Thoughts on Imaginary Numbers
I take issue with some scientist's position on imaginary mass as "undefined". I think it is rather clearly defined. The only difference is that it cannot be defined as a "real" mass. But this does not mandate that it cannot be defined at all.
To the point, in mathematics, imaginary numbers are not necessarily undefined. It only requires that the definition includes reference to the fact that such a quantity is imaginary in the nomenclature of mathematics. Therefore, in keeping with the way we use math in theoretical physics, the term "imaginary number", for instance, does not imply that such numbers do not exist (or must be viewed as undefined, or as incapable of being defined). We merely use the label "imaginary number" to distinguish a member of the set of all imaginary numbers from other sets of numbers (real, prime, etc.), and therefore, by extension, employ similar terminology to distinguish between various measurable and/or calculable physical quantities, which works because most quantities are understood by comparing numerical values to some previously-defined well-understood zero reference, and in accordance with the strictures of dimensional analysis.
I would caution that some scientists let old-school thinking cloud their judgment in this area, as it has been standard practice for ages to assume that any theory implying things like imaginary mass must be set aside as nonsense, and to assume something is wrong with the calculations, experimental apparatus, or something else; but only because ideas that do not fit with "mainstream" scientific truths (which are not literal truths at all, but staunchly-held assumptions) are usually rejected out of hand -- at times so much so that many tend to go by what they have been taught (usually a pronouncement, rule-of-thumb, or other position expressed by some long-dead person of high regard) regardless even of what experiments may actually be telling them.
Here is an example of how out-dated precepts dominate science. I recall a physics course in which we studied the Schroedinger equation, whose solution can be either positive or negative, although only the positive solutions are considered "real" (in the literary sense). And, in response to my question as to whether or not any negative solutions are used for anything, my professor stated flatly that the negative solutions are always ignored -- because, according to him, they have no meaning. Yet, it seemed to me (and still does) that the reason for these negative solutions is that Mother Nature is simply accounting for the kinds of tachyons to which the Schroedinger eqation can be applied. This professor, though, old-school all the way, would have nothing to do with the idea of tachyons. So, when I foolishly offered my opinion, he did not take it as the epistemological observation I meant it to be, but (being that kind of teacher) took my opinion as calling into question his all-knowing pronouncement. And as I was already on bad terms with him, for asking too many "why" questions (which he does not like), this just made it worse for me. But, setting that aside, his attitude illustrates how an assumption can be viewed as if it were a fact, even if it has never actually been proven experimentally.
Thus, in theoretical physics, we need to remember that we are using the language of mathematics, or else make clear in the text that our meaning is literary, not mathematic, if that is what we intend. So, we should be sure to make it explicit whether we mean mathematical imaginary or literary imaginary, since that can make a great deal of difference in how the ideas we want to convey are interpreted (because, technically speaking, mathematicians borrowed the word "imaginary" from spoken language, and attached their own meaning to it). Even so, I still have to object to the implication, in either case, that an imaginary mass is undefined.
Consider. We can examine genuine undefined quantities to which we can compare the nature of an imaginary mass. For example, in a numeric ratio, if the denominator is zero, then the ratio is said to be "undefined" -- for division by zero indeed makes no logical sense. But taking the square-root of a number (even of zero, or of negative numbers) can always be defined, because it will always make some sort of sense, mathematically, in the appropriate set of numbers. How else could we use the imaginary unit, (-1)^(1/2), to solve so many practical problems involving complex numbers (such as when engineers need to find the solutions to differential equations, or to understand electromagnetic fields, and so on)? It is just that we have to give certain kinds of numbers, and therefore calculable quantities, their own special definitions (imaginary, irrational, abstract, etc.).
Consequently, I hold that a slight modification of wording is sometimes warranted; to replace the word "undefined" with the word "unusual", in order to have a more accurate and up-to-date statement on mathematically imaginary mass.
And here is one instance where different mathematically imaginary numbers are needed. In the standard scenario on tachyons, the particle is described by using the imaginary-unit as an operator that turns an ordinary mass into an imaginary mass, -im. And this works fine, as long as we are only providing an explanation of what tachyons are. But if we want to do calculations, theoretical physics models, and other work where we wish to present standard complex numbers (involving the imaginary-unit) alongside tachyonic complex numbers (similarly defined), then it seems to me there can arise a great deal of confusion, if we use the same imaginary-unit in both sets of complex numbers, and are
not providing some sort of representations theory that lets us depict these two types of complex numbers differently from each other.
Therefore, I devised a way to distinguish between bradyonic and tachyonic complex numbers, using my new imaginary-unit, which I originally called the "imagination-unit", but which is now called "Richter's Tachyonics Operator" (a name provided by a friend). [Search: "Alt-Science: The Tachyonics Operator".]
In short, my imagination-unit is simply an operator that transforms standard quantities, variables, or symbols into tachyonic (superluminal) analogs of themselves. Thus, it can be used to change the standard imaginary-unit itself into a tachyonic imaginary-unit, which results in the two desired sets of complex numbers. Using these, then, we are able to compare the things that exist and events that happen in ordinary space-time to those that may occur on the other side of lightspeed, with little or no confusion.
Comments welcome. E-mail: HKurtRichter@yahoo.com