ approximation, not merely restricting to connected Feynman diagrams.