Abstract
Although describing very different physical systems, both the Klein–Gordon equation for tachyons (m2<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 m2<0 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 m2>0 Klein–Gordon equation, and provide a unified framework for the scalar products of the three equations.
Highlights
Sánchez VillaseñorRecently, it was shown by the authors of [1] that the scalar tachyonic representation of the Poincaré group can be realized unitarily and irreducibly on suitable, oscillatory solutions of the Klein–Gordon equation with negative squared mass, m2 = −κ 2 < 0: Received: 31 May 2021Accepted: 16 July 2021∂μ ∂μ φ − κ 2 φ = 0, Published: 20 July 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- (1)where the usual signature (+, −, −, −) and the units in which h = c = 1 are assumed.The key ingredient in order to achieve that is the non-local scalar product given by the following: iations. hφ, φit =Licensee MDPI, Basel, Switzerland. κ2
It was shown by the authors of [1] that the scalar tachyonic representation of the Poincaré group can be realized unitarily and irreducibly on suitable, oscillatory solutions of the Klein–Gordon equation with negative squared mass, m2 = −κ 2 < 0: Received: 31 May 2021
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affilwhere the usual signature (+, −, −, −) and the units in which h = c = 1 are assumed
Summary
Note that the ball of momenta |~p | < κ is cut out Both components of the wave functions for positive and negative energies (φ+ (~p ), φ− (~p )) are needed in the same irreducible representation with support on the one-sheet hyperboloid: the sets of the form (φ+ (~p ), 0) or (0, φ− (~p )) are not invariant subspaces under Lorentz transformations. Supports a unitary and irreducible representation of the 4D −Euclidean group, but a similar, non-local scalar product is required as follows: R. where Jα (z) is the Bessel functions of the first kind. Gordon and Helmholtz equations, the projections onto those spaces, and the unitarity of the representations
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