Stay tuned for a new article on my mathematical invention, The
Imagination Unit, also referred to as Richter's Tachyon Operator.

I intend to provide a formal representations theory that explains how
the imagination-unit can be used to represent tachyonic variables.

The purpose for employing such an operator is as follows.

Given a particle of mass m, moving at velocity v, which may or may
not equal lightspeed c, there are three relativistic cases;
(1) v < c, so that m is real (such as for electrons, protons, etc.),
(2) v = c, so that m is zero (such as for massless photons), and
(3) v > c, so that m is imaginary (such as for tachyons).

This above a commonly-used introduction to the idea of tachyons.

The standard representation of a tachyon, therefore, is obtained by
multiplying the mass m by the standard imaginary-unit, i, to make it
a pure imaginary, im, where i = (-1)^(1/2) ; i^2 = -1 .

One problem with the standard representation, however, is that we
have no way to distinguish between different types of tachyons.
All such tachyons are essentially the same, and could travel at any
velocity above lightspeed, up to, and including, infinite speed.

Also, the standard imaginary-unit, i, is used in other ways that are
not associated with tachyons. For instance, due to the manner in
which the imaginary-unit comes about in the representations of waves,
it appears as an operator in the Schroedinger equation of Quantum
Mechanics, which is used to describe the behavior of a particle with
wavelike characteristics, which are themselves represented, in turn,
by a wavefunction. But the use of i in the Schroedinger equation
does not imply that the particle is a tachyon. So, if we have a
particle of mass m, described using the Schroedinger equation, but we
also wish to discuss a tachyon of mass im in the same context, then
how are we to know that the i in the Schroedinger equation is used
differently than the i in the definition of the tachyonic mass, im?

We could use two different symbols for i, but that does not solve the
problem of the confusion caused by using two different applications
of the same operator, (-1)^(1/2), in the same context.

To solve this problem, we need an entirely new imaginary-unit; one
with a different definition than the standard imaginary-unit, i.

To the point, we can use an operator i^i that is defined as causing
the mass m to be transformed into its tachyonic analog. Thus, if m
denotes the mass of a standard particle, then (i^i)m is its tachyonic
analog, so that im no longer indicates a tachyon, but is simply the
pure imaginary obtained from m, by multiplying m by i.

To illustrate the significance of this representation, consider the
complex mass M obtained as the sum of a real mass m and an imaginary
mass im, defined according to the equation denoted;
M = m + im.
Here, m is the real component of M, and im is the imaginary component
of M, but im is not tachyonic.
A corresponding tachyonic version of M is defined;
(i^i)M = (i^i)(m + im) = (i^i)m + (i^i)im .

Now, the sum M + (i^i)M is a special case, called a "supercomplex"
mass. It is the sum of the standard complex mass and the tachyonic
complex mass. The concept of such a supercomplex mass would not be
possible without the use of a tachyonic transformation operator, such
as the new imaginary-unit, i^i, which I call the "imagination-unit".

Now, in Quantum Mechanics, all particles are described using complex
variables. Employing the imagination-unit to define tachyons allows
us to discuss ordinary particles and tachyons in the same theoretical
context without changing the meaning of the standard imaginary-unit.