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Journal of Physics Conference Series

Degeneracy of the eigenvalues of hermitian matrices

W.J. Caspers

Published Mar 1, 2008• 4 citations
Publisher

Abstract

Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.

DOI
10.1088/1742-6596/104/1/012032

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Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.
This paper presents a parametrization of a degenerate density matrix. The problem needs to be approached first with a diagonalized form (the spectral representation) to deal with degeneracy. Such a form is useful for this parametrization in that the conditions to be a density matrix from a Hermitian matrix are applied only to a diagonal eigenvalue matrix, not a unitary matrix. Those conditions can be satisfied by parametrizing eigenvalues with squared spherical coordinates in dimension of the matrix. Degeneracy in eigenvalues brings symmetries between a eigenvalue matrix and a unitary matrix, which are realized in a form of a commuting unitary matrix, called a commutant. The associated redundant parameters in a unitary matrix have to be eliminated. It is realized in this paper that degrees of degeneracies can be defined as the total number of possible pairs of the same eigenvalues and one degree of degeneracy corresponds to one phase and one two dimensional rotation in a unitary matrix or a commutant. In this way all the degeneracies are identified and assigned to one phase-one rotation block. Therefore, a unitary matrix or a commutant is a product of these blocks and a general diagonal phase matrix. In physics a unitary matrix is often parametrized by rotation and phase matrices, called an angular representation here. There are many possible different phase configurations. It is often not a trivial matter whether a given parametrized unitary matrix is general. A simple diagram will be introduced to illustrate how to transform one phase configuration to another.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
"This record is a duplicate. The updated and complete version available at https://doi.org/10.5281/zenodo.19842968" Within the Relational-Informational Model (RIM), where gauge fields emerge from the Uhlmann curvature of the observer’s reduced density matrix ρObs, I show that the number of fermion generations is determined by the topology of the degeneracy locus—the set of spacetime points where eigenvalues of ρObs coincide. By the von Neumann–Wigner non-crossing theorem, pairwise eigenvalue de- generacy in a Hermitian family requires three independent real conditions (codi- mension 3). In four-dimensional spacetime, each degeneracy condition therefore defines a one-dimensional submanifold—a degeneracy string. For the SU(3) sector of RIM, the spectrum of ρObs contains three eigenvalues {p1,p2,p3}, admitting ex- actly 3 2 = 3 independent pairwise degeneracies. Consequently, three topologically distinct degeneracy strings exist, each carrying a distinct sector of the Uhlmann holonomy. I identify these three sectors with the three fermion generations.An explicit qutrit toy model demonstrates that a closed loop in parameter space encircling all three degeneracy points produces a Z3 monodromy—a cyclic permuta- tion of eigenvalues—whose braid structure provides an information-geometric origin for the CKM mixing matrix and CP violation. The SU(2) sector contributes no additional generations (since 2 2 = 1) but instead generates the isospin doublet structure within each generation. This result constitutes a sharp, falsifiable prediction: the number of fermion generations is exactly three, as a topological consequence of pairwise degeneracies in a three-level quantum system embedded in four-dimensional spacetime. The prediction is consistent with precision electroweak data (Nν = 2.984 ±0.008 from LEP) and current LHC bounds on fourth-generation quarks.
In the framework of quantum theory, we present one theorem and three corollaries regarding the direct connection between constants of motion of a physical system and degeneracies of its energy eigenvalues. It is shown that this connection emerges when there exist quantum operators which commute with the Hamiltonian, but not with each other. Further it is shown that if the commutator of these operators is a nonvanishing constant number then (a) all the eigenvalues of the system are degenerate, and (b) the degree of degeneracy is infinite. A number of examples are discussed including the parity degeneracy of the hydrogen atom and the infinite degeneracy of the Landau levels of a charged particle in a constant magnetic field.