PARTICLES BEYOND THELIGHT BARRIERDespite almost universal belief that particlescannot move faster than light, one can think ofnegative-energy particles traveling backward in timewhose speed exceeds that of light. Experimentersare looking for these faster-than-light particles.

OLEXA-MYRON BILANIUK and E. C. GEORGE SUDARSHAN

FOR MANY DECADES now the view hasprevailed that no particle could pos-sibly travel with a velocity greaterthan the velocity of light in vacuum,c = 3 X 108 m/sec. It is generallyheld that this limitation is a directconsequence of the special theory ofrelativity. Albert Einstein himself hassaid so in his original paper on rela-tivity.

Some time ago we reexamined thispoint1 and concluded that existence ofsuperluminal particles is in no wayprecluded by Einstein's theory. Onthe contrary, it is this very theory thatsuggests the possibility. Other phys-icists, notably Yakov P. Terletskii2

and Gerald Feinberg,3 have reiteratedthis same conclusion.

Around the turn of the century,shortly before Einstein had publishedhis revolutionary paper on the specialtheory of relativity, Arnold Sommer-feld4 examined the problem of accel-erating particles to velocities greaterthan c. He concluded that at suchvelocities particles would have to be-have in a patently absurd fashion:Upon loss of energy they would haveto accelerate! Einstein's theory hasremedied this seemingly inadmissiblestate of affairs. It predicted that themass of a particle would get infinitelylarge upon approaching the velocityc Thus it became obvious that no

particle could be accelerated past the"light barrier."

This argument still stands. To as-sail it is a futile task. Yet it is possi-ble to circumvent it by asking the fol-lowing question: Is the accelerationprocess the only way by which fastparticles are produced? Certainlynot! Take photons or neutrinos.They do travel with a velocity equal toc without ever having been acceleratedfrom a slower speed. In fact there

are no slow photons or neutrinos.When they are created in atomic ornuclear processes they take off rightaway with the velocity of light. Con-versely, the only way to slow themdown is to make them disappear.

A third class?

For this reason photons and neutrinosmust be regarded as a class of objectsdifferent from "normal" particles suchas electrons or nucleons. Let us

Olexa-Myron Bilaniuk, a native of theUkraine, attended the University ofLouvain, Belgium and got his PhD atthe University of Michigan. After fiveyears at the University of Rochesterhe joined the Swarthmore College fac-ulty. Besides his interest in relativ-istic dynamics, Bilaniuk is an experi-mental nuclear physicist. He is fluentin seven languages, has a commer-cial pilot's license, skis and holds asailplane-soaring award.

E. C. George Sudarshan was born andraised in India, where he won his BSand MA from Madras University. Atthe University of Rochester he earnedhis PhD, left for two years as a Har-vard fellow, and then joined the Ro-chester faculty. For the last five yearshe has been at Syracuse University.Sudarshan works in quantum theory offields and particle physics and is knownfor his pioneering work in the V—Atheory of weak interactions.

PHYSICS TODAY • MAY 1969 • 43

henceforth refer to the normal ones asClass I, and to photons and neutrinosas Class II. Whereas Einstein's theoryencompasses both of these classes, onlyfor Class I does it require c to be theunattainable limiting velocity. Fol-lowing this line of thought, we haveinquired whether Einstein's theorycould possibly accommodate yet athird class of objects, that is, super-luminal particles.

Would existence of such particlesreally violate any basic maxims ofphysics, such as conservation of en-ergy or causality? If so, it would in-deed be senseless to speculate furtherabout such particles. But if their ex-istence would not lead to any contra-dictions, one should be looking forthem.

There is an unwritten precept inmodern physics, often facetiously re-ferred to as Gell-Mami's totalitarian

principle, which states that in physics"anything which is not prohibited iscompulsory." Guided by this sort ofargument we have made a number ofremarkable discoveries, from neutrinosto radio galaxies.

Several such searches are in prog-ress now. Because theory does notexclude the possibility that a mag-netic analog to the electric charge canexist, physicists persist in their questfor the magnetic monopole.5 A simi-lar search is on for the "quark," afundamental particle having 1/3 ofthe electronic charge, whose existenceis suggested by recondite symmetriesof elementary particles.

In each of the above cases, the pos-sible existence of the new particle wasfirst suggested by the logical exten-sion of the regularities or symmetriesgoverning the known physical world.The special theory of relativity pro-

A TARDYON'S POCKET DICTIONARY TO THE WORLD OF TACHYONS

Subluminal—slower than lightSuperluminal—faster than lightUltraluminal—faster than c2/V,

where V is the relative velocityof two laboratory systems

Tachyon or meta-particle—a faster-than-light particle

Tardyon—any slower-than-light par-ticle

Luxon—photon or neutrino

Meta-relativity — special-relativitytheory of tachyons

Proper mass (m,,)—In analogy toproper length and proper time,this term should be used insteadof the term "rest" mass. Theterm proper mass is appropriatefor all classes of particles, includ-ing luxons, whose proper massis zero, and tachyons, whoseproper mass is imaginary. (Itdoes not make sense to speak ofthe "rest" mass of a tachyon.)

Meta-mass (m *)—absolute value ofthe proper mass of a tachyon(m,, = /m+)

Reinterpretation principle—interpre-tation of negative-energy tachyonspropagating backward in time aspositive-energy tachyons propagat-ing forward in time. This rein-terpretation invalidates causalityobjections to the possibility of ex-istence of faster-than-light signalsand permits construction of a con-sistent theory of tachyons.

Transcendent tachyon—infinite-ve-locity tachyon having finite mo-mentum (p = m* c) and zero en-ergy. Such a tachyon is every-where on the supercircle of itstrajectory. A transcendent stateof a tachyon is analogous to thequantum-mechanical state ofrest, in which a definite value ofmomentum implies total uncer-tainty of position.

vides for just such an extension. Wehave called it "meta-relativity."

Geometrical representation

Consider a particle having energy £and momentum components p r , p]f andpz. According to special theory ofrelativity, the quantities must obey:

£ O OO OO OO c\ i

2 - p?c- - p,rc~ - p:-c~ = m(*c*If the same particle is observed fromanother frame of reference its energyand its momentum components willbe different, but their values will stillsatisfy the above equation. This prop-erty is called invariance of the en-ergy-momentum four-vector under aLorentz transformation. Without lossof generality, we may assume that theparticle moves along the x axis, sothat py and pz are both zero. It nowbecomes possible to represent the sit-uation graphically. The equation

describes a hyperbola in the coordi-nate system of ps and E (black curvein figure 1). Each point on the up-per branch of the black curve fur-nishes the energy E and the momen-tum pr that the particle will have ina given frame of reference. It canbe shown that the slope of the hyper-bola at that point gives the velocityof the particle in the correspondingreference frame,

vd. — dE dpr

Because this slope is everywheresmaller than c, we see that even bychoosing a frame in which the en-ergy is incredibly large, its velocitywill still not exceed the value of c.This means, conversely, that even ifwe could supply a particle with an ar-bitrarily large amount of energy, itsvelocity with respect to us would stillbe bounded by the value of c. Let usalso note that transform as we may,we will not succeed in getting ontothe lower branch of the black hyper-bola. So much the better, as a par-ticle with negative energy is not ob-servable.

A point of particular interest on theupper black curve is that for whichpr — 0. This point corresponds to aframe in which the particle is at rest.Even though its momentum vanisheswe see that the energy of the particleremains at a finite value of E = m0C".We call it the proper energy, or restenergy, of the particle; m0 is its propermass.

In the special case in which m0 *s

44 • MAY 1969 • PHYSICS TODAY

locity dependence is contained in theexpression of the relativistic mass,

4

RELATIVISTIC-ENERGY (E) VS. MOMENTUM (p,) dependence for three classesof particles. Slope of a curve at any given point gives the velocity of the particlewhose energy and momentum are furnished by the coordinates of the point. Be-cause the slope of the red curve is, in absolute value, everywhere greater than c,this curve represents the as-yet-undiscovered tachyons. The gray curve depicts thenonrelativistic dependence where energy is simply E — p2/2m<>. It has been shiftedup to show that for small velocities it coincides with the black hyperbola. —FIG. 1

zero the two branches of our hyper-bola degenerate into two intersectinglines,

(E + pxc) (E - pxc) = 0

drawn in pink in figure 1. Take apoint on the pink line in the upperright-hand quadrant. The coordinatesof this point give us the energy E andthe momentum px of a photon in thecorresponding reference frame. Whenwe change to another frame the en-ergy and momentum of the photon willassume new values, as in the case ofnormal particles. But the velocity will

not; the slope of the line is every-where c. This geometrical representa-tion shows clearly that there can beno vantage point with respect to whichthe photon would appear to be atrest.

Now we are almost ready for ourgeneralization. But before we pro-ceed with it, let us take a look at therelativistic dependence of energy onvelocity. Because in relativity theorythe energy of a particle is given by

E = mc2 (1)

and c is a universal constant, the ve-

m = mo[l - (tVc)2]-1/? (2)

Here m0 is the proper (or rest) massof the particle and v is its velocity inour frame of reference. Because en-ergy is a measurable quantity, the ex-pression for m must yield a real num-ber so that the value that me2 givesfor E remains itself a real number. Aswe have already seen, the velocity ofparticles whose behavior is describedby the black hyperbola is alwayssmaller than c. This means that theexpression under the square-root signabove is always positive for these par-ticles. Inasmuch as m() is also givenby a real number, the relativistic massm, and hence the total energy E, arealso real. (Note that the square rootprovides a plus as well as a minussign for the energy. These signs cor-respond to the upper and the lowerbranch of the hyperbola, respectively.So far we consider only the upperbranch of the curve to have physicalsignificance.)

Equation 1 has general validity; soit must also hold for protons. If weknow the energy £ of a photon, wecan compute its relativistic mass asm = E/c2. But we have just seen thatthe relativistic mass is also given byequation 2. In the case of photons vis equal to c and we seem to be introuble; we are dividing by zero. Theonly way out is to require that parti-cles traveling with the velocity of chave zero proper mass. We know thisconclusion to be in full accord with ex-perimental observations.

We have just seen again that parti-cles traveling at subluminal and thosetraveling at luminal velocities are twoentirely distinct kinds of objects. Theformer can never reach the velocity ofc, the latter can have nothing but c.Yet even though these two kinds ofparticles are so drastically different,they are governed by the same rela-tivistic laws.

A hypothetical situation

Let us for a moment imagine a hypo-thetical (and highly improbable, butnot inconceivable) situation in whichthe entire formalism of the specialtheory of relativity is based solely onobservations of slower-than-light parti-cles, that is normal (Class I) particleshaving a nonvanishing proper massra0.

One could imagine that under suchcircumstances some nonconfonning

PHYSICS TODAY • MAY 1969 45

E

\

/ /

REINTERPRETATION OF NEGATIVE ENERGIES. An objection to tachyon ex-istence points out that a superluminal particle, given by S in one frame, will be given byS' in another, in which it would appear to carry "negative energy" as shown at leftin the E vs. p, diagram. The space-time diagram (right) reveals however that insuch a system the sequence of events involving this particle would also appear as"negative" (cW), Thus the event corresponding to the point P takes place beforethe event corresponding to the point O, when the events are referred to the cf axis.So one can think of "negative-energy" particles that have been "absorbed first andemitted later" simply as positive-energy particles that have been emitted first andabsorbed later—a perfectly normal situation. —FIG. 2

physicist might start reasoning as fol-lows. Although the theory of relativ-ity excludes the possibility of acceler-ating particles to the velocity of c,does it also preclude the emission, dur-ing some as yet unknown process, ofa new kind of particle (Class II) hav-ing the velocity of c?

He would next take a closer lookat relativistic laws and would discoverthat such particles are indeed admis-sible, but only under the conditionthat they possess no proper mass. Tosome of his colleagues such a condi-tion may appear rather far fetched.

What if one imagines a laboratoryattached to such a particle, travelingwith the same velocity c as the parti-cle? In this lab the particle would beat rest. But then it ought to vanishbecause its proper mass (rest mass)is required to be zero.

How can it be? Our imaginaryphysicist would answer: The situa-tion is simple, no lab can travel rela-tive to us with the velocity of c. All

46 • MAY 1969 • PHYSICS TODAY

observers are Class I objects and assuch are restricted to subluminal veloc-ities. Conditions under which sucha conceptual difficulty appears cannever arise.

After this introduction the readermay have guessed which question weare about to ask next. Although rela-tivity precludes acceleration of parti-cles to velocities greater than c, is itnot possible that a new kind (ClassIII) of particle having a velocitygreater than c is emitted during someas yet undiscovered process?

Imaginary proper mass

To answer this question we must firstlook at equations 1 and 2. In orderfor the energy E to be given by a realnumber the expression for m mustagain furnish a real number. But howcan it when a negative number is un-der the square-root sign? The onlyway m can be made real is by requir-ing that the proper mass m0 be imag-inary; so it is expressed as some real

number m times the square root ofminus one,

m0 =

Now the expression for relativisticmass may be rewritten as

mzzm* [ ( u / c ) 2 - 1 ]" 1 / 2

As long as v remains greater than c,the new expression for m will remainreal as required. The price we hadto pay for having E remain real wasto require the proper mass of the hy-pothetical faster-than-light particle tobe imaginary. Let us refer to its ab-solute value m# as the meta-mass.

Because no physical significance canbe attached to an imaginary number,skeptics may wish to reject the no-tion of an imaginary proper mass rightoff. They should not be so hasty. Aswe have already seen, all observersare confined to subluminal velocities.Consequently there are no observers inwhose frame of reference a super-luminal particle would be at rest.

Proper mass of a superluminal parti-cle is not an observable physical quan-tity; it is a parameter devoid of anyimmediate physical significance. Assuch this meta-mass may well be de-noted by an imaginary number. Phys-ics abounds in such parameters.What must remain real is the energyand the mass of the particle as theyappear to an observer. Both thesequantities will be real if we assumethe meta-mass to be an imaginary pa-rameter.

Next we must inquire if our gen-eralization is in accord with the re-quired invariance of the energy-mo-mentum vector discussed earlier. Inour simplified two-dimensional ver-sion this invariance is expressed asE2 — px

2c2 — m02c4. We have seen

the graphical representation of this ex-pression both for subluminal particlesfor which m{)

2 is a real positive num-ber, and for photons for which m0

2

is zero (figure 1). How do our hypo-thetical superluminal particles, withtheir imaginary proper mass, fit intothe picture?

Making the picture complete

It is a remarkable aspect of the gen-eralization that we not only find theconcept of an imaginary mass con-sistent with the energy-momentumequation and its graphical representa-tion, but that we actually need it tomake the picture complete. We knowfrom analytic geometry that the fam-ily of hyperbolas described by theequation

x2 — y2 — constant

includes not only those curves forwhich the constant is positive, and thedegenerate case where the constant iszero, but also all those curves forwhich the constant is negative. Byadmitting the possibility of existence

i of particles with imaginary propermass, we provide physical meaning tothe red hyperbola in figure 1. Thiscurve corresponds to our superluminalmeta-particles. Because their propermass m{) is given by im#, m0

2 be-i comes — m^2, which is a real nega-

tive number. The red hyperbola isthus a graphical representation of theenergy-momentum invariance of thehypothetical superluminal particles,

£ 2 _ p ^ 2 c 2 _ _ 7?1+2C4

Let us note that the slope of this curveis everywhere greater than c. Thisis another way of saying that if parti-cles with an imaginary proper mass

do exist, their velocity could neverbe less than c. Let us henceforth referto these particles as tachyons, a namesuggested in 1967 by Feinberg and de-rived from the Greek word raxta,meaning swift. (In contrast, let usrefer to all subluminal particles astardyons. So as not to leave lightlikeparticles—photons and neutrinos-nameless, let us call them luxons.)

Our skeptic friends cannot be ex-pected to give up that easily though.They may point out that [1 — (v/c ) 2 ] 1 / 2 turns up not only in the ex-pression for mass but also in expres-sions for length, time interval and soon. Because these quantities are allphysically measurable, they must begiven by real numbers. The only wayto make these numbers real is again topostulate that proper lengths andproper lifetimes of tachyons areimaginary parameters, just as theirproper mass. Because none of thequantities that have to be symbolizedby imaginary parameters are acces-sible to measurement, there can be noquarrel with their imaginary charac-ter.

The reinterpretation principle

A much more serious objection to thepossible existence of tachyons is thatfor certain observers these particles ap-pear to have negative energy. This isso because the red hyperbola in figure1 crosses over into the region ofE < 0. Because the explanation ofthis question provides the key to theresolution of other even more seriousobjections, let us examine this problemin some detail.

Suppose that we are at rest in asystem S. We are observing a tach-yon that has been emitted by a sourceand absorbed a while (and manymiles) later by a sink. While intransit the tachyon had a velocity vrelative to us. This same particle hasa different velocity, say u, relative toanother system S' that travels with re-spect to S with a velocity w.

It can be shown that the point rep-resenting the state of the tachyon inthe system S' will pass into the nega-tive energy region when the productvw becomes greater than c2. But un-der these very circumstances anotherremarkable thing happens. To theobserver in the S' system the "nega-tive-energy" particle will appear tohave been absorbed first and emittedlater. This can be seen at right infigure 2.

In this graph two sets of reference

axes are drawn, the black unprimedaxes of the system S and the grayprimed axes of the system Sr. Eachof the primed axes makes an angle <p— tan~1(u?/c) with the correspond-ing unprimed axis, as required by thetheory of relativity. Points O and Prepresent the coordinates (positionand time) at which the tachyon isemitted and absorbed. In the un-primed system the event at P takesplace after the event at O. In theprimed system the sequence is re-versed.

Such sequence reversal will takeplace every time the point P falls be-low the x' axis. We were intriguedwhen we noticed that this occurs un-der exactly the same conditions underwhich the energy turns negative, thatis, when the product vw becomesgreater than c2. As we shall see inthe following, it is the interpretationof this coincidence of sign reversalsthat provides the key to a consistenttheory of faster-than-light particles.

It is precisely by putting togetherthe two quizzical concepts of "nega-tive-energy" particles traveling "back-ward in time" that the resolution ofthe difficulty is found. We call itthe reinterpretation principle. A"negative-energy" particle that hasbeen absorbed first and emitted lateris nothing else but a positive-energyparticle emitted first and absorbedlater, a perfectly normal situation.(In a sense, this line of reasoning is"antiparallei" to the Stiickelberg-Feyn-man interpretation of positrons asnegative-energy electrons runningbackward in time.0)

The two observers will disagree onthe direction in which the particletravels but there is nothing disquiet-ing about this disagreement. Classi-cal relativity is full of examples ofsimilar disagreements, notably with re-spect to simultaneity. The chief re-quirement of the theory is for thelaws of physics to remain the samefrom system to system. There is norequirement whatever that eventsshould have the same appearance.

Now you see it, now you dont

Keep in mind that events need nothave the same appearance, particu-larly when collisions of tachyons withnormal particles (tardyons) are en-visaged. Depending on the referenceframe from which the collision isviewed, the process may look com-pletely dissimilar although in eachsystem the collision is governed by

PHYSICS TODAY • MAY 1969 • 47

the same basic laws of conservationof energy and momentum.

To one observer the tachyon mayappear to approach the target particle,collide with it and then bounce off it.To another observer this same pro-cess may appear as two tachyons ap-proaching the target, colliding with itsimultaneously and vanishing by trans-mitting all of their energy and momen-tum to the target particle.

Such a difference in appearance ofthe same event will result every timethe collision leads to a "negative-en-ergy" tachyon. Because such a tach-yon travels "backward in time," thecollision looks like a simultaneous fu-sion of two tachyons with the targetparticle, rather than like a "billiard-ball" collision where we have two par-ticles before and two after the collision.In other words, the number of tachy-ons before and after a collision neednot be the same. They may appear ordisappear, depending on the relativemotion of the observer.

Infinite-energy source?

What we have just said can be usedto refute another major objection tothe existence of tachyons. That thehyperbola describing the energy-mo-mentum invariance of tachyons (redcurve in figure 1) passes into the re-gion of negative energy has been heldto imply the existence of an infinitesource of energy. The argument goeslike this. Two tachyons of zero en-ergy collide simultaneously with a tar-get particle having energy E{). In thecollision process each tachyon acquiresan amount of negative energy —Er.For energy to be conserved, the targetparticle must now have an energy Eo

+ 2Ej. Tachyons are thus viewed asvehicles for "pumping" energy into anordinary particle by themselves ac-quiring negative energy. Becausethere is no limit to the amount ofnegative energy they could assume,there should then be no limit, accord-ing to the argument, to the positiveamount of energy that could be ob-tained from them.

This objection is fallacious becauseit ignores the sequence reversal thatinvariably accompanies the appear-ance of negative energies. In theabove collision we would not see twotachyons approaching the target andthen two tachyons receding. The two"receding" tachyons would also ap-pear as approaching, each carryingpositive energy +EV By taking se-quence reversal into account, the ob-

jectionable source of infinite amountsof energy disappears.

No communication with the past

The most subtle of all objectionsagainst faster-than-light particles wasraised as early as 1917 by Richard C.Tolman in his book The Theory ofRelativity of Motion.7 This objectionis quite basic.

In a simplified form the argumentgoes as follows. Suppose an observerA in a reference system S has a sourceof infinitely fast particles. At a timet = t0 he sends out a burst n of theseparticles to an observer B in a systemS' that is receding from S at a uni-form rate to. Let us assume that thereception of the signal in S' triggersa similar burst TL> of particles that nowtravel with infinite velocity relative toS'. Signal T^ will arrive in S at atime t1 that is earlier than t0.

This sequence of events is a causalloop: The effect (reception of signalT-2 by A) takes place before the cause(sending of signal n by A). It can beshown that the same argument appliesnot only to infinite velocities, but toall velocities greater than c2/w, wherew is the relative velocity of the twoobservers.

It must have been this argumentthat provided inspiration for ReginaldBuller's limerick:

There was a young lady namedBright

Whose speed was far faster thanlightShe went out one dayIn a relative way

And returned the previous night.

This objection, although in principlequite profound, still suffers from thesame defect as the preceding one. Itfails to take into account the relativesequence reversal accompanying prop-agation of ultraluminal signals whosevelocity is greater than c2/w>. Whenthis effect is introduced into the argu-ment, the causal circle breaks down.There is no more back and forth ex-change of signals. In each case theobserver believes that he is sendingout both signals. Similar reinterpre-tation invalidates the difficulties en-countered by Helmut Schmidt8 in hisinvestigation of the causality objectionagainst waves propagating at super-luminal velocities.

Nonetheless, there are more so-phisticated versions of the causalityargument that can be resolved onlyby a close scrutiny of the hypotheti-

cal process of tachyon emission andabsorption. Such considerations leadfrom macrocausality to microcausalitywhere thermodynamic fluctuations andquantum-mechanical uncertaintiesplay a decisive role. The relevanceof the second law of thermodynamicsto causality arguments regarding thepossibility of existence of superluminalsignals was analyzed by Terletskii9 in1960. Because he ignored the possi-bility of reinterpreting "antisignals"propagating backward in time as posi-tive-energy signals propagating for-ward in time, he concluded that anysuperluminal signals could only ariseto the extent of stochastic entropyfluctuations. This same flaw mars hisotherwise most valuable monographParadoxes in the Theory of Relativity.2

Summing up, the causality objec-tions against the logical consistency ofthe tachyon hypothesis are at best notcompelling, and possibly altogetherunfounded. Still, much room forthought remains.

Accelerate upon loss of energy

Soon after we had shown that the ar-guments against the possibility of ex-istence of superluminal particles areeither flawed or inconclusive, severalenterprising physicists initiated asearch for these particles.10 To get aninsight into some of the avenues ofapproach open to experimenters, andto be able to appreciate the difficultiesthey are facing, let us take a closerlook at some of the surmised proper-ties of tachyons.

Let us start by plotting the energyE and the momentum p of a particleas a function of its velocity. In thisway we shall be able to get a com-parative view of properties of each ofthe three classes of particles, the sub-luminal, the luminal and the hypo-thetical superluminal particles. In theenergy graph of figure 3 we first notethe black U-shaped curve bounded onboth sides by the pink lines at v= c. This black curve corresponds totardyons (normal particles) whose en-ergy tends to infinity as they approachthe velocity of light. Because theseparticles are also observable at rest,their proper mass is given by a realnumber. These are the particles weare all made of.

The two vertical lines represent theluxons (photons and neutrinos) withzero proper mass whose velocity infree space is confined to the value ofc. The domain of the tachyons be-gins beyond the pink lines. Variation

48 • MAY 1969 • PHYSICS TODAY

Luxons

- 3 - 2 -1

RELATIVISTIC ENERGY VS. VELOCITY. Tardyons have energy even at rest.Their energy is seen to increase indefinitely as the limiting velocity of c is ap-proached. The vertical pink lines, which represent luxons, tell us that no matterwhat energy one of these particles may have, its speed will be c. The red curvereveals the singular behavior of the tachyons. Upon loss of energy they wouldaccelerate; when energy is imparted to them they would slow down. To retardthem to their lowest possible speed, which is c, an infinite amount of energy wouldhave to be provided. —FIG. 3

of energy with velocity is describedby the red curve. Inspection of thiscurve leads to some rather uncommonconclusions. Because the red curverises monotonically as the velocity ofc is approached, we are forced toconclude that energy must be sup-plied to a tachyon, rather than takenaway from it, if it is to be sloweddown. If it is to be decelerated toits lowest possible velocity, which isthe velocity of light, an infiniteamount of energy would have to beimparted to it. Conversely, upon lossof energy a tachyon will accelerate.This is a corroboration of the resultdeduced by Arnold Sommerfeld4

around the turn of the century. This

conclusion, which seemed so absurdat that time, is seen here to be anatural consequence of the relativistictheory.

''Transcendent" tachyons

It was also Sommerfeld who, at thatearly stage, had inferred from Max-well's electromagnetic theory thatelectrically charged faster-than-lightparticles would spontaneously radiateelectromagnetic waves.4 This meansthat soon after their emission chargedtachyons are likely to dissipate all oftheir energy. In so doing they wouldattain in the limit the state of infinitevelocity. Because the red curve goesto zero as the velocity v approaches

infinity, we are led to another re-markable result: Infinitely fasttachyons carry no energy.

To better comprehend the featuresof this "transcendent" state, let uscompare the energy-velocity curve offigure 3 with the corresponding curvein the momentum-velocity graph offigure 4. As the velocity increases, themomentum tends to a finite nonzerovalue. This can also be seen in figure1 where the point corresponding tothe state of zero energy falls on theapex of the red hyperbola. In a sense,we have here an analog to the staterepresented by the apex of the blackhyperbola, which denotes the stateof rest (zero momentum) of a normalparticle. In a closed universe thisanalogy goes even further. The in-finitely fast tachyon is present every-where on the supercircle of its trajec-tory, which corresponds to the quan-tum-mechanical situation of a particleat rest, with the ensuing total un-certainty as to its position.

For all we know, the world may becrisscrossed by tachyons, just as it isswarming with the elusive neutrinos.If and how these meta-particles maybe detected no one can say; muchmore work regarding possible tachyoninteractions is needed if the search fortachyons is to become less fortuitous.

A transcendent (infinitely fast)tachyon should be particularly diffi-cult to detect, as it has no energyor momentum to spare. It may ap-pear that it could signal its previousexistence by giving all of its mo-mentum to a stationary particle whoserecoil would then be observed. Butthis process is forbidden because torecoil the absorbing particle needs notonly momentum but also energy.

The requisite of energy and mo-mentum conservation can be met onlyunder very restricting kinematic con-ditions when the absorbing particle isitself in motion as shown in figure 5.In this case the absorption of asingle neutral zero-energy tachyonwould look very much like an elasticcollision of the absorbing particle withan infinitely massive object: The en-ergy of the particle would remain un-altered, but the direction of its mo-tion would abruptly change. Shouldthe tachyon have been carrying anelectric charge, its absorption would beaccompanied moreover by a corre-sponding change in the charge of theabsorbing particle.

Notice that in the above process thetransfer of momentum occurs at in-

PHYSICS TODAY • MAY 1969 • 49

finite speed. In classical physics suchinstantaneous "action-at-distance" hasa simple model: a rigid body! Hencetachyons, in a sense, reintroduoe theconcept of rigid bodies into relativisticquantum mechanics.

First experimental search

It appears that a natural place tolook for tachyons is the immediatevicinity of possible sources of theseparticles. Radioactive beta decay isone such source. Although a satisfac-tory theory exists that accounts formost of the details of this process,there is still room for conjecturing that,to ever so small a degree, tachyonsmay be participating in it, along withbeta rays (nuclear electrons) andneutrinos. Prompted by our 1962paper,1 Tors ten Alvager and PeterErman, physicists at the Nobel In-stitute in Stockholm, carried out in theyears 1963-65 an extensive searchfor charged meta-particles using astrong thulium-170 beta source. Afirst object of their search was a par-ticle whose meta-mass would equal themass of a normal electron. To sepa-rate such "meta-electrons," should anybe emitted, from the normal electrons,they deflected the charged radiation bymeans of a magnet (a double-focus-ing beta spectrometer) onto a semi-conductor counter. The magnetselected particles of a definite mo-mentum, and the counter recordedtheir energy. At equal momentum,electrons and meta-electrons are ex-pected to carry different amounts ofenergy. It was hoped that this dif-ference could be used to detect thepresence, if any, of the meta-elec-trons. The results of this investiga-tion were negative.11

Cerenkov radiation

Another experiment, again broachedby Alvager, and carried out togetherwith Michael Kreisler at the Prince-ton-Pennsylvania Accelerator, followsthe approach that we suggested inour first paper on meta-relativity.1

Charged meta-particles are to be de-tected by recording the electromag-netic waves they are expected to emit.This Cerenkov radiation is analogousto the bow wave of a ship, or to theshock wave of a supersonic airplane.The direction of propagation of thisradiation relative to the particle tra-jectory is uniquely determined by theparticle velocity. Moreover, in avacuum only tachyons can travel fasterthan light; hence they emit Cerenkov

50 • MAY 1969 • PHYSICS TODAY

radiation. Thus an unambiguousidentification of electrically chargedsuperluminal particles should be pos-sible.

An essential improvement on ouroriginal idea was the introduction byAlvager of a strong electric field atthe presumed tachyon source. Thisfield is directed so as to supply emittedtachyons with energy and thereby en-able them to radiate over a longerportion of their trajectory. A firstseries of measurements, in which theattempt was made to producetachyons in lead by bombarding itwith gamma rays from a 5-milli-curie cesium-134 source, was re-cently concluded.12 Again no traceof tachyons was found. Further ex-periments along these lines are underway with improved apparatus.

Because no Class-II particles (pho-tons and neutrinos) carry a charge, itis quite possible that Class III par-ticles are also all electrically neutral.A neutral particle can be detected onlyindirectly, and the difficulties whichexperimenters face in trying to espyneutral meta-particles are formidable.Notwithstanding, a project whose aimis to ferret out neutral tachyons isbeing carried out by Alvager at In-diana State University. His setup isingeniously simple. It parallels theearly work by Irene Joliot-Curie andFrederic Joliot on the 'Very pene-trating" gamma rays that later turnedout to be neutrons. Working belowthe threshold for neutron production,Alvager is looking for a penetratingradiation component other than neu-trons.

Another type of experiment, onewhich does not rely on the interactionof tachyons with matter for their de-tection, may soon be undertaken atArgonne and Brookhaven by a groupof seasoned high-energy experimentersusing their powerful missing-massspectrometer.13 High-energy protonsstriking a deuteron target produce(among other things) He3 plussomething else. If that "somethingelse" is a tardyon or a luxon, He3 re-coils are restricted to a limited forwardrange of angles. Should He3 nucleibe detected in the spectrometer whenit is set at a larger angle, the missingreaction product can only be atachyon.

Quantization

The hypothesis of faster-than-lightparticles must also be approached onanother plane. It is important to

make sure that meta-particles are con-sistent not only with relativity theorybut also with quantum theory. Inother words, we must ask the ques-tion if there is room for tachyons inthe existing theories of the subatomicworld.

None of these theories carry a re-striction on particle velocity as such.In fact they provide for the possi-bility of propagation of microscopicdisturbances with arbitrarily highspeeds, or admit solutions involvingspacelike quanta. If these solutionsare rejected, this is done solely ongrounds of their alleged incompati-bility with relativity theory and thecausality principle. We hope to haveshown that these reasons are not com-pelling and hence that it is reasonableto try to devise a quantum theory ofsuperluminal particles. The appar-ent paradoxes of faster-than-light par-ticles will reappear in any such at-tempt and they must be satisfactorilyresolved. Such a quantum theorywould be expected to parallel ourclassical theory of tachyons.

The first one to have attempted towork out the details of a relativisticquantum theory of faster-than-lightparticles was Sho Tanaka at KyotoUniversity.14 Because he was notaware of the possibility of reinterpret-ing negative-energy particles as posi-tive-energy particles propagatingforward in time, he conceived offaster-than-light particles as virtualobjects only. Moreover, the energyand momentum of the tachyon fieldin his theory do not transform likecomponents of a four-vector, whichmakes his interesting work unaccept-able as a relativistic theory oftachyons.

More recently, using our reinterpre-tation of negative energies, Feinberghas modified and imaginatively inno-vated Tanaka's work.3 Nonetheless,some questions as to the invarianceof the energy-momentum four-vectorremain. Michael Arons of the CityUniversity of New York and one ofus (Sudarshan) have proposed a novelapproach that appears to obviatethese difficulties.15 This approach hasbeen used to construct a quantum fieldtheory of interacting tachyons.16

Still, much exploratory theoreticalwork remains to be done, particularlyon the interactions in which tachyonsmay participate. The question of theexistence of faster-than-light particlesis closely linked with the possibilitythat we may find new, as yet undis-

RELATIVISTIC MOMENTUM VS. VELOCITY. The momentum of a tardyontends to infinity as particle velocity approaches c. Luxons may have any momentumvalue, but their velocity will always remain at c. As the velocity of a tachyon in-creases indefinitely, its momentum approaches the limiting value of p — m*c. —FIG. 4

Momentumof absorbed tachyon

TACHYON ABSORPTION. To satisfy energy and momentum conservation when atranscendent tachyon is absorbed by an ordinary particle, the vector diagram represent-ing the process must be isosceles (because the transcendent tachyon can only changethe direction, and not the magnitude, of the momentum of the absorbing particle).Such an absorption would look very much like an elastic collision between the absorb-ing particle and an infinitely massive object. —FIG. 5

covered, interactions characteristic ofsuperluminal particles.

Although we think that super-luminal particles do exist, the onlyunequivocal way to ascertain that theydo is by actually detecting them.So far experimenters must grope verymuch in the dark, as nothing isknown about the interactions in whichtachyons are likely to participate.Further work on the part oftheoreticians may help to make thetask of the experimenters more man-ageable.

We find the question of existence ofthe hypothetical superluminal par-ticles to be a challenging new fron-tier. Regardless of the outcome ofthe search for tachyons, investigationsin this field must invariably lead to adeeper understanding of physics. Iftachyons exist, they ought to be found.If they do not exist, we ought to beable to say why not.

References

1. O. M. P. Bilaniuk, V. K. Deshpande,E. C. G. Sudarshan, Am J. Phys.30,718 (1962).

2. Ya. P. Terletskii, Paradoksy teoriiotnositelnosti, Nauka Press, Moscow,1966; English translation: Para-doxes in the Theory of Relativity,Plenum Press, New York (1968).

3. G. Feinberg, Phys. Rev. 159, 1089(1967).

4. A. Sommerfeld, K. Akad. Wet.Amsterdam Proc. 8, 346 (1904).

5. H. H. Kolm, PHYSICS TODAY 20,no. 10,69 (1967).

6. E. C. G. Stuckelberg, Helv. Phys.Acta 14, 588 (1941) and 15, 23(1942). R. P. Feynman, Phys. Rev.74, 939 (1948) and 76, 749 (1949).

7. R. C. Tolman, The Theory of Rela-tivity of Motion, Univ. of CaliforniaPress, Berkeley (1917), pp. 54-55.

8. H. Schmidt, Zeits. Phys. 151, 365(1958) and 151,408 (1958).

9. Ya. P. Terletskii, Doklady Akad.Nauk SSSR 133, 329 (1960); Eng-lish translation: Soviet Physics—Dok-lady 5, 784 (1961).

10. T. Alvager, J. Blomqvist, P. Erman,1963 Annual Report of the NobelResearch Institute, Stockholm (un-published ).

11. T. Alvager, P. Erman, 1.965 AnnualReport of the Nobel Research Insti-tute, Stockholm (unpublished).

12. T. Alvager, M. N. Kreisler, Phys.Rev. 171, 1357 (1968).

13. B. Maglic and R. Schluter, privatecommunications.

14. S. Tanaka, Progr. Theor. Phys.(Japan) 24, 177 (1960).

15. M. E. Arons, E. C. G. Sudarshan,Phys. Rev. 173, 1622 (1968).

16. J. Dhar, E. C. G. Sudarshan, Phys.Rev. 174, 1808 (1968). •

PHYSICS TODAY • MAY 1969 • 51