doubly special relativity theories as different bases of κ-poincaré algebra

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Physics Letters B 539 (2002) 126–132 www.elsevier.com/locate/npe Doubly special relativity theories as different bases of κ -Poincaré algebra J. Kowalski-Glikman 1 , S. Nowak Institute for Theoretical Physics, University of Wroclaw, Pl. Maxa Borna 9, Pl-50-204 Wroclaw, Poland Received 8 March 2002; received in revised form 28 May 2002; accepted 1 June 2002 Editor: P.V. Landshoff Abstract Doubly Special Relativity (DSR) theory is a theory with two observer-independent scales, of velocity and mass (or length). Such a theory has been proposed by Amelino-Camelia as a kinematic structure which may underline quantum theory of relativity. Recently another theory of this kind has been proposed by Magueijo and Smolin. In this Letter we show that both these theories can be understood as particular bases of the κ -Poincaré theory based on quantum (Hopf) algebra. This observation makes it possible to construct the space–time sector of Magueijo and Smolin DSR. We also show how this construction can be extended to the whole class of DSRs. It turns out that for all such theories the structure of space–time commutators is the same. This results lead us to the claim that physical predictions of properly defined DSR theory should be independent of the choice of basis. 2002 Elsevier Science B.V. All rights reserved. PACS: 02.20.Uw; 02.40.Gh; 03.30 1. Introduction About a year ago, in two seminal papers [1,2] (see also [3]) G. Amelino-Camelia proposed a theory with two observer-independent kinematical scales: of ve- locity c and of mass κ . 2 There are two major motiva- E-mail addresses: [email protected] (J. Kowalski-Glikman), [email protected] (S. Nowak). 1 Research partially supported by the KBN grant 5PO3B05620. 2 In his papers Amelino-Camelia uses the scale of length λ instead of the scale of mass κ , however, since the construction presented there describes the energy–momentum sector of the theory, it seems more natural to use the scale of mass. It should be also noted that it was the scale of mass (more precisely of momentum, κc), and not of length, that has been shown explicitly to be observer-independent. tions for such extension of special relativity. The first stems from the quest for quantum gravity in which the Planck length is supposed to play a fundamen- tal role. For example, in loop quantum gravity the area and volume operators have discrete spectra, with minimal value proportional to the square and cube of Planck length, respectively [4]. The basic observation is that if one regards the Planck length as a fundamen- tal, intrinsic characteristic of the space–time structure, this length should be the same for all observers and thus one immediately finds himself in conflict with FitzGerald–Lorentz contraction. The only solution of this problem is to modify the principles of special rel- ativity so as to incorporate the existence of this second observer-independent scale. It was indicated [1,2] that a possible candidate for such a theory might be a the- 0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)02063-4

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Page 1: Doubly special relativity theories as different bases of κ-Poincaré algebra

Physics Letters B 539 (2002) 126–132

www.elsevier.com/locate/npe

Doubly special relativity theories as different bases ofκ-Poincaré algebra

J. Kowalski-Glikman1, S. Nowak

Institute for Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9, Pl-50-204 Wrocław, Poland

Received 8 March 2002; received in revised form 28 May 2002; accepted 1 June 2002

Editor: P.V. Landshoff

Abstract

Doubly Special Relativity (DSR) theory is a theory with two observer-independent scales, of velocity and mass (or length).Such a theory has been proposed by Amelino-Camelia as a kinematic structure which may underline quantum theory ofrelativity. Recently another theory of this kind has been proposed by Magueijo and Smolin. In this Letter we show that boththese theories can be understood as particular bases of theκ-Poincaré theory based on quantum (Hopf) algebra. This observationmakes it possible to construct the space–time sector of Magueijo and Smolin DSR. We also show how this construction can beextended to the whole class of DSRs. It turns out that for all such theories the structure of space–time commutators is the same.This results lead us to the claim that physical predictions of properly defined DSR theory should be independent of the choiceof basis. 2002 Elsevier Science B.V. All rights reserved.

PACS:02.20.Uw; 02.40.Gh; 03.30

1. Introduction

About a year ago, in two seminal papers [1,2] (seealso [3]) G. Amelino-Camelia proposed a theory withtwo observer-independent kinematical scales: of ve-locity c and of massκ .2 There are two major motiva-

E-mail addresses:[email protected](J. Kowalski-Glikman), [email protected] (S. Nowak).

1 Research partially supported by the KBN grant 5PO3B05620.2 In his papers Amelino-Camelia uses the scale of lengthλ

instead of the scale of massκ , however, since the constructionpresented there describes the energy–momentum sector of thetheory, it seems more natural to use the scale of mass. It shouldbe also noted that it was the scale of mass (more precisely ofmomentum,κc), and not of length, that has been shown explicitlyto be observer-independent.

tions for such extension of special relativity. The firststems from the quest for quantum gravity in whichthe Planck length is supposed to play a fundamen-tal role. For example, in loop quantum gravity thearea and volume operators have discrete spectra, withminimal value proportional to the square and cube ofPlanck length, respectively [4]. The basic observationis that if one regards the Planck length as a fundamen-tal, intrinsic characteristic of the space–time structure,this length should be the same for all observers andthus one immediately finds himself in conflict withFitzGerald–Lorentz contraction. The only solution ofthis problem is to modify the principles of special rel-ativity so as to incorporate the existence of this secondobserver-independent scale. It was indicated [1,2] thata possible candidate for such a theory might be a the-

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)02063-4

Page 2: Doubly special relativity theories as different bases of κ-Poincaré algebra

J. Kowalski-Glikman, S. Nowak / Physics Letters B 539 (2002) 126–132 127

ory based on deformed Poincaré symmetry, for exam-ple theκ-Poincaré theory [5–9]. This claim was thenproved to be correct: it has been explicitly shown inthe papers [10,11] that theκ-Poincaré theory in the so-called bicrossproduct basis indeed predicts the exis-tence of an observer-independent maximal mass. Thesecond motivation comes from some puzzling obser-vations of ultra-high energy cosmic rays, whose ex-istence seems to contradict the standard understand-ing of astrophysical process likee+–e− productionin γ γ collisions and the photopion production in thescattering of high energy protons with soft photons.It turns out that both these effects might be explainedif one assumes that the threshold condition becomesdeformed, possibly as a result of existence of a newfundamental length (or mass) scale (see [12] and ref-erences therein). It is exciting to note that it is likelythat this effect will be a subject of numerous experi-mental tests in the near future.

Most of the works on the relativity theory withtwo observed independent kinematical scales, dubbed“Doubly Special Relativity” (DSR), has been donein the framework of (or strongly motivated by) alge-braic construction based on the quantum (Hopf)κ-Poincaré algebra, being a deformation of the standardPoincaré algebra of special relativity. Recently how-ever Magueijo and Smolin [13] have proposed a seem-ingly completely different DSR. The relation betweenpredictions of these two theories was thoroughly an-alyzed in [14]. The existence of two DSRs raises anobvious question how many theories of this kind mayexist. In this Letter we show that from the quantum al-gebraic point of view both above mentioned DSR the-ories are in fact completely equivalent, and might beconsidered as representations of the sameκ-Poincaréalgebra in different bases. Moreover, one can easilyconstruct different yet representations of this algebra,each of whose would correspond to a different DSRtheory. Therefore, in what follows we would use thenotion of different bases instead of different DSRs.

The plan of this Letter is the following. In Section 2we present three bases ofκ-Poincaré algebra: the bi-crossproduct one, lying behind the first DSR construc-tion, the Magueijo–Smolin one, and the classical one,whose algebraic sectors is identical with the standardPoincaré algebra, and we derive the transformationsrelating them. In Section 3 we make use of the co-algebraic sector of these bases to derive the space–

time non-commutative structure and to extend it tothe whole of the phase space. Section 4 is devoted toremarks concerning the picture resulting from mathe-matical constructions presented in Sections 2 and 3.

2. Lorentz algebra and energy–momentum sector

One of the main assumption in construction of DSRis that Lorentz subalgebra of theκ-Poincaré algebra isnot to be deformed. This assumption is motivated bythe fact that one wants to work with Lorentz structurethat integrates to a group, and not to a quasigroup,and it restricts the possible choices of bases severely(note that this postulate is not satisfied by the so-calledstandard basis of theκ-Poincaré algebra [5]). We,therefore, assume that the three rotation generatorsMi

and three boost generatorsNi satisfy

[Mi,Mj ] = i εijkMk, [Mi,Nj ] = i εijkNk,

(1)[Ni,Nj ] = i εijkMk.

One also assumes that the action of rotations is notdeformed and that generators of momenta commute.Taking these postulates as a starting point, we can de-fine the (deformed) action of the Lorentz algebra onenergy–momentumsector. Our starting point would bethe bicrossproduct basis in which the resulting alge-bra is a quantum algebra, i.e., in addition to the al-gebra of commutators (which is usually non-linear) itpossesses additional structures: co-product� and an-tipodeS. However, one should remember that quan-tum algebra structure is built on an enveloping algebra,which means that one is entitled to make any transfor-mations among generators (and not only the linear oneas in the case of Lie algebras.) This leaves, of course, alot of freedom in the choice of energy and momentumgenerators.

2.1. The bicrossproduct basis

Since, as said above the action of rotations is stan-dard it is sufficient to write down only the commuta-tors of deformed boost generators with momenta. Onegets [6]

[Ni,pj ] = i δij

2

(1− e−2p0/κ

) + 1

2κ�p2

)

(2)− i1

κpipj ,

Page 3: Doubly special relativity theories as different bases of κ-Poincaré algebra

128 J. Kowalski-Glikman, S. Nowak / Physics Letters B 539 (2002) 126–132

and

(3)[Ni,p0] = i pi.

One can easily check that the first Casimir operator ofthe algebra (1)–(3) reads

(4)m2 =(

2κ sinh

(p0

))2

− �p2 ep0/κ .

It follows that for positiveκ the three-momentum isbounded from above�p2 � κ2 and the maximal valueof momentum corresponds to infinite energy [10,11].

The quantum algebra (Hopf) structure in this basisis provided by the following co-products� andantipodesS, of whose only the explicit form of themomenta co-products

�(pi) = pi ⊗ 1 + e−p0/κ ⊗ pi,

(5)�(p0) = p0 ⊗ 1 + 1 ⊗ p0,

will be relevant to what follows. Taking these formulasas a starting point, we can now turn to analysis ofanother bases.

2.2. Magueijo–Smolin basis

In the recent paper [13] Magueijo and Smolin pro-posed another DSR theory, whose boost generatorswere constructed as a linear combination of the stan-dard Lorentz generators and the generator of dilatation(but in such a way that the algebra (2) holds.) In thisbasis the commutators of four-momentaPµ and boostshave the following form

(6)[Ni,Pj ] = i

(δij P0 − 1

κPiPj

),

and

(7)[Ni,P0] = i

(1− P0

κ

)Pi.

It is easy to check that the Casimir for this algebra hasthe form

(8)M2 = P 20 − �P 2

(1− P0/κ)2.

The question arises as to if this basis is equivalentto the bicrossproduct one. The answer is affirmative,indeed one easily checks that the relation between

variablesPµ andpµ is given by

(9)pi = Pi,

p0 = −κ

2log

(1− 2P0

κ+ �P 2

κ2

),

(10)P0 = κ

2

(1− e−2p0/κ + �p2

κ2

).

Let us note in passing that the formula above showsthat the maximal momentum in the bicrossproductbasis (�p2 = κ2, p0 = ∞) corresponds to the maximalenergy in the Magueijo–Smolin basis,P0 = κ .

Using formulas above one can without difficultypromote this algebra to the quantum algebra. Thisamounts only in using the homomorphisms (9), (10)to define the new co-products. They read

(11)(Pi) = Pi ⊗ 1+(

1− 2P0

κ+ �P 2

κ2

)1/2

⊗ Pi,

(12)

(P0) = P0 ⊗ 1+ 1⊗ P0 − 2

κP0 ⊗ P0 + 1

κ2�P 2 ⊗ P0

+ 1

κ

(1− 2P0

κ+ �P 2

κ2

)1/2∑Pi ⊗ Pi.

2.3. The classical basis

There is yet another basis which we will presenthere for comparison (this basis was first described in[8,15–17].) In this basis, which we call the classicalone, the boosts-momenta commutators together withthe Lorentz sector form the classical Poincaré algebra,to wit

(13)[Ni,Pj ] = i δij P0, [Ni,P0] = i Pi .

The Casimir for this basis equals, of course the one ofspecial relativity, to wit

(14)M2 =P20 − �P2.

The classical generatorsPµ are related to the bi-crossproduct basis generators by the formulas

(15)P0 = κ sinhp0

κ+ ep0/κ �p2

2κ,

(16)Pi = ep0/κ pi

Page 4: Doubly special relativity theories as different bases of κ-Poincaré algebra

J. Kowalski-Glikman, S. Nowak / Physics Letters B 539 (2002) 126–132 129

and one can easily compute the expression for co-product

(17)

�(P0) = κ

2

(K ⊗ K − K−1 ⊗ K−1)

+ 1

(K−1 �P2 ⊗ K + 2K−1Pi ⊗Pi

+ K−1 ⊗ K−1 �P2),(18)�(Pi ) =Pi ⊗ K + 1 ⊗Pi ,

where

K = ep0/κ = 1

κ

[P0 + (

P20 − �P2 + κ2)1/2

].

3. The space–time non-commutativity

In the preceding section we investigated the energy–momentum algebras. Now it is time to explain the rel-evance of the co-product (quantum) structure of thesealgebras. Briefly, this structure makes it possible to ex-tend an energy–momentum algebra to the whole of aphase space, i.e., the space describing both energy–momentum and space–time sectors in self consistentway. One should stress that this is the only way tointerconnect the space–time and energy–momentumsectors in a systematic way. It turns out that the co-algebra of the energy–momentum sector is in one toone correspondence with algebraic sector of space–time algebra and vice versa. Observe that in order toconstruct such a correspondence we need one moredimensionful parameter, which we identify with thePlanck constant (in the following we will use the con-vention in whichh = 1.)

There is a general procedure how to constructthe space–time commutator algebra from energy–momentum co-algebra, which consists of the follow-ing steps [6,18]:

1. One defines the bracket〈�, �〉 between momentumvariablesp,q and position variablesx, y in anatural way as follows

(19)

〈pµ,xν〉 = −iηµν, ηµν = diag(−1,1,1,1).

2. This bracket is to be consistent with the co-product structure in the following sense

〈p,xy〉 = 〈p(1), x〉〈p(2), y〉,

(20)〈pq,x〉 = 〈p,x(1)〉〈q(2), x(2)〉,where we use the natural notation for co-product

�t =∑

t(1) ⊗ t(2).

It should be also noted that by definition

〈1,1〉 = 1.

One sees immediately that the fact that momentacommute translates to the fact that positions co-commute

(21)�xµ = 1 ⊗ xµ + xµ ⊗ 1.

Then the first equation in (20) along with (19)can be used to deduce the form of the space–timecommutators.

3. It remains only to derive the cross relations be-tween momenta and positions. These can be foundfrom the definition of the so-called Heisenbergdouble (see [18]) and read

(22)[p,x] = x(1)〈p(1), x(2)〉p(2) − xp.

As an example let us perform these steps in thebicrossproduct basis [6,18]. It follows from (20) that

〈pi, x0xj 〉 = − 1

κδij , 〈pi, xj x0〉 = 0,

from which one gets

(23)[x0, xi] = − i

κxi.

Let us now make use of (22) to get the standardrelations

(24)[p0, x0] = i, [pi, xj ] = −i δij .

However, it turns out that this algebra contains onemore non-vanishing commutator, namely,

(25)[pi, x0] = − i

κpi .

Of course, the algebra (23)–(25) satisfies the Jacobiidentity.

3.1. Magueijo–Smolin basis

To find the non-commutative structure of spacetime in Magueijo–Smolin basis we start again with

Page 5: Doubly special relativity theories as different bases of κ-Poincaré algebra

130 J. Kowalski-Glikman, S. Nowak / Physics Letters B 539 (2002) 126–132

Eq. (19)

〈Pµ,Xν〉 = −iηµν.

Let us now turn to the next step, Eq. (20). It is easy tosee that the only terms in (11), (12), which are relevantfor our computations are the bilinear ones, so we canwrite

(Pi) = 1 ⊗ Pi + Pi ⊗ 1− 1

κP0 ⊗ Pi + · · · ,

(P0) = 1 ⊗ P0 + P0 ⊗ 1 − 2

κP0 ⊗ P0

+ 1

κ

∑Pi ⊗ Pi + · · · .

It follows immediately that the only non-vanishingcommutators in the position sector are

(26)[X0,Xi] = − i

κXi.

Now we can use Eq. (22) to derive the form of theremaining commutators. Since this computation is abit tricky, let us present the necessary steps.

[P0,Xi ] =∑

j

⟨1

κ

√1− 2P0

κ+ �P 2

κ2 Pj ,Xi

⟩Pj

+ Xi〈1,1〉P0 − XiP0

= 1

κ

∑j

〈Pj ,Xi〉Pj

(we made use of the fact that the only terms linear inmomenta have non-vanishing bracket with positions)from which it follows immediately that

(27)[P0,Xi ] = − i

κPi

and by employing the same procedure we obtain theremaining commutators

(28)[P0,X0] = i

(1− 2P0

κ

),

(29)[Pi,Xj ] = −i δij ,

(30)[Pi,X0] = − i

κPi .

Of course, as it is easy to check, the algebra abovesatisfies the Jacobi identity.

3.2. The classical basis

We again start with the duality relation

〈Pµ,Xν〉 = −iηµν,

and to get the commutators in the position sector asabove we take the part of the co-product up to thebilinear terms

(Pi ) = 1 ⊗Pi +Pi ⊗ 1 + 2

κP0 ⊗Pi

+ 1

κPi ⊗P0 + · · · ,

(P0) = 1 ⊗P0 +P0 ⊗ 1 + 1

κ

∑Pi ⊗Pi + · · ·

which leads again to

(31)[X0,Xi] = − i

κXi .

Then by employing the same method as in the preced-ing subsection we find the space–time commutators

(32)[P0,X0] = i

2

(K + K−1 − 1

κ2�P2 K−1

)

(33)[P0,Xi ] = − i

κPi ,

(34)[Pi ,Xj ] = −i Kδij ,

(35)[Pi ,X0] = 0.

Let us summarize our results. We found the trans-formations in the energy–momentum sector for the bi-crossproduct, Magueijo–Smolin, and classical basesof κ-Poincaré algebra. Next we extended this con-struction to the space–time sector. This examplesstrongly suggest that the following two claims holdunder mild and physically acceptable assumption con-cerning the change of bases (one must assume that inany basis an algebra becomes the standard Poincaréalgebra in the leading order inκ expansion).

(a) Any algebra consisting of the undeformedLorentz sector with the standard action of rotationsand generic action of boosts on commuting four-momenta can be equipped with theκ-Poincaré quan-tum algebraic structure and

(b) This structure can be extended to the whole ofthe phase space, and the non-commutative structureof the space–time sector is always (i.e., in any basis)

Page 6: Doubly special relativity theories as different bases of κ-Poincaré algebra

J. Kowalski-Glikman, S. Nowak / Physics Letters B 539 (2002) 126–132 131

given by

(36)[x0, xi] = − i

κxi.

The precise formulation of these statements and theirproof are presented in [19].

4. Physical interpretation and open problems

Till now we have been dealing with mathematics,it is time therefore to turn to physics. In view ofremarks at the end of preceding section the situationis as follows. Instead of having a number of distinctDSR theories we have to do with a whole class of suchtheories, which are related to each other by redefinitionof momenta

(37)p0 → f(p0, �p2), pi → g

(p0, �p2)pi

restricted only by the requirement that the action of ro-tations is preserved, each of whose can be interpretedas a basis ofκ-Poincaré quantum algebra. If we are tobase a physical theory on such a mathematical struc-ture, we have clearly two choices either to find somephysical and/or mathematical motivation to single outone particular basis, or to define the physical theory insuch a way that its physical predictions are indepen-dent of the choice of a basis. From this perspective thefact that one of the most fundamental prediction of thetheory, namely the non-commutativity in the space–time sector (36) is invariant under the change of basisis very encouraging. However, one should note that insuch a case, the “momentum” variables do not have adirect physical meaning. This does not seem very sur-prising after all. Indeed, in special relativity as well asin non-relativistic mechanics energy and momentumare defined so as to be conserved in physical processes.On the other hand energy/momentum conservation isdirectly related to the homogeneity of space–time.Here we have to do with non-commutative space–timeand the presence of length scale strongly suggest thatspace–time is homogeneous only at scales much largerthan this scale. It is not clear how to define the en-ergy/momentum conservation (and thus to answer thequestion what is thephysicalenergy and momentum)for a non-commutativespace–time, but it is certain thatany such definition must include information concern-ing the space–time non-commutativity.

One should also note that there are at least threeproblems, which must be solved before any physicalapplication of deformed Poincaré symmetry could beseriously considered.3

1. The co-algebra structure is not co-commutative inany basis. This is a virtue ofκ-Poincaré becauseit shows that this algebra differs fundamentallyfrom the standard Poincaré algebra. However, thissuggests that physical particles transforming un-derκ-Poincaré algebra have some braid statistics,which might be inconsistent in 3+ 1 space–timedimensions. This problem (to our knowledge) hasnever been investigated, but it is encouraging tonote that deformed space–time symmetries alge-bras has been successfully applied to describe asystem of bosons [20].

2. There is an apparent similarity between the alge-bras (2), (3) or (6), (7) and the algebras consid-ered in [21], which suggests that the boost opera-tors in some DSR bases might be not self-adjointbut only symmetric. This problem is currently be-ing studied and the results will be presented else-where [22].

3. It may seem that the universal space–time alge-bra (36) breaks explicitly the spacial translationalinvariance. Indeed this algebra is not invariant un-derxi → xi + ki , whereki arec-numbers. How-ever, since space–time co-ordinates do not com-mute, there is no reason to expect that translationsdo, and in fact the algebra (36) is invariant if onegets non-commutative translationsxµ → xµ +χµ, where[xµ,χν] = 0, and[χ0, χi] = −i/κ χi .We expect that proper understanding of the non-commutative translations will help understandingboth the physical momenta and the conservationlaws.

Acknowledgement

We would like to thank J. Lukierski for usefulcomments.

3 We thank the anonymous referee for his suggestion to state andbriefly discuss these problems.

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132 J. Kowalski-Glikman, S. Nowak / Physics Letters B 539 (2002) 126–132

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