Perturbation Theory and Lie Algebras

Abstract

An approach to perturbation theory is considered based on the formula exp (—iB) A exp (iB) = A exp (iB]), where [AB] = AB − BA is the commutator. The operators A constitute a linear space A and the operators B considered are such that B] take A into itself. The present discussion considers the case where A is finite dimensional with coordinates c1, ···, cn; i.e., A is isomorphic with the set of n-dimensional vectors c. Under these circumstances sufficient conditions for the basic formula are given in terms of the ``analytic vectors'' of Nelson. The set, B, of B's available can be considered closed under the processes of taking linear combinations and forming the commutator. Thus B is a Lie algebra. Exponentiation leads to a Lie group of operators U, and A and A′ are said to be B equivalent if A′ = U−1AU. For A finite dimensional each B is associated with an n × n matrix, b, which specifies the operation B] relative to the vectors c. Effectively then, B is finite dimensional. The matrices b form a Lie algebra with a corresponding Lie group of matrices u such that A and A′ are B equivalent if and only if the corresponding two vectors c and c′ are u images; i.e., c′ = u c. Computationally, therefore, the set of A′ equivalent to a given A is obtained by considering the orbit of a given vector c under the Lie group; i.e., the set of u c. A neighborhood A′ of a given A consists of those operators A′ in the form A + δA where δA is arbitrary except for a restriction on the size of its coordinates, ci. Given A, a neighborhood A′ can be found for which one can obtain by a well-known construction on the orbits a set of functionally complete and functionally independent invariants for B equivalence. Computationally global and rational invariants are desirable and these can be obtained in the form of similarity invariants, provided that a can be mapped onto a set of n × n matrices a in such a way that if c corresponds to a, then b c corresponds to a′ = [ab]. If the sets A and B are identical such a mapping is immediately available and if, in addition, the corresponding Lie algebra is semisimple, in general, given A, the A's in some neighborhood are each equivalent to an A″ which is a function of A. This corresponds to a case in which a very simple perturbation of A levels occurs. Two examples are discussed.