@misc{10498/25645,
year = {2021},
month = {7},
url = {http://hdl.handle.net/10498/25645},
abstract = {Although describing very different physical systems, both the Klein-Gordon equation for tachyons (m(2) < 0) and the Helmholtz equation share a remarkable property: a unitary and irreducible representation of the corresponding invariance group on a suitable subspace of solutions is only achieved if a non-local scalar product is defined. Then, a subset of oscillatory solutions of the Helmholtz equation supports a unirrep of the Euclidean group, and a subset of oscillatory solutions of the Klein-Gordon equation with m(2) < 0 supports the scalar tachyonic representation of the Poincare group. As a consequence, these systems also share similar structures, such as certain singularized solutions and projectors on the representation spaces, but they must be treated carefully in each case. We analyze differences and analogies, compare both equations with the conventional m(2) > 0 Klein-Gordon equation, and provide a unified framework for the scalar products of the three equations.},
organization = {We thank the Spanish Ministerio de Ciencia e Innovacion (MICINN) for financial support (FIS2017-84440-C2-2-P), and V.A. acknowledges financial support from the State Agency for Research of the Spanish MCIU through the `Center of Excellence Severo Ochoa' award for the Instituto de Astrofisica de Andalucia (SEV-2017-0709). J.G. acknowledges financial support from the Spanish MICINN (PGC2018-097831-B-I00).},
publisher = {MDPI},
keywords = {unitary and irreducible representation},
keywords = {Poincaré group},
keywords = {Euclidean group},
keywords = {non-local scalar product},
keywords = {tachyonic scalar field},
title = {Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation},
doi = {10.3390/sym13071302},
author = {López Ruiz, Francisco Felipe and Guerrero, Julio and Aldaya, V.},
}