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Item 5.

TOP QUARK                                      updated: 18-APR-1993 by SIC
---------

The top quark is the hypothetical sixth fundamental strongly
interacting particle (quark).  The known quarks are up (u), down (d),
strange (s), charm (c) and  bottom (b).  The Standard Model requires quarks
to come in pairs in order to prevent mathematical inconsistency due to
certain "anomalous" Feynman diagrams, which cancel if and only if the
quarks are paired.  The pairs are (d,u),(s,c) and (b,?).  The missing
partner of the b is called "top".

In addition, there is experimental evidence that the b quark has an
"isodoublet" partner, which is so far unseen.  The forward-backward
asymmetry in the reaction e+ + e- -> b + b-bar and the absence of
flavor-changing neutral currents in b decays imply the existence of the
isodoublet partner of the b. ("b-bar", pronounced "bee bar", signifies the
b antiquark.)

The mass of the top quark is restricted by a variety of
measurements. Due to radiative corrections which depend on the top quark
circulating as a virtual particle inside the loop in the Feynman diagram,
a number of experimentally accessible processes depend on the top quark
mass.  There are about a dozen such measurements which have been made so
far, including the width of the Z, b-b-bar mixing (which historically gave
the first hints that the top quark was very massive), and certain aspects
of muon decay.  These results collectively limit the top mass to roughly
140 +/- 30 GeV.  This uncertainty is a "1-sigma" error bar.

Direct searches for the top quark have been performed, looking for
the expected decay products in both p-p-bar and e+e- collisions.  The best
current limits on the top mass are:
(1) From the absence of Z -> t + t-bar, M(t) > M(Z)/2 = 45 GeV.
This is a "model independent" result, depending only on the fact that the
top quark should be weakly interacting, coupling to the Z with sufficient
strength to have been detected at the current resolution of the LEP
experiments which have cornered the market on Z physics in the last several
years.
(2) From the absence of top quark decay products in the reaction p
+ p-bar -> t + t-bar -> hard leptons + X at Fermilab's Tevatron collider,
the CDF (Collider Detector at Fermilab) experiment.  Each top quark is
expect to decay into a W boson and a b quark.  Each W subsequently decays
into either a charged lepton and a neutrino or two quarks.  The cleanest
signature for the production and decay of the t-t-bar pair is the presence
of two high-transverse-momentum (high Pt) leptons (electron or muon) in the
final state.  Other decay modes have higher branching ratios, but have
serious experimental backgrounds from W bosons produced in association with
jets.  The current published lower limit on M(t) from such measurements is
91 GeV (95% confidence), 95 GeV (90% confidence).  However, these limits assume
that the top quark has the expected decay products in the expected branching
ratios, making these limits "model dependent," and consequently not as
"hard" as the considerably lower LEP limit of ~45 GeV.  Unpublished results
from CDF and D0 now claim lower top mass limits of 108 GeV and 103 GeV for
the respective detectors, presumably at 95% confidence.  These numbers
will probably change by the time they make it into print.

The future is very bright for detecting the top quark.  LEP II, the
upgrade of CERN's e+e- collider to E >= 2*Mw = 160 GeV by 1994, will allow
a hard lower limit of roughly 90 GeV to be set.  Meanwhile, upgrades to
CDF, start of a new experiment, D0,  and upgrades to the accelerator
complex at Fermilab have recently allowed higher event rates and better
detector resolution, should allow production of standard model top quarks of
mass < 150 GeV in the next two years, and even higher mass further in the
future, at high enough event rate to identify the decays and give rough mass
measurements.  There have already been a few unpublished "candidate" events
from CDF and D0, which, if verified, would be the first direct evidence of
the top quark, with mass in the vacinity of 130 GeV.

References: Phys. Rev. Lett. _68_, 447 (1992) and the references therein.

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Item 6.

Tachyons                                        updated: 22-MAR-1993 by SIC
--------

There was a young lady named Bright,
Whose speed was far faster than light.
She went out one day,
In a relative way,
And returned the previous night!

-Reginald Buller

It is a well known fact that nothing can travel faster than the
speed of light. At best, a massless particle travels at the speed of light.
But is this really true?  In 1962, Bilaniuk, Deshpande, and Sudarshan, Am.
J. Phys. _30_, 718 (1962), said "no".  A very readable paper is Bilaniuk
and Sudarshan, Phys. Today _22_,43 (1969).  I give here a brief overview.

Draw a graph, with momentum (p) on the x-axis, and energy (E) on
the y-axis.  Then draw the "light cone", two lines with the equations E =
+/- p. This divides our 1+1 dimensional space-time into two regions.  Above
and below are the "timelike" quadrants, and to the left and right are the

Now the fundamental fact of relativity is that E^2 - p^2 = m^2.
(Let's take c=1 for the rest of the discussion.)  For any non-zero value of
m (mass), this is an hyperbola with branches in the timelike regions.  It
passes through the point (p,E) = (0,m), where the particle is at rest.  Any
particle with mass m is constrained to move on the upper branch of this
hyperbola.  (Otherwise, it is "off-shell", a term you hear in association
with virtual particles - but that's another topic.) For massless particles,
E^2 = p^2, and the particle moves on the light-cone.

These two cases are given the names tardyon (or bradyon in more
modern usage) and luxon, for "slow particle" and "light particle".  Tachyon
is the name given to the supposed "fast particle" which would move with v>c.

Now another familiar relativistic equation is E =
m*[1-(v/c)^2]^(-.5).  Tachyons (if they exist) have v > c.  This means that
E is imaginary!  Well, what if we take the rest mass m, and take it to be
imaginary?  Then E is negative real, and E^2 - p^2 = m^2 < 0.  Or, p^2 -
E^2 = M^2, where M is real.  This is a hyperbola with branches in the
spacelike region of spacetime.  The energy and momentum of a tachyon must
satisfy this relation.

You can now deduce many interesting properties of tachyons.  For
example, they accelerate (p goes up) if they lose energy (E goes down).
Futhermore, a zero-energy tachyon is "transcendent," or infinitely fast.
This has profound consequences.  For example, let's say that there were
electrically charged tachyons.  Since they would move faster than the speed
of light in the vacuum, they should produce Cerenkov radiation. This would
*lower* their energy, causing them to accelerate more!  In other words,
charged tachyons would probably lead to a runaway reaction releasing an
arbitrarily large amount of energy.  This suggests that coming up with a
sensible theory of anything except free (noninteracting) tachyons is likely
to be difficult.  Heuristically, the problem is that we can get spontaneous
creation of tachyon-antitachyon pairs, then do a runaway reaction, making
the vacuum unstable.  To treat this precisely requires quantum field theory,
which gets complicated.  It is not easy to summarize results here.  However,
one reasonably modern reference is _Tachyons, Monopoles, and Related
Topics_, E. Recami, ed. (North-Holland, Amsterdam, 1978).

However, tachyons are not entirely invisible.  You can imagine that
you might produce them in some exotic nuclear reaction.  If they are
charged, you could "see" them by detecting the Cerenkov light they produce
as they speed away faster and faster.  Such experiments have been done.  So
far, no tachyons have been found.  Even neutral tachyons can scatter off
normal matter with experimentally observable consequences.  Again, no such
tachyons have been found.

How about using tachyons to transmit information faster than the
speed of light, in violation of Special Relativity?  It's worth noting
that when one considers the relativistic quantum mechanics of tachyons, the
question of whether they "really" go faster than the speed of light becomes
much more touchy!  In this framework, tachyons are *waves* that satisfy a
wave equation.  Let's treat free tachyons of spin zero, for simplicity.
We'll set c = 1 to keep things less messy.  The wavefunction of a single
such tachyon can be expected to satisfy the usual equation for spin-zero
particles, the Klein-Gordon equation:

(BOX + m^2)phi = 0

where BOX is the D'Alembertian, which in 3+1 dimensions is just

BOX = (d/dt)^2 - (d/dx)^2 - (d/dy)^2 - (d/dz)^2.

The difference with tachyons is that m^2 is *negative*, and m is
imaginary.

To simplify the math a bit, let's work in 1+1 dimensions, with
coordinates x and t, so that

BOX = (d/dt)^2 - (d/dx)^2

Everything we'll say generalizes to the real-world 3+1-dimensional case.
Now - regardless of m, any solution is a linear combination, or
superposition, of solutions of the form

phi(t,x) = exp(-iEt + ipx)

where E^2 - p^2 = m^2.  When m^2 is negative there are two essentially
different cases.  Either |p| >= |E|, in which case E is real and
we get solutions that look like waves whose crests move along at the
rate |p|/|E| >= 1, i.e., no slower than the speed of light.  Or |p| <
|E|, in which case E is imaginary and we get solutions that look waves
that amplify exponentially as time passes!

We can decide as we please whether or not we want to consider the second
sort of solutions.   They seem weird, but then the whole business is
weird, after all.

1)      If we *do* permit the second sort of solution, we can solve the
Klein-Gordon equation with any reasonable initial data - that is, any
reasonable values of phi and its first time derivative at t = 0.  (For
the precise definition of "reasonable," consult your local
mathematician.)  This is typical of wave equations.  And, also typical
of wave equations, we can prove the following thing: If the solution phi
and its time derivative are zero outside the interval [-L,L] when t = 0,
they will be zero outside the interval [-L-|t|, L+|t|] at any time t.
In other words, localized disturbances do not spread with speed faster
than the speed of light!  This seems to go against our notion that
tachyons move faster than the speed of light, but it's a mathematical
fact, known as "unit propagation velocity".

2)      If we *don't* permit the second sort of solution, we can't solve the
Klein-Gordon equation for all reasonable initial data, but only for initial
data whose Fourier transforms vanish in the interval [-|m|,|m|].  By the
Paley-Wiener theorem this has an odd consequence: it becomes
impossible to solve the equation for initial data that vanish outside
some interval [-L,L]!  In other words, we can no longer "localize" our
tachyon in any bounded region in the first place, so it becomes
impossible to decide whether or not there is "unit propagation
velocity" in the precise sense of part 1).    Of course, the crests of
the waves exp(-iEt + ipx) move faster than the speed of light, but these
waves were never localized in the first place!

The bottom line is that you can't use tachyons to send information
faster than the speed of light from one place to another.  Doing so would
require creating a message encoded some way in a localized tachyon field,
and sending it off at superluminal speed toward the intended receiver. But
as we have seen you can't have it both ways - localized tachyon disturbances
are subluminal and superluminal disturbances are nonlocal.

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Item 7. Special Relativistic Paradoxes - part (a)

The Barn and the Pole                   updated 4-AUG-1992 by SIC
---------------------                   original by Robert Firth

These are the props.  You own a barn, 40m long, with automatic
doors at either end, that can be opened and closed simultaneously by a
switch. You also have a pole, 80m long, which of course won't fit in the
barn.

Now someone takes the pole and tries to run (at nearly the speed of
light) through the barn with the pole horizontal.  Special Relativity (SR)
says that a moving object is contracted in the direction of motion: this is
called the Lorentz Contraction.  So, if the pole is set in motion
lengthwise, then it will contract in the reference frame of a stationary
observer.

You are that observer, sitting on the barn roof.  You see the pole
coming towards you, and it has contracted to a bit less than 40m. So, as
the pole passes through the barn, there is an instant when it is completely
within the barn.  At that instant, you close both doors.  Of course, you
open them again pretty quickly, but at least momentarily you had the
contracted pole shut up in your barn.  The runner emerges from the far door
unscathed.

But consider the problem from the point of view of the runner.  She
will regard the pole as stationary, and the barn as approaching at high
speed. In this reference frame, the pole is still 80m long, and the barn
is less than 20 meters long.  Surely the runner is in trouble if the doors
close while she is inside.  The pole is sure to get caught.

Well does the pole get caught in the door or doesn't it?  You can't
have it both ways.  This is the "Barn-pole paradox."  The answer is buried
in the misuse of the word "simultaneously" back in the first sentence of
the story.  In SR, that events separated in space that appear simultaneous
in one frame of reference need not appear simultaneous in another frame of
reference. The closing doors are two such separate events.

SR explains that the two doors are never closed at the same time in
the runner's frame of reference.  So there is always room for the pole.  In
fact, the Lorentz transformation for time is t'=(t-v*x/c^2)/sqrt(1-v^2/c^2).
It's the v*x term in the numerator that causes the mischief here.  In the
runner's frame the further event (larger x) happens earlier.  The far door
is closed first.  It opens before she gets there, and the near door closes
behind her. Safe again - either way you look at it, provided you remember
that simultaneity is not a constant of physics.

References:  Taylor and Wheeler's _Spacetime Physics_ is the classic.
Feynman's _Lectures_ are interesting as well.

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Item 7. Special Relativistic Paradoxes - part (b)

The Twin Paradox                                updated 17-AUG-1992 by SIC
----------------                                original by Kurt Sonnenmoser

A Short Story about Space Travel:

Two twins, conveniently named A and B, both know the rules of
Special Relativity.  One of them, B, decides to travel out into space with
a velocity near the speed of light for a time T, after which she returns to
Earth. Meanwhile, her boring sister A sits at home posting to Usenet all
day.  When A finally comes home, what do the two sisters find?  Special
Relativity (SR) tells A that time was slowed down for the relativistic
sister, B, so that upon her return to Earth, she knows that B will be
younger than she is, which she suspects was the the ulterior motive of the
trip from the start.

But B sees things differently.  She took the trip just to get away
from the conspiracy theorists on Usenet, knowing full well that from her
point of view, sitting in the spaceship, it would be her sister, A, who
was travelling ultrarelativistically for the whole time, so that she would
arrive home to find that A was much younger than she was.  Unfortunate, but
worth it just to get away for a while.

What are we to conclude?  Which twin is really younger?  How can SR
give two answers to the same question?  How do we avoid this apparent

Much of the confusion surrounding the so-called Twin Paradox
originates from the attempts to put the two twins into different frames ---
without the useful concept of the proper time of a moving body.

SR offers a conceptually very clear treatment of this problem.
First chose _one_ specific inertial frame of reference; let's call it S.
Second define the paths that A and B take, their so-called world lines. As
an example, take (ct,0,0,0) as representing the world line of A, and
(ct,f(t),0,0) as representing the world line of B (assuming that the the
rest frame of the Earth was inertial). The meaning of the above notation is
that at time t, A is at the spatial location (x1,x2,x3)=(0,0,0) and B is at
(x1,x2,x3)=(f(t),0,0) --- always with respect to S.

Let us now assume that A and B are at the same place at the time t1
and again at a later time t2, and that they both carry high-quality clocks
which indicate zero at time t1. High quality in this context means that the
precision of the clock is independent of acceleration. [In principle, a
bunch of muons provides such a device (unit of time: half-life of their
decay).]

The correct expression for the time T such a clock will indicate at
time t2 is the following [the second form is slightly less general than the
first, but it's the good one for actual calculations]:

t2          t2      _______________
/           /      /             2 |
T  =  | d\tau  =  | dt \/  1 - [v(t)/c]              (1)
/           /
t1          t1

where d\tau is the so-called proper-time interval, defined by

2         2      2      2      2
(c d\tau)  = (c dt)  - dx1  - dx2  - dx3 .

Furthermore,
d                          d
v(t) = -- (x1(t), x2(t), x3(t)) = -- x(t)
dt                         dt

is the velocity vector of the moving object. The physical interpretation
of the proper-time interval, namely that it is the amount the clock time
will advance if the clock moves by dx during dt, arises from considering
the inertial frame in which the clock is at rest at time t --- its
so-called momentary rest frame (see the literature cited below). [Notice
that this argument is only of a heuristic value, since one has to assume
that the absolute value of the acceleration has no effect. The ultimate
justification of this interpretation must come from experiment.]

The integral in (1) can be difficult to evaluate, but certain
important facts are immediately obvious. If the object is at rest with
respect to S, one trivially obtains T = t2-t1. In all other cases, T must
be strictly smaller than t2-t1, since the integrand is always less than or
equal to unity. Conclusion: the traveling twin is younger. Furthermore, if
she moves with constant velocity v most of the time (periods of
acceleration short compared to the duration of the whole trip), T will
approximately be given by      ____________
/          2 |
(t2-t1) \/  1 - [v/c]    .             (2)

The last expression is exact for a round trip (e.g. a circle) with constant
velocity v. [At the times t1 and t2, twin B flies past twin A and they
compare their clocks.]

Now the big deal with SR, in the present context, is that T (or
d\tau, respectively) is a so-called Lorentz scalar. In other words, its
value does not depend on the choice of S. If we Lorentz transform the
coordinates of the world lines of the twins to another inertial frame S',
we will get the same result for T in S' as in S. This is a mathematical
fact. It shows that the situation of the traveling twins cannot possibly
lead to a paradox _within_ the framework of SR. It could at most be in
conflict with experimental results, which is also not the case.

Of course the situation of the two twins is not symmetric, although
one might be tempted by expression (2) to think the opposite. Twin A is
at rest in one and the same inertial frame for all times, whereas twin B
is not.  [Formula (1) does not hold in an accelerated frame.]  This breaks
the apparent symmetry of the two situations, and provides the clearest
nonmathematical hint that one twin will in fact be younger than the other
at the end of the trip.  To figure out *which* twin is the younger one, use
the formulae above in a frame in which they are valid, and you will find
that B is in fact younger, despite her expectations.

It is sometimes claimed that one has to resort to General
Relativity in order to "resolve" the Twin "Paradox". This is not true. In
flat, or nearly flat space-time (no strong gravity), SR is completely
sufficient, and it has also no problem with world lines corresponding to
accelerated motion.

References:
Taylor and Wheeler, _Spacetime Physics_  (An *excellent* discussion)
Goldstein, _Classical Mechanics_, 2nd edition, Chap.7 (for a good
general discussion of Lorentz transformations and other SR basics.)

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