The Tachyonics Operator vs. The Standard Imaginary Unit
As mentioned, relativistic mass (M) is related to rest-mass (m) by the equation;
M = Rm = m/[(1 - [(v/c)^2])^(1/2)] .
Consider again a tachyon of mass tM, with correspondingly the same
amount of moving mass. The tachyonic mass, tM, can be represented by
describing it as an imaginary analog of the standard mass;
tM = -iM ,
where i is the standard imaginary-unit, defined;
i = (-1)^(1/2) , so that i^2 = -1 .
Note that the minus-sign accompanying i, in this definition, is absolutely mandatory for presenting a rigorous definition of tM. In such cases, the imaginary-unit (i) is used in an algebraic sense, as an operator that, when multiplied to a real number, is understood - by convention - to imply that the given real number is viewed as an imaginary number. To go any further on this topic, however, it is necessary to lay some groundwork.
[Readers sufficiently familiar with the theory behind complex and imaginary numbers, as presented below, may wish to proceed to the next subheading.]
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, we write;
(-X)^(1/2) = [(-1)X]^(1/2) = ... = i[X^(1/2)] .
(-25)^(1/2) = [(-1)(25)]^(1/2) = ... = i5 .
Now, the sums of real and imaginary numbers form a set called the "complex numbers", which, obviously, includes the set of all real numbers and the set of all imaginary numbers. To explain how complex numbers work, let x and y denote real numbers, and let iy denote an imaginary number, so that z denotes the sum of x and iy, written;
z = x + iy ,
where z is a complex number, x is the real-number part or "real component" of z, and y is the imaginary-number part or "imaginary component" of z.
Using function notation, let Re be a function of z that gives a real number, Re(z), and Im 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 must be true;
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 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 that can be 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 that z* is the "conjugate" of z.
The product of z and z* is the square of the modulus of z, according to the proof;
z*z = (x - iy)(x + iy) = (x^2) - xiy + xiy - [(iy)^2]
= (x^2) - [(-1)(y^2)] = (x^2) + (y^2) = |z|^2 .
The ratio, z/Z, of two complex numbers, z and Z, is 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 important equations, which should be covered briefly as follows.
An equation of the form
a(x^2) + bx + c = 0
is called a "quadratic equation" in standard form, where x is a variable, and a, b, and c are arbitrary constants. Equations of this form are used to solve so many real-world problems that a full accounting of them would fill sets of encyclopedias. So, we need not go into numerous examples here, although at least one example would be appropriate.
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, 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, arrising from the fact that the height of the object is decreasing), then we can write a quadratic equation in standard form, as follows;
(1/2)g(t^2) + vt + (h - H) = 0 .
When rearranged to isolate h, this equation gives height, h, after time t has elapsed;
h = -(1/2)g(t^2) - vt + H .
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 move on to point out how complex numbers come into the context of quadratics.
Assume that a quadratic equation in standard form exists, denoted;
a(x^2) + bx + c = 0 .
Let s = d^(1/2) ,
where d = (b^2) - 4ac ,
to establish a convenient abbreviation.
The given quadratic has a solution x 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 sometimes allowed that s could be the square-root of a negative number (i.e., the term s could be an
imaginary number). 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 "-b + s" and "-b - s" . That is, if d is positive, then it is said that the quadratic
equation has "two distinct real roots". However, if d = 0, then s = 0, so that -b + s = -b - s = -b , and there is only one real root, called a "double root", because it satisfies both possibilities given 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 given quadratic equation has no real roots. In such cases, the quadratic equation can be referred to as "irreducible", in venues where only distinct real roots and/or double roots are considered.
Otherwise, for negative determinants, the possiblities for x can be denoted as follows.
Possiblity 1 is; x = (-b + si)^(1/2) ,
Possibility 2 is; x = (-b - si)^(1/2) ,
si = [d^(1/2)][(-1)^(1/2)] = [(-1)d]^(1/2) = (-d)^(1/2) .
This shows how the imaginary-unit can be introduced in quadratic equations.
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, has provided us with very useful mathematical tools; for example, in giving us various means of solving quadratic equations that have negative determinants.
Algebraically, of course, complex numbers obey a special set of rules, as follows.
Let A, B, C, and D denote real numbers. Then the following realtionships hold true.
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 be plotted on the plane;
P = (x,y) . If we change y to yi, so that the y-axis becomes an imaginary axis, then P 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 by the formula denoted;
z = x + yi = (x,yi) ,
can be represented by a point in a plane that is formed by using the real and imaginary number-lines as the coordinate axes for the plane. Such a plane is called the "complex plane", and, with respect to this plane, the complex number z can always be denoted by the ordered-pair (x,yi).
Now, because complex numbers are actually ordered-pairs of numbers, then they can also be used to represent vectors in the plane. And here is how that is 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, as follows.
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 a, called "polar coordinates", so that
z = x + yi = (x,yi) = (r,a) .
In that case, knowing (from trigonometry) that "r" and "a" are related to x and y in the standard plane by the identities;
x = r(cosa) and y = r(sina) ,
we can, by substitution, determine a trigonometric representation of z, with respect to the complex plane, and write;
z = r[(cosa) + i(sina)] ,
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 "a" is commonly referred to as the "amplitude" of z.
Another useful application of the imaginary-unit is in the representation of waveforms.
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, then the frequency F, period T, and wavelength L are related using the following formula;
F = 1/T = v/L .
Here, let A be a constant, called the "central maximum", which is the maximum value of y. 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 a 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 be obtained by introducing a quantity k, referred to as the "phase constant", the "phase delay",
or simply the "phase", and by defining D as the ratio of k over K, denoted; D = k/K .
Then the sine-wave can be represented by plotting the following formula graphically;
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 ,
K(x - vt) = Kx - Kvt = Kx - wt ,
then, alternatively, we can write;
y = A cos(Kx - wt + k) .
Unfortunately, dealing with sinusoidal waves using trigonometric functions can get tedious. The more efficient way to do the same thing is to convert to complex notation.
From trigonometry, we have the following relationship, using the base e of natural logarithms;
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 introduce a new funtion y', defined;
y' = A[e^(ik)]e^[i(Kx - wt)] = A'e^[i(Kx - wt)] ,
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 makes for a much faster work when many waves are to be handled (a common task in physics and engineering). It is referred to as "complex notation", and is used because it is quicker than manipulating 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, we can proceed to the introduction of a new imaginary-unit; one that implies a wholly new kind of operation.