canonical realizations of doubly special relativity

September 5, 2007 19:23 WSPC/142-IJMPD 01063

International Journal of Modern Physics DVol. 16, No. 7 (2007) 1133–1147c© World Scientific Publishing Company

CANONICAL REALIZATIONS OF DOUBLYSPECIAL RELATIVITY

PABLO GALAN∗ and GUILLERMO A. MENA MARUGAN†

Instituto de Estructura de la Materia, CSIC,Serrano 121-123, 28006 Madrid, Spain

∗[email protected]†[email protected]

Received 20 December 2006Communicated by J. Pullin

Doubly special relativity is usually formulated in momentum space, providing the explicitnonlinear action of the Lorentz transformations that incorporates the deformation ofboosts. Various proposals have appeared in the literature for the associated realizationin position space. While some are based on noncommutative geometries, others respectthe compatibility of the space–time coordinates. Among the latter, there exist severalproposals that invoke in different ways the completion of the Lorentz transformationsinto canonical ones in phase space. In this paper, the relationship between all thesecanonical proposals is clarified, showing that in fact they are equivalent. The generalizeduncertainty principles emerging from these canonical realizations are also discussed indetail, studying the possibility of reaching regimes where the behavior of suitable positionand momentum variables is classical, and explaining how one can reconstruct a canonicalrealization of doubly special relativity starting just from a basic set of commutators.In addition, the extension to general relativity is considered, investigating the kind ofgravity’s rainbow that arises from this canonical realization and comparing it with thegravity’s rainbow formalism put forward by Magueijo and Smolin, which was obtainedfrom a commutative but noncanonical realization in position space.

Keywords: Doubly special relativity; generalized uncertainty principle; gravity’s rainbow.

PACS Number(s): 03.30.+p, 04.50.+h, 11.30.Cp, 04.60.-m

1. Introduction

Most approaches to quantum gravity support the existence of a fundamental scale(expected to be of Planck order) that would act as a threshold for the inset ofquantum effects, so that beyond it the classical picture of space–time should bereplaced by a genuine quantum description (see, e.g. Refs. 1 and 2). For instance, inloop quantum gravity the classical smooth geometry of the space–time is replaced bya discrete quantum geometry at small scales, with a minimal nonzero eigenvalue forthe area operator that is indeed of the Planck order.3–6 Similarly, different analyses

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1134 P. Galan and G. A. Mena Marugan

of scattering processes in string theory suggest the existence of a minimal length.7–10

The possible emergence of this fundamental scale seems to pose a problem if Lorentzsymmetry has to be approximately valid (at least in some asymptotic regions) inthe effective theory that describes the flat space–time limit of quantum gravity.Since, in general, lengths are not invariant under Lorentz transformations, differentinertial observers would disagree on the value of the scale at which the quantumdiscreteness of space–time begins to manifest itself. In this sense, the existence ofa fundamental scale would contradict the relativity principle of inertial frames.

This apparent inconsistency may nonetheless be overcome by modifying specialrelativity so as to preserve an additional scale, apart from the one provided bythe speed of light.11–17 Since the action of the Lorentz transformations must nowleave invariant two dimensionful quantities, the family of theories that incorporatethis suggestion is generically called doubly special relativity (DSR).11 DSR theoriesare usually formulated in momentum space because they lead to modified disper-sion relations,18 which imply interesting observational consequences.11,19–24 Thenew scale is an energy and/or momentum scale, and its invariance is possible onlybecause the action of the Lorentz group becomes nonlinear in momentum space.

In order to determine the corresponding action of the Lorentz group in positionspace, various proposals have been discussed in the literature.25–29 A class of pro-posals is based on the use of noncommutative geometries. Namely, they introducespace–time coordinates that are not mutually compatible in the quantum theory, asit happens e.g. in the case of κ-deformed Minkowski space–time.17,25 Nevertheless,it is perfectly feasible to implement DSR theories in position space while keep-ing the commutativity of the space–time geometry.18,26–32 For example, Kimberly,Magueijo and Medeiros26 proposed a realization that preserves the Lorentz invari-ance of the contraction between energy–momentum and position. This requirementensures that free field theories based upon DSR admit planewave solutions.26 Such aposition space realization was later reformulated and generalized to the frameworkof general relativity by Magueijo and Smolin, including the effect of curvature.30

The resulting formalism describes geometries that depend explicitly on the energy–momentum of the particle that probes them (in the language of the renormalizationgroup, geometry “runs”30). For this reason, this proposal is called gravity’s rainbow.

There exists another class of proposals that, while maintaining the use of a com-mutative geometry for the description of the space–time, are based (in one way oranother) on a canonical implementation of the Lorentz transformations rather thanon the planewaves requirement of Ref. 26. Probably, the first of these proposals wasmade by Mignemi,27 who demanded covariance on phase space when studying thetransformation law for commuting coordinates in DSR. Nonetheless, instead of pur-suing this alternative, he finally focused his study on noncommutative geometries.Another proposal for a canonical implementation appeared in a work by the presentauthors.28 This proposal employed (the existence of) a nonlinear map that trans-forms the physical energy–momentum into an auxiliary one on which the Lorentzaction becomes linear.33 The realization of DSR in position space was deduced bycompleting this map into a canonical transformation, so that the symplectic form on

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Canonical Realizations of Doubly Special Relativity 1135

phase space remains invariant.28 Similarly, Hinterleitner29 proposed a prescriptionfor DSR in position space such that the complete Lorentz transformation on phasespace turned out to be canonical. On the other hand, Cortes and Gamboa31 ana-lyzed the modification of quantum commutators in DSR under the assumption thatone may adopt commuting space–time coordinates corresponding to the generatorsof translations in the auxiliary energy–momentum, so that they form altogether acanonical set.

Although all these proposals were put forward independently, it is clear that theyare interrelated. For instance, Hossenfelder pointed out recently18 that it is possibleto use a generalized uncertainty principle to find the nonlinear map between thephysical and the auxiliary energy–momentum. This map, together with the resultsof Ref. 33, suffices to fix the DSR theory. The main aim of the present work is to dis-cuss in detail the connection between the different canonical realizations suggestedfor DSR. In spite of the confusion that seems to exist in the literature, we will provethat they are actually equivalent. Moreover, extending Hossenfelder’s analysis, wewill study in detail the generalized uncertainty principles associated with this kindof realization. In particular, we will provide an explicit construction of the bijectivemap between DSR theories and sets of modified commutators between physical (orauxiliary) energy–momentum and auxiliary (or respectively physical) position vari-ables. We will also investigate the possibility that there exist limiting regimes inwhich these mixed (auxiliary and physical) elementary phase space variables reacha classical behavior. Finally, following exactly the line of reasoning defended byMagueijo and Smolin30 to extend the DSR realization in position space to generalrelativity, we will discuss the gravity’s rainbow formalism that arises from a canon-ical realization. We will not consider here other more tentative suggestions for theextension of DSR to general relativity, like e.g. that explored in Ref. 29.

The organization of the rest of the paper is as follows. In Sec. 2, we briefly reviewsome basic aspects of DSR theories formulated in momentum space. We prove inSec. 3 that the different canonical realizations of DSR in position space that haveappeared in the literature are in fact equivalent. Section 4 deals with the modifiedquantum commutators that correspond to canonical realizations of DSR. In Sec. 5,we summarize the proposal for a gravity’s rainbow and compare the formalism putforward by Magueijo and Smolin with the corresponding counterpart obtained froma canonical realization of DSR. As an important example, we consider in Sec. 6 theSchwarzschild solution to the modified Einstein equations obtained in these twotypes of gravity’s rainbow. Finally, Sec. 7 contains the conclusions.

2. DSR in Momentum Space

In DSR, the nonlinear action of the Lorentz transformations in momentum spacecan be easily captured in a nonlinear invertible map U between the physicalenergy–momentuma Pa := (−E, pi) and an auxiliary energy–momentum that

aLowercase Latin letters from the beginning and the middle of the alphabet denote Lorentz andflat spatial indices, respectively.

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transforms like the usual momentum variables in standard special relativity, Πa :=(−ε, Πi).33 Representing the conventional linear action of the Lorentz group by L,the nonlinear Lorentz transformations are given by16,33

La(Pb) =[U−1 L U

]a(Pb), (1)

where the symbol “” denotes composition. Demanding that the standard action ofrotations not be modified, the map U is totally determined by two scalar functions g

and f .16,25 Using a notation similar to that of Refs. 28 and 32, U can be expressedb

Pa = U−1a (Πb) ⇒

E = g(ε, Π),

pi = f(ε, Π)Πi

Π.

(2)

Here, Π is the (Euclidean) norm of the auxiliary momentum Πi. Each admissiblechoice of functions f and g leads to a different DSR theory. We remember that themap U−1 must be invertible in its range. In addition, in order to recover standardspecial relativity, U−1 must reduce to the identity (i.e. g ≈ ε, f ≈ Π) in the regionof negligible energies and momenta compared to the scale of the DSR theory. Thisscale corresponds to the limit of the functions f and/or g when the auxiliary energy–momentum reaches infinity (on the mass shell ε2−Π2 = µ2, where µ is the Casimirinvariant of the auxiliary momentum space). So, at least one of the two consideredfunctions must have a finite limit.

3. Canonical Implementation of DSR in Position Space

For the realization of DSR in position space we will focus on a family of proposalsthat, in the context of commutative geometry, are based on a canonical implemen-tation of DSR.18,27–29,31,32

Let us start by considering the proposal introduced by Mignemi in Ref. 27.Mignemi investigated the Hamiltonian formalism for a free particle with physicalenergy–momentum Pa and associated commuting space–time coordinates xa. FromEq. (1), the nonlinear action of the Lorentz transformations in momentum space is

Pa → P ′a = La(Pb). (3)

Here, we have conveniently adapted all expressions to our notation.c Mignemi iden-tifies then the position space as the cotangent one to the momentum space. He

bOur definition of the functions f and g coincides with that introduced in Refs. 28 and 32 butdiffers in general from other conventions found in the literature. Although our convention simplifiesthe calculations, when comparing our results with those of other works, it is important to takeinto account this possible discrepancy.cMignemi denotes the action of the deformed boosts by W , rather than using the symbol L forall Lorentz transformations in DSR. In addition, he calls pa the physical energy–momentum andqa the space–time coordinates conjugate to it, whereas we denote these physical variables by Pa

and xa, respectively.28,32 We reserve the notation qa for the auxiliary space–time coordinatesconjugate to the auxiliary energy–momentum Πa.

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also introduces implicitly the demand that the origin of the space–time coordinatesremain invariant under all Lorentz transformations. Appealing to covariance, hededuces the following transformation of coordinates27:

xa → x′a =(

∂P ′a

∂Pb

)−1

xb := La(xb, Pc). (4)

Note that, from Eq. (3), ∂P ′a/∂Pb = ∂La/∂Pb. The above transformation in position

space is linear in the coordinates xa, but depends nontrivially on the physicalenergy–momentum.

Soon after Mignemi’s proposal, Hinterleitner29 suggested independently that therealization of DSR in position space can be determined by completing the Lorentztransformations into canonical ones on phase space. Although he restricted hisanalysis to two-dimensional space–times for simplicity, we will reproduce his argu-ments in four dimensions. He considered canonical transformations between phasespace coordinates in two different inertial frames, (xa, Pa) and (x′a, P ′

a), imposingthat their restriction to momentum space coincided with the corresponding Lorentztransformations in DSR. The canonical transformation is then totally fixed if onerequires that its action on the space–time coordinates be homogeneous (i.e. theorigin be left invariant). Calling Pa = Pa(P ′

b), it is straightforward to conclude thatx′a = (∂Pb/∂P ′

a)xb, so that (employing the inverse function theorem) one arrivesprecisely at transformation (4). Thereby, we see that Mignemi’s and Hinterleitner’sproposals are in fact the same. Let us note that, in spite of the fact that the twoproposals have appeared separately in the literature, their equivalence is actuallyobvious once the condition of covariance is understood as the invariance of the sym-plectic form dxa ∧ dPa, which in turn implies that the considered transformationsmust be canonical.

On the other hand, in Ref. 28 we proposed to find the realization of DSR inposition space by demanding that the extension of the nonlinear map (2) to phasespace preserve the form dqa ∧ dΠa. Remember that qa are the auxiliary space–time coordinates, canonically conjugate to Πa.28,32 The above requirement assignsto the system new, modified space–time coordinates xa that are conjugate to thephysical energy–momentum Pa, so that the relation between (xa, Pa) and (qa, Πa)is a canonical transformation. Demanding that the origins of the two different setsof space–time coordinates coincide, one then gets

qa =∂Pb

∂Πaxb. (5)

With this relationship between auxiliary and physical coordinates one can fullydetermine the nonlinear action of the Lorentz group both in position and momen-tum spaces. It is given by

L(xa, Pa) = [U−1 L U ](xa, Pa). (6)

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Here, L represents the standard linear action of the Lorentz group on phase space,while U is the canonical map between physical and auxiliary variables determinedby U and Eq. (5).

Since U is a canonical transformation, so is its inverse, U−1. Moreover, thestandard Lorentz action L corresponds as well to a canonical transformation, whichmaps auxiliary variables from an inertial frame to another one. Hence, the composi-tion L = U−1LU provides a canonical transformation. In addition, its restrictionto momentum space reproduces Eq. (1). Thus, L can be regarded as the canoni-cal extension to phase space of the DSR transformation of the energy–momentum.Besides, L leaves the origin of the space–time coordinates invariant (because so doU and L). Therefore, our proposal leads exactly to the same Lorentz transforma-tion on physical phase space that was put forward by Hinterleitner. In this way, ourcanonical realization of DSR turns out to be equivalent to those of Mignemi andHinterleitner.

The main distinction between our proposal and the ones suggested by Mignemiand Hinterleitner is that the latter introduce directly the canonical transformationbetween physical variables on phase space as measured by two different observers,related by means of a nonlinear Lorentz transformation, whereas our realizationof DSR relies on the canonical map between auxiliary and physical variables. Inother words, Mignemi and Hinterleitner provide the nonlinear map L appearing inEq. (6), while our proposal gives the map U .

Still in the context of commutative geometry, Cortes and Gamboa31 analyzedthe modification of the quantum commutation relations in the framework of DSR.They considered space–time coordinates interpretable as the generators of transla-tions in the auxiliary energy and momentum variables (−ε, Πi). These space–timecoordinates are therefore canonically conjugate to the auxiliary energy–momentumand, according to our notation (which differs from that of Ref. 31), they can beidentified with qa. Similarly, one can consider space–time coordinates that generatethe translations in the physical energy–momentum (−E, pi). These are what wehave called the physical space–time coordinates xa. The relation between (qa, Πa)and (xa, Pa) is obviously a canonical transformation. In particular, one can thenstraightforwardly assign commutators to the complete set of mixed (i.e. half auxil-iary and half physical) phase space variables (qa, Pa) by multiplying their Poissonbrackets just by an imaginary factor i (with the Planck constant set equal to theunity). In this way, one recovers the result of Cortes and Gamboa, i.e.

[qa, Pb] = i∂Pb

∂Πa, (7)

the rest of commutators vanishing. Finally, remember that ∂Pb/∂Πa = ∂U−1b /∂Πa

from Eq. (2).In Ref. 18 Hossenfelder adopts a very similar viewpoint. The wave-vector ka

introduced in that work is what we have generically referred to as physical energy–momentum, Pa, and is canonically conjugate to the physical coordinates xa. This

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wave-vector is related to the auxiliary energy–momentum via the nonlinear map U .The commutators considered by Hossenfelder are those corresponding to the mixedphase space variables (xa, Πa). Regarded as fundamental commutators, obtaineddirectly from the Poisson brackets, they are then given by

[xa, Πb] = i∂Πb

∂Pa, (8)

with ∂Πb/∂Pa = ∂Ub/∂Pa.In conclusion, although the literature contains several proposals for the canonical

realization of DSR in phase space, we have seen that they are indeed equivalent,leading to the same transformations of the space–time coordinates. We expect thatour discussion helps to clarify this issue definitively.

4. Modified Commutation Relations

At the end of the previous section, we have shown that a canonical realizationof DSR yields modified commutation relations if one chooses as elementary phasespace variables any of the two mixed sets (qa, Pa) or (xa, Πa) (formed by bothauxiliary and physical variables). We now want to discuss the different behaviorsof the generalized uncertainty principles obtained with these two sets. We will alsoprovide a specific recipe to obtain the map U that determines the DSR theorystarting from the modified basic commutators.

4.1. Fundamental commutators

We first regard as elementary variables the set formed by the auxiliary space–timecoordinates and the physical energy–momentum, (qa, Pa), i.e. we adopt the choicemade in Ref. 31. From Eqs. (2) and (7), we obtain the following nonvanishingcommutators:

[q0, E] = −i∂g

∂ε, [q0, pi] = −i

∂f

∂ε

Πi

Π,

[qi, E] = i∂g

∂ΠΠi

Π, [qi, pj] = i

∂f

∂ΠΠi

ΠΠj

Π+ i

f

Π

(δij −

Πi

ΠΠj

Π

).

(9)

We can view these relations as a generalized uncertainty principle, noticing thatthe usual Heisenberg relations are recovered for energies and momenta well belowthe DSR (Planck) scale, because g ≈ ε and f ≈ Π in this limit. It is also worthpointing out that all the operators on the right-hand side of these expressionscommute, because they are determined only by functions of the energy–momentum,with no contribution from the position variables qa. In terms of these fundamentalcommutators, we can also define the commutator of other phase space functions.

In order to simplify our expressions and make more clear the difference betweenthis choice of fundamental commutators and the one considered by Hossenfelder,

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we will restrict our attention in this subsection to a subfamily of DSR theories inwhich the physical energy is just a function of the auxiliary one, with no depen-dence on the auxiliary momentum, namely, E = g(ε). These theories have receivedspecial attention in the literature.18,30,31,34 As an immediate consequence of thefact that (∂g/∂Π) = 0, the commutator [qi, E] becomes equal to zero. Moreover, letus consider DSR theories of the so-called DSR2 and DSR3 types, whose physicalenergy is bounded from above. We remember that DSR3 theories have a boundedphysical energy but an unbounded physical momentum, whereas DSR2 theoriespossess an invariant scale both in the physical energy and momentum. In thesetypes of theories, one has that (∂g/∂ε) → 0 when ε approaches infinity, and hencethe commutator [q0, E] vanishes for infinitely large auxiliary energy.

On the other hand, the physical momentum is bounded from above not only intheories of the DSR2 type, but also in the so-called DSR1 theories, whose physicalenergy is nonetheless unbounded. In a theory of the DSR1 or DSR2 type, thepartial derivatives ∂f/∂ε and ∂f/∂Π may both vanish as ε and Π tend to infinity(on the mass shell). We note, however, that while this vanishing is allowed, it isnot a necessary consequence of the boundedness of the physical momentum (onthe mass shell), because the function f depends on both ε and Π. If (∂f/∂ε) → 0when the auxiliary energy–momentum approaches infinity, the commutator [q0, pi]becomes zero in that limit. Similarly, if (∂f/∂Π) → 0, the commutators [qi, pj ] arenegligible when the auxiliary energy–momentum gets large, because for theoriesof the DSR1 or DSR2 type one also has that (f/Π) → 0 at infinity. Thus, theoperators representing the auxiliary spatial coordinates would commute with thoserepresenting the physical momentum.

We therefore conclude that all the modified phase space commutators may van-ish for DSR2 theories when the auxiliary energy–momentum tends to infinity (onthe mass shell). It is then possible to find DSR theories in which the behavior of thephase space variables (qa, Pa) becomes classical when one approaches the region ofinvariant energy and momentum scales.

Let us now analyze the choice of fundamental commutators made byHossenfelder,18 which correspond to a set of elementary phase space vari-ables formed by the physical space–time coordinates and the auxiliary energy–momentum. With the restriction to the subfamily of DSR theories with E = g(ε),we obtain

[x0, ε] = −i1

∂εg, [x0, Πi] = i

1∂εg

1∂Πf

∂f

∂ε

Πi

Π,

[xi, ε] = 0, [xi, Πj ] = i1

∂Πf

Πi

ΠΠj

Π+ i

Πf

(δij −

Πi

ΠΠj

Π

).

(10)

In order to make our expressions more compact, we have employed a notation ofthe type ∂εg := ∂g/∂ε for partial derivatives that appear in denominators.

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As we have commented above, (∂g/∂ε) → 0 when the auxiliary energyapproaches infinity in DSR2 and DSR3 theories. Thus, [x0, ε] explodes in this limitand there is a total lack of resolution for simultaneous measurements of x0 and ε.On the other hand, for a generic DSR theory, [x0, Πi] may either vanish or tend toinfinity depending just on the functional form of the partial derivatives of f andg, with no a priori restrictions. Finally, for theories of the DSR1 and DSR2 type,the ratio Π/f (and possibly 1/∂Πf as well) tends to infinity for large auxiliaryenergy-momenta (on the mass shell). So, for these theories, [xi, Πj ] diverges. Wehence see that all the nontrivial fundamental commutators may diverge for certainDSR2 theories, with a complete loss of resolution when one reaches the regime ofthe invariant (physical) energy and momentum scales. This result strongly contrastswith the conclusions obtained for the alternative choice of modified commutators(9) that we have analyzed previously.

One can also study the behavior of the commutators when the energy of thesystem is restricted to be a certain function of the momentum. This interdependencemakes the expressions of the commutators change “on shell.” After the reductionto energy surfaces, and for example with the choice of modified commutators (9)made by Cortes and Gamboa, one obtains the generalized uncertainty principle:

[qi, pj ] = idf

dΠΠi

ΠΠj

Π+ i

f

Π

(δij −

Πi

ΠΠj

Π

). (11)

This expression is similar to the last one in Eq. (9), but the partial derivativeof f with respect to Π has been replaced by a total derivative. This replacementsimplifies the analysis of the emergence of a classical behavior. For DSR1 and DSR2theories, it is straightforward to see that the ratio f/Π and the total derivativedf/dΠ tend to zero when the auxiliary momentum approaches infinity. Therefore,the commutators of the operators that represent the auxiliary (spatial) positionvariables and the physical momentum vanish in this limit, which corresponds to theregime in which the physical momentum reaches the value of the invariant scale.

Although it may seem initially counterintuitive, the emergence of a regime withthis kind of classical behavior poses in fact no inconsistency. If one measures theauxiliary position with infinite resolution then, according to the standard Heisen-berg principle, one must have a complete uncertainty in its conjugate momentum,which is the auxiliary one. However, since DSR1 and DSR2 theories map infiniteauxiliary momenta into a finite physical momentum scale, the uncertainty in thelatter of these momenta can be kept under control, yielding a vanishing productwith the uncertainty in the auxiliary position.

4.2. Derivation of a DSR theory from a generalized uncertainty

principle

To conclude this section, we will derive explicit formulas for the nonlinear mapU that characterizes the DSR theory using a set of modified commutators. These

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formulas, together with Eq. (9) or (10), prove that the relevant information presentin a canonical realization of a DSR theory and in a generalized uncertainty principleis equivalent. We will study explicitly the case of the fundamental commutators[qa, Pb]. The analysis can be straightforwardly extended to the alternative case ofthe commutators [xa, Πb].

Note that the functions f and g that determine the DSR theory, and that in ageneric situation depend on both ε and Π, appear in the expressions (9) as partialderivatives. Therefore, given a generalized uncertainty principle of the form (9), wewill obtain the map U (or equivalently U−1) by mere integration. The reconstructionof an arbitrary function z(ε, Π) from its partial derivatives u := ∂z/∂ε and v :=∂z/∂Π is immediate (assuming the integrability condition ∂u/∂Π = ∂v/∂ε):

z(ε, Π) =∫ Π

0

v(ε, Π) dΠ +∫ ε

0

u(ε, 0) dε. (12)

The function z represents either f or g, and we have employed that f(0, 0) =g(0, 0) = 0 to fix an additive integration constant. It then finally suffices to realizethat

∂g

∂ε= −q0, E, ∂g

∂Π= qi, EΠi

Π,

∂f

∂ε= −q0, PiΠi

Π,

∂f

∂Π= qi, PjΠi

ΠΠj

Π,

(13)

where the above Poisson brackets ·, · are the straight classical counterpart of thefundamental commutators (9) (namely, these brackets are obtained by replacingenergy–momentum operators — that commute among themselves — by classicalvariables, and removing an imaginary factor i).

5. Gravity’s Rainbow

As we commented in the introduction, Magueijo and Smolin put forward a general-ization of DSR that attempts to incorporate the effect of curvature and, therefore,provide an extension to general relativity.30 This generalization is called gravity’srainbow and rests on a realization of DSR in position space that is compatiblewith the commutativity of the geometry. The realization selected by Magueijo andSmolin is not a canonical one; instead, it is constructed from a specialization ofthe requirement (introduced by Kimberly, Magueijo and Medeiros26) that free fieldtheories in DSR continue to admit planewave solutions. This specialization canbe formulated as the invariance under Lorentz transformations of the contractionbetween the energy–momentum and an infinitesimal space–time displacement.30 Inour notation, this condition can be expressed in the form Πadqa = Padxa. Wehave denoted the physical coordinates by xa to emphasize that their expressions interms of auxiliary phase space variables will now differ from those obtained with acanonical realization of DSR.

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From the above requirement, Magueijo and Smolin derived an energy dependentset of orthonormal frame fields. The expressions of the one-forms that they obtaineddepend nontrivially on the auxiliary energy–momentum,

dx0 =ε

gdq0, dxi =

Πf

dqi. (14)

Owing to this dependence, a rainbow of metrics emerge in the formalism, each par-ticle being associated with a different metric according to its energy–momentum.30

Actually, from Eq. (5) we see that a canonical realization of DSR leads alsoto space–time displacements that depend explicitly on the energy and momentum(which can be understood as corresponding to the probe used by the observerto test the geometry). In this sense, canonical DSR theories can be extended aswell to a type of gravity’s rainbow formalism.28,32,34 In the following, we restrictagain our attention to the subfamily of DSR theories with E = g(ε). It is notdifficult to see then that the requirement of invariance of the symplectic formdqa ∧ dΠa = dxa ∧ dPa leads to the following one-forms,28 rather than to thosein Eq. (14):

dx0 = detU

(∂f

∂Πdq0 +

∂f

∂ε

Πi

Πdqi

),

dxi = detU

(∂g

∂ε

ΠiΠj

Π2dqj

)+

Πf

(dqi − ΠiΠj

Π2dqj

).

(15)

Here, detU = 1/[(∂g/∂ε)(∂f/∂Π)]. The analysis carried out by Magueijo andSmolin in Ref. 30 can then be generalized by substituting relations (14) withEq. (15), and will not be repeated here. Instead, in the next section we consider aparticularly interesting example of the solutions to the modified Einstein equationsthat arise in this type of gravity’s rainbow formalism, namely, the generalization ofthe Schwarzschild solution.

6. Modified Schwarzschild Solution

Since we want to study spherically symmetric solutions of the modified Einsteinequations, we will impose a natural symmetry reduction on the physical phase space(of the test particle that probes the geometry), restricting the physical momen-tum to be parallel or antiparallel to the spatial position. Therefore, for each givenphysical momentum, εi

jkpidxj = 0.34 Here, εijk is the Levi–Civita symbol. Sincepi/p = Πi/Π, this condition can be rewritten as dxi = dxjΠjΠi/Π2. With thisreduction, we find from Eq. (15)

dx0± =

1∂εg

dq0, dxi =1

∂Πfdqi, (16)

where we have defined

x0± := x0 ± 1

∂εg

∂f

∂εx, x :=

√xixi. (17)

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The sign in the definition of x0± depends on whether pi and dxi are parallel or

antiparallel. Note that the new coordinate x0±, although not canonically conjugate

to the energy, differs from the canonical time x0 only in the inclusion of a radialshift vector that is constant in space–time.

Apart from this constant shift, we notice from Eqs. (14) and (16) that, in thetwo realizations of DSR that we are considering (the canonical one and that ofRef. 26), the geometry is affected just by two independent scalings that are constantin the space–time, but depend on the auxiliary energy–momentum. They consistof a time dilation and a conformal transformation of the spatial components. Theassociated (spherically symmetric) gravity’s rainbow formalisms can be exploredsimultaneously if one adopts the generic notation dq0 = G(ε)dx0±, dqi = F (ε, Π)dxi,with x0± designating x0 for the case of the gravity’s rainbow elaborated by Magueijoand Smolin. Explicitly,

G(ε) :=g(ε)ε

, F (ε, Π) :=f(ε, Π)

Π, (18)

for Magueijo and Smolin, whereas for a canonical realization

G(ε) :=∂g

∂ε(ε), F (ε, Π) :=

∂f

∂Π(ε, Π). (19)

One can now easily derive the modified Schwarzschild solution following thesame line of reasoning that was explained in Ref. 30. In terms of the physicalcoordinates (called energy independent by Magueijo and Smolin), we can genericallyexpress a spherically symmetric metric as30

ds2 = −A(x)[G(ε)]2(dx0±)2 + [F (ε, Π)]2(B(x)dx2 + x2dΩ2). (20)

Translated into auxiliary space–time coordinates (energy dependent in Magueijoand Smolin’s terminology), we obtain

ds2 = −A(q|ε, Π)(dq0)2 + B(q|ε, Π)dq2 + q2dΩ2, (21)

where

q :=√

qiqi, q = F (ε, Π)x,

A(q|ε, Π) := A[x(q, ε, Π)], B(q|ε, Π) := B[x(q, ε, Π)],(22)

and the round metric on S2 is not changed. According to Birkoff’s theorem, thefunctions A and B must satisfy A(q|ε, Π) = B(q|ε, Π)−1 = 1 − C(ε, Π)/q for everyfixed auxiliary energy–momentum.

Taking into account the relationship between q and x, we see that, in order torecover the independence of the function A(x) on the energy–momentum, the ratioC(ε, Π)/F (ε, Π) must be a genuine constant, not only with respect to its space–timedependence, but also with respect to ε and Π.30 On the other hand, in the regime oflow energies and momenta F (ε, Π) approaches the unity and C(ε, Π) coincides with2M , where M is the Schwarzschild mass of general relativity and we are setting G,

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the effective Newton constant in the low energy limit, equal to one. Therefore, wemust have C(ε, Π) = 2MF (ε, Π).

This completely determines the modified Schwarzschild solution. In physicalcoordinates, the explicit expressions in the gravity’s rainbow formalisms proposedby Magueijo and Smolin, on the one hand, and arising from a canonical realization,on the other hand, are, respectively,

ds2 = −(

1 − 2M

x

)(g

ε

)2

(dx0)2 +(

1 − 2M

x

)−1(f

Π

)2

dx2

+(

f

Π

)2

x2dΩ2;

ds2 = −(

1 − 2M

x

)(∂g

∂ε

)2

(dx0±)2 +(

1 − 2M

x

)−1(∂f

∂Π

)2

dx2

+(

∂f

∂Π

)2

x2dΩ2.

(23)

In particular, notice that the horizon is placed at the same location as in generalrelativity, x = 2M . The corresponding notions of null and spatial infinities and ofasymptotic flatness in the context of DSR can be introduced as explained in Ref. 35.

7. Summary and Conclusions

We have studied several proposals for the realization of DSR in position space thatare based on a canonical implementation in phase space. We have shown that,although these proposals have been discussed separately in the literature, they areactually equivalent. Some of these proposals have been formulated as the require-ment that the action of the Lorentz transformations on the physical space–timecoordinates be such that the total transformation in the resulting phase space becanonical.27,29 In another case, the demand of a canonical transformation has beenintroduced instead on the nonlinear map that relates the physical phase spacewith an auxiliary one where the Lorentz transformations have the standard linearaction.28,32 Finally, in other cases the realization in position space has been encodedin a generalized uncertainty principle, with modified commutators that correspondeither to the physical energy–momentum and the space–time coordinates that gen-erate translations in the auxiliary energy–momentum31 or, alternatively, to theopposite choice of auxiliary versus physical variables.18

We have also analyzed the behavior of those two different kinds of modifiedcommutation relations. For the first kind, namely, when one considers auxiliaryspace–time coordinates and physical energy–momentum, we have shown that allphase space commutators may vanish in the limit of large auxiliary energy andmomentum (on the mass shell) for certain DSR theories of the so-called DSR2type. Therefore, it is in principle possible to approach a classical regime for thementioned elementary phase space variables when the physical energy–momentum

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tends to the invariant scale of the DSR theory. In contrast, for the same type of DSRtheories, the alternative choice of physical space–time coordinates and auxiliaryenergy–momentum as elementary variables leads to modified commutators thatgenerally explode when the system approaches the invariant energy–momentumscale. In terms of these elementary variables, there is a complete loss of resolutionin the considered limit. In addition, we have shown how to construct the nonlinearmap U that determines the DSR theory from the sole knowledge of a given setof modified phase space commutators. In this construction, it has been essentialthat both the fundamental commutators and certain projections of them in themomentum direction are not affected by factor ordering ambiguities, because theyare defined exclusively in terms of functions of the energy–momentum.

Finally, we have also discussed the possibility of generalizing the gravity’s rain-bow formalism put forward by Magueijo and Smolin in order to find an extensionto general relativity of the canonical realization of DSR theories. We have shownthat the generalization is possible and we have compared the gravity’s rainbow thatresults from the canonical realization with that elaborated in Ref. 30 starting froma different proposal for the realization of DSR in position space.26 In both cases,one actually arrives to a rainbow of metrics that depend nontrivially on the energy–momentum of the test particles. We have studied in detail the particular example ofthe generalized Schwarzschild solution to the modified Einstein equations. The mainmodification with respect to the standard solution in general relativity consists ofa time dilation and a spatial conformal transformation, both of them dependenton the energy–momentum. We have deduced the explicit form of the dilation andconformal factors in the case of the canonical realization, showing that they gener-ally differ from those obtained by Magueijo and Smolin. This difference may haveimportant consequences for black hole thermodynamics.34

Acknowledgments

The authors are grateful to L. Garay and C. Barcelo for conversations. P. Galangratefully acknowledges the financial support provided by the I3P framework ofCSIC and the European Social Fund. This work was supported by funds providedby the Spanish MEC Project No. FIS2005-05736-C03-02.

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canonical realizations of doubly special relativity

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