Special Matrices of Mathematical Physics

The following sections are included:

• Preface

• Contents

We here recall some basic concepts on matrices, with the main purpose of fixing notation and terminology. Only ideas and results necessary to the discussion of the particular kinds of matrices we shall be concerned with will be considered. Matrix Theory in general is covered in Gantmacher's classic [1].

Stochastic matrices can be introduced in many ways, according to different fields of interest. We shall present them, and the Markov chains they relate to, as a means of describing time evolution of physical systems whose future history is completely determined by the present state, without any influence of the past.

The entries of stochastic matrices are probabilities, real numbers in the closed interval [0,1]. Consequently, stochastic matrices are non-negative matrices in the generic case. In many cases the matrices turning up are strictly positive. Positive and non-negative matrices govern Markov chains of different brands.

One of the successes of the stochastic approach has come from its application, by Richard Kerner, to the problem of glass transition. Glasses are complicated, amorphous systems, for which there is, to the moment, no complete theory.

A beautiful illustration of the stochastic method come from its application to Selenium glasses, modified by the presence of Arsenic or Germanium. The resulting model successfully describes simple glasses with small concentrations of modifiers.

There is no complete, clear-cut classification of the stochastic matrices, not even a well-established terminology. We shall say below that they can be reducible or irreducible, regular or not regular, that they are eventually primitive or imprimitive, but these properties do not always exclude each other. Roughly speaking, there are two main tentative classifications. One limited to the irreducible cases, another more general. It so happens that the limitation to irreducible matrices would exclude many cases of physical interest, specially those involving transients. In this chapter a general description of the standard notions will be presented. We shall, in particular, see how the properties of a matrix appear encapsulated in its spectrum.

The wealth of possibilities in the evolution of a Markov process is very great. A microscopic state can remain as an actor indefinitely, or revisit the scene from time to time, or still vanish from it. Some systems stick to equilibrium in a compulsive, step-by-step way, others refuse it from the start. Equilibrium itself can be stable or not. In the long run, a chain can attain one equilibrium distribution, or many, or none. It can also circulate along several distributions indefinitely. And, among those chains which do arrive at an equilibrium state, a few do recover some memory of things past.

Making a Fourier transformation is equivalent to diagonalizing a circulant matrix, and an inverse Fourier transformation takes a diagonal matrix into a circulant matrix. In this way, circulant matrices — and their diagonalized counterparts — underpin elementary harmonic analysis. Though the formalism may attain its greatest interest in the continuum case, we shall content ourselves with the discrete finite case, in which the central roles played by certain special matrices are easier to ascertain. There is a direct relationship between those matrices and the Weyl–Heisenberg groups. Consequently, in our policy of always using some Physics steersman, Quantum Mechanics on phase space will be the main character on the scene. In the discrete case all the most important properties, with the exception of those involving topological and measure-theoretical questions, turn up in a very simple, algebraic way. Everything can be deduced from the properties of a single matrix, the shift operator…

The following sections are included:

• Bases

• Double Fourier transform

• Random walks

The real Mechanics of Nature is Quantum Mechanics, and classical structures should come out as survivals of those quantal characteristics which are not completely “erased” in the process of taking the semiclassical limit. It is consequently amazing that precisely those quantal structures backing the fundamental Hamiltonian formalism of Classical Mechanics, in particular the symplectic structure, be poorly known. We shall now see how spaces of matrices can be endowed with a differential geometry. And how, applied to those matrices which are basic to Quantum Mechanics, this differential structure leads to a quantum symplectic space, from which the classical phase space inherits its makeup.

The following sections are included:

• Matrix differential geometry

• The symplectic form

• The quantum fabric
  • Braid groups
  • The Yang–Baxter equation
  • Quantum tapestry

One of the main tasks of Mathematical Physics is to provide new tools, adapted to the new resources continuously turning up. The recent advances in computer algebraic manipulations have brought to the forefront a renewed interest in closed expressions. We present now an organizing tool, whose main quality is precisely to provide closed expressions for a variety of mathematical and physical results, ranging from the characteristic classes of Differential Geometry to the virial series in the theory of real gases. This organizing tool is the multi-variable polynomial introduced by Eric Temple Bell in the 1930s…

The following secions are included:

• Definition and elementary properties
  • Formal examples

• The matrix representation
  • An important inversion formula

• The Lagrange inversion formula
  • Inverting Bell matrices
  • The Leibniz formula
  • The inverse series

• Developments
  • A useful pseudo-recursion
  • Relations to orthogonal polynomials
  • Differintegration
  • Homogeneity degree

The determinant of a matrix A can be written as a Bell polynomial in the traces of powers of A. This leads to many important classical relationships: between different kinds of symmetric functions, between the coefficients and the roots of a polynomial, between the invariants of a Lie algebra, between the characteristic classes and characters of a differentiable manifold.

The applications of some matrix functions, like the exponential e^A, are well-known. Those of other functions are less popular. The realpower function A^t, applied to a Bell matrix B[g], yields the continuous iterate g^<t> of g and thereby helps bridging the gap between the two current approaches to dynamical systems: the continuous one, based on differential equations, and the discrete one, founded on the iteration of maps. A theory for the eigenvectors of B[g] turns up as a bonus.

The following sections are included:

• Microcanonical ensemble
  • Phase space volumes
    • Cycle indicator polynomials
  • Towards the canonical formalism

• The canonical ensemble
  • Distribution functions
    • Relativistic gases, starting
    • Quantum correlations

• The grand canonical ensemble
  • The Mayer formula
    • Relativistic gases, continued
  • Only connected graphs matter

• Braid statistics

• Condensation theories
  • Mayer's model for condensation
  • The Lee–Yang theory

• The Fredholm formalism

The following sections are included:

• Appendix A: Formulary
  • General formulas
  • Algebra
  • Stochastic matrices
  • Circulant matrices
  • Bell polynomials
    • Orthogonal polynomials
    • Differintegration, derivatives of Bell polynomials
  • Determinants, minors and traces
    • Symmetric functions
    • Polynomials
    • Characteristic polynomials and classes
  • Bell matrices
    • Schröder equation
    • Fredholm theory
  • Statistical mechanics
    • Microcanonical ensemble
    • Canonical ensemble
    • Grand canonical ensemble
    • Ideal relativistic quantum gases

• Bibliography

• Index