The Imagination Unit

or

How to Describe Things Made of Tachyons



A Representations Theory by H. Kurt Richter



Part 1


About Tachyons

Because of certain implications of Einstein's theory of Special
Relativity, some physicists have predicted the existence of particles
that always travel faster-than-light (FTL).
Here is one way that this prediction comes about.
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 = m(c^2) ,
where c is the lightspeed constant (approximately 3 x 10^8
meters/second).

Suppose now that we roll the basketball across the court; setting it
in motion with respect to the the stationary floor of the court. It
can then be viewed as existing in a different frame of reference,
local to itself; one that is moving relative to the stationary frame
of the floor.

To relate these frames, we apply certain transformation equations;
explained as follows.

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 will begin counting the 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 two reference-frames will be related according to the following
formulas, referred to as 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 R is an operator that allows us to calculate the relative value
of a quantity for a moving object from the corrresponding value taken
while the object is at rest. To explain rather briefly, the operator
R is a multiplicative operator that involves the ratio of velocity v
over the lightspeed constant c, and it can be defined;
R = 1/[(1 - [(v/c)^2])^(1/2)] = (1 - [(v/c)^2])^(-1/2) .

Since Einstein made extensive use of the Lorentz transformations in
his theory of Special Relativity, involving the operator R, then R
can be called the "Relativity Operator".

Here, 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)] .
Notice therefore that, because the ratio v/c is under a square-root
symbol, then there is only one relationship between v and c that
makes sense for a real basketball; v must be less than 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
three fundamental cases for M:
(1) v < c makes R real, which means that M is a real-valued non-zero
quantity.
This is the case for all subatomic particles with positive rest-mass
(electrons, protons, etc.).
(2) v = c makes R undefined, because, by convention, the square-root
of zero is zero, but the inverse of zero is undefined (or else it is
infinity). This is the case for massless photons.
(3) v > c makes R imaginary, which means that M is an imaginary
quantity.
This is the case for particles with negative rest-mass (particles
that travel FTL).

All of the subatomic particles cataloged by physicists as having
mass, as far as we can tell, have positive rest-mass, including real
and virtual paticles with measureable or calculable rest-mass. The
scalar energy E and vectorial momentum P are both defined using the
rest-mass m;
E = R[m(c^2)] and P = R(mV) ,
where V is vectorial velocity, defined; | 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, here v = c, so that
R = 1/[(1 - [(v/c)^2])^(1/2)] = 1/ [(1 - (1^2))^(1/2))]
= 1/[(1 - 1)^(1/2)] = 1/[0^(1/2)] = 1/0 .
Alternatively, yet remaining mathematically rigorous, we can say
instead that the inverse of 0 is undefined, and maintain that the
rest-mass of a photon is 0; which means that all photons are massless
particles, made entirely of energy. [Note: If mass is 0, then the
relativistic formula for mass gives the ratio 0/0, which is just 0;
i.e., the infinity implied by R gets canceled-out.]

Particles with negative rest-mass that always travel FTL are
called "tachyons". They will have reversed causality (negative
time), as compared to that of ordinary massive particles and/or
massless photons, and their rest-mass is both imaginary and
negative. But we will delve into this case in Part 3, and in much
greater detail than has been provided for the other two cases
above.

Notice, as well, 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 real mass up to lightspeed.
And because we do not have access to infinite amounts of energy, and
we 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 spacetime barrier.

Thus, many physicists have assumed (logically) that nothing "real"
exists on the other side of c.
Unfortunately, this has also caused some to conclude that tachyons
cannot be created, not even by a Big Bang like the one that initiated
our universe. Hence, some of these scientists insist that tachyons
do not exist, although that claim has yet to be proven conclusively,
and is not justfied by logical inferrence (it amounts to a personal
preference in the interpretation of Einstein's theory of Special
Relativity, not a mathematically rigorous interpretation).

To be clear, the relativity operator, R, does not suggest that
nothing FTL can exist at all. It does indicate that it would take
infinite amount of energy to accelarate a real mass up to c (because
of the ratio v/c), but it does not forbid objects that already travel
at FTL speeds from existing on the other side of c. 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
proposed that there was a brief period of superluminal expansion of
all the energy associated with the first moments of the Big Bang, and
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 so low.

Also, while we depict tachyons as having imaginary mass
mathematically, we must remember that words
like "imaginary", "abstract", and other terms employed by
mathematicians, in math contexts, are labels for different types of
numbers, chosen to distinguish between those numbers. But such
labels do not necessarily imply that the numbers do not exist.
So, to label a tachyon's mass as "imaginary" does not imply non-
existence - because we are using the 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, which is the ratio of the circumference
over the diameter of any size 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
as 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. 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.
And likewise, the base e of natural logarithms is labeled using the
same terminology, for the very same reasons.

Thus, because an imaginary number can always be represented as the
product of i and any real number, we can state that all imaginary
numbers can 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
particles, theoretically speaking, this does not mean that they
cannot or do not exist, somewhere.

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 tardyon (slower-
than-light particle) up to c.

By the way, tardyons are also referred to as "bradyons", from the
Greek word "bradus", meaning "slow". In this context, then, we can
refer to massless particles (such as photons of light) as "luxons".

Consider the emission of a bradyon from some composite body, and
suppose that we have taken a high-speed video of the event. Next,
imagine the emission of a tachyonic analog of the bradyon.
If we could "see" the emission of the tachyon, as viewed from our
ordinary (bradyonic) frame of reference, it would look as if the
tachyon came from infinity and was absorbed by the body.
In other words, viewing a video of the emission of tachyonic analogs
of bradyons, it would practically look as if we were running the
video of the ordinary process in reverse.

For a more in-depth explanation of tachyons, click "Tachyons" at
www.TachyonicsSociety.com, or, for a long list of sources, do a
Google search using the keyword "tachyon".

In what follows, a non-standard method of representing tachyons is
provided, where yet another operator - a new kind of imaginary-
unit, inspired by the standard imaginary-unit, i - will be used to
imply a transformation across the lightspeed barrier. However,
because this operator is based on the standard imaginary-unit, it is
best, for the broadest understanding of this new operator, to explain
in sufficient detail the standard imaginary-unit first.


More to come.

[Note: This is a re-post of a previously posted article.]