Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation

• Main Authors: Francisco F. López-Ruiz, Julio Guerrero, Victor Aldaya
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Symmetry
• Subjects: unitary and irreducible representation; Poincaré group; Euclidean group; non-local scalar product; tachyonic scalar field
• Online Access: https://www.mdpi.com/2073-8994/13/7/1302

Although describing very different physical systems, both the Klein–Gordon equation for tachyons (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>m</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>) 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>m</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> supports the scalar tachyonic representation of the Poincaré 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> Klein–Gordon equation, and provide a unified framework for the scalar products of the three equations.