particles beyond the light barrier

7/30/2019 Particles beyond the light barrier

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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 SUDARSHANFO R 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 10 8 m / s e c . I t is generallyheld that this limitation is a directconsequence of th e special theory ofrelativity. Alb ert Ein stein himself hassaid so in his original paper on rela-tivity.Some time ag o 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. O ther phy s-icists, notably Yakov P. Terletskii2and 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! Eins tein's theory hasremedied this seemingly inadmissiblestate of affairs. I t predicted that themass of a particle would get infinitelylarge upon app roac hing the ve locityc Thus it became obvious that no

particle could be accelerated past th e"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 produc ed? 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 w ay 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. L et us

Olexa-Myron Bilaniuk, a native of theUkraine, attended the University ofLouvain, Belgium and got his PhD atthe University of Michiga n. 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 fluen tin seven languages, has a commer-cial pi lot '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 facu lty. 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 VAtheory of weak interactions.

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henceforth refer to the normal ones asClass I, and to photons and neutrinosas Class II. W herea s Einstein's theoryencompasses both of these classes, onlyfor Class I does it require c to be th eunattainable l imiting velocity. Fol-lowing this line of thought, we haveinquired whether Einstein's theorycould possibly accommodate ye t 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 furtherabou t such particles. But if their ex-istence would not lead to an y contra-dictions, one should be looking forthem.There is an unwrit ten 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 ofth e electronic charge, whose existenceis suggested by recondite symmetriesof elementary particles.In each of the above cases, th e pos-sible existence of th e new particle wasfirst suggested by th e logical exten-sion of the regularities or symmetriesgoverning th e known physical world.T he special theory of relativity pro-

A TARDYO N'S POCKET DICT IONARY TO THE WORLD OF TACHYO NSSubluminalslower than l ightSuperluminalfaster than lightUltraluminalfaster than c2/V ,where V is the relative velocityof two laboratory systemsTachyon or meta-particlea faster-than-l ight particleTardyonany slower-than-light par-ticleLuxonphoton or neutrinoMeta-relativity special-relativitytheory of tachyonsProper mass (m,,)In analogy toproper length an d proper time,this term should be used insteadof the term "re st " mass. The

term proper mass is appropriatefo r al l classes of particles, includ-in g 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 *) abs olute value ofthe proper mass of a tachyon(m,, = /m + )Reinterpretation principleinterpre-tation of negative-energy tachyonspropagating backward in t ime aspositive-energy tachyons propagat-ing forward in t ime . This rein-terpretation inval idates causali tyobjections to the possibility of ex-istence of faster-than-light signalsand permits construction of a con-sistent theory of tachyons.Transcendent tachyoninfinite-ve-locity tachyon having finite mo-mentum (p = m* c) an d zero en-ergy. Such a tachy on is every-where on the supercircle of itstrajectory. A transcende nt stateof a tachyon is analogous to thequantum-mechanical state ofrest, in which a definite value ofmomentum impl ies total uncer-tainty of posit ion.

vides for just such an extension. Wehave called it "meta-relativity."Geometrical representationConsider a particle having energy and momentum components p r , p]f andp z . According to specia l theory ofrelativity, the quantities must obey:O O O O O O O c\ i2 - 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-e rgy-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 graphica lly. Th e equation

describes a hyperbola in the coordi-nate system of ps and E (black curvein figure 1) . Ea ch point on the up-per branch of the black curve fur-nishes th e energy E and the momen-tu m 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 dprBecause this slope is everywheresmaller than c, w e 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, th at even ifwe cou ld sup ply a particle with an ar-bitrarily large am ou nt 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 mu ch the better, as a par-ticle with negative energy is not ob-servable.

A po int of pa rticu lar interest on theupper black curve is that for whichp r 0. This poin t corresponds to aframe in wh ich the particle is at rest.Even though its momentum vanisheswe se e that the energy of the particleremains at a finite value of E = m0C".W e call it the pro pe r en ergy, or restenergy, of the particle; m 0 is its propermass .

In the special case in which m 0 *s44 MAY 1969 PHYS ICS TODA Y


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locity dependence is contained in theexpression of the relativistic mass,

RELATIVISTIC-ENERGY (E ) VS. MO ME NTU M (p ,) dependence for three classesof particles. Slope of a curve at any given po int gives the veloc ity of the p articlewhose energy and mom entum 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 ta chyon s. The gray curve d epicts thenonrelativistic dependence where energy is simply E p2/2m. It has been shiftedup to show that for small velocities it coincides w ith 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. Ta ke apoint on the pink line in the upperright-hand quad rant. Th e coordinatesof this point give us the energy E andthe momentum px of a photon in thecorresponding reference frame. W h e nwe change to another frame the en-ergy and momentum of the photon willassume new values, as in the case ofnormal particle s. Bu t th e velocity will

not; the slope of the line is every-where c. This geometrical representa-tion shows clearly that there ca n beno vantage point with respect to whichth e photon would appear to be atrest .Now we are almost ready for ou rgeneralization. But before w e 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, th e ve-

m = m o [ l - ( t V c ) 2 ] - 1 /? (2)Here m 0 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 mean s 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 ene rgy. Th ese 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.)

Eq uation 1 has general validity; soit mu st also hold for prot ons . If weknow the energy of a photon, weca n compute its relativistic mass asm = E/c 2. But we have just seen thatthe relativistic mass is also given byequa tion 2. In the case of photons vis equal to c and we seem to be introuble; we are dividing by zero. Th eonly way out is to require that parti-cles traveling with the velocity of chave zero prope r mass. W e 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. Th eformer can never reach the velocity ofc, the latter can have nothing but c.Yet even though these two kinds ofparticle s are so drastica lly different,they are governed by the same rela-tivistic laws.

A hypothetical situationL et 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 massra 0.One could imagine that under suchcircumstances some nonconfonning

PHYSIC S TODAY MAY 1969 45


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E

\

/ /

REINT ERPR ETA TION OF NEG ATIV E EN ER GIE S. 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 -tim e diagram (rig ht) reveals however that insuch a system the seque nce of events involving this particle w ould 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 "neg ative-energy" particles that have bee n "absorbed first andemitted later" simply as positive-energy particles that have been emitted first andabsorbed latera 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-in g some as yet unknown process, ofa new kind of particle (Class II) hav-ing the v elocity 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 i t be? Our imaginaryphysicist would answ er: Th e situa-tion is simple, no lab can travel rela-tive to us with the velocity of c. All46 MAY 1969 PHYS ICS 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 we

are 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 massTo 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 num ber. But howcan it when a negat ive number is un-der the square-root sign? The onlyway m can be made real is by requir-ing that the proper mass m 0 b e imag-inary; so it is expressed as some real

number m times th e square root ofminus one,

m0 =Now the expression for relativisticmass may be rewrit ten as

m z z m * [ ( u / c ) 2 - 1 ] " 1 / 2As long as v remains greater than c,the new expression for m will remainreal as required. T he price we hadto pay for having E remain real wasto require the proper mass of the hy-po the tica l faster-tha n-light particle tobe im agin ary. Le t 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 im ag ina ry p rop er mass rightoff. Th ey sh ould no t be so hasty. Aswe have already seen, all observersare confined to subluminal velocities.Consequently there are no observers inw ho se fra me of referen ce a super-lum ina l p arti cle wo uld b e at rest.


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Proper mass of a super luminal parti-cle is not an obse rvab le phy sical qua n-tity; it is a parameter devoid of an yimmediate 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 -dim ens iona l ver-sion this invariance is expressed asE2 px2c2 m02c4. W e have seenthe graphical representation of this ex-pression both for subluminal particlesfor which m {)2 is a real posit ive num-ber, and for photons for which m 02is zero (figure 1) . How do our hypo-thetical superluminal particles, withtheir imaginary proper mass, fit intothe picture?Making the picture completeIt is a remarkable aspect of the gen-eralization that we not only find theconcept of an imaginary mass con-sistent with the energy-momentumequation an d its graphical representa-tion, but that we actually need it tomake the picture comp lete. W e knowfrom analytic geometry that the fam-ily of hyperbolas described by th eequation

x2 y2 constantincludes not only those curves forwhich the constant is positive, and th edegenerate case where the constant iszero, but also all those curves forwhich the constant is neg ative. Byadmitting the possibility of existenceof particles with imagina ry prop ermass, we provide physical meaning tothe red hyperb ola in figure 1. Thiscurve corresponds to our superluminalmeta-particles. Beca use their prop ermass m {) is given by im#, m 02 be- m^2, which is a real nega-tive num ber. Th e red hyperbola isthus a graphical representation of theenergy-momentum invariance of th ehypothetical superluminal particles,

2 _ p ^ 2 c 2 _ _ 7 ? 1 + 2 C 4Let us note that th e slope of this curveis everywhere g reater tha n c. Thisis another way of saying that if parti-cles with an imaginary proper mass

do exist, their velocity could neverbe less than c. L et us henceforth referto these particles as tachyons, a namesuggested in 1967 by Feinberg and de-rived from the Greek word raxta,me anin g swift. (In contrast, let usrefer to al l subluminal particles astardyons. So as not to leave lightlikeparticlesphotons and n e u t r i n o s -nameless, le t 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. T he 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 thequantit ies that have to be symbolizedby imaginary parameters are acces-sible to measurement, there can be noquarrel with their imaginary charac-ter.Th e reinterpretation principleA much more serious objection to th epossible 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. W e are observing a tach-yo n that has been emitted by a sourceand absorbed a while (and manymiles) later by a sink. W hile 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.

I t ca n be shown that the point rep-resenting the state of th e 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 th e 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 tw o sets of reference

axes are drawn, th e black unprimedaxes of the system S and the graypr imed axes of the system S r. Eachof th e primed axes makes an angle


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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-

tu m 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," th ecollision 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-on s before and after a collision neednot be th e same. The y may appear ordisappear, depending on the relativemotion of the observer.Infinite-energy source?W h a t we have just said can be usedto refute another major objection tothe existence of tachyon s. Th at thehyperbola describing th e 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. T he argument goeslike this. Tw o 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 E r.For energy to be conserved, the targetparticle must now have an energy Eo+ 2E j. 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 th eabove 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 pastT he most subtle of all objectionsagainst faster-than-light particles wasraised as early as 1917 by Richard C.Tolman in his book Th e 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 ou t a burst n of theseparticles to an observer B in a systemS' that is receding from S at a uni-form rate to . L et us assume that th ereception of th e signal in S' triggersa similar burst TL> of particles that nowtravel with infinite velocity relative toS' . Signal T^ will a rrive 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) . I t can beshown that the same argument appliesno t only to infinite velocities, but toall velocities greater than c2/w , wherew is the relative velocity of the tw oobservers.It must have been this argumentthat provided inspiration for ReginaldBuller's limerick:

There was a young lady namedBrightWhose speed was far faster thanlightShe went out on e da yIn a relative way

And returned the previous night.This objection, although in principle

quite 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>. W h e nthis 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 causali tyargument that can be resolved onlyby a close scrutiny of the hypothet i -

ca l process of tachyon emission andabsorption . Such considerations leadfrom macrocausality to microcausalitywhere thermodynamic fluctuations andquantum-mechanical uncertaintiesplay a decisive role. Th e 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"propagat ing 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 sam e flaw mars hisotherwise most valuable monographParadoxes in the Theory of Relativity.2Summing up, the causality objec-tions against the logical consistency ofthe tachyon hypothesis are at best notcompelling, and possibly altogetherun fou nd ed . Still, mu ch room forthought remains.Accelerate upon loss of energySoon after w e had shown that the ar-guments against the possibility of ex-istence of superluminal particles areeither flawed or inconclusive, severalent erp risin g physic ists 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.

L et 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 lum inal and the hypo-thetica l superlum inal 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 veloc ity of light. Because theseparticles are also observable at rest,their proper mass is given by a realnum ber. The se are the particles weare all ma de of.

The two vertical lines represent theluxons (photons and neutrinos) withzero pro pe r mass who se velocity infree space is confined to the value ofc. T he dom ain of the tachyons be-gins be yo nd th e pin k lines. Variation

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Luxons

- 3 - 2 - 1

RELATIVISTIC ENERGY VS. VELO CITY. 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 rep resent luxons, tell us that no m atterwhat energy one of these particles may have, its speed will be c. The red curvereveals the singular behavior of the tachyo ns. 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. F IG. 3

of energy with velocity is describedby the red curve. Inspection of thiscurve leads to some ra ther uncommonconclusions. Beca use the red curverises mono tonically as th e velocity ofc is approached, we are forced toconclude that energy must be sup-plied to a tachyon, ra ther 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 th e resultdeduced by Arnold Sommerfeld4around the turn of the century . This

conclusion, which seemed so absurdat that time, is seen here to be anatural consequence of the relativistictheory.''Transcendent" tachyonsI t 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 th e velocity v approaches

infinity, we are led to another re-markable res ult: Infinitely fasttachyons carry no energy.To bet te r comprehend the featuresof this " t ranscendent" state, le t uscompare th e energy-veloci ty curve offigure 3 with th e corresponding curvein the momentum-veloc i ty graph offigure 4. As th e velocity increases, th emomentum tends to a finite nonzero

value. This can also be seen in figure1 where the point corresponding tothe state of zero energy falls on th eapex of th e re d hyperbola. In a sense,we have here an analog to the staterepresented by th e apex of the blackhyperbola, which denotes th e stateof rest (zero momentum) of a normalparticle. In a closed universe thisanalogy goes even further. Th e 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 w e know, the world may becrisscrossed by tachyons, just as it isswarming with the elusive neutrinos.If and how these meta-particles m aybe 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-cul t 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 m o-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 restrict ing kinematic con-ditions when the absorbing particle isitself in motion as shown in figure 5.In this case the absorption of asingle neut ra l zero-energy tachyonwould look very much like an elasticcollision of the absorbing particle withan infinitely massive object: T he en -ergy of the particle would remain un-altered, bu t th e direction of its mo-tion would abruptly change. Shouldth e tachyon have been carrying anelectric charge, its absorption would beaccompanied moreover by a corre-sponding change in the charge of theabsorbing pa rt icle.Notice tha t in th e above process thetransfer of momentum occurs at in-PHYSICS TODAY MAY 1969 49


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finite speed. In classical physics suchinstantaneous "action-at-distance" hasa simple model: a r igid body! Hencetachyons, in a sense, reintroduoe theconcept of rigid bodies into relativisticquantum mechanics .First experimental searchI t appears that a natural place tolook for tachyons is the immediatevicinity of possible sources of theseparticles. Radioactive beta decay ison e such source. Althoug h 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, tachyonsm ay 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 norm al electron. To sepa-rate such "meta-electrons," should anybe emitted, from the normal electrons,they deflected the charged radiation bymeans of a magne t (a double-focus-in g beta spect rometer ) onto a semi-conductor counter . The mag netselected 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 negat ive .11

Cerenkov radiationAnother experiment, again broachedby Alvager, and carried out togetherwith Michael Kreisler at the Prince-ton-Pennsylvania Accelerator, followsthe approach that we suggested inou r first paper on meta-relativity.1Charged 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 l ight; hence they emit Cerenkov50 MAY 1969 PHYS ICS TODAY

radiation. Thus an unam biguousidentification of electrically chargedsuperluminal particles should be pos-sible.An essential improvement on ouroriginal idea was th e introduction byAlvager of a strong electric field atthe presume d 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 theat tempt was made to producetachyons in lead by bombarding i twith gamma rays from a 5-milli-curie cesium-134 source, was re-cently concluded.12 Again no traceof tachyons was found. Fu rther 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. Wo rking belowthe threshold for neutron production,Alvager is looking for a penetra t ingradiation 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 . 1 3 High-energy protonsstriking a deuteron target produce(among other th ings) He 3 plussomething else. If that "somethingelse" is a tardyon or a luxon, He 3 re -coils are restricted to a limited forwardrange of angles. Should H e 3 nucleibe detected in the spectrometer whenit is set at a larger angle, the missingreaction product can only be atachyon.QuantizationThe hypothesis of faster-than-lightparticles must also be approached onanother plane. I t is impo rtant to

make sure that meta-particles are con-sistent not only with relativity theorybut also with quan tum 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 qua nta. If these solutionsare rejected, this is done solely ongrounds of their alleged incompati-bility with relativity theory and thecausali ty principle. W e hope to haveshown that these reasons are not com-pelling and hence that it is reasonableto try to devise a qu an tum theory ofsupe rlum inal particles. The appar-ent paradoxes of faster-than-light par-ticles will reappear in any such at-t empt and they must be satisfactorilyresolved. Such a quantum theorywould be expected to parallel ourclassical theo ry of tachyons.

The first one to have attempted towork out the details of a relativisticquan tum theory of faster-than-lightparticles was Sho Tanaka at KyotoUniversity.1 4 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. Mo reover, 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 CityUn ivers ity of N ew York and one ofus (Su darsh an) h ave proposed a novelap pro ac h tha t appe ars 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 parti cipa te. Th e question of theexistence of faster-than-light particlesis closely linked with the possibilitythat we may find new, as yet undis-


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RELATIVISTIC MO MEN TUM VS. VELO CITY. 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 ABS OR PTION . To satisfy energy and momentum conservation when atranscendent tachyo n 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, interact ions characterist ic ofsuperluminal part icles.Although we think that super-luminal particles do exist, the onlyunequivocal way to ascertain that theydo is by actually de tec t ing them.So far experimenters must grope verymuch in the dark, as nothing isknown about the interactions in whichtachyons are likely to part icipate.Further work on the part oftheoret icians may help to make thetask of the experimenters more man-ageable .W e find the question of existence ofthe hypothet ical superluminal par-ticles to be a challenging new fron-tier. Reg ardless of the outco me ofth e search for tachyons, investigationsin this field must invariably lead to adeeper understan ding of physics. Iftachyons exist, they ought to be found.If they do not exist, we ought to beable to say why not.

References1. 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(19 42) . 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 PhysicsDok-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 (1 968 ).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 (19 68 ).

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particles beyond the light barrier

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