Understanding the Imagination-Unit

The complete treatise on Richter's Tachyonics Operator.

By H. Kurt Richter, founder: Tachyonics Society of America

Part 1: Overview

This post introduces a mathematical operator I originally called the "Imagination Unit", inspired by the standard imaginary-unit, but which is now called "Richter's Tachyonics Operator", and amounts to a representations theory in which a new kind of imaginary-unit is used to represent tachyonic/superluminal quantities.

The purpose for employing such an operator is to remove confusion wrought by using the standard negatively-signed imaginary-unit, in ordinary space-time, that does not imply superluminality, in the same context as specifications of tachyonic quantities, in which the same symbolism does imply superluminality. The negatively-signed imaginary-unit implying superluminality comes from the Relativity Operator, R, which Einstein derived from the Lorentz Transformations, and used in his theory of Special Relativity (SR). [Note: Einstein typically used the Greek latter alpha, rather than R, for this operator.]

The Relativity Operator is defined; R = [1 - (v/c)^{2}]^{(-1/2) }, where v is velocity and c is the vacuum constant of lightspeed. Note, then, that when v > c, any ordinary quantity or variable R operates on becomes a negatively-signed imaginary.

Given a particle of mass m, moving at velocity v, which may or may not equal c, there are three relativistic cases, from SR; (1) v < c, so m is positive real (e.g., electrons, protons, etc.); said to be "bradyonic", (2) v = c, so m is zero (e.g., photons, luxons, ...); said to be "massless", and (3) v > c, so that m is a negative imaginary, such as for particles called "tachyons".

This is an example of an introduction to the idea of the putative class of particle called the "tachyon", for which v > c, and time is therefore negative (tachyons have reversed causality), compared to bradyons -- if we count bradyons as positive real particles.

The standard representation of a tachyon, therefore, is obtained by multiplying the real mass m by a negative imaginary-unit, -i, to make the result a pure imaginary, -im. One problem with this representation, however, is that we have no way to distinguish between different types of tachyons; they could travel at any velocity above lightspeed, including infinite speed. And that is not meaningful in a physical sense, since an actual detection experiment would likely only confirm the existence of tachyons that do not or cannot travel infinitely fast. So, we could use a way to limit the range of tachyon velocities.

Also, the negative 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, used to describe the behavior of a bradyon with wavelike characteristics (where the bradyon itself is described using a wave-function). But the unit obtains a negative sign in a certain rearrangement of terms in the time-dependent Schroedinger equation, but is not implying superluminality. So, if a bradyon described using such a version of the equation is spoken of in the same context as a tachyon of mass -im, how are we to know that the -i in the rearranged Schroedinger equation is applied differently than the -i used to define the tachyon itself? And it does'nt help to attempt an explanation in accompanying text. [Print Reference: Modern Physics for Scientists and Engineers, by Thornton & Rex, from Saunders College Publishing, 1993, pg. 209. Online, simply search "Schroedinger equation", and note how the terms in the time-dependent form can be arranged so that a negative sign gets placed on the imaginary-unit.]

There are, of course, negative solutions to the Schroedinger equation, which are usually ignored as nonsense, but which are actually indicative of tachyons with wave-particle duality. But I would also like to describe tachyons that do not have wave characteristics, where the Schroedinger equation reduces to a linear reference (the equation of a line).

In any case, we could use two different symbols for -[(-1)^{1/2}] , but that does not remove the confusion caused by having two interpretations of the same operator, (-1)^{1/2 }, in the same discussion. To solve that issue, then, I devised a new kind of imaginary-unit; one with a different definition than the standard imaginary-unit. It does not remove or replace the standard imaginary-unit, but does help eliminate possible confusion.

To the point, we can use an operator, i^{i}, defined as causing a 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 exact superluminal analog, so that -im no longer necessarily indicates a tachyon, but is simply the pure imaginary obtained from m by multiplication with -i, being non-specific by itself (could be bradyonic or tachyonic). This does not, however, remove the need for R, since it still applies, and works the same as it always has. We merely use i^{i }to keep from having to repeatedly explain things using two applications of the imaginary-unit in text accompanying equations employed to describe subatomic particles.

To illustrate, consider the complex mass M obtained as the sum of a real mass m and a standard imaginary mass im, defined; 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 thus defined;

i^{i}M = i^{i}(m + im) = i^{i}m + i^{i}(im) .

This also works with

M = m - im,

where

i^{i}M = i^{i}m - i^{i}(im) = i^{i}m + i^{i}(-im),

with the -im bradyonic (that is, the -i does not imply superluminality).

Now, the sum M + i^{i}M is a special case we can call a "super-complex" mass, which concept would not be possible without the use of a tachyonic transformation operator, such as my tachyonics operator, i^{i}. The only extra requirement for understanding this new operator properly is to supply a transformation equation somewhere in context, which defines the operator as implying a transformation across the lightspeed barrier (indicated by the constant, c), and where integration is used to establish one-to-one correspondences between velocities ranging from relative-zero speed to lightspeed and velocities ranging from lightspeed to infinite speed, exclusively.

In Quantum Mechanics, particles can be described using complex variables. Using the tachyonics operator to describe tachyonic variables allows us to discuss known particles and tachyons in the same quantum-mechanical context without two interpretations of the negatively-signed imaginary-unit. And the defining equation for the tachyonics operator helps to place limits on tachyons, the same way natural limits exist for bradyons. Hence, applying i^{i} to designate tachyonic quantities further implies the existence of a complete superluminal universe, because it invokes the Light Cone of SR, in which space-time is divided into four separate bradyonic and tachyonic regions (two regions each, for past and future), thus implying a tachyonic universe co-existing with the visible universe.

In other words, my Tachyonics Operator can be used to establish a superluminal number system in analogy to the standard number system, including a tachyonic analog (i^{i}i) of the standard imaginary-unit (i). As a result, by logical inference, it suggests the existence of a superluminal universe taken in direct analogy to the visible universe. Theoretically, then, this may also indicate the existence of superluminal substructure for the visible universe; meaning, all ordinary particles could be composed of very small tachyons.

The Imagination-Unit (continued)

Part 2: Tachyons and Special Relativity

At this point, in the interests of clarity, I should discuss SR in a little more detail. Readers familiar with SR, and the notion of tachyons, can skip this part.

Consider an ordinary object at rest; for example, a basketball at rest on a basketball court. It has a rest-energy E and a rest-mass m, related by the well-known equation

E = mc^{2} ,
where c is the lightspeed constant (approximately 3 x 10^{8} meters/second).

Suppose we next roll the basketball across the court; setting it in motion with respect to the stationary surface of the court. It can then be viewed as existing in a different frame, local to itself; moving relative to the stationary frame of the court. And to relate these frames, we can apply transformation equations to the variables associated with various quantities (mass, velocity, ...) specified initially in either frame.

Orient a set of Cartesian coordinate axes so that the ball's center-of-gravity starts at the origin O, fixed relative to the floor, where we begin counting time t at t = 0, and the ball's center-of-gravity, with a mere push, can be made to move in the positive x-direction at a constant velocity v, without obstruction, so that the values of y and z are always zero.

Next, let x, y, and z denote the spatial parameters, and t the time parameter, for the stationary reference-frame, but let x', y', z', and t' denote the corresponding respective parameters for the moving reference-frame (the one moving with the ball), and where the x-axis and the x'-axis lie on the same infinitely-long line in space. Then the reference-frames will be related according to the Lorentz transformations;
x' = R(x - vt) ,
x = R(x' + vt') ,
y' = y ,
z' = z ,
t' = R[t - (vx/c^{2})] ,
t = R[t' + (vx'/c^{2}] ,
where the Relativity Operator, R = 1/{[1 - (v/c)^{2}]^{1/2}}, allows us to calculate the relative value of a quantity for a moving object from the corresponding value at rest.

If M denotes the basketball's moving mass, and m is its rest-mass, then we have;

M = mR = m[(1 - [(v/c)^2])^(-1/2)] = m/{[1 - (v/c)^{2}]^{1/2}}.

Notice therefore that, because the ratio v/c is part of the expression in R of which we take a square-root, then there is only one relationship between v and c that makes sense for a real basketball with positive time; v < c.

Suppose now, however, that we let M denote the mass of a real or a virtual subatomic particle, instead of a basketball. Then there are the three fundamental cases for M;

v < c, for positive real bradyons, v = c, for massless photons, and v > c, for negative imaginary tachyons.

Most of the subatomic particles cataloged by physicists as having mass, as far as we can tell, have positive rest-mass, including both real and virtual particles with mass. [Note: The neutrino may be the first exception to this rule to be recognized.] The scalar energy E and vector momentum P are defined using the real rest-mass m;

E = R(mc^{2})

and

P = R(mV) ,

where V is vector velocity; | V | = v .

Of note is the fact that the second case, for massless photons, actually works-out to make R an infinity if we embrace the mathematical convention that the inverse of 0 is infinity;
1/0 = (infinity) .
This occurs because, if v = c, then
R = 1/[(1 - 1^{2})^{1/2}] = 1/(0^{1/2}) = 1/0 .
Alternatively, yet remaining mathematically rigorous, we can say instead that the inverse of 0, in such cases, is "undefined", and maintain that the rest-mass of a photon is 0; which means all photons are massless particles, made entirely of energy.

Contrastingly, tachyons are particles with negative rest-mass that always travel faster-than-light, and have reversed causality (negative time), compared to bradyons. And their rest-mass is both imaginary and negatively signed.

I must now go into greater detail on this than has been provided for the other two cases.

Notice that the relativity operator, R, dictates what happens when you try to accelerate a real mass up to lightspeed. It works-out that M approaches infinity as v approaches c. In other words, it would take an infinite amount of energy to accelerate a bradyonic mass up to lightspeed. And because we do not have access to infinite energy, and do not observe infinite energy expended anywhere in the universe at large, then the lightspeed constant represents a kind of universal speed-limit. It is, by all accounts, a space-time barrier.

Thus, many physicists assumed (logically) that nothing "real" exists on the other side of lightspeed. Unfortunately, this has also caused some to conclude that tachyons cannot be created, even by a Big Bang like the one that initiated our universe. Hence, some people continue to insist that tachyons do not and cannot exist.

To be clear, the relativity operator, R, does not mandate that nothing faster-than-light (FTL) can exist, somewhere. It does indicate that it would require infinite energy to accelerate a real mass up to c, but it does not forbid objects that already travel at FTL speeds from existing on the other side of the lightspeed barrier. Nor is it necessary to get tachyons by accelerating real masses to and beyond c. In the cosmological Big Bang idea called "Inflation Theory", it is said that there was a period of superluminal expansion for all the energy associated with the first moments of the Big Bang. It is therefore entirely possible that many particles of various kind were created that retained the superluminal velocities of the energies out of which they were formed, at that time. Furthermore, because of its reversed causality, a tachyon's energy decreases as its velocity increases, with its zero-energy state at infinite speed. So, it is reasonable to think that higher-speed tachyons were easily created, because the required energy would be extremely low.

Also, while we depict tachyons as having imaginary mass, mathematically, we must remember that words like "imaginary", "abstract", and other terms employed in math contexts are labels for different types of numbers and numerical quantities, chosen to distinguish between them. But such a label does not necessarily imply that imaginary quantities do not exist. Thus, to label a tachyon's mass as "imaginary" does not imply non-existence for tachyons, because we are using the strict mathematical meaning of the word "imaginary", not its common literary meaning.

Interestingly, the standard imaginary-unit, i, can be defined in terms of two well-known irrational transcendental numbers. One of these is the value of Pi (the ratio of the circumference over the diameter of any size of perfect circle), and is often given the approximate value of 3.14. The other is the base e of natural logarithms, defined as the limit as n approaches infinity of the n-th power of the sum of 1 and 1/n, for any integer n. It is also defined using the following expansion;

e = 1 + 1/n! + 1/2! + 1/3! + ... + 1/n! + ... ,

which is commonly approximated as 2.72.
The relationship between i, Pi, and e is that i equals ln(-1) divided by Pi, denoted;
i = (-1)^{1/2 }= [ln(-1)]/(Pi) ,
where ln(-1) is the logarithm, to base e, of negative unity.

Now, Pi is referred to as "irrational" and "transcendental" because its decimal expansion is non-recurring and infinite (apparently). In fact, to date, though computers have been used to calculate its value to several million decimal places, we have yet to find its final digit, or to identify a recurring pattern. And the base e of natural logarithms is labeled using the same terminology, for similar reasons. Thus, because an imaginary number can always be represented as the product of i and any real number, we can state that they can also be defined in terms of these two irrational transcendental numbers -- although no-one would insist that Pi or e do not actually exist.

Consequently, just because we think of tachyons as imaginary, theoretically speaking, this does not mean that they cannot or do not exist.

To understand how tachyons work, be aware that it would take an infinite amount of energy to slow a tachyon down to c, just as it would take an infinite amount of energy to speed a bradyon up to c. And if we could see the emission of a tachyon from a composite body, as viewed from a bradyonic frame, it would appear as if the tachyon came from an infinite or very far-off distance and was completely absorbed by that body. That is, if we have a video of the ordinary emission of a bradyon from the body, the analogous ejection of a tachyonic analog of the bradyon would look much like we had merely run the video of the ordinary process in reverse.

The Imagination-Unit (continued)

Part 3: The Standard Imaginary-Unit

In a subsequent part, my non-standard method of representing tachyons is explained in detail, where the Tachyonics Operator -- a new kind of imaginary-unit -- is used to imply a transformation across the lightspeed barrier. However, because this new operator was inspired by the standard imaginary-unit, it is best, for the broadest understanding, to explain the standard imaginary-unit sufficiently, along with a few of its applications.

As mentioned, the relativistic mass M of a bradyon in motion can be related to the same particle's rest-mass m by the equation;

M = mR = m/{[1 - (v/c)^{2}]^{1/2}} .

Consider, then, a tachyon of mass M_{t} , with correspondingly the same amount of mass. The tachyon mass, M_{t} , can be represented by describing it as an imaginary analog of M;

M_{t} = -iM ,

where i is the standard imaginary-unit. i = (-1)^{1/2}, so that i^{2} = -1 .

Note that the minus-sign accompanying i, in this definition of M_{t} , is mandatory for having an empirical definition of the tachyonic mass, M_{t} . In such cases, the standard imaginary-unit is used algebraically as an operator that, when multiplied to any real quantity, is understood to imply that the real quantity is evaluated instead as a perfectly analogous imaginary quantity. But to go any further on this topic, it is necessary to lay some groundwork, so that later statements will be readily understood. [Readers familiar with complex and imaginary numbers can skip this part too.]

The imaginary-unit comes about as a natural consequence of considering certain numbers that cannot be categorized as "real". For instance, no real number x is such that x^{2} = -1. We can, however, imagine another kind of number, i, defined specifically as the square-root of -1, so that i^{2} = -1. Thus, if X is a positive real number, and we want to find the square- root of its negative, then we can always write;

(-X)^{1/2} = [(-1)X]^{1/2} = [(-1)^{1/2}](X^{1/2}) = i(X^{1/2}) .

For example,
(-25)^{1/2} = [(-1)(25)]^{1/2} = [(-1)^{1/2}](25^{1/2}) = i5 .

Now, all the sums of real and imaginary numbers form a set called "complex numbers", which includes the set of all real numbers and the set of all imaginary numbers. That is, if we let x and y denote real numbers, and we let iy denote an imaginary number, with z the sum of x and iy, according to the equation; z = x + iy ,

then z is a complex number, while x is referred to as the "real-number part" or "real component" of z, and y is referred to as the "imaginary-number part" or "imaginary component" of z. We can also represent this using function notation, where Re is a function of z that gives a real number Re(z), and Im is a function of z that gives an imaginary number Im(z), so that

z = x + iy = Re(z) + Im(z) ,

where Re(z) = x , and Im(z) = iy .

Consequently, if x is nonzero but iy = 0, then z is real. On the other hand, if iy is nonzero but x = 0, then z is referred to as a "pure imaginary". Of course, whenever z = 0, then one of the following mutually exclusive cases holds; Case 1: x = 0 and y = 0 simultaneously, or Case 2: iy = -x , where x and y are each nonzero.

Interestingly, because complex numbers are essentially the same as ordered pairs of numbers, then the following definitions also hold for almost all complex numbers.

The absolute-value |z| of a standard complex number z, and which absolute-value is called the "modulus" of z, is a real number obtained using the Pythagorean theorem;

|z| = |x + iy| = (x^{2} + y^{2})^{1/2 }.

Letting z denote a complex number, defined as a sum, so that z = x + iy ,
and letting z* denote another complex number, defined as the corresponding difference, so that z* = x - iy ,
where z* employs the same values of x and y as does z,
we say formally that z* is the "conjugate" of z.

The product of z and its conjugate, z*, is the square of the modulus of z,

according to the following proof;

z*z = (x - iy)(x + iy) = x^{2} - xiy + xiy - (iy)^{2} = x^{2} + 0 - (i^{2})(y^{2}) = x^{2} + y^{2} = |z|^{2} .

The ratio, z/Z, of two complex numbers, z and Z, is in fact a real number, obtained by multiplying the numerator and denominator by the conjugate of the denominator, which is the same as dividing the product Z*z by the squared modulus of Z. Denoted;

z/Z = (Z*z)/(Z*Z) = (Z*z)/(|Z|^{2}) .

One tremendously useful application of complex numbers is their appearance in the solutions to quadratic equations, which should be covered briefly as follows.

An equation of the form
ax^{2} + bx + c = 0
is referred to as a "quadratic equation", in standard form, where x is a variable, and a, b, and c are constants. Equations of this form are used to solve so many problems that a full accounting of them would fill an encyclopedia. And therefore, examples are easily had in the literature, and online. One such example is appropriate in this discussion.

If, in the given equation, the constant "a" is half the acceleration g due to gravity near the surface of the Earth, and "x" is changed to time t, with "b" as the initial velocity v of a falling object, dropped from an initial height H, thus reaching a lower height h in the time t, and we let c = h - H (because we will need a negative value for this difference, arising from the fact that the height of the object is decreasing), then we can write a quadratic equation, in standard form, describing the situation as follows;

(1/2)g(t^{2}) + vt + (h - H) = 0 .

When rearranged to isolate height, h, we can calculate h after the time t has elapsed;
h = H - (1/2)g(t^{2}) - vt .
This, then, is an excellent example of how quadratic equations crop up in real-life situations; in this case, should we need to know the height of a falling object at some time during its fall. We can move on now to point out how complex numbers come into play specifically in the solving of certain quadratic equations.

Again, suppose there is a quadratic in standard form; ax^{2} + bx + c = 0 .
Here, let s = d ^{1/2 }, where d = b^{2} - 4ac , to establish a convenient abbreviation.
Such an equation has a solution x that can be obtained as follows.

Possibility 1 is; x = (-b + s)/(2a) ,

Possibility 2 is; x = (-b - s)/(2a) ,

where s = d^{1/2} = (b^{2} - 4ac)^{1/2} .

The difference d, in the term s, is called the "discriminant" of the quadratic equation, and, due simply to the fact that s is the square-root of a difference, then it is allowed that it could be the square-root of a negative number (i.e., s could be imaginary). In particular, if d is positive, then s is real, and therefore x comes in two distinct and real versions, called the "roots" of the quadratic equation, corresponding to terms "-b + s" and "-b - s" . That is, if d is positive, then it is said that the quadratic has "two distinct real roots".
However, if d = 0, then s = 0, so that -b + s = -b - s = -b . In that case, there is only one real root, called a "double root" because it satisfies both possibilities for x above. Such a root is readily obtained by writing; x = -b/(2a) .
Alternatively, if d is negative, then s is an imaginary number, and the quadratic equation has no real roots. In such cases, it can be referred to as "irreducible", in venues where only distinct real roots and/or double roots are considered valid. Otherwise, for negative determinants, the possiblities for x can be described as follows.

Possiblity 1 is; x = (-b + si)^{1/2} ,

Possibility 2 is; x = (-b - si)^{1/2} ,

where si = (d^{1/2})[(-1)^{1/2}] = [(-1)d]^{1/2} = (-d)^{1/2} ,

showing how the imaginary-unit (i) can be introduced in the context of quadratics.

The invention of complex numbers, which hinge on the notion of imaginary numbers, the basic understanding of which, in turn, is made clear by the definition and applications of the standard imaginary-unit, i, provides very useful mathematical tools; for example, in giving means of solving quadratic equations that have negative determinants.

Algebraically, of course, complex numbers obey special rules. To explain them, then, let A, B, C, and D denote real numbers, and note that the following relations typically hold.

A + Bi = C + Di if and only if A = C and D = B .

(A + Bi) + (C + Di) = (A + C) + (B + D)i .

(A + Bi) - (C + Di) = (A - C) + (B - D)i .

(A + Bi)(C + Di) = (AC - BD) + (AD + BC)i .

(A + Bi)/(C + Di) = [(AC + BD)/(C^{2} + D^{2})] + [(BC - AD)/(C^{2} + D^{2})]i .

Graphically, we have yet another set of rules, as follows.

Consider the standard x,y-plane, and let an ordinary point P_{o} be plotted on the plane;
P_{o} = (x,y) .
If we change y to yi, so that the y-axis becomes an imaginary axis, then P^{o} becomes the point indicated by plotting the complex number z as a point in this plane, so that
z = (x,yi) .
That is, a complex number z, defined using the formula;

z = x + yi = (x,yi) ,

can be represented by a point in a plane formed by using the real and imaginary number-lines as the coordinate axes of the plane. Such a plane is called the "complex plane", and therefore the complex-number z can always be denoted by the ordered-pair (x,yi).

Because complex numbers are also ordered-pairs of numbers, then they can be used to represent vectors in the plane. And here is an example of how that can be done.

If we stipulate that the point z is at the location indicated by the arrow of a directed line-segment from the origin O to z, within the complex plane, then the modulus |z| of z can be interpreted as the magnitude of a vector represented by this directed line-segment.

In that case, let "r" denote the magnitude (length) of the vector, and let "a" indicate the angle the vector makes with the x-axis. Then r is defined formally;

r = |z| = [x^{2} + (yi)^{2}]^{1/2} = [x^{2} + (-1)y^{2}]^{1/2} = (x^{2} - y^{2})^{1/2} ,

and we can specify z using the two variables, r and ~~0~~, called "polar coordinates", so that

z = x + yi = (x,yi) = (r,~~0~~) .

Knowing from trigonometry that r and ~~0~~ are related to x and y of the standard plane by the identities

x = r(cos~~0~~) and y = r(sin~~0~~) ,

we can next, by substitution, determine a trigonometric representation for z, with respect to the complex plane, and write;

z = r[(cos~~0~~) + i(sin~~0~~)] ,
which is called the "polar form" of the complex number z.

We must remember, of course, that r is also the modulus of z. Furthermore, angle ~~0~~ is commonly referred to as the "amplitude" of z.

This illustrates how the imaginary-unit occurs in vector analysis, but another application is in the representation of sinusoidal waves.

Consider the graph of a sine-wave in the x,y-plane, with a period T and wavelength L, and where the sine-wave is pictured as propagating along the x-axis to the right, so that y is the amplitude of the wave (its distance above or below the x-axis) at a given instant of time t, making y a function (f) both of x and of t, denoted; y = f(x,t) . If v is the speed of the wave-front, then the frequency F, period T, and wavelength L are related using the formula;

F = 1/T = v/L .

Here, let A be a constant called the "central maximum", which is the maximum y value. Since a sine-wave can be used to represent a steady oscillation, a perfect circular orbit, or other such harmonic motion, then we can introduce another constant K of the motion, called the "wave number", and relate it to the value of Pi (approximated as 3.14), so that 2(Pi) corresponds exactly to one cycle, according to the formula;

K = 2(Pi)L = 2(Pi)/(Tv) .

Now, any central maximum, A, approaching the y-axis from the left will be located some distance D (on the x-axis) from the y-axis, at time t. However, since the values of K and of D always vary proportionally with respect to each other, then D can also be obtained by introducing a quantity k, called the "phase constant", the "phase delay", or simply the "phase", and by defining D as the ratio of k over K; D = k/K .
Then the sine-wave can be represented graphically by plotting the formula;

y = f(x,t) = A cos[K(x - vt) + k] .

On the other hand, since uniform circular motion can be represented as the number of radians swept-out per unit time, using the angular frequency w, defined;

w = 2(pi)F = Kv ,
so that
K(x - vt) = Kx - Kvt = Kx - wt ,

then we can also write;
y = A cos(Kx - wt + k) .

Unfortunately, dealing with sinusoidal waves using trigonometric functions gets tedious. The more efficient way to deal with waves is to convert to complex notation.

From trigonometry, we have the following relationship, using the base e of natural logs;

e^{iV} = cos(V) + i[sin(V)] , for any arbitrary or "dummy" variable V. Thus, letting

V = Kx - wt + k ,

we can write;

e^{i(Kx - wt + k)} = cos(Kx - wt + k) + i[sin(Kx - wt + k)] ,

where the real (Re) and imaginary (Im) components can be defined;

cos(Kx - wt + k) = Re(e^{i(Kx - wt + k)}) , and i[sin(Kx - wt + k)] = Im(e^{i(Kx - wt + k)}) .

Suppose, however, that only the real component is needed, or, otherwise, the imaginary component is zero. Then we can define y using only the real component, as follows;

y = A cos(Kx - wt + k) = Re(Ae^{i(Kx - wt + k) }) .

Next, we can introduce a new funtion y', defined;

y' = Ae^{ik}e^{i(Kx - wt)} = A'e^{i(Kx - wt) },

where
A' = Ae^{ik} ,
so that the phase k can be temporarily "absorbed" into a more compact representation, wherein the real component is denoted;

y = Re(y') .

This sort of representation is useful when many waves are to be handled. It is referred to as "complex notation", and is used primarily because it is quicker and easier to deal with exponents than to manipulate sine and cosine functions. And it has been explained here as another example of how the standard imaginary-unit, i, has practical applications in real-world situations. Having learned something about imaginary numbers, therefore, we can proceed to the detailed explanation of the new imaginary-unit.

This ends the section on the standard imaginary-unit. It was not meant to be exhaustive, but only sufficient to describe the standard unit, and how it comes into play in different situations. Of note is the fact that engineers sometimes replace the "i" with a "j" because of the confusion that can be introduced when dealing with equations having a different application of the unit than occurs in the same context. My contention, therefore, is that no prohibition exists against the invention of other kinds of imaginary-units, in addition to the use of different symbols for the same unit, to mitigate confusion when two or more applications of the standard unit are employed. One such case is in the descriptions used for the putative class of subatomic particles called "tachyons" discussed in physics, where a negatively-signed imaginary-unit is required to accurately define tachyonic variables. But a negatively-signed standard imaginary-unit does not necessarily and does not always imply superluminality. It occurred to me, then, that a new sort of imaginary-unit could well be devised, to help eliminate possible confusion.

The Imagination-Unit (continued)

Part 4: Comments on Relativistic Imaginaries

Reconsider the Relativity Operator, R, defined; R = [1 - (v/c)^{2}]^{(-1/2) },
and, once more, let M denote a moving mass, with m the corresponding rest-mass, so

M = Rm = m[1 - (v/c)^{2}]^{(-1/2) }.

If v > c, the case for tachyons, then R is an imaginary number, making M imaginary.

A commonly-used definition of a tachyonic mass M_{t} has it that M_{t} = -iM , for some tachyonic mass taken as a direct analog of a given bradyonic mass M. This would, for instance, be the kind of definition physics professors first give to undergraduate students. And that is perfectly understandable, considering the way tachyons are presented in the literature. [See entry "Tachyons" by physicist Gerald Feinberg in the Encyclopedia of Physics by Lerner and Trigg, from VCH Publishers. I have the 2nd Edition, published in 1991, in which the entry is on page 1246 of that edition. Online, just search "tachyons".] But this "standard" definition leaves room for confusion whenever standard complex quantities and tachyonic complex quantities are discussed in the same context.

Allow me to explain this situation more specifically by giving a notational scheme that lets us look at how the Relativity Operator works, without having to plug a bunch of actual numbers into the equation, just to see what it does with them.

Let Q indicate the absolute-value of the difference 1 - (v/c)^{2}, denoted; Q = | 1 - (v/c)^{2} | ,
and let the following notation convention be observed;

Q+ = (+1)Q whenever v < c ,

Q0 = (0)Q = 0 whenever v = c ,

Q- = (-1)Q whenever v > c .

Then, because R = [1 - (v/c)^{2}]^{(-1/2) }, let us indicate the three cases of R;

Case 1: R = R+ = (Q+)^{(-1/2) }if, and only if, v < c .

Case 2: R = R0 = 0 if, and only if, v = c (assuming 1/0 is undefined; not infinity).

Case 3: R = R- = (Q-)^{(-1/2) }if, and only if, v > c .

In the last case, for tachyons, we have;

R- = (Q-)^{(-1/2) }= [(-1)Q]^{(-1/2) }= ... = 1/[i(Q^{(1/2)}] = (1/i)(Q^{(-1/2)}) .

However, 1/i = -i , according to the following proof;

1/i = 1/[(-1)^{(1/2)}] = (-1)^{(-1/2) }= (-1)^{[(1/2) - 1]} = [(-1)^{(1/2)}][(-1)^{(-1)}] = i[1/(-1)] = i/(-1) = -i ,

because 1/(-1) = [(-1)/(-1)][1/(-1)] = [(-1)1]/[(-1)(-1)] = (-1)/1 = -1 .

Consequently, if v > c , then R becomes;

R- = (Q-)^{(-1/2)} = (1/i)(Q^{(-1/2)}) = -i(Q^{(-1/2)}) .

Thus, the relativistic tachyonic mass M_{t} , where v > c, is properly defined;

M_{t} = (R-)m = -i(Q^{(-1/2)})m ,

while the corresponding bradyonic mass M , where v < c, continues to be defined as usual, but we can also write;

M = (R+)m = (Q^{(-1/2)})m .

Hence, we can legitimately write M_{t} = -iM , when deriving Mt using the Relativity Operator, R, but we cannot write M_{t} = iM , in such cases, because the sign is wrong. Tachyonic mass must involve a negatively-signed imaginary-unit, or it is not actually tachyonic. [A positively-signed imaginary mass is just an imaginary bradyon mass.]

We see that, because of the importance of keeping track of the sign on the imaginary-unit in the standard derivation of tachyonic mass, M_{t} , we must adopt special rules on the symbols we employ (i.e., we must use a notation convention), and which allows us to represent tachyonic mass in terms of some bradyonic mass, while maintaining sufficient rigor to assure accurate conceptualization. With that necessity established, I will address in my next post the main source of confusion caused by this definition of M_{t} ; particularly that a negatively-signed standard imaginary-unit does not always imply superluminality.

The Imagination-Unit (conclusion)

Part 5: The Imagination-Unit Detailed

As demonstrated, we can use the relativistic mass M of an ordinary particle to define a corresponding perfectly-analogous tachyonic-mass M_{t} , writing;

M_{t} = -iM ,

where

M = mR = m[1 - (v/c)^{2}]^{(-1/2) },

with m as the ordinary particle's rest-mass.

We have also seen why the negative sign on the imaginary-unit is necessary for correct representation of tachyonic mass, compared to the associated bradyonic mass. However, despite thereby placing the definition of a tachyon's mass on formal footing, this creates confusion when complex quantities associated with both M and M_{t} are discussed in the same context -- especially when we try to use both kinds of quantities in one formula.

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 in the equations from somewhere other than R; say, when employing vector velocity, as with the formula for momentum, and this bradyon goes in the opposite direction to the original bradyon. This can happen, let's say, if the oppositely-moving imaginary mass, iM, is for a particle traveling near-to but slower-than lightspeed, and we require that the imaginary-unit, i, is interpreted according to its common convention of implying that iM is merely a standard imaginary; such as, in quantum physics, when we discuss processes involving massive virtual particles (for instance, the neutral Z-particle and/or the charged W-particles that mediate the weak-nuclear interactions).

How 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 potential for confusion brought about by having two different interpretations of the same symbol, i; one for bradyons and another for tachyons. And it is impractical to keep having to explain the difference in the accompanying text.

To solve that problem, I introduce a new imaginary-unit, as an operator i^{i} , originally called the "imagination-unit", which is defined as transforming any ordinary quantity or variable it operates on into the exact tachyonic analog of itself.

That is, multiplying the imagination-unit to a standard quantity and/or symbol is defined as imposing a transformation across the lightspeed barrier, so that it is understood (by a new convention) 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 a regular basis in mathematics, physics, and engineering.

The Relativity Operator still applies, as do the Lorentz Transformations. And this would not help much if velocity restrictions are not specified, as well. So, we further define the imagination-unit, i^{i} , to involve an exclusive evaluation between c and infinite-speed.

That is, for example, instead of writing M_{t} = -iM , we can define a tachyonic-mass using an evaluation formula;

[v = infinity]

M_{t} = i^{i}M = M | ,

[v = c]

where the brackets indicate exclusivity (evaluation between the enclosed values, but not at those values).

Representing tachyons this way allows us to discuss bradyons with negative imaginary mass, -iM, in the same context as tachyons, defined by i^{i}M, without fear of the confusion
that would be possible with two interpretations of the standard imaginary-unit, (-1)^{1/2 }.

Case in point, as noted, when we want the tachyonic analog of a bradyon mass, but the mass itself appears in a formula that involves giving it a negative sign and an imaginary-unit, displayed together as "-iM", instead of the standard positive bradyonic mass, though it is not tachyonic. We would not want -i to imply that this mass is tachyonic. So, write;

[v = infinity]

-i(M_{t}) = i^{i}(-iM) = -iM | ,

[v = c]

which reads: Negative imaginary tachyonic mass -i(M_{t}) is equal to the tachyonic analog of the standard negatively-signed imaginary mass, -iM, which analog is equal to -iM evaluated between c and infinite-speed, exclusively.

We do, of course, continue to relate motion involving the bradyonic masses M and -iM to their respective tachyonic analogs using the Lorentz transformations, since all tachyonic analogs are, by definition, in reference-frames that always move relative to all bradyonic reference-frames (and as long as the tachyons do not move at infinite-speed in either type of frame). That is, the Relativity Operator, R, is not removed or replaced. It still applies. But it does not place an upper limit on the speed of a tachyon, and that brings in runaway solutions (infinities) that cannot be allowed in experimental physics situations.

The mass of a tachyon that moves at infinite speed can be defined, but that must be done quite separately, in a different though related manner, and given 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 meaningless exercises. Infinite-velocity tachyons can certainly be imagined, and thus described using pure mathematics, but they must be considered as having applications only in metaphysical terms. A simple representations scenario, in that case, would be to define an infinite-speed operator, I^{i} , as implying evaluation at infinite velocity.

Now, the transformation across the lightspeed barrier is best understood by inspection of the Velocity Spectrum, denoted as

I^{i}v > i^{i}v > i^{i}c > c > v > (v = 0_{rel}) > (iv = 0_{abs} = i~~v~~) < (~~v~~ = 0_{rel}) < ~~v~~ < ~~c~~ < i^{i}~~c~~ < i^{i}~~v~~ < I^{i}~~v~~ ,

where

I^{i}v is infinite-speed,

i^{i}v is any superluminal velocity considered exclusively between the tachyonic analog of lightspeed, i^{i}c, and infinite-speed, I^{i}v,

c is the lightspeed constant,

v is bradyonic velocity between relative-zero speed 0_{rel} and lightspeed c, also exclusively,

and 0_{abs} is an absolute-zero velocity (a standard pure imaginary),

while corresponding values for antiparticles are shown to the right of 0_{abs} .

Note that tachyonic lightspeed, i^{i}c, can be defined; i^{i}c = (1.00...001)c , where the exact number of zeros to the right of the decimal-point is an empirical unknown -- making this version of tachyonic-c both an irrational and a transcendental imaginary-number.

Considering first only regular bradyons and tachyons, but no antiparticles, one-to-one correspondences across the lightspeed barrier, associating bradyonic 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^{i}v.

That is, the evaluations associated with i^{i} are understood using integration whenever a spread of real quantities, in standard space-time, must be related to the corresponding
spread of their tachyonic analogs in superluminal space-time.

A similar tactic is employed for antiparticles.

Obviously, it is not always necessary to use the imagination-unit to describe any kind of tachyon. The operator is provided as an option when complex bradyonic quantities and complex tachyonic quantities are treated in the same context, and a method is needed to eliminate confusion between representations of the two. It is also used to place limits on tachyonic analogs of bradyons, corresponding to the natural limitations of bradyons.

In conclusion, "Tachyonics" is the label for the overall study of tachyons. And it is for that effort that I devised the imagination-unit, also referred to as "Richter's Tachyonics Operator", for the reasons I have stated in this article. Note, however, that while the operator can be used on any variable or quantity, to obtain a direct superluminal analog, it can also be used as the basis for postulating and describing tachyons not in analogy to known bradyons or luxons. The reason for that is the possibility that there exists such tachyons on the other side of lightspeed, and once scientists of the future start making practical use of different kinds of tachyons, they will have need of valid representations for them. It is hoped, therefore, that my ideas will be taken seriously at that time.

EOF