Professor Gernot Akemann

Research Interests

Random Matrix Theory and its applications in Particle Physics, in particular Quantum Chromodynamics.

Matrix models have a long history with many applications in todays physics and mathematics. Their general strategy is to replace a complicated, interacting physical theory by a simple, zero-dimensional model. The only ingredients are global, anti-unitary symmetries such as time reversal or complex and charge conjugation. The Physics to be described (Hamiltonian, Dirac operator, etc.) is then replaced by a random matrix with the same symmetries, and a Gaussian distribution of its matrix elements in the simplest case.

The solution of the matrix model under consideration is then a well defined mathematical problem that can be performed exactly, for infinite or sometimes finite matrix size. It has turned out that in some cases this approach does not yield a merely phenomenological model, but represents a precise map onto an effective theory of the relevant degrees of freedom. An example for this is Quantum Chromodynamics (QCD), the theory of strong interactions of quarks and gluons. In the last years I have become particularly interested in this map to matrix models and the role of a chemical potential for the quarks. At low energies in the confined phase with broken chiral symmetry QCD is well described by an effective theory of its Pseudo-Goldstone bosons: Chiral Perturbation Theory. In a particular finite volume and zero mass limit it maps to a random matrix model description.

A prominent tool to solve QCD and to which matrix models can partly compare are Lattice simulations, where QCD is solved numerically on a space-time lattice. A chemical potential makes the spectrum of the Dirac operator to be described by matrix models complex. This seriously challenges the numerical techniques in Lattice QCD. For that reason I have become very interested in matrix models with complex eigenvalues, trying to construct and solve such models. While this program in not yet completed for all 3 symmetry classes of orthogonal unitary and symplectic models, we have seen already an impressive agreement with Lattice QCD data, extending results for real spectra at zero chemical potential.

The techniques used for this purpose are (complex) orthogonal polynomials and their asymptotic, saddle-point methods and exact evaluations of integrals over unitary groups. Other mathematical questions that appear in this context are characteristic polynomials and determinantal formulas for them, as well as the theme of universality. Do matrix models provide a unique answer when deforming the Gaussian distribution of matrix elements? The latter is not dictated by the underlying Physics and I am very interested in heuristic and exact proofs of this feature, being crucial for their predictability. Some of these methods and questions are also studied by my colleagues in the mathematical physics group, providing a unifying theme of our group.