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

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.

Effects of Compactification on Free Massive Scalar Fields in Five-Dimensional Space-Time with an Extra Time Dimension: Analysing Some Results

“…It has been noted in many other works that the mass terms of the generated Kaluza-Klein tower of higher modes in the compactification of timelike extra dimension have negative signs in contrast with those found in the compactification of an extra spacelike dimension. In Section 4, we pointed out in detail how the resulting imaginary masses lead to the necessity for handling the case using various methodologies reported in the literature under the topic called tachyonic Klein-Gordon (TKG) theory (see [14] and references therein). In this paper, we provide detailed calculation procedures for two reasons: 1) to make it easier for those who are new in this field to grasp the subject and 2) to promote new ideas regarding the compactification technique other than the familiar infinite Fourier series expansion approach originally introduced in Kaluza-Klein theory more than a hundred years ago.…”

“…Klein-Gordon equation with imaginary masses has been discussed in the literature even until recently (for example see [14] and [15]). Such an equation is sometimes called a tachyonic Klein-Gordon (TKG) equation [14].…”

“…Klein-Gordon equation with imaginary masses has been discussed in the literature even until recently (for example see [14] and [15]). Such an equation is sometimes called a tachyonic Klein-Gordon (TKG) equation [14]. In the literature, a hypothetical particle moving faster than light is known as a tachyon (which has not been found experimentally) whereas a field having an imaginary mass is known as a tachyonic field.…”

“…Lopez-Ruiz, Guerrero, and Aldaya [14] investigate mathematically the TKG equation and the conventional real-mass KG equation (m 2 > 0) in connection with the Helmholtz equation providing a unified framework for the scalar products of the three equations. This work further reduces the gap between real versus imaginary mass of free scalar field equations in the mathematical sense.…”

“…Our work in this paper starts with an extra time S. Zahir dimension in a modified Minkowski space as well. Then, compactification of the extra time dimension using the KK technique [6] led to the emergence of a tower of imaginary mass scalar fields (called TKG fields as per the nomenclature used in the literature [14]). It is worth pointing out that extra timelike dimension has been considered in curved space involving a complex brane world scenario (see [16] and references therein), too.…”