es. Weopriatevelocity,the boost

Physics Letters A 323 (2004) 345–350

www.elsevier.com/locate/pla

Velocity of particles in Doubly Special Relativity

M. Daszkiewicz, K. Imilkowska, J. Kowalski-Glikman∗,1

Institute for Theoretical Physics, University of Wroclaw, PL Maxa Borna 9, Pl-50-204 Wroclaw, Poland

Received 24 June 2003; received in revised form 15 January 2004; accepted 12 February 2004

Communicated by P.R. Holland

Abstract

Doubly Special Relativity (DSR) is a class of theories of relativistic motion with two observer-independent scalinvestigate the velocity of particles in DSR, defining velocity as the Poisson bracket of position with the apprHamiltonian, taking care of the non-trivial structure of the DSR phase space. We find the general expression for four-and we show further that the three-velocity of massless particles equals 1 for all DSR theories. The relation betweenparameter and velocity is also clarified. 2004 Elsevier B.V. All rights reserved.

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1. Introduction

There is a rapidly growing interest in a reseaprogramme, which might be called “Quantum SpecRelativity”, i.e., a class of theories of kinematithat differ in their predictions from that of SpeciRelativity in the regime of ultra-high energies. Thepossible differences might be understood as tracestill unknown Quantum Gravity theory present evin the regime of negligible gravitational field. Itfeasible that predictions of such generalizationsSpecial Relativity, like dependence of the speedmassless particles on momentum they carry, mighexperimentally tested in the near future experimen

* Corresponding author.E-mail addresses: marcin@ift.uni.wroc.pl (M. Daszkiewicz),

kaim@ift.uni.wroc.pl (K. Imilkowska), jurekk@ift.uni.wroc.pl(J. Kowalski-Glikman).

1 Partially supported by the KBN grant 5PO3B05620.

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.02.046

Two alternative classes of “Quantum Special Rativity” have been recently attracting attention.one, one argues that Lorentz invariance is brokehigh energies due to string [1], loop-quantum grity [2] effects, or due to existence of the cosmoloical preferred frame [3]. In the second, proposedAmelino-Camelia [4,5], called Doubly Special Reltivity (DSR), one assumes that the ten-dimensionagebra of physical symmetries (rotations, boosts,translations) is still present, but is deformed in a wso as to possess two observer-independent scalethis Letter we will concentrate on this second possiity only.

The construction of the first specific model of DScalled DSR1, presented in [6] and [7] borrowed afrom the earlier investigations in quantum Hopf defmations of Poincaré algebra, the so-calledκ-Poincaréalgebra (see, e.g., [8,9]). It turned out however, tthere exist another DSR models: the first one, canowadays DSR2 was formulated by Magueijo a

.

346 M. Daszkiewicz et al. / Physics Letters A 323 (2004) 345–350

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Smolin [10], and it was soon realized that there exiwhole class of DSR theories [11,12].

In the majority of the (especially early) literatudevoted to DSR, Doubly Special Relativity is definonly as a theory based on energy–momentum seThis isnot what we understand by DSR in this LetteIn our view, a DSR theory is defined by a setcommutators describing the whole of the phase spof the system. There are two systematic and equivaways of deriving such a set in any particular mode2

The first is based on the Hopf algebra structure thatbe built on the energy–momentum algebra, and theof the so-called Heisenberg double construction [114]. Equivalently, one can get the same phase spacmaking use of the geometric, de Sitter picture of DS[15,16].

Doubly Special Relativity is a theory of partickinematics, and the proper understanding of the ccept of velocity in such a theory is of course an imptant step towards full understanding of it. The firsttempt to analyze the notion of velocity has been malready in the early days of theκ-Poincaré theory in[17], from the DSR perspective this problem has beinvestigated, among others, in [18–23].

The starting point of our investigations reporthere consists of two major assumptions: that velois defined as the Poisson bracket of position wdeformed relativistic Hamiltonian (see also [17]), athat to compute this bracket one must take into accothe non-trivial phase space structure of DSR theorWe show that in all DSR theories four velocititransform as standard Lorentz vectors, and thatthree velocities of massless particles equal one. Tgeneral statement will be justified in Section 3; befturning to that, we present our method on the specexample of DSR1.

2. Particle velocity in DSR1

Before starting our investigations let us state cleawhat our assumptions are. We define four velocas the Poisson bracket, based on a DSR phase s

2 Some other methods of deriving consistent phase spaceexist, of course. We are not aware, however, of any non-trialternative to the procedure we make use of.

e

structure, of positions with an appropriate Hamtonian. Let us explain why we decided to use Poisbrackets instead of commutators, usually employethe DSR literature. The reason is quite simple:use of commutators implies that the relevant objeare operators acting on some Hilbert space. Howeworking with the formal commutator algebra is equalent to phrasing results in terms of Poisson brackwith an appropriate symplectic structure dictatedthe commutator algebra. Since we will make usethe expression for three-velocity as a function of fovelocitiesvi = xi/x0 whose meaning forx0 being anoperator is not clear, it is just safer to work with Poson brackets.3 For this reason we will confirm our discussion to classical particles and not quantum mawaves. In our investigations in this section we will alpostulate a particular form of relativistic HamiltoniaWe will argue that such form is natural in the next stion devoted to general properties shared by all Dtheories.

In all the DSR theories (contrary to the statemethat can be sometimes found in the literature),Lorentz algebra of rotationsMi and boostsNi isexactly the same as in Special Relativity:4

[Mi,Mj ] = εijkMk, [Mi,Nj ] = εijkNk,

(1)[Ni,Nj ] = −εijkMk.

In the DSR1 theory the momenta transform unaction of boosts as follows [9,17]:

(2)

[Ni,pj ] = δij

(κ

2

(1− e−2p0/κ

) + 1

2κ�p 2

)− 1

κpipj ,

and

(3)[Ni,p0] = pi.

The first Casimir of this theory equals

(4)C =(

2κ sinh

(p0

2κ

))2

− �p 2ep0/κ = m2.

3 Note, however, that even though in DSR the positionsxµ

do not commute (see (5) below) it is well known that thdifferentials are commuting [24]. Thus the velocity formula abois, presumably, well defined even as an operator expression.

4 Let us stress again that[∗,∗] denotes the Poisson bracket anot the commutator.

M. Daszkiewicz et al. / Physics Letters A 323 (2004) 345–350 347

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Let us now turn to description of the phase spaceDSR1. As for all DSR theories we have

(5)[x0, xi] = − 1

κxi, [xi, xj ] = 0.

As shown in [16] there are infinitely many phaspaces compatible with the DSR1 boost transfortions (3), (4) and the brackets (5). Here we will dscribe only the two most simple ones for whosecross brackets take the following form:

[p0, x0] = −1, [pi, x0] = 1

κpi,

[pi, xj ] = δij e−2p0/κ − 1

κ2

( �p 2δij − 2pipj

),

(6)[p0, xi] = − 2

κpi

or, following [9,17],

[p0, x0] = 1, [pi, x0] = − 1

κpi,

(7)[pi, xj ] = −δij , [p0, xi] = 0.

We choose the Hamiltonian to be

(8)H = κ2 coshp0

κ− �p 2

2ep0/κ .

This Hamiltonian has the largeκ limit

H ∼ κ2 + 1

2

(p2

0 − �p 2) + · · · ,i.e., it reduces in this limit to the standard Hamiltoniof relativistic particle (up to the irrelevant constashift). Let us now define the four-velocities in thstandard way as the bracket

(9)u0 ≡ x = [x0,H], ui ≡ xi = [xi,H].In calculating the brackets in (9) one should carefutake care of the non-trivial phase space structureDSR1 (6) or (7) (in accordance with the schepresented in [17])

[x0,H] ≡ ∂H∂p0

[x0,p0] + ∂H∂pi

[x0,pi ],

(10)[xk,H] ≡ ∂H∂p0

[xk,p0] + ∂H∂pi

[xk,pi].

The second set of Hamilton equations is quite simp

(11)pµ = 0,

because in all DSR theories the Hamiltonianpends on momenta only, and momenta have vanisbracket among themselves. Notice that this propguarantees that in DSR free motion is uniform. Oshould note at this point that for this reason it seethat the so-called “twisted phase spaces”, investigrecently in [25], which would lead to non-uniform motion of free particles, are likely not to be physical.

It is easy to derive the expression for four velocfollowing our general prescription (10). We find

u0 = κ sinhp0

κ+ �p 2

2κep0/κ ,

(12)ui = piep0/κ,

in the case of phase space (6) and

u0 = −κ sinhp0

κ− �p 2

2κep0/κ ,

(13)ui = −piep0/κ ,

for the phase space (7).5 We see that the four velocitiediffer by the overall sign, but that in both casesexpression for three-velocity is exactly the samereads

(14)vi = ui

u0= pi

(κ

2

(1− e−2p0/κ

) + �p 2

2κ

)−1

.

Using the expansion of the mass-shell conditionmassless particles (cf. Eq. (4))

(15)0 = κ2(1− e−p0/κ)2 − �p 2,

we find

(16)v(m=0)i = pi

κ

(1− e−p0/κ

)−1.

Using (15) again we find that the speed of massparticles

(17)v(m=0) = ∣∣v(m=0)i

∣∣ = 1,

as in Special Relativity. Of course, there are deviatifrom Special Relativistic results in the case of massparticles. Indeed, in the massive case we have withhelp of (4)

(18)v(m)i = pi

[κ(1− e−p0/κ

) − m2e−p0/κ

2κ

]−1

.

5 In this particular case our result agrees with the one reporte[19,22].

348 M. Daszkiewicz et al. / Physics Letters A 323 (2004) 345–350

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Note that the speed-of-light of a massive particle, ithe maximal speed of massive particle carrying infinenergy equals again

(19)v(∞) = 1,

becausep0 = ∞ corresponds to| �p | = κ , as it followssimply from (4).

It is worth noticing that if one derives the transfomation rules for four velocities (12), (13) under actiof boosts, using expressions (2), (3), one finds that ttransform as standard Lorentz four vectors of SpeRelativity. This is not an accidental property of DSRin fact it holds for all DSR theories. Let us turn therfore to general formulation and properties of such tories.

3. Velocity in DSR theories—generalities

The results derived in the previous section turnto be valid not only for DSR1, but in fact for aDSR theories. In order to see that let us recallgeometric, de Sitter space formulation of the Dtheories presented in [15]. The starting point herea five-dimensional manifold of Minkowski signature

(20)ds2 = gAB dηA dηB = −dη20 + dη2

i + dη24.

In this space the four-dimensional de Sitter spacembedded by

(21)−η20 + η2

1 + η22 + η2

3 + η24 = κ2.

We split the ten-dimensional algebra of isometriesde Sitter space (21) into the six-dimensional Lorealgebra (1) and the remainder, which we identify wpositionsxµ satisfying (5). The remaining brackets a

[Mi,ηj ] = εijkηk, [Mi,η0] = 0,

(22)[Mi,η4] = 0,

[Ni,ηj ] = δij η0, [Ni,η0] = ηi,

(23)[Ni,η4] = 0,

[x0, η4] = 1

κη0, [x0, η0] = 1

κη4,

(24)[x0, ηi ] = 0,

[xi, η4] = 1

κηi, [xi, η0] = 1

κηi,

(25)[xi, ηj ] = 1δij (η0 − η4).

κ

The main result of [15], based on the previousvestigations reported in [11] and [12] is that any DStheory can be represented as a particular coordisystem on the de Sitter space (21), i.e., the mapfrom the space of physical momentapµ to ηA satisfy-ing (21). For example, in the case of the DSR1 theowith phase spaces (6), (7) this mapping takes the f

η0 = ±(

κ sinhp0

κ+ �p 2

2κep0/κ

),

ηi = ±(pie

p0/κ),

(26)η4 = κ coshp0

κ− �p 2

2κep0/κ .

Let us observe now that in view of Eqs. (21) and (2κη4 is the most natural candidate for relativisHamiltonian. Indeed, it is by construction Lorentinvariant, and reduces to the standard relativistic pacle Hamiltonian in the largeκ limit. Indeed, using thefact that forpµ small compared toκ , in any DSR the-ory ηµ ∼ pµ + O(1/κ) we have

κη4 = κ2

√1+ p2

0 − �p 2

κ2

(27)∼ κ2 + 1

2

(p2

0 − �p 2) + O

(1

κ2

).

Then it follows from Eqs. (24), (25) thatηµ =[xµ, κη4] can be identified with four velocitiesuµ.The Lorentz transformations of four velocities athen given by Eq. (23) and are with those of SpeRelativity. Moreover, since

(28)u20 − �u2 ≡ C = m2

by the standard argument the three velocity equvi = ui/u0 and the speed of massless particle equaLet us stress here once again that this result is Dmodel-independent, though, of course, the relabetween three-velocity of massive particles and enethey carry depends on a particular DSR modeluses.

Thus if in the time-of-flight experiment (see, e.[26] for recent review), will measure time gap betwearrivals of photons emitted from a distant sourand carrying different energies, this would falsify tconstruction of the DSR models presented above.

Note finally that as the result of Eq. (28) and tdefinition of three velocity asvi = ui/u0, the rule

M. Daszkiewicz et al. / Physics Letters A 323 (2004) 345–350 349

als,ntlyngd

eds istureticleons, for

orthets

the

ketsting

enta

bydionven

al-re-ormn-

n-es.

i.e.,alideirvestheayp-ack-tter

aneaveys-

entbetoistoofwillier.

le

2,

gr-

p-

r-

p-

s.

t.

p-

of adding the latter in any DSR theory is identicwith the standard rule of Special Relativity. It ihowever, an open question if this can be consisteextended to the rule of addition of momenta, followithe considerations of Lukierski and Nowicki [27] anJudes and Visser [28].

4. Comments and concluding remarks

The main result of our investigations reporthere is that if the phase space of DSR theorieconstructed in the way suggested by geometric picof de Sitter space, the speed of massless parequals 1. There is a number of simple observatione can make. First, in the scheme adopted hereany DSR theory the velocity is just

(29)vi = ui

u0= ηi(p)

η0(p),

which provides the velocity–momentum relation fan arbitrary DSR theory, since in any DSR theoryvariablesηµ are functions of momenta, given by idefinition. It follows also that the boost parameterξ isrelated to velocity in exactly the same way as inSpecial Relativity:

(30)tanhξ = v.

This can be easily seen by realizing that the brac(23) are equivalent to the equations (for boost acin 3rd direction, say)

(31)dη3

dξ= η0,

dη0

dξ= η3,

from which (30) immediately follows. To obtain thcorresponding equation for dependence of momepµ on rapidity in a particular DSR theory, defineda particular functionsηA(pµ), one uses Eq. (31) anthe Leibnitz rule. But this does not change the relat(30) where the right-hand side can be taken a gifunction of momenta.

It is interesting to compare our result with the cculation of the group velocity of wave packets psented in [20,21]. In these papers the authors perftheir computations using essentially only the nocommutative structure ofκ-Minkowski space–time(5), the particular form of non-commutative differetial calculus, and a natural ordering of plane wav

Their result, that the group velocityv(g) = ∂p0/∂p

(where the derivative is taken on the mass-shell,assuming that Eq. (4) holds), should be therefore vfor all DSR theories. The clear disagreement of thresult with the one presented in this Letter deserfurther studies, since it seems to indicates that inframework of DSR the behavior of matter waves mdiffer from that of particles. Specifically, this discreancy means that either naively constructed wave pets could not represent point particles, so that the laare represented by a non-linear combination of plwaves, or that one of these two notions (linear wpacket and/or point particles) just does not make phically sense in the DSR theories.

To conclude, let us stress that the final judgmon possible dependence of velocity on energy willmade by near future experiments. We would likenote, however, that if it turns out that our resultcorrect, i.e., if in the DSR theories we still havedo (as in Special Relativity) with universal speedphysical signals carried by massless particles, thisconstitute an important information, making it easto formulate Doubly Special Relativity operationally

Acknowledgement

We would like to thank J. Lukierski for his valuabcomments on the draft of this Letter.

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