Completeness of non-normalizable modes

TL;DR

This paper proves the completeness of certain non-normalizable mode sets in brane-world gravity, demonstrating their role in constructing propagators and challenging the notion that normalizability is essential for physical relevance.

Contribution

It establishes the completeness of non-normalizable mode bases in brane-world models and shows they can be used to construct finite-residue propagators, expanding the understanding of mode spectra.

Findings

Non-normalizable modes form complete bases in certain brane-world scenarios.

Propagators can be constructed with non-normalizable modes as poles with finite residues.

Normalizability is not a necessary condition for a mode's physical relevance.

Abstract

We establish the completeness of some characteristic sets of non-normalizable modes by constructing fully localized square steps out of them, with each such construction expressly displaying the Gibbs phenomenon associated with trying to use a complete basis of modes to fit functions with discontinuous edges. As well as being of interest in and of itself, our study is also of interest to the recently introduced large extra dimension brane-localized gravity program of Randall and Sundrum, since the particular non-normalizable mode bases that we consider (specifically the irregular Bessel functions and the associated Legendre functions of the second kind) are associated with the tensor gravitational fluctuations which occur in those specific brane worlds in which the embedding of a maximally four-symmetric brane in a five-dimensional anti-de Sitter bulk leads to a warp factor which is divergent. Since the brane-world massless four-dimensional graviton has a divergent wave function in these particular cases, its resulting lack of normalizability is thus not seen to be any impediment to its belonging to a complete basis of modes, and consequently its lack of normalizability should not be seen as a criterion for not including it in the spectrum of observable modes. Moreover, because the divergent modes we consider form complete bases, we can even construct propagators out of them in which these modes appear as poles with residues which are expressly finite. Thus even though normalizable modes appear in propagators with residues which are given as their finite normalization constants, non-normalizable modes can just as equally appear in propagators with finite residues too -- it is just that such residues will not be associated with bilinear integrals of the modes.