Introducing "The Imagination Unit"

As demonstrated, we can use the real mass M to define a tachyonic mass tM, writing;

tM = -iM ,

where M = m[(1 - [(v/c)^2])^(-1/2)] , with m as bradyonic rest-mass.

We have also shown why the negative sign on the imaginary-unit is necessary for correctness, in such cases. Yet, despite thereby placing the definition of tachyon mass on a formal footing, this representation can create some confusion when complex quantities associated with both M and tM are discussed in the same mathematical contexts; especially when the two appear in the same formulas.

For example, suppose there is yet another particle with an imaginary mass, iM, with the same amount of mass as M, but which is bradyonic, not tachyonic, and we get a negative sign from somewhere other than R; say, when employing vector velocities (as with the formula for momentum), and this tardyon goes in the opposite direction to the original. This can happen if the oppositely-moving imaginary mass, iM, is for a particle traveling near-to but slower-than lightspeed, and we want the imaginary unit, i, to be interpreted according to its common convention of implying that iM is merely a standard imaginary; such as, in quantum physics when discussing processes involving massive virtual particles (e.g., neutral Z-particle and/or charged W-particles mediating weak-nuclear interactions). How then do we distinguish between bradyonic and tachyonic -iM?

We could, and should, assign a different symbol to denote the tachyonic -iM, but that does little to eliminate the potential for confusion wrought by having two different interpretations of the same imaginary-unit, i; one applied to tardyons, and another to tachyons. To solve that problem, we can introduce a new imaginary-unit, an operator I originally called the "imagination-unit", which transforms any ordinary quantity into its tachyonic analog. That is, applying I multiplicatively to any standard quantity and/or symbol is defined -- according to a new convention, to go along with the new unit -- as imposing a transformation across the lightspeed barrier of space-time, so it is understood to project that quantity or symbol into superluminal space-time, where causality is reversed, all velocities are FTL, and all objects therein can be referred to as "actual imaginaries", to distinguish them from the standard imaginaries that we deal with on an everyday basis in mathematics, physics, and engineering.

Naturally, this would not help much if velocity restrictions are not specified. So, we further define the imagination-unit, I, to imply evaluation between lightspeed and infinite speed exclusively. That is, for example, instead of writing tM = -iM , we define the tachyonic mass tM by evaluating M exclusively between [v = infinity] and [v=c], for the representation of the evaluation set equal to tM = IM . [The brackets indicate evaluation between the enclosed values, but not at those values.]

Representing tachyons in this way allows us to discuss a standard particle with negative imaginary mass, -iM, in the same context as tachyonic particles, defined by IM, without fear of the confusion that would be possible with two interpretations of the standard imaginary-unit, i.

Case in point, suppose we want the tachyonic analog of a bradyon, but it originally has a negative imaginary mass, -iM, though it is not tachyonic. We would not want -i to imply that this mass, or anything else, is tachyonic.

So, we merely write [v = infinity] as the limit at infinity to an evaluation of

-i(tM) = I(-iM) = -iM ,

with [v = c] as the lower limit.

The evaluation representation should state: "Negative imaginary tachyonic mass -i(tM) is equal to the tachyonic analog of the standard negative imaginary mass, I(-iM), which is equal to the standard negative imaginary mass -iM evaluated specifically and exclusively between c and infinity".

We can, of course, relate motion involving the tardyonic masses M and iM to their tachyonic analogs using the Lorentz transformations, because all tachyonic analogs are, by definition, in reference-frames that are always moving relative to all tardyonic reference-frames, as long as the tachyons do not move at infinite speed in either frame.

The mass of a tachyon that moves at infinite speed can be defined, but that must be done quite separately, in a different manner, and given only as a side-note, because such a tachyon cannot be treated satisfactorily in any rigorous particle-physics setting, due to the fact that the presence of an infinite velocity turns all equations involving it into exercises.

Infinite-velocity tachyons can, of course, be imagined, and described using pure mathematics, but they must be considered as having applications in metaphysics.

These ideas are best understood by inspecting the Velocity Spectrum, denoted;

I(inf. v)>{ Iv > Ic > [ c > v > (v=rel(0))]} (antiparticle v expansion here; opposite order),

where I(inf. v) is infinite speed, Iv is any velocity between tachyonic lightspeed Ic and infinite speed (exclusively), c is the lightspeed constant, v is any ordinary velocity between v = rel(0) (relative zero velocity) and c, also exclusively; iv = abs(0) can denote a zero reference between the particle vs. antiparticle representation of the spectrum (bars or underlines can be used to denote corresponding antiparticle velocities, off to the right).

Note that a theoretical absolute-zero velocity is a standard pure imaginary.

Now, tachyonic lightspeed, Ic, can further be defined; Ic = (1.00...001)c , with the exact number of zeros to the right of the decimal-point an empirical unknown; making tachyonic lightspeed both an irrational and transcendental number (which makes sense).

Considering first only standard particles [everything to the left of rel(0)], and no antiparticles, when dealing with complicated systems, one-to-one correspondences across the lightspeed barrier, c, associating standard variables with their tachyonic analogs, can be realized by integrating with respect to velocities on the other side of c, exclusive of c and I(inf. v). That is, the evaluations associated with "I" can be accomplished using integration, whenever a spread of real quantities must be related to a corresponding spread of tachyonic analogs in superluminal space-time.

A similar tactic is employed for the antiparticles [everything to the right of rel(0)].

Finally, it is not always necessary to use the imagination-unit to describe things made of tachyons, but it is given here as a viable option when complex bradyonic quantities and complex tachyonic quantities are treated in the same context, and a method is needed to eliminate confusion between the two.