EIGENBUNDLES, QUATERNIONS, AND BERRY’S PHASE

Daniel Henry Gottlieb

Abstract. Given a parameterized space of 
square matrices, the associated set of
eigenvectors forms some kind of a structure 
over the parameter space. When is that 
structure a vector bundle? When is there a 
vector field of eigenvectors? We answer those 
questions in terms of three obstructions, using a
Homotopy Theory approach. We illustrate 
our obstructions with five examples. One of
those examples gives rise to a 4 by 4 
matrix representation of the Complex
Quaternions. This representation shows the 
relationship of the Biquaternions with low
dimensional Lie groups and algebras, Electro-
magnetism, and Relativity Theory. The
eigenstructure of this representation is very 
interesting, and our choice of notation
produces important mathematical expressions 
found in those fields and in Quantum
Mechanics. In particular, we show that the Doppler 
shift factor is analogous to Berry’s Phase.