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Michael N: Kreisler
Particles?
A review of the hypotheses about the nature of
tachyons and of experimental searches for them
During the last few years there has
been an effort to search for tachyons ---
particles that travel faster than the
speed of light. I hope to show here
that there is, in fact, some justifica-
tion for a search for particles that
would seem to violate all we have
learned about special relativity-and
for the very modest investment that
has been devoted to the question.
The experiments that have been
performed to look for these particles
will be reviewed, and I will avoid,
for the most part, any lengthy dis-
cussion of the wealth of recent the-
oretical papers in which the debate
about the existence of these particles
still rages.
It has become almost traditional in
this subspecialty to begin with a well-
known limerick :
A certain young lady named Bright
Could travel much faster than light.
She departed one day
In a relative way
And returned on the previous night.
Now that we have observed tradi-
tion, we turn to a more serious con-
sideration of these weird particles.
As is well known, the expression for
the energy E of a normal particle
of rest mass mo which is traveling
with a velocity v is given by
.. . . . . . . . .. ... . .. ... ..... .................
-... .... ...
Michael N. Kreisler is an Associate Professor of
physics at the University of Massachusetts at Am-
herst. He received his Ph.D. in high-energy physics
particle research from Stanford University in
1966 and taught physics at Princeton Uni-
versity before joining the faculty at the University
of Massachusetts. His research has concentrated on
measurements of. high-energy neutron interactions,
;
studies of multipion resonances, and searches for
rare decays of elementary particles. This work has
been supported in part by the National Science
: Foundation. Address: Department of Physics and
Astronomy, University of Massachusetts, Am-
E = -
V1 - (v/c)2
moc 2
.
where c is the speed of light. This
expression indicates that accelerating
a particle to speeds equal to or greater
than the speed of light requires an
infinite amount of energy and there-
fore should be impossible. It is this
fact that led Einstein to state that
"velocities greater than that of light
have no possibility of existence" (1).
In addition, this fact and the appar-
ent problems that faster-than-light
particles would create in special rel-
ativity -- in particular causal para-
doxes-have been strong enough the-
oretical arguments to deter any in-
vestigations in the area. We shall
return to these paradoxes below.
Although theorists, including Som-
merfeld (2), had considered such
particles as early as 1904 (in pre-
special relativity days), it was not
until the work of Bilaniuk, Deshpande,
and Sudarshan in 1962 and then
Feinberg in 1967 that the subject
became of interest again. Bilaniuk,
Deshpande, and Sudarshan (3) count-
ered the first objection regarding
infinite energy input by noting that
we are all quite happy with the exis-
tence and creation of photons and
neutrinos, both of which always travel
at the speed of light. Their proposal
was to postulate the existence or crea-
tion of particles with velocities always
greater than c, thereby circumventing
the infinite energy requirement.
The possibility of these new particles
is rather appealing because their
existence would indicate an inter-
esting symmetry-namely,. there
would be three allowed types of par-
ticles, classified by their velocities:
2. Particles such as photons and
massless particles which only exist
if |v| = c always
3. Particles with |v| > c always
Feinberg (4) introduced the name
tachyons, from the Greek word mean-
ing swift, for the third type of particles.
This name has become quite fashion-
able, and, as an interesting aside,
its quick acceptance has led to other
proposals for new names for normal
particles-bradyons, from the Greek
for slow, and tardyons, a name with
an obvious derivation. While per-
haps amusing, these names are not
very useful and we will avoid them.
Countering the objections of causal
paradoxes is not as simple as merely
postulating new particles. In order
to discuss the problem, we must first
examine the paradoxes implied by
the existence of tachyons. In standard
fashion, as shown in Figure 1, we
consider two coordinate - frames S
and S' which have a common x axis.
The frame S' moves at a constant
velocity v (lu) < c) in the +x direc-
tion relative to S. Now, let us assume
that an observer in S sees a tachyon
created at point A at time tA. The
tachyon travels with a velocity +u
to point B where it is absorbed at
time tg. For this observer, the dis-
tance and time separations were both
positive:
At = tB - tA > 0
Ax = xB - XA > 0
Using the standard Lorentz trans-
formation, we can calculate what an
observer in the S', or moving, frame
sees :
Δχ' = Υ(Δχ - Δt) =
v(Ax -„ Ax) = VAX (1 _2)
1. Normal particles, which travel
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the particular event. Carrying these
arguments further, if the tachyons
Y (At - At- ) = yAt (1.
-
are charged or carry any other quan-
tum numbers, the observer in S' must
see an anti-tachyon traveling from
B to A (see Fig. 1).
Al' = Y (At - 2 Ax) =
where y = (1 - 2%)
-1/:
Clearly, the spatial separation Ax
or Ax' is positive in both frames. How-
ever, if the velocity of the tachyon
is chosen such that uv > c2, the time
separation for the moving observer
is negative! Apparently a paradox
exists, because the observer sees the
tachyon absorbed before it was
created !
The paradox can be resolved by
using the "reinterpretation principle,"
which is merely the statement that,
when going from one inertial frame to
another, it is essential that the form
of physical laws be invariant. How-
ever, there is no requirement that
the description or interpretation of
a particular phenomenon be the same.
So long as there is no violation of a
physical law in either frame, ob-
servers in different frames could inter-
pret a given series of events differently.
In the case described above, the
physical process was the passage of a
tachyon between points A and B.
The paradox is resolved if the observer
in the S' frame interprets the process
as the creation of the tachyon at B and
absorption at A. The observers then
agree on the physical process but
disagree about the interpretation of
Before considering a more difficult
paradox, there are certain character-
istics of tachyons that must be men ..
tioned. The calculations that were
'sketched above indicate that the
sign of the fourth component of a
tachyon four-vector can be changed
by a Lorentz transformation. In other
words, those tachyons that are travel-
ing backward in time in a given frame
also appear to have negative energies.
Another interesting possibility that
exists if there are tachyons is the de-
cay of a normal, moving particle
into itself plus a tachyon. Such de-
cays without tachyons are forbidden,
owing to the requirements of simul-
taneous energy and
momentum
conservation. We will use these prop-
erties of tachyons in the discussion
of the following paradox.
Consider our two observers again,
one moving at a velocity v with re-
spect to the other (see Fig. 2). These
two observers, A and B, agree on the
following course of action: A, the
stationary observer, will send a
tachyon to B at 12:00 noon his (A's)
time unless he has received a tachyon
from B before noon. B, upon receiving
the tachyon from A, will immediately
send a tachyon back to A. In the
happy event that their relative velocity
am 'the velocity of the tachyons are
such that the Lorentz transformation
between the frames does not reverse
the sign of t or E, everything is fine.
That is, A sends his tachyon out at
noon and gets a return signal some-
time later.
However, if we are not so lucky, the
tachyon emitted by B will be traveling
backward in time as viewed from the
stationary frame. It will therefore
arrive at A before noon. A will detect
it and not send out his tachyon. Why
then did B send one back?
. This paradox may be resolved (4)
if we examine A's detector, which,
for this purpose, can be an atom or
a proton. When the detector absorbs
a positive-energy tachyon, its energy
increases and either the proton moves
or the atom goes into an excited state.
If the observer wants to be sensitive
only to positive-energy tachyons, his
detector must consist of stationary
protons or atoms in the ground state.
Such detectors are not able to absorb
negative-energy tachyons, and the
paradox would not arise. If he wants
to be sensitive to both positive and
negative energies, he must choose,
for example, a proton with some
nonzero kinetic energy. The signal
that a negative-energy tachyon had
been absorbed would be a sudden
loss of energy by the proton (for ex-
ample, it could suddenly come to rest).
However, the observer would in
principle be unable to distinguish that
absorption of a negative-energy
tachyon from the spontaneous emis-
sion of a positive-energy one. For
that reason, he would assume that
at 11:00 his detector spontaneously
emitted a tachyon and would not
attribute it to a signal from B. There-
fore, the paradox is explained.
The resolution of such simple ex-
amples does not mean that appar-
ently unresolvable paradoxes cannot
be invented. In fact, arguments re-
garding the existence of tachyons
have filled many journal pages in
recent months. However, as there
was no compelling argument against
their existence and since a good ex-
perimental result is usually worth
more than a journal of theoretical
speculations, Torsten Alväger and
I (5) decided to see if the question
·of tachyons was amenable to experi-
ment. If we were fortunate enough
Z
NO
Z'
S
S'
NG-
NO
A
..
T+
T -.
B
B
Figure 1. An example of a possible paradox. sorbed at B. In frame S' (right). the moving
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riow uo taunyous penaver
5 4 .
4. 1 mivic puszung paiduux.
Before an experimental search could
be conducted, it was first necessary
to determine the properties that
tachyons should exhibit. In what
ways do they differ from normal par-
ticles? Do present experimental re-
sults put stringent limits on their
existence? We present a partial list
of the properties of tachyons.
1. The relativistic expressions for
the energy and momentum of a par-
ticle of rest mass m and traveling at
velocity u are
mc2
E =
{1 - (u/c) 2 } 1/2
[P]=
{1 - (u/c) 2} 1/2
mu
.
If |u| is greater than c, the "rest mass,"
m, must be an imaginary quantity
if the observable quantities E and |p|
are to remain real. Since a tachyon
rest mass is unobservable, this choice
is allowed. We will use the notation
m = iu, where u is a real number.
Thus for tachyons :
E
=
2
Mc 2
- 1>
1/2
(p) =
{})" - }"
The relation between energy and
momentum is then E2. = |p|2;2 --
ic4 instead of the same expression
with a plus sign, which holds for
normal particles. If the discussion
s
£
restricted
to positive-energy
achyons, the bounds on the energy
and momentum are
0<E<œ and pc <|p| <œ
These relations indicate several re-
narkable properties of tachyons: (1)
achyons can exist with zero total
energy and with finite momentum;
2) infinite velocities are possible;
nd (3) when a tachyon loses energy,
t accelerates !
.. A tachyon appears to be a tachyon
n all Lorentz frames. In Figure 3
he algebra of velocity addition is
resented. If the velocity is greater
han c in one frame, the Lorentz
ransformation to any other frame
caves the velocity greater than c.
A will send B a tachyon at 12:00 his (A's) time unless B sends him a tachyon signal
before noon. B will only send A a tachyon after he receives one from A.
A's clock
I haven't received a signal from B.
I'll send him a tachyon.
12
7
9
3
6
Ah ha! Here's A's tachyon.
I'll send him back one.
12
A
B
9
9
6
The tachyon travels backward in time as viewed from A's coordinate frame.
Here's B's signal. I won't send
him a tachyon at noon.
11
12
B
7
9
3
6
Why did B send A a tachyon?
total energy, they can in principle
be created with zero energy input.
One is then led to expect spontaneous
tachyon production independent of
the value of u. However, Feinberg
(4), who has shown that it is possible
to include tachyons in the formalism
of relativistic quantum mechanics,
claims that tachyons most probably
obey Fermi-Dirac statistics. In that
event, spontaneous production would
be severely limited because all the
energy states possible to reach via
spontaneous creation would be filled,
inasmuch as the exclusion principle
allows at most a single particle obey-
ing Fermi-Dirac statistics in each avail-
able freely specified quantum state.
4. It is kinematically allowed for a
tachyon to decay into itself plus a
photon. This type of decay is not
permitted for normal particles since
it is impossible to satisfy hath maman
taneously. For charged tachyons, it
is in fact possible to calculate both
the energy spectrum of the photons
emitted and a total decay rate. This
process yields many of the same fea-
tures as Cerenkov emission, which
for normal particles occurs when the
velocity of particle propagation in
a medium exceeds the velocity of
light in that medium. For tachyons,
of course, the velocity is always greater
than the speed of light-even in a
vacuum. Due to the similarity, we
will refer to this process of photo-
emission as Čerenkov emission in a
vacuum.
In order to derive an expression for
the rate of energy loss by this process,
we must impose a cut-off on the radia-
tion energy spectrum-namely, we
assume that no photon can carry
away from the tachyon enough energy
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-
B
A

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is flat from zero to the energy of the
tachyon-that is, all photon energies
from zero to the full energy of the
tachyon are equally probable. The
energy-loss rate per unit path length is
for a tachyon of charge Ze and energy
E. With this expression it is possible
to determine the distance a tachyon
would travel before its energy dropped
to less than 1 eV. Surprisingly, in-
dependent of the initial energy, the
distance is very small. Typically, if
the initial energy is approximately
uc2, the distance is a fraction of a milli-
meter!
This result has an important experi-
mental consequence. Because these
objects lose all their energy so quickly,
it is highly unlikely that they would
have been observed in any previous
experimental studies. Standard de-
tection devices such as scintillation
counters or bubble chambers would
not have found tachyons. All such
devices require a particle to deposit
energy in order to be detected.
5. Tachyons cannot be "stopped"
by interactions with matter. But can
they be captured by a nucleus or by
an electron? If this has an appreciable
probability, any experiment looking
for such objects would be affected
drastically. Feinberg (4) claims that
it is not at all clear whether such a
process could occur. We have at-
tempted to estimate the magnitude
of this capture effect using a fairly
simple model, in which the tachyons,
if captured, enter bound orbits around
the capture centers (5). Even if all the
electrons in lead could serve as cap-
ture centers, the mean free path in
Figure 3. Velocity addition for tachyons. If
a tachyon is traveling with a velocity u in
one Lorentz frame, what is its velocity in an-
other frame moving at a velocity v with re-
spect to the first frame?
u +
(7 - 1)|v||u|v
02
- YV
u' =
(1 - v(Au ) .
y = (1 - 02/c2) - 1/2
4/2
c2
= 1 + ( 2)
-
- 1
1 - 02 / 02 .
1
model, would be on the order of 104
meters! We then feel confident that
the probability of capture is rather
small.
As indicated above, the existence of
tachyons cannot be ruled out by their
nonappearance in conventional par-
ticle detectors. However, it is inter-
esting to see what limits can be set
on their production by studying ex-
isting experimental data. To do this,
we examined photoreactions-those
induced by photons-because they
are well understood, both theoretically
and experimentally. We compared
the total cross section for photons
interacting with lead with both the
sum of experimentally observed par-
tial cross sections and with the the-
oretical total cross-section predictions.
In the low-energy region the total
cross section is quite large (at 0.4
MeV, it is ~70 barns), and the agree-
ment between theory and experiment
is quite good: ~ ± 2%. This small
an uncertainty would still allow a
very large cross section for other
processes. If it were all due to ta-
chyons, these measurements place
an upper limit of only ~ 1 barn on
tachyon production. Our conclusion
thus was that there was in principle
no reason not to have tachyons and
there was no overwhelming evidence
against their existence.
How to look for tachyons
There are basically two types of ex-
periments that can be performed to
hunt for these particles. The first
utilizes the tachyon's spacelike four-
vector. In other words, in a reaction
such as
A+ B->C+X
the momenta and energies of A, B,
and C are measured. These quantities
coupled with energy and momentum
conservation determine the square
of the mass of the X particle, without
any direct observations on X. If X
is a tachyon, the square of the mass
is negative, enabling us to make a
unique identification.
The second method, suitable partic-
ularly for charged tachyons, in-
volves detecting the Čerenkov radia-
tion emitted in a vacuum. Unfortu-
nately, this technique is not easily
implemented. In regions close to the
production point, there will be large
Gamma
source
T --
Pb
¥T+
-- 9kV
Plate system
Face of photo'
multiplier
Figure 4. Schematic diagram of a charged
tachyon experiment. Tachyons are produced
in the lead shield surrounding the radioactive
source and travel to the high-voltage plates,
where they are detected by the emission of
Čerenkov radiation in a vacuum (from 5).
cesses. Far from the production point,
in relatively low background areas,
the tachyons will, in general, have
radiated away almost all their energy,
making detection rather difficult.
For reasons of simplicity and cost,
T. Alväger and I (5) chose to utilize
the second technique. In order to
overcome the problems just men-
tioned, we made the additional as-
sumption that charged tachyons in-
teract with electrostatic fields. In
particular, we assumed that tachyons
could gain energy in the same manner
as normal particles. Therefore, it
would be possible to increase a ta-
chyon's energy to any desired value
at any point along its path. The rate
of change of energy along its path is
then
AF _ _ 2m2 E2 + Ze El
h2c2
where & is the electric field and we
have assumed E < pc2. The first term
in this expression is the rate at which
energy is radiated away in the form
of Cerenkov light, while the second
is the gain of energy in the field.
Clearly, the tachyon will reach a
stationary energy state when it is
emitting energy at the same rate it is
gaining. For energy levels in the few
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f un CtCivslauc
1
1
fields required are only several hun-
dred volts/cm and the levels are
reached very quickly --- typically in a
small fraction'of a millimeter. With sta-
tionary energy states of a few electron
volts, the Cerenkov radiation will be
partially in the visible range. The
detection problem is then trivial,
because standard photomultiplier
tubes can be used.
A schematic drawing of experimental
apparatus using this technique is
shown in Figure 4. A cesium 134
source (emits photons of 797 and
605 keV) was used to produce the
tachyons in a lead shield surrounding
the source. If tachyons were produced,
they would travel through some addi-
tional lead shielding and then pass
between two parallel plates situated
in vacuum and held at 9 kV voltage
difference. The phototube looked at
the region between the plates. The
Čerenkov radiation is expected to be
emitted at ~90° with respect to the
direction of motion so that the photo-
tube is located at the optimum angle.
The electric field was chosen to place
the radiation in the sensitive region
of the phototube. The detection
technique was rather simple; the
pulse height was recorded for all
events with measurable pulses. The
majority of the events were triggers
due either to dark current in the
photomultiplier or to light from small
corona points on the plates. Data
were taken under various conditions-
with and without the source and
with and without the high voltage.
The number of photons in the sensi-
tive region of the spectrum which will
reach the phototube per tachyon
can be calculated. Since, on the aver-
age, all the tachyons pass through
the same field, the existence of ta-
chyons should yield a peak in the
pulse height spectrum. Figure 5 shows
the pulse height spectrum with the
no-source data subtracted and with
the position for a tachyon peak in-
dicated. Clearly, there is no evidence
for abundant tachyon production.
Assuming that a peak with a height
of at least 0.1 counts/sec was the
minimum "tachyon signal" detect-
able in this apparatus, we found that
the photoproduction
cross section
for tachyons in lead by 800 keV pho-
tons was less than 3 × 10-30 cm2.
This limit (shown in Table 1) is valid
for charges on the tachinne hatwenn
Expected position
of tachyons
: 0.10
Counts/sec
0.05
0.00
9
-0.05
-
1
0
10
20
30
40
50
60
Channel number
Figure 5. The observed pulse-height spec-
trum, showing the expected position for a
tachyon peak (from 5).
charge on the tachyon is too large,
the stationary levels yield light which
falls below the sensitive region of the
phototube; if the charge is too small,
it takes a long time to reach a station-
ary level, thereby greatly reducing
our detection efficiency. It is inter-
esting to note that the limit is valid
for all masses u, since tachyons can
exist with zero total energy whenever
'p'c = pc2.
Although this experiment laid to
rest any qualms about the existence of
huge fluxes of these particles, the
prospect of looking for them was quite
appealing. In efforts to improve on
the first experiment, we were joined
by M. Davis (8, 9). The ground rules
for the second-generation experiment
were straightforward; it had to be
inexpensive and not require a long
operating time. The major problem
limiting the first experiment was
the relatively large counting rate
due to corona discharge and to dark
current in the phototube. A simple
way to avoid these problems would
be to use two detectors and place
their signals in coincidence. This
would also avoid the necessity of
pulse-height analysis. A schematic
drawing of the experimental setup is
shown in Figure 6. The idea is the
same as in the first experiment-
namely, tachyons are produced in
lead by ~1.2 MeV photons from
a Co60 source and then travel through
two identical detectors consisting of
parallel plates in a vacuum.
In order to reduce corona discharges,
the plates were covered with opaque
construction paper (see Fig. 7). This
innovation, which proved very suc-
cessful, involved an additional cost
of hitmantonales within the Lordand
two detectors were counted for 104
seconds each, with and without the
source. In each state, we observed
7 counts, a number consistent with
the expected accidental rate. This
yielded a counting rate for tachyons
of less than
4.8 X 10-4 counts/sec
implying that the photoproduction
cross section is less than
1.67 X 10-33 cm2 at 1.2 MeV
What does this limit mean? To a
physicist, it is instructive to note that
this upper limit is more than 108
times smaller than electron-positron
pair production at the same energy.
In terms of a mean free path for pho-
tons, a photon could travel through
11,000 miles of lead before it had
any noticeable probability of produc-
ing a tachyon.
Although these experiments were the
first to address this problem, recently
there have been others, employing
different techniques, that should be
discussed. Two bubble chamber ex-
periments (10, 11) have attempted
to look for tachyons using the missing
mass technique. For normal particles,
the square of the missing mass. is al-
ways greater than zero. For single
tachyons, the mass squared is always
negative, and when a pair of tachyons
is created, the pair may have either
a positive or negative mass squared.
Therefore, if one examines an inter-
action and calculates the mass squared
of missing particles, only tachyons
or tachyon pairs would appear to
have negative values. Great care
must be taken, of course, to ensure
that such values are not introduced
artificially by measurement errors
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9
T-
Gamma
source
T+
Pb
Face of
photo.'
multiplier
Plate
system
Evacuated
chamber
Figure 6. A second-generation detector. Two
identical devices are placed in coincidence
(from 8).
The first of the two experiments, by
Baltay and his colleagues (10), merely
looked for negative invariant masses
for unseen or neutral particles. The
experimenters examined reactions in
which either K- mesons or antiprotons
were stopped in a bubble chamber.
In particular, they searched for can-
didates for the following reactions:
K-+ p-Aº + T°;
K-p-A° + T° + Tº
+ p +++- To;
PPT+T- To + To
where To and Tº are unseen neutral
particles, hopefully tachyons. This
particular set of reactions is advan-
tageous as it is not necessary to make
any assumptions about the interaction
of tachyons with matter. The only
assumption is that negative values
of the square of the mass are not sup-
pressed with respect to positive ones
in the case of tachyon pairs. The
measurement is quite simple: the
momenta of all the visible particles
are measured, and the mass squared
of any missing particles is calculated.
The mass squared is then plotted,
and all very low-mass or negative-
value events are examined carefully.
The data for one reaction is shown
in Figure 8. There are some "tachyon
candidates" in the data sample; how-
ever, in all cases, a careful examination
of each questionable event showed
that the apparent negative values were
incorrect and had been caused either
by measurement errors or by addi-
tional effects that had not been in-
cluded in the reconstruction process
(for example, scattering of one of the
particles after the interaction of in-
terest). The results of the re-analysis
of the "borderline" events is also
shown in Figure 8. The lack of ta-
chyon candidates indicates that the
probability of producing tachyons
in these reactions is ~2,000 times
less likely than producing T° mesons.
À
8
Figure 7. The improved detector. Note the corona (from 9). (Copyright, 1969, The
nnannie constructin
A
EXIt CAPtiltitai iCoulS are given
in Table 1.)
The other major bubble chamber
experiment, by Danburg and his
colleagues (11), required the assump-
tion that charged tachyons would
leave tracks in a bubble chamber
similar to those of normal particles.
This relies on the assumptions that
Čerenkov radiation does not occur
and that ionization energy loss does.
The experiment consisted of a search
for events in which charged pairs of
tachyons are produced. Since each
member of the tachyon pair can be
examined, this technique should de-
tect tachyons (subject to the correct-
ness of the major assumption) inde-
pendent of the sign of the mass squared
of the tachyon pair. As in the previous
experiment, no candidates survived
careful re-examination, yielding an
upper limit on the production cross
section (see Table 1).
----
Other techniques
A slightly different approach has
been . used by Murthy (12), at the
Tata Institute. He argues that ta-
chyon production might occur in
high-energy cosmic-ray interactions-
extensive air showers. Once a primary
particle interacts in the atmosphere
and a shower develops, the major
components of the shower-electrons
and photons-travel at v = 0,
thereby defining the shower front.
Heavier particles tend to travel more
slowly and therefore arrive later than
the shower front. Some early quark
searches used that fact to look for
heavy quarks. Tachyons, on the other
hand, would arrive before the showers.
For example, if a tachyon were pro-
duced at 2 km above the surface, it
would arrive ~2 usec before the
shower front; the full range of the
time difference is 0 to 20 usec.
The experiment consisted of trigger-
ing on a potential tachyon signal and
waiting for ~20 usec for the arrival
of an extensive air shower. The po-
tential tachyon signal could be created
by charged tachyons radiating Čeren-
kov light in a vacuum or by either
neutral or charged tachyons inter-
acting with liquid scintillator. The
total rate for the arrival of showers
following such potential tachyon sig-
nals is completely accounted for by
accidental coincidences. In addition,
the tima distribution
- of the
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.

Declassified in Part - Sanitized Copy Approved for Release 2014/01/09 : CIA-RDP79-00999A000200010093-3
andom coincidences. Since there is no
:vidence for tachyon production, the
ambimit on tachyon production in exten-
hive air showers relative to electron
umproduction is ~10-4 to 10-5.
wouj
mbA fairly interesting search for tachyon-
ticleike objects has recently been com-
theleted by Bartlett and Lahana (7).
occin order to justify their search, they
doelote that ordinary particles are either
cardheutral or electrically charged and
irs hat those that move at v = c are
cacheutral. Therefore, there is a certain
n bymmetry if tachyons are either neu-
I diral or magnetically charged. The
recexperiment, then, is a search for
indmagnetic monopoles that are traveling
arefaster than c. Although this appears
viouto be a rather uneconomical ap-
iveproach-namely, to hope not only
athat there are tachyons but that they
croare monopoles as well-there are
some theoretical arguments (13) that
charged tachyons would, in fact,
exhibit all of the properties of mag-
netic monopoles.
hà
thThe experiment is analogous to the
taearlier searches with the appropriate
ijinterchanging of & and H fields.
s-The two phototubes detect the Če-
harrenkov light from radiation in the
hermagnetic field and are sensitive to
ajoobjects with magnetic monopoles
ron.between 1/10 and 4 times the size of
ca Dirac monopole. Working with
onta 20,000 curie, Co60 source, the re-
toresearchers found no candidates. This
haryields extremely good limits on the
arkproduction cross sections, which were
fortypically less than 10-36 cm2.
her
ers:An experiment (14) has been con-
ro-ducted at Brookhaven National Labo-
, itratory to search for the emission of
thenegative-energy tachyons by protons.
theIn the reaction proton -> proton +
tachyon, the emission of a negative-
energy tachyon appears the same as
er-the absorption of a positive-energy
nd one. A proton at rest would therefore
val suddenly move if the reaction has
o- occurred. This process can be searched
ed for by examining a bubble chamber
in- with no incident particles; any pro-
er: tons that were suddenly "inspired"
er- to move would leave tracks. The
he experiment yielded
many
such
irs tracks -- all of which could be ex-
g -; plained by gamma emission from
by; radioactive materials near the bubble
n, chamber. Taking one event as an
l -! upper limit, the researchers found
2348 events
(m°) 2
150|
100
50
1
0
-0.15
-0.10
-0.05
0
0.05
0.10
(m2º)2 in (BeV)2
Remeasurements of
events with m22 ≤0.014
(m7º)2
15
10-
5
0
-0.15
-0.10
-0.05
0
0.05
0.10
(mº)2 in (BeV)2
Figure 8. Results of a search for neutral
tachyons and tachyon pairs. The upper
graph indicates some tachyon "candidates"
greater than 1021 years. Similar life-
times can also be obtained from a
consideration (14) of experiments on
baryon conservation and measure-
ments of the heat flux from the earth.
Philosophical considerations
All of these experiments have one
major limitation: since they produce
upper limits on the production and/or
existence of tachyons, they never seem
to satisfy the skeptic. The skeptics-or
true believers-give arguments which
ask: "But suppose the cross section
is really only a factor of ten below
the current limit?" Usually there
is no satisfactory answer to such ques-
tions, although the skeptics should
be encouraged to perform the experi-
ments themselves. However, there
does exist a method which, in princi-
which do not stand up to a refined analysis,
as shown in the lower graph (from 10).
argument is presented as a philosophi-
cal end to this discussion of tachyons.
L. Parker (13) suggests that there are
two types of coordinate systems: (1)
the normal, or subluminal, frames in
which ordinary particles behave prop-
erly and tachyons travel with |u| >
c; and (2) superluminal ("faster-than-
light"') frames-relative to which
tachyons behave as normal particles.
In a world with only one spatial di-
mension, it can be shown that (1) in
the superluminal frame, tachyons
have real masses, and in such frames
it is possible to construct a quantum
field theory completely similar to that
for subluminal particles in subluminal
systems; (2) the mathematical trans-
formations involved in going between
the frames are entirely symmetric.
Therefore, in a superluminal frame,
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Number of events/0.002 (BeV)2

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Table 1. Summary of searches for tachyons.
Type of search.
Comments
Typical results
Čerenkov radiation ---
.1e <Z <2e
Photoproduction cross section < 3 × 10-30 cm2; 800 KeV
photons on lead
(5)
Čerenkov radiation-
photomultipliers
.5€ < Z < 1.9€
Photoproduction cross section < 1.7 X 10-33 cm2; 1.2 MeV
photons on lead
(8)
Missing mass squared- neutral
bubble chamber
typical result:
Probability for the reaction : K- + p -> Aº + tachyon
(10)
K- and p stopping
Probability for the reaction: K- + p 1º + To
<2 × 10-3
Missing mass squared ---
bubble chamber
Assume charged tachy- Production cross section for charged tachyon pairs < 2 X (11)
ons leave tracks in bub-
10-31 cm2
ble chamber
K- interactions at 2
GeV/c
Cosmic ray-extensive
air showers
Tachyons arrive before
shower front
Occurrence of tachyons in cosmic ray showers
< 10-4
(12)
Occurrence of electrons in cosmic ray showers
Tachyon-like magnetic
monopoles
Interchange e and H
Photoproduction cross section
(7)
on lead
< 6 × 10-37 cm2
on H2O
< 2 × 10-36 cm2
Bubble chamber -- emis-
sion of negative energy
tachyons
Examine bubble cham- Lifetime for decay p -> p + negative energy tachyon is
ber with no incident
greater than 2 × 1021 years.
particles
(14)
ordinary particles appear to be tachy-
ons.
These statements and others can be
summarized by introducing the ex-
tended principle of relativity-namely,
all the laws of physics have the same
form relative to each of the frames.
Therefore, the laws governing tachy-
ons and subluminal particles should
be interchanged under the transforma-
tion, but the total structure of the
laws should have the same form.
Clearly, since photons have the same
properties in both coordinate systems,
the laws describing them should be
form-invariant under the transforma-
tion. This then leads to the following
observation.
Consider photon-photon (Yy) scat-
tering. This process will appear the
same in both coordinate frames. In
each frame the process will depend
on the contributions from intermediate
virtual particle-antiparticle pairs. If
the probability of yy scattering in the
subluminal system, calculated with-
out considering tachyons, is P, then
the probability of yy scattering in the
superluminal system, calculated with-
out considering non-tachvons. is also
no interference between the two
frames, the total probability for YY
scattering, including both effects, is
2P. The situation is a clear one in
principle. If there are tachyons, the YY-
scattering cross section should be
twice as large as predicted without
tachyons.
Ignoring the question of our ability to
calculate the cross section to a factor
of two, or the question of whether
quantum electrodynamics would be
thrown out before tachyons were
acknowledged, the feasibility of such
an experiment is in doubt. Unfor-
tunately, the predicted cross sections
(15) are very low. In the visible
region, the cross section is ~10-65
cm2, implying that experiments are
impossible. There is some possibility
that at the energies becoming avail-
able with high-energy accelerators
the experiment could be performed.
However, it is doubtful if such experi-
ments will be useful in tachyon
searches because the calculation of the
expected rates increases in difficulty
with increasing energy. Clearly, the
limits on the existence of tachyons
have been pushed quite low. This
author. at least. would he verv sur-
References
1. A. Einstein. 1905. Annalen der Physik 17:
sec 10.
2. A. Sommerfeld. 1904. Konink. Acad.
Wetensch. Amsterdam Proc. 7:346.
3. O. M. Bilaniuk, V. K. Deshpande, and
E. C. G. Sudarshan. 1962. Am. J. Phys.
30:718.
4. G. Feinberg. 1967. Phys. Rev. 159:1089.
5. T. Alväger and M. N. Kreisler. 1968.
Phys. Rev. 171:1357.
6. This choice is by no means the only one.
An excellent summary of the present
theoretical arguments regarding Ceren-
kov radiation by tachyons can be found
in (7).
7. D. F. Bartlett and M. D. Lahana. 1972.
Phys. Rev. D 6:1817.
8. M. B. Davis, M. N. Kreisler, and T.
Alväger. 1969. Phys. Rev. 183:1132.
9. M. N. Kreisler. 1969. The Physics Teacher
7:391.
10. C. Baltay, G. Feinberg, N. Yeh, and R.
Linsker. 1970. Phys. Rev. D 1:759.
11. J. S. Danburg, G. R. Kalbfleisch, S. R.
Borenstein, R. C. Strand, V. Vander-
Burg, J. W. Chapman, and J. Lys. 1971.
Phys. Rev. D 4:53.
12. P. V. Ramana Murthy. 1971. Leit.
Nuovo Cimento. 1:908.
13. L. Parker. 1969. Phys. Rev. 188:2287.
14. J. S. Danburg and G. R. Kalbfleish.
1972. Phys. Rev. D 5:1575.
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Refer-
ences
photomultiplier