Dispersion relations in finite-boost DSR

We find finite-boost transformations DSR theories in first order of the Planck length $l_p$, by solving differential equations for the modified generators. We obtain corresponding dispersion relations for these transformations, which help us classify the DSR theories via four types. The final type of our classification has the same special relativistic dispersion relation but the transformations are not Lorentz. In DSR theories, the velocity of photons is generally different from the ordinary speed c and possess time delay, however in this new DSR light has the same special relativistic speed with no delay. A special case demonstrates that any search for quantum gravity effects in observations which gives a special relativistic dispersion relation is consistent with DSR.

DSR theories have been proposed for quantum gravity (QG) modifications to Einstein's special relativity [1][2][3][4][5]. The transformations of the energy and momentum under finite boosts have been obtained in [6]. In this paper, we continue this study by finding the finite-boost DSR transformations to leading order of the Planck length from the solutions of the differential equations, and by looking at the observational consequences of these transformations. We obtain the corresponding dispersion relations, which allow us to classify DSR theories to four types with each type resulting in a different specific kind of DSR to first order of the Planck length.
In fact, the first differential equation for DSR was given by Amelino-Camelia [2]. However, despite this paper being one of the pioneering works in DSR, the proposal and differential equations were specific and did not contain all DSR theories to first order. Also, by starting from κ-Poincaré boost generators [7], Amelino-Camelia and colleagues obtained differential equations for a specific DSR theory. They solved the differential equations and found finite transformations for the generators [3]. These finite transformations are important but they are not the most general. We generalize their method and results.

Maguijio and Smolin obtained a different realization of DSR theories by non-linear action of the
Lorentz group on the energy-momentum space [4]. Also, they gave a procedure for finding corresponding transformations for any given modified dispersion relation [5]. This modified dispersion should leave the Planck scale invariant. Motivated by the DSR proposal, we starting from the boost formula in the κ-Poincaré and DSR theories [7,8], take its leading order in Planck length, and find a generalized boost by adding some free parameters β i to this first order boost (β i are real numbers and i = 0, ..., 4.). The generalized boost parametrizes the DSR theories to first order. By using this boost, we obtain differential equations which govern the evolution of the energy and momentums in momentum space. These differential equations are some extensions of the differential equations like d 2 p 0 dξ 2 − p 0 = 0, and dp 2 dξ = 0 in special relativity. With perturbative methods, we obtain solutions of the aforementioned differentials, which lead us to the finite-boost DSR transformations in (3+1) dimensional momentum space.
We show that the Amelino-Camelia (AC) [1], and the Maguejo-Smolin (MS) [4] DSR theories in first order of the Planck length l p , are special cases of our DSR finite-boost transformations.
Our study is in the same direction as [11][12][13][14], i.e. 'beyond special relativity theories' and can be regarded as a logical continuation and extension. It may also prove useful for investigating the geometrical nature of DSR [15].
Other approaches, such as the generalized uncertainty principle (GUP) is related to DSR [18].
There is a result of recasting the commutators between p and x by adding modifications involving the Planck energy [16,17]. As is well known, the DSR theories are obtained by adding the Planck scale as an invariant to the boost sectors of the Poincaré group. Thus, a general expression for the first order DSR finite transformations can assist in finding a consistent GUP theory to first order in the Planck energy.
Observations of quantum gravity effects for astrophysical and cosmological observations are challenging, and finding these tiny effects are notoriously difficult in practice [25,26]. On the theoretical side, there many approaches for including quantum gravity effects (see e.g. [19,20] for a review). Finite-boost transformation DSR theories and classifications of first order DSR theories (by use of a dispersion relation) provide more possibilities for investigating quantum gravity effects in observations. The freedom comes from the use of adjustable parameters in the finite-boost transformations.

II. LORENTZ TRANSFORMATIONS FROM THE DIFFERENTIAL EQUATIONS
The Lorentz transformations in momentum space are where Λ ν µ are the components of the Lorentz matrices. The p ′ µ are the energy and momenta components for a particle in the primed inertial system which moves with velocity v with respect to the unprimed inertial system. For the infinitesimal Lorentz matrices, we have where ω µν are the components of the anti-symmetric [ω µν ] matrices. In the (1+1)-dimensional case, we have Here, ξ is the rapidity parameter. Thus, the explicit form of the infinitesimal Lorentz transformations are: From which, we can write the first order differential equations for the p 0 and p 1 components as The second order differential equations which contains only p 0 and p 1 are: The first equation of the differential equations in Eq. (6) in the primed system has the solution where C 1 and C 2 are constants, and can be determined from the initial conditions p ′ 0 (ξ = 0) = p 0 and dp ′ 0 dξ (ξ = 0) = p 1 . For the p 2 and p 3 components, the infinitesimal transformations Eq. (4) give the first order differential equations In the primed system, the solution of the first differential equations in Eq. (8) will be where C 1 is a constant. By using the initial condition p ′ 2 (ξ = 0) = p 2 , one finds that this constant should be zero. Therefore, from Eq. (7) and Eq. (9), we find the well-known Lorentz transformations in the energy momentum space as In the next section, we will extend this method for finding solutions of the finite-boost DSR in first order of Planck length transformations which are nonlinear extensions of the system of differential equations in Eq. (6).

A. DSR Boosts
The generator of the DSR theories in (3+1) dimensions is given by It originates from the κ-Poincaré group [7][8][9]. In leading order of l p , this generator will be In fact, this generator represents only one class of DSRs. In order to include a wider range of theories, we modify this generator by generalizing to preserve parity and time-reversal symmetry: where β 0 , ..., β 4 are arbitrary real numbers. We parameterize the extended κ-Poincaré symmetry in the first order of the Planck length by these real numbers. One cannot obtain MS-DSR and other DSR theories from Eq. (12), but one can obtain them from Eq. (13). This generalization gives us DSR theories to the first order in the Planck length. We also have extended freedom in finding more general dispersion relations to match observational results. In other words, Eq. (13) generalizes first order κ-Poincaré symmetry. This extension obeys the group structure and does not violate the relativity principle [11].
By using the general generator Eq. (13), we can write the infinitesimal transformation in the first order of the Planck length l p [10,11], as B.
Generalization of the first order κ-Poincaré symmetry Majid and Ruegg [7], demonstrated that the κ-Poincaré algebra is a semi-direct product of the To obtain the corresponding algebra for this extended group, we rewrite the general generator in Eq. (13) for any arbitrary directioñ We find the commutators of this generator with p µ and the commutators of this generator with M i (the generators of the rotations). We also find the commutators of this generator with itself.
This modified algebra is given in the following, The other commutators are and the rotations of momentums which remain unmodified are The algebraic structure of our extended κ-Poincaré group is important for studying symmetries of this group.

C. Differential Equations and Finite-Boost Transformations
From Eq. (14), we find differential equations which govern the evolution of energy and momentums in momentum space, and we solve the differential equations for the p 0 and p 1 components together and for the p 2 and p 3 components together. For the p 0 and p 1 components we solve the second order equations which are more simple, but for the p 2 and p 3 components it is best to find the solutions of the first order differential equations. For the p 0 and p 1 components the first order differential equations are By introducing the constants we obtain the second order differential equations, For the p 2 and p 3 components the differential equations are Solutions of the differential equations in Eq. (24) and Eq. (25) give us the finite-boost DSR transformations for all orders of the rapidity parameter ξ, but only in the first order of the Planck length. Solutions of these differential equations are given in Appendix A 1.
The finite-boost transformations to the first order in l p are where for convenience. These transformations are finite-boost DSR transformations to first order in l p but for all orders of rapidity ξ. We confirm they are the same as the transformations in [6], which have been obtained using commutators.

D. Dispersion Relation and its classifications
Taking p µ = (p 0 , p) = (m, 0) for the initial p µ in the transformations in Eq. (26), we obtain general expressions for the cosh ξ and sinh ξ as Here, m is the rest mass of the particle. Using cosh 2 ξ − sinh 2 ξ = 1, we find the corresponding dispersion relation for the transformations of Eq. (26) to first order in l p as Expressing this dispersion relation in terms of β i yields  a rotation (Wigner rotation). These properties have been checked in [6] for the generalized boost in Eq. (13). Here, we check and confirm in detail some of these properties.
First we check combination property for the infinitesimal transformations Eq. (14). We assume another infinitesimal transformations like Eq. (14), but from p ′ µ to p ′′ µ with rapidity parameter ξ ′ in the same direction which is given by Combinations of the infinitesimal transformations in Eq. (14) and Eq. (31) will give us other similar infinitesimal transformations from p µ to p ′′ µ in the same direction with rapidity parameter ξ ′′ , which is the sum of two rapidity parameters, In the finite-boosts transformations case, as given by Eq. (26), we have expressions that include cosh ξ, sinh ξ, cosh 2 ξ, sinh 2 ξ, and cosh ξ sinh ξ, which yield the same transformations after two successive boosts in the same direction and the rapidity parameters satisfy Eq. (32) as in the infinitesimal transformations case.
For two perpendicular boosts, as mentioned, we have a new boost and a rotation. In special relativity this rotation is the well-known Wigner rotation θ W as discussed in [21,22] and is given by In DSR theories, the rotation part of the Poincaré group are unmodified and only the boosts are changed. Therefore no correction will be added to the Wigner rotation [6].
The interesting consequences of the Wigner rotation in special relativity have been studied before in [21][22][23]. An important effect is related to the spin of a composite system such as a proton which is a composite system of quarks. The sum of the spins for a composite system can violate Lorentz invariance. In fact, spin is related to the Poincaré group, or, to the κ-Poincaré group and its extensions, such as the extension in subsection III B. The spin S of a moving particle with mass m can be defined by transforming its Pauli-Lubinski 4-vector w µ = (1/2)ǫ νρσµ J ρσ p ν to its rest frame by a rotationless DSR boost D(p), where D(p)p = (m, 0), and (0, S) = D(p)w/m. Under an arbitrary DSR transformationΛ, the spin and momentum of a particle will be transformed via where is the Wigner rotation [23]. By using Eq. (35), we can show that the Wigner rotation for DSR is equal to the Lorentz case.
In the primed moving frame, the proton is boosted with a rotationless DSR transformation along its spin direction. Each quark spin will be affected by a Wigner rotation, and these rotations can change the vector spin sum of the quarks and antiquarks. This changing of the spin sum has consequences for the parton model of the proton. However, the proton spin in the moving frame will be the same as in the rest frame [23]. Since the Wigner rotation in DSR is the same as the special relativistic case, there are no new consequences (i.e. no new kinds of spin) of the Wigner rotations besides the usual special relativistic effects.
The group of the DSR theories is the κ-Poincaré group [7][8][9], and our extension of this group in Eq. (13) in first order Planck length, is an extension of the κ-Poincaré symmetry. The commutators between generators for every two boosts in one direction, or in two different directions satisfy Eq. (19), which shows that combinations of two boosts will give another boost, or a boost and a rotation. We do not check these properties in full detail; however, they can be tested with the transformations in Eq. (26) by long but straightforward calculations.
We have tested the group property and find the generator in Eq. (13) satisfies an extension of the κ-Poincaré algebra as given in subsection III B. This is assurance that the finite-boost transformations in Eq. (26) are not just reparameterizations of the compact Lie groups as discussed in [30].
and we can compute the constants α 1 to α 3 from the relations in Eq. (27) as By putting these values in Eq. (29) we find the dispersion relation of the MS-DSR to the first order in l p , Transformations for this DSR in first order are given in Appendix A 2.

B. AC-DSR
For the AC-DSR we have and we can compute the constants α 1 to α 3 by use of the relations in Eq. (27) as By using these values, we find dispersion relation of the AC-DSR in the first order of l p as and transformations for this DSR are given in Appendix A 3.
Using the conditions α 1 = α 3 , and α 2 = 0 in Eq. (29) one finds a modified dispersion relation with a cubic term, Transformations for this type of DSR are in Appendix A 4.

D. Special Relativistic Dispersion Relation
The final interesting example utilizes α 1 = α 3 , and α 2 = 0 in Eq. (29). These conditions lead us to the unmodified special relativistic dispersion relation In terms of the β i parameters these conditions will be which is valid for many general DSR theories in first order of Planck length. By putting the conditions of Eq. (44) into Eq. (27) we find Here, we have taken B ≡ β 0 , which is the free parameter. There also two other free parameters β 2 and β 4 , which are taken to be zero for the following transformations. By using the values of α 1 to α 3 parameters in Eq. (26), we find the following transformations

V. OBSERVATIONAL CONSEQUENCES OF THE FIRST ORDER DSR
The proposal for observation of quantum gravity effects in gamma ray bursts was given initially by Amelino-Camelia and his colleagues in [24]. They assumed a deformed dispersion relation for the photons in the form of where E QG is an effective quantum gravity scale, and f is a model dependent function of E/E QG .
One can find an upper limit on the scale with observations and it may be different from the Planck energy M ≃ 1. 22 × 10 19 GeV. This dispersion relation to first order in E/E QG is where η is some number. The proposal is that the velocity of the gamma ray bursts should be different from c, which is given by Moreover, a signal which is coming from distance D with energy E will be received with time delay different than the ordinary case with the speed c. A linear correction to the velocity of high energy photons like Eq. (49) has been suggested in many papers for testing quantum gravity effects [25][26][27][28][29][30][31][32].
We find the velocity of photons and time delay for the photons in our formalism for finite-boost DSR transformations. By using the dispersion relation in Eq. (29), the velocity of the photons will and the time delay will be As can be seen, the η parameter in Eq. (48) has been replaced by (α 1 + α 2 − α 3 ) in the finite-boost DSR transformations. To be clear, the distance D is a cosmological distance that depends on the redshift z, matter density Ω m , cosmological constant density Ω Λ , and Hubble constant H 0 . The time delay is Here, we take Ω m = 0.3, Ω Λ = 0.7, H 0 = 72 km s −1 Mpc −1 for the cosmological parameters [31].
The time delay in Eq. (52) and Eq. (53) is the only Lorentz invariance violating (LIV) term for time delay between a high energy astrophysical photon source and receiver. The general expression for time delay is ∆t = ∆t LIV + ∆t int + ∆t spe + ∆t DM + ∆t grav , where ∆t int is the time delay due to the fact that photons with high and low energies do not leave the source simultaneously. ∆t spe is a result of the special relativistic effects if the photons have non-zero mass and ∆t DM is due to dispersion by the line-of-sight of free electron content, which is non-negligible especially for low energy photons. Finally, ∆t grav is due to the gravitational potential contribution along the photons propagation paths for possible violation of Einstein's equivalence principle [32].

A. Special Relativistic dispersion relation and null results in observations for QG Effects
For α 1 = α 3 , and α 2 = 0, which gives the special relativistic dispersion in Eq. (43), the velocity of the photons will be the unchanged special relativistic c, and the time delay in Eq. (52) will be zero.
Therefore, the transformations in Eq. (46) (and also all transformations in the special relativistic dispersion relation family such as Eq. (A10)) are different from the Lorentz transformations in first order of the Planck length, but the velocity of the photons are the same as special relativity.
These photons will be received in the same time as in the usual special relativistic case. Simply put, as has been done e.g. in [25,26], measuring the velocity and time delay of GRB photons is not sufficient for investigating quantum gravity effects to first order. There are two possible extensions and applications of these finite-boost DSR transformations.
First, the solutions of the extended second order differential equations can be investigated for a better understanding of their dynamics and behavior [33,34]. Second, the possible extensions of the Poincaré or κ-Poincaré group may be fruitful for investigations (e.g. dual DSR theory [35,36] For obtaining the finite-boost transformations, first we solve the differential equations for the p 0 and p 1 components. In the primed inertial system, the form of Eqs. (24) are the same. We introduce the perturbative solutions of these differential equations in first order of the Planck length l p as The unknown function A(ξ) should also satisfy the following additional differential equation d 2 A dξ 2 − A = (a 0 p 2 1 + a 1 p 2 0 ) sinh 2 ξ + (a 0 p 2 0 + a 1 p 2 1 ) cosh 2 ξ + 2(a 0 + a 1 )p 0 p 1 sinh ξ cosh ξ.
For the p 2 and p 3 components, we take where A 1 , ..., A 6 , are unknown functions of p 0 , ..., p 3 . By putting these solutions in the differential equations for p 2 and p 3 in Eq. (25), we can find the unknown functions A 1 , ..., A 6 , which lead us to the transformations for the p 2 and p 3 components.

(A5)
If we expand the expressions for the transformations and dispersion relation of the MS-DSR theory which has been given in [4], we will find the same expressions as in Eq. (A5) and Eq. (38) to first order in the Planck length.