Random Matrix Theory (RMT)

A broad range of applications

Since the 60's, random matrix ensembles are simple toy models for a large variety of physical and non physical systems. For example RMT helped to understand what quantum chaos is. Moreover new phenomena in mesoscopic systems, in particular in condensed matter theory where correctly described by RMT. Also in mathematics RMT had a strong impact like number theory or representation theory. Other applications can be found in econo physics, studies of earth quakes and statistical physics in general.

Universality

Their broad applicability is due to the universality of statistics of levels as well as of eigenvectors. This universality can be found in systems whose scales vary over many orders like the level statistics of the Hydrogen atom and the vibrations of a metallic plate like a drum.

Analytical results

Random matrix ensembles are attractive because of their simple structures. These structures make analytical results accessible. Such results would remain in the dark for the original systems under investigation.

Global symmetries are the key

Although the physical systems are very often too complex due the high number of degrees of freedom or the complexity of the equations describing them they may reduce to their global symmetries in certain limits. If this happens then it is quite possible that the system can be described by a random matrix model.

Orthogonal Polynomial Theory

Orthogonal polynomials in quantum mechanics

Orthogonal polynomials are well known in quantum mechanics as solutions of the Schrödinger equation of integrable systems. To name the most popular ones, the Hermite polynomials are the diagonalizing basis of the quantum harmonic oscillator. However there are much more orthogonal polynomials like the Laguerre polynomials or the Legendre polynomials. Each of them are orthogonal with respect to their own particular

measure.

Application to RMT

In RMT these polynomials achieved a particular popularity as mathematical tools. They are unbelievably efficient when calculating level correlations. On the other hand, they are intimately related to random matrix ensembles themselves. Lately we also discovered interesting relations of structures found in orthogonal polynomial theory and in Susy with the aid of RMT.

Skew-orthogonal polynomials

The concept of orthogonal polynomials can be extended to an antisymmetric bilinear product. The corresponding polynomials are known as skew-orthogonal polynomials. These polynomials are also related to random matrix ensembles and, thus, have their own advantage as a mathematical tool.

Mixing of both kinds of polynomials

Recently we discovered a mixing of both types of polynomials, orthogonal and skew-orthogonal ones. They correspond to a particular type of random matrices interpolation random matrix ensembles of different symmetries. Such random matrix models became the focus of interest, especially in applications to lattice QCD.

Supersymmetry (Susy) and Supergroups

Susy in particle Physics

Supersymmetry was originally introduced in particle physics. It states that each bosonic particle has a fermionic counterpart and vice versa. Although an experimental verification of this assumption is still lacking it is nevertheless a generalization of the standard model with interesting advantages.

Susy in RMT

In addition to Susy in particle physics it was also introduced in other fields of physics like General Relativity Theory and quantum mechanics.

Particularly fruitful was the application of Susy in RMT. In RMT, Susy is purely a mathematical tool. Its overwhelming success is born out in a drastic reduction of integration variables when going from integrals over ordinary random matrices to integrals over supermatrices. Moreover the space of supermatrices naturally incorporates structures which were found long ago in RMT but their relation to Susy were unknown. We made this connection and got a new perspective of these two theories.

Integration in superspaces

The definitions of integrals in superspaces are of particular interest. Especially the change of coordinates is not as simple as in ordinary spaces. When changing coordinates boundary terms, also known as Efetov-Wegner terms, may occur. The ultimate reason for these terms is the integration over anti-commuting (Grassmann) variables which is equivalent to differential operators. Up to now compact, explicit formulas are missing for an arbitrary change of coordinates which is a big disadvantage of the Susy approach in RMT. However for particular cases we lifted the darkness and got new insights.

Supergroups

Supergroups are the supersymmetric generalization of ordinary groups. They are intimately related to RMT, too. Since the last few decades supergroups are in the focus of representation theorists. A particular interesting kind of classifying representations are with help of Gelfand-Tzetlin coordinates. These coordinates are particularly useful for some group and supergroup integrals often encountered in RMT.

Statistics of Correlation matrices

Correlation of time series

Correlation matrices can be found in many fields of physics as well as far beyond physics. Correlations of empirical data as in climate research,

finance, signal transmission and medicine are often made with time series analysis. These time series can be the prices of stocks, the temperature at different places, the electrical current of channels, etc.

Wishart RMT

John Wishart was one of the first who modeled correlation matrices. The Wishart random matrix ensemble was named in honour of him. This kind of random matrix model serves as a benchmark model for empirical correlation matrices. Recently we were able to derive an exact, compact formula of the marginal probability density function which was up to then only known in terms of infinite sums over Jack polynomials. Hence, our approach can be considered as the first successful resummation of these infinite sums.

RMT and Quantum Chromodynamics (QCD)

Relation to RMT

For many systems the connection to RMT is only given by observations of the analytical results with empirical data. This is not the case in QCD which is the theory of the strong interaction. In the microscopic limit chiral RMT exactly agrees with the epsilon-regime of QCD, i.e. the effective Lagrangian of the zero-dimensional Goldstone bosons (the three pions in a two flavor theory) can be also derived from a chiral random matrix ensemble.

The main idea

The main idea is the derivation of explicit non-perturbative results in chiral RMT which are not accessible in QCD due to the forbiddingly complexity of the Yang-Mills action. Then the low energy constants can be extracted by fitting lattice simulations to the analytical results. Since the number of low-energy constants is finite one can predict other quantities. This procedure was amazingly successfully applied in the past.

RMT and Lattice QCD

Lately QCD at finite lattice spacing became the focus of interest. Luckily there are random matrix ensembles corresponding to lattice QCD in the infrared limit, too. In particular staggered fermions and Wilson fermions were modeled by RMT. Quite recently we were able to derive all eigenvalue correlations of the Wilson-Dirac operator in the epsilon-regime. Comparisons to lattice data were astoundingly good for some quantities while for others strong deviations from the analytical predictions were found.

Theory of Relativity and Quantization

History

The concept of relativity goes back to Galilei and was adopted by Newton. They assumed an absolute time. Two observers in different inertial frames of references can compare their measurements obtained from one and the same system if they transform their result by a Galilei transformation. This theory of relativity was modified by Einstein in two steps. The first one describes the observation of a maximal speed of any information given by the speed of light in vacuum which is in each inertial system the same. The change from one inertial system to another one is then given by a Poincaré transformation. In the second step he unified the gravitational force with the change of frames of reference and, thus, traced gravitation back to an inertial force.

Quantum field theory

Quantum mechanics and quantum field theory is one of the most successful theories in physics. Since its introduction many theorists made a large effort to unify it with Einstein's theory of relativity, in particular gravitation. Furthermore there are many approaches to quantize a classical system. Each of them has there own advantages and disadvantages.

Schwinger's action principle

In the late 50's Schwinger introduced a generalization of Hamilton's action principle. It should be the exact counter part of the variational principle known in classical field theory but now in quantum field theory. His hope was a unified action principle yielding the commutation relations of the fields, the quantum field equations and the evolution equation of the wavefunctions. He aimed at avoiding a quantization of a classical system and to start right away from a quantum system. Besides he also looked at a unification of his theory with General Relativity Theory which was only of limited success. In my diploma thesis I modified his principle to obtain a background independent variational principle and will hopefully publish some of these results.

Selected Articles

1. 1.M. Kieburg: Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT, preprint-arXiv:1202.1768 (2012)
   
   

1. 2.M. Kieburg, K. Splittorff, J.J.M. Verbaarschot: The Realization of the Sharpe-Singleton Scenario, preprint-arXiv:1202.0620 (2012)
   
   

1. 3.M. Kieburg, J.J.M. Verbaarschot, S. Zafeiropoulos: On the Eigenvalue Density of the non-Hermitian Wilson Dirac Operator, Phys. Rev. Lett. 108, 022001 (2012), preprint-arXiv:1109.0656
   
   

1. 4.C. Recher, M. Kieburg, T. Guhr: On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices, Phys. Rev. Lett. 105, 244101 (2010), preprint-arXiv:1006.0812
   
   

1. 5.M. Kieburg, T. Guhr: A new approach to derive Pfaffian structures for random matrix ensembles, J. Phys. A 43, 135204 (2010), preprint-arXiv:0912.0658
   
   

1. 6.M. Kieburg, T. Guhr: Derivation of determinantal structures for random matrix ensembles in a new way, J. Phys. A 43, 075201 (2010), preprint-arXiv:0912.0654, selected for Journal of Physics A; Mathematical and Theoretical Highlights of 2010 Collection