The Category of Ladders

Abstract

In the first part of this paper we introduce the category of ladders and develop the underlying theory. A ladder consists of a sequence of vector spaces (V _n ) and linear operators ${(A^+_n), (A^-_n)}$ acting between these vector spaces in ascending and descending direction. Unlike as in classical quantum mechanics ladders are defined as objects of a category, the corresponding notion of ladder homomorphisms allows to perform a mathematically structural and rigorous analysis of ladder theory. We allow dependence on n for spaces and operators in the ladder. The job in ladder theory is to find SIE-subladders, on which the Intrinsic Endomorphisms ${A^-_n A^+_n}$ and ${A^+_n A^-_n}$ act as Scalars α _n . A fundamental ladder theorem will provide conditions on the (generalized) commutators or anticommutators assuring the existence of SIE-subladders. The second part contains examples of ladders from classical quantum mechanics, such as the Heisenberg ladder, the Dirac ladder the ladder for the Lie algebra ${{\bf sl}(2,\mathbb{C})}$. Whereas these classical examples are distinguished by constance of operators and spaces, we then show how the generalized ladder theory allows to handle deformations: h-discretization, q-discretization and “periodization”. Other examples come from orthogonal polynomials. The Legendre, Laguerre and Bessel ladder are presented. Ladder theory allows a certain “anticommutator factorization” of the relevant second order differential operators. In a final section we apply ladder theory to ladders that are at the same time complexes. This enables us to give a transparent structural proof of Hodge’s theorem. The idea of ladder operators and factorization is well-known in quantum mechanics and quantum field theory. The book of Shi-Hai Dong (Factorization method in quantum mechanics, Springer, Dordrecht, 2007) contains a good survey of applications in physics, it provides historical background and links to sources in the physics literature. Note that our approach is independent—not only with respect to notation—of that in Shi-Hai Dong (Factorization method in quantum mechanics, Springer, Dordrecht, 2007).