The Tachyonics Operator and the Velocity Spectrum
The traditional method of describing a tachyon is to take the result of inputting a superluminal velocity in the formula for the Relativity Operator, a , and using a negatively-signed but otherwise standard imaginary-unit, i , to express the result. That is, if mr is a real mass, then an imaginary mass m , for a tachyon, can be obtained as a direct analog of the real mass, by writing; m = -imr . [Reference: Encyclopedia of Physics, 2nd Ed., by Lerner & Trigg, VCH Publishing, pg. 1246.]
However, due to the implication that there exists a superluminal universe, we can imagine that it exhibits its own number system, incompatible with the standard number system (although, there is a one-to-one correspondence). In that case, the standard imaginary-unit ( i = sqrt (-1) ; i^2 = -1 ) becomes inadequate to describe the kind of tachyon I wish to employ to explain quantum gravity (though the traditional method works fine in most contexts involving tachyons).
One problem is that a negatively-signed imaginary-unit does not automatically imply superluminality. There are many instances of negative imaginaries in the equations physicists use to study natural phenomena. So, it becomes confusing if we also use the same symbolism to define tachyons -- and it is bothersome to repeatedly have to explain representations in texts accompanying the equations. I solve this problem by defining tachyons using an operator, ii , inspired by the standard imaginary-unit, but which is used only to designate tachyonic quantities. This frees-up the standard imaginary-unit, so that negative imaginaries can be treated in the usual fashion, without confusing them with implied superluminality.
My new operator, originally called the "imagination-unit", can be referred to as the "tachyonics operator", and its application is very simple. Multiplying the new operator to any real quantity, variable, or symbol transforms it into a superluminal analog of itself. All that's required is that a transformation equation is given in the context of a discussion involving tachyons (so, if no tachyons, ignore it).
In other words, if mr is a bradyon's mass, then mt = iimr is the imaginary mass of a tachyon taken in perfect analogy to mr . However, the transformation must be limited, so that unwanted infinities do not nullify the result. Thus, we write;
[ v > c ]
mt = iimr = mr | , tt < 0 .
[ v < infinity ]
This formula simply states that a tachyonic mass results from multiplying the tachyonics operator to a bradyonic mass, which involves evaluating the given bradyonic mass in tachyonic spacetime (between lightspeed and infinite speed, exclusively), where reversed causality holds for the tachyonic analog.
An example exercise is to use integration to establish a scaling format for the superluminal analogs of bradyonic quantities, such as velocities in general. Note too that the tachyonics operator can be used to define an entire number system for tachyonic spacetime -- including the specification of a superluminal analog of the standard imaginary-unit; it = ii i .
A convenient way to understand how the transformation works is to inspect the velocity spectrum, involving all possible theoretical velocities; where bars indicate antiparticle velocities, bradyonic and lightspeed velocities are within the braces, v0 is a relative-zero velocity, while absolute-zero velocity is an imaginary quantity (denoted by the standard imaginary-unit, i ), the superluminal velocities (in one-to-one correspondence with bradyon velocities) are designated by the tachyonics operator, ii , and infinite velocities are outside the purview of experimental testing (at this time).
One other point could be raised here, regarding the lightspeed constant, which acts as a barrier between bradyonic and tachyonic spacetimes. It clearly stands to reason that a superluminal universe would have an analog of the lightspeed constant (if it has analogs of everything else), and thus superluminal lightspeed could be defined as, let's say iic = 1.000...0001c , where the exact number of zero-valued decimal-places is undetermined. And in that case, an extra place must be made in the velocity spectrum for iic , since it could be an issue if that has a bearing on an experiment. We should therefore consider the superluminal photonic tunneling experiments, in which velocities for the tunneling photons are calculated at significant multiples of c . Such experiments could in-fact provide the theoretical lower-limit on the possible superluminal analog of c .
Gü nter Nimtz, of the University of Koln, Germany, for instance, has been doing photonic tunneling for about two decades, and is an often-cited authority on the subject; noting his own and other physicist's determinations of speeds at which different categories of photons (radio, microwave, etc.) engage in superluminal tunneling (which seems to depend largely on two factors -- the frequency of the photons, and the specific nature of the apparatus used by various investigators). Superluminal photon velocities cited in the most easily obtained online resources I consulted for this article (but dated only through '03) ranged from 1.4c to 8c . [The actual speeds I found in the literature were: 1.4c, 1.7c, 2c, 4.7c, and 8c. Reference: psiquadrat.de/downloads/nimtz03.pdf . Consider also hal.archivesouvertes.fr/docs/00/24/69/30/PDF/ajp-jp1v4p565.pdf . ]
The purpose of getting into this is (1) to establish the concept of tachyons for the present discussion, (2) to specify the velocity spectrum, with the goal of using it to understand a logical transformation scheme that relates bradyonic quantities to their exact superluminal analogs (e.g., setting-up a scaling, gauge-based, or data-spread format for one-to-one correspondences), and (3) to have a springboard for postulating tachyons with properties not in analogy to known particles.
To be continued in next post.