Non-commutative space-time of Doubly Special Relativity theories

Jerzy Kowalski-Glikman; Sebastian Nowak

REVIEW

Non-commutative space-time of Doubly Special Relativity theories

High Energy Physics - Theory · Submitted 29 Apr 2002 · 23 references

Pith's verdict: The basis-independence result and the Snyder identification are real and check out; the abstract's 'unique' outruns what §III actually proves.

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arXiv IDhep-th/0204245 (version 1)

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CitationJerzy Kowalski-Glikman, Sebastian Nowak. "Non-commutative space-time of Doubly Special Relativity theories." arXiv:hep-th/0204245 (2002). https://pith.science/paper/6JJ2M522

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Worth your time if you work on DSR, with one caveat about the abstract's wording. What is actually new: Lukierski–Nowicki derived the κ-deformed phase space only in the bicrossproduct basis. This paper does it in three bases (bicrossproduct, Magueijo–Smolin, classical), shows the position–position commutator [x0, xi] = −(i/κ) xi comes out the same in all of them, and then identifies the resulting algebra — [X0, X̃i] = −iℓ² Ni, [X̃i, X̃j] = iℓ² εijk Mk with undeformed Lorentz action (eqs. 56–59) — with Snyder's 1947 non-commutative Minkowski space. That identification is the real payoff: it turns the 'which basis is physical' mess into a statement that the space-time arena is fixed even though the energy-momentum sector is not. I went through the three explicit computations and the general argument in §III; the algebra is right. Soft spots, in order of weight. The abstract says the space-time 'is unique' (s3). What §III shows is basis-independence for bases reachable from the classical one by eq. (42) with the 1/κ expansion (43)–(44); the sentence 'any DSR basis can be obtained from the classical basis in this way' is asserted, not proved. That is the load-bearing gap, and it is a scope problem rather than an error. Second, uniqueness is uniqueness within the Heisenberg double construction; a different phase-space construction could give a different algebra. Third, the ℓ observer-independence check (eq. 68) is a consistency check on covariance that was built in from the start — fine, but closer to definitional than to a new prediction, and the paper half-acknowledges this. On the pairing: I don't think = −iη is a real worry. It is the canonical evaluation pairing, and Lorentz invariance plus the compatibility condition (8) pins it up to a rescaling of x, which is a change of variables, not a different algebra. The reader's concern there is weaker than stated. One correction to the record: 'no limitations found' is wrong — the conclusions list open problems (particle dynamics, field theory, the plethora of momentum spaces) explicitly. Who should read it: anyone working on κ-Poincaré, DSR phenomenology, or Lorentz-covariant non-commutative geometry. It is short, clean, and useful. Send it out, and ask for the scope of 'unique' to be tightened.

Referee report

Significance

If the result holds, it is a genuinely useful structural observation: it relocates the physical content of DSR from the ambiguous energy-momentum sector to a unique configuration space, and it connects the DSR programme to the older Snyder non-commutative geometry and to the minimal-length/generalized-uncertainty literature. The paper earns credit for doing the explicit algebra in three separate bases rather than asserting basis-independence abstractly, and for a general argument that, while sketchy, is checkable step by step; the identification with Snyder's algebra is a concrete, falsifiable statement rather than a slogan. The claim that the deformation parameter ℓ is observer-independent, if established beyond infinitesimal boosts, would give the DSR programme a second observer-independent scale in the space-time sector, which is the paper's stated motivation.

Major objections (5)

[§III, eqs. (42)–(45)] The basis-independence of the space-time commutator, eq. (45), is proved only for bases reachable from the classical basis by the transformation (42) with the 1/κ expansion (43)–(44). The sentence 'we see that any DSR basis can be obtained from the classical basis in this way' is supported only by a counting argument — two free functions α, β in (42) versus two independent functions A, C in (39)–(40) after (41) eliminates B — and no existence or uniqueness proof for the resulting differential equations is given. Since the abstract's headline claim (abstract:s3) is that the DSR space-time is unique, this exhaustiveness step is load-bearing and should either be proved (for instance by exhibiting the flow generated by (41)) or the claim should be explicitly qualified to the class (42).
[§II, footnote 1; abstract] Footnote 1 acknowledges that a different κ-Poincaré theory exists in which the deformation is taken along a null direction, and that this case is excluded from the analysis. The abstract's unqualified statement that 'the space-time of the DSR theory is unique' therefore overstates what is shown; the uniqueness result should be stated for the timelike-deformation case, with the null-plane case either treated or explicitly declared out of scope in the abstract and conclusions.
[§V, eqs. (56)–(59)] The algebra (56)–(59) is the central object of the paper, but its consistency is asserted rather than shown: the text states that the algebra (48) together with (14), (15), (11)–(13) 'satisfies Jacobi identity', and the Snyder-form algebra (56)–(59) is presented without an explicit Jacobi check involving the position–position commutator [X̃i, X̃j] = iℓ²εijk Mk and the boost–position commutators (58). Because the whole claim of a consistent non-commutative Minkowski space with undeformed Lorentz symmetry rests on this, an explicit verification (or a demonstration that it follows automatically from the Heisenberg-double construction) should be included.
[§VI, eq. (68)] The observer-independence of ℓ is verified only to first order in the infinitesimal boost parameter ε. The DSR requirement is that ℓ be the same for any inertial observer, which requires the transformation law to hold for finite boosts; the paper should either extend the check to finite transformations or state clearly that the result is established only to leading order in 1/κ. Relatedly, the 'minimal length' interpretation of ℓ is imported from the one-dimensional generalized-uncertainty-relation literature ([21]), and the paper itself notes that its three-dimensional extension differs from that of [21]; the strength of the minimal-length reading should be calibrated accordingly.
[§I and §VI (scope of the construction)] The uniqueness established is uniqueness within the Heisenberg-double construction with the pairing (7) and the co-commutative position co-product (9). A different phase-space construction (for example a Drinfeld double, or a pairing not proportional to ημν) could in principle produce a different space-time algebra, and the paper does not argue otherwise. This scope restriction should be stated where the uniqueness claim is made, so that the abstract and conclusions do not read as uniqueness over all possible DSR phase-space constructions.

Minor objections (6)

[§II.1, physical mass formula] The displayed expression for m²_phys immediately below eq. (5) is garbled in typesetting (the exponent and the outer square are ambiguous), and the reader cannot reconstruct the intended formula; it should be re-typeset and checked against [13].
[§II.1, footnote 2] The definition of physical mass as 1/m_phys = lim_{p→0}(1/p) dp0/dp should be stated with the momentum variable and the limit prescription made explicit, since p0(p) is basis-dependent.
[§VI, eqs. (61)–(63)] The identification with Snyder's algebra would be easier to verify if the conventions were stated: Snyder's paper uses different metric and generator normalizations, and the sign of the [X0, X̃i] term relative to Ni depends on those conventions. A short dictionary would help the reader confirm that the match is exact rather than up to sign.
[§IV, eqs. (51)–(52)] In deriving (51) the bracket with Xj picks out only the linear-in-momentum term of α(P0, P̄²)Pk; it would be worth saying so explicitly, since otherwise it is not evident that the higher-order terms in the expansion (50) cannot contribute.
[Title and throughout] There are numerous typesetting errors that should be corrected: the title reads 't heories', the affiliation reads 'Wroc/suppress law' (presumably Wrocław), and the text contains 'defoted' for 'devoted', 'the the parameter m', 'the speed o light', and broken spacing in 'SP ACE-TIME', 'propos ed', 'descr ibe', 'inﬁni tely', 'space-t ime', and 'stru c-ture'.
[§VII, conclusions] The conclusions mention that the algebra (57) is the 'simplest possible extension' of the string-motivated [Xμ, Xν] = iΘμν with central Θμν, but do not note that in (57) the right-hand side is not central — it involves the Lorentz generators. This contrast with the θ-deformed case deserves one explicit sentence, since it is the feature that makes the algebra compatible with undeformed Lorentz symmetry.

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The paper

The paper claims

Doubly Special Relativity's space-time is the same in every energy-momentum basis, and it is Snyder's 1947 non-commutative Minkowski space.

The reading

Doubly Special Relativity proposes a kinematics with two observer-independent scales, the speed of light and a mass (or length) scale, as a flat-space remnant of quantum gravity. The trouble is that the energy-momentum half of the theory is wildly ambiguous: the $k$-Poincaré algebra can be written in infinitely many bases, related by non-linear redefinitions of momenta, and each basis gives different predictions for dispersion relations, maximal momenta, and the like. This paper argues that the ambiguity is confined to that sector. Using the co-product of the algebra together with the Heisenberg-double construction of the deformed phase space, the authors derive the commutators among space-time coordinates and the way boosts and rotations act on them, and find that both are independent of which energy-momentum basis one started from. The resulting space-time turns out to be exactly the non-commutative Minkowski space proposed by Snyder in 1947, with an undeformed Lorentz action and a length parameter that stays the same for every inertial observer. If this holds, DSR has a unique arena in which to formulate particle and field dynamics, even though its momentum space is not unique.

What carries the argument

The engine is the co-product of the $k$-Poincaré Hopf algebra combined with the Heisenberg double. The co-product $Δ(p_μ)$ says how momenta compose in multi-particle states; paired with the fixed bracket $<p_μ, x_ν> = -iη_{μν}$, it fixes the position commutators through $<p, xy> = <p_{(1)}, x><p_{(2)}, y>$. The cross-commutators between momenta and positions then follow from the Heisenberg-double relation $[p, x] = x_{(1)}<p_{(1)}, x_{(2)}>p_{(2)} - xp$. Because the leading correction terms in the co-product are governed by the same expansion coefficients in every basis, the resulting $[x_0, x_i]$ comes out basis-independent. The final step is a change of variables, $X̃_i = X_i - N_i/κ$, which absorbs the boost generators and reveals the Snyder algebra with its invariant $X_0^2 - X̃_iX̃_i$.

Extended reading notes

Core claim

The central claim is that the space-time sector of Doubly Special Relativity is basis-independent. The authors first show that any algebra satisfying the mild structural conditions of a $k$-DSR algebra can be reached from the classical basis by a two-function redefinition of momenta, and that each such algebra can be promoted to a Hopf algebra with a co-product. They then compute the position commutators from the co-product using the pairing $<p_μ, x_ν> = -iη_{μν}$ and the Heisenberg-double rule, and find $[x_0, x_i] = -i x_i/κ$ in every basis, with the boost action on positions likewise universal. Rewriting positions as $X̃_i = X_i - N_i/κ$ turns this into $[X_0, X̃_i] = -iℓ^2 N_i$ and $[X̃_i, X̃_j] = iℓ^2 ε_{ijk} M_k$, with $[N_i, X̃_j] = iX_0$ and $[N_i, X_0] = iX̃_j$ — precisely Snyder's algebra, whose invariant quadratic form is $X_0^2 - X̃_iX̃_i$. The authors conclude that this algebra, not any particular momentum basis, should be taken as the starting point for DSR physics.

Load-bearing premise

The uniqueness claim rests on the choice of the fixed pairing between momenta and positions, $<p_μ, x_ν> = -iη_{μν}$, together with the Heisenberg-double rule; the paper calls this pairing natural but does not show that no other admissible pairing would produce a different space-time algebra.

Editorial extensions

If this is right

The DSR program acquires a unique configuration space, so a particle Lagrangian and second-order dynamics can be built on it rather than on an arbitrarily chosen momentum basis.
The deformation parameter becomes a genuine length scale ℓ rather than an arbitrary momentum-space parameter, and the paper checks that ℓ is unchanged to first order under an infinitesimal boost.
Because the Lorentz sector is undeformed, the non-commutative Minkowski space preserves ordinary Lorentz symmetry, unlike the constant-$Θ$ non-commutativity familiar from string-inspired models.
The one-dimensional reduction of the Snyder commutator $[P_i, X̃_j] = -iδ_{ij} - iℓ^2P_iP_j$ reproduces the deformed commutator long used to argue for a minimal length in generalized uncertainty relations.
The mismatch between a unique space-time and a plethora of momentum spaces becomes an explicit problem: something must select which momentum variables are the physical ones.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the space-time algebra is the real invariant content of DSR, then the physical momentum should be defined dynamically — through particle motion or field theory on this space-time — rather than by picking a favoured algebraic basis, which suggests the basis debate is a red herring for phenomenology.
The construction leans on one chosen pairing; a natural check is whether the Heisenberg double with any other admissible Lorentz-covariant pairing yields the same position commutators, or whether 'unique' really means 'unique given this pairing'.
Since the Snyder algebra is essentially the algebra of the Lorentz group acting on a de Sitter-like momentum space, one might expect the DSR phase space to inherit a curved momentum geometry, connecting this result to the relative-locality programme.
The minimal length appearing here is a quantum-mechanical one, so a sharp test would come from a quantum field theory built on this space-time: any observable deviation from standard dispersion would have to survive the undeformed Lorentz symmetry.

Original abstract

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there is infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the $\kappa$-Poincar\'e quantum (Hopf) algebra. Then we use the co-product of this algebra and the known construction of $\kappa$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space-time of the DSR theory is unique and equivalent to the theory with non-commutative space-time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space-time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space-time, its intrinsic length parameter $\ell$ becomes observer-independent.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there is infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the $\kappa$-Poincar\'e quantum (Hopf) algebra. Then we use the co-product of this algebra and the known construction of $\kappa$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space-time of the DSR theory is unique and equivalent to the theory with non-commutative space-time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space-time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space-time, its intrinsic length parameter $\ell$ becomes observer-independent.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity.	background 1.00	0.06	0.03	asserted 0.98
s1	We observe that there is infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the $\kappa$-Poincar\'e quantum (Hopf) algebra.	result 0.95	0.75	0.06	asserted 1.00
s2	Then we use the co-product of this algebra and the known construction of $\kappa$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR phase space.	method 0.98	0.92	0.07	asserted 1.00
s3	Next we show that contrary to the ambiguous structure of the energy momentum sector, the space-time of the DSR theory is unique and equivalent to the theory with non-commutative space-time proposed long ago by Snyder.	result 1.00	0.88	0.75	asserted 1.00
s4	This theory provides non-commutative version of Minkowski space-time enjoying ordinary Lorentz symmetry.	result 0.58	0.44	0.16	asserted 1.00
s5	It turns out that when one builds a natural phase space on this space-time, its intrinsic length parameter $\ell$ becomes observer-independent.	result 0.98	0.83	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (23)

[21]

A. Kempf, G. Mangano, R. Mann, Phys. Rev. D52, 1108 (1995) [ hep-th/9412167]; A. Kempf, G. Mangano, Phys. Rev. D55, 1108 (1997) [ hep-th/9612084]

View on Pith Pith review arXiv 1995
[1]

G. Amelino-Camelia, gr-qc/0204051 and references therein

View on Pith Pith review arXiv
[2]

G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002), [gr-qc/0012051]

View on Pith Pith review arXiv 2002
[3]

G. Amelino-Camelia, Phys. Lett. B 510 , 255, (2001), [hep-th/0012238]

View on Pith Pith review arXiv 2001
[4]

G. Amelino-Camelia, gr-qc/0106004, Proceedings of the 37th Karpacz Winter School on Theoretical Physics, to appear

View on Pith Pith review arXiv
[5]

J. Kowalski-Glikman, Phys. Lett. A 286 , 391 (2001), [hep-th/0102098]

View on Pith Pith review arXiv 2001
[6]

N.R. Bruno, G. Amelino-Camelia, and J. Kowalski- Glikman, Phys. Lett. B 522 , 133 (2001), [hep-th/0107039]

View on Pith Pith review arXiv 2001
[7]

J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. B264, 331 (1991)

View on Pith 1991
[8]

S. Majid and H. Ruegg, Phys. Lett. B334, 348 (1994)

View on Pith 1994
[9]

H.S. Snyder, Phys. Rev. 71, 38 (1947)

View on Pith 1947
[10]

H. Ruegg in Integrable Systems, Quantum Groups, and Quantum Field Theories, L.A. Ibort and M.A. Rodr ´iguez eds., NATO ASI series, vol. 409, Kluwer Academic Pub- lishers, Dordrecht, 1993; J. Lukierski, H. Ruegg and V.N. Tolstoy, in Quantum Groups: Formalism and Ap- plications, Polish Scientiﬁc Publishers, (1995), p. 259

View on Pith 1995
[11]

P. Kosi´ nski, J. Lukierski, P. Maslanka, J. Sobczyk, Mod. Phys. Lett. A10, 2599 (1995)

View on Pith 1995
[12]

J. Magueijo and L. Smolin, hep-th/0112090

View on Pith Pith review arXiv
[13]

J. Kowalski-Glikman and S. Nowak, hep-th/0203040

View on Pith Pith review arXiv
[14]

J. Lukierski and A. Nowicki, hep-th/0203065

View on Pith Pith review arXiv
[15]

J. Lukierski and A. Nowicki, Proceedings of Quantum Group Symposium at Group 21, (July 1996, Goslar) Eds. H.-D. Doebner and V.K. Dobrev, Heron Press, Soﬁa, 1997, p. 186

View on Pith 1996
[16]

A. Nowicki, in Proceedings of IX Max Born Sympo- sium New Symmetries in the Theories of Fundamental Interactions, Polish Scientiﬁc Publishers, (1997), p. 43 [math.QA/9803064]

View on Pith arXiv 1997
[17]

P. Ma´ slanka, J. Math. Phys. 34, 6025 (1993)

View on Pith 1993
[18]

J. Lukierski, Proceedings of Quantum Group Symposium at Group 21, (July 1996, Goslar) Eds. H.-D. Doebner and V.K. Dobrev, Heron Press, Soﬁa, 1997, p. 173

View on Pith 1996
[19]

J. Lukierski, H. Ruegg and W.J. Zakrzewski, Ann. Phys. 243, 90 (1995)

View on Pith 1995
[20]

J. Lukierski, Proceedings of the III International Wor k- shop on Classical and Quantum Integrable Systems, JINR (1998), [ hep-th/9812063]

View on Pith Pith review arXiv 1998
[22]

M.R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2002) [ hep/th-0106048]

View on Pith 2002
[23]

H.S. Snyder, Phys. Rev. 71, 68 (1947)

View on Pith 1947

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Papers citing it · 28 sentences without a confident label

Uses its method or tool 3 papers

Following this algebraic procedure [ref] one obtains that the phase space Poisson brackets associated to the coproduct of momenta in the bicrossproduct basis of $κ$-Poincaré is {x̃^0, x̃^i} = -x̃^i/Λ , {x̃^0, p_0} = -1 , {x̃^0, p_i} = p_i/Λ , {x̃^i, p_j} = -δ^i_j , {x̃^i, p_0} = 0 .

Spacetime from locality of interactions in deformations of special relativity: the example of $\kappa$-Poincar\'e Hopf algebra · The case of $κ$-Poincaré Hopf algebra · 0.91
The “pairing” operation of Hopf algebras formalism allows us to determine the modified Heisenberg algebra for a given coproduct [ref] .

Beyond special relativity and the notion of spacetime · Noncommutative spacetime in k-Poincaré Hopf algebra · 0.87
Then, we can change the basis of [ref], [ref]-[ref] by the following transformation (see e.g. [ref] )

Towards new relativistic doubly $\kappa$-deformed D=4 quantum phase spaces · Doubly $κ$-deformation directly from Yang model · 0.85

Compares with it 1 paper

The boost transformation for the two momentum variables $(p, q)$ [Eqs. [ref]-[ref]] is related with the coproduct of boosts of $κ$-Poincaré in the bicrossproduct basis [ref] Δ(N_i) = N_i ⊗I + e^-P_0/Λ ⊗N_i + 1/Λ ϵ_ijkP_j ⊗M_k ,

Spacetime from locality of interactions in deformations of special relativity: the example of $\kappa$-Poincar\'e Hopf algebra · Lorentz transformation in the bicrossproduct basis from locality · 0.88

Cites it as background 19 papers

For this purpose we return here to the simple perspective and formalization of the first studies [ref] which noticed that some known deformations of relativistic symmetries, some of the ones studied within the “doubly-special relativity" research programme (see, e.g., Refs. [ref] ), could result in the property that coincidences of events established by nearby observers might not appear as coincidences of events in the coordinatization of spacetime by distant observers.

Modeling transverse relative locality · Introduction · 0.80
It has been shown to describe the deformed symmetries of so-called $κ$-Minkowski spacetime [ref] [ref] described by the commutation relations of the $𝔞𝔫(3)$ Lie algebra: [equation] where the deformation parameter $1/κ$ corresponds to an invariant length.

Complex scalar field in \kappa-Minkowski noncommutative spacetime · Introduction · 0.85
These include $κ$-Minkowski spacetime and the associated momentum space of the $κ$-Poincaré algebra, and also Snyder spacetime (both examples are reviewed in [ref] , and they were considered in relation with the calculation of photon time delays in Refs. [ref] and [ref] , respectively).

Does a deformation of special relativity imply energy dependent photon time delays? · A general model for photon time delays in noncommutative spacetimes · 0.96

Discussion

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[1] [21]

A. Kempf, G. Mangano, R. Mann, Phys. Rev. D52, 1108 (1995) [ hep-th/9412167]; A. Kempf, G. Mangano, Phys. Rev. D55, 1108 (1997) [ hep-th/9612084]

View on Pith Pith review arXiv 1995

[2] [1]

G. Amelino-Camelia, gr-qc/0204051 and references therein

View on Pith Pith review arXiv

[3] [2]

G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002), [gr-qc/0012051]

View on Pith Pith review arXiv 2002

[4] [3]

G. Amelino-Camelia, Phys. Lett. B 510 , 255, (2001), [hep-th/0012238]

View on Pith Pith review arXiv 2001

[5] [4]

G. Amelino-Camelia, gr-qc/0106004, Proceedings of the 37th Karpacz Winter School on Theoretical Physics, to appear

View on Pith Pith review arXiv

[6] [5]

J. Kowalski-Glikman, Phys. Lett. A 286 , 391 (2001), [hep-th/0102098]

View on Pith Pith review arXiv 2001

[7] [6]

N.R. Bruno, G. Amelino-Camelia, and J. Kowalski- Glikman, Phys. Lett. B 522 , 133 (2001), [hep-th/0107039]

View on Pith Pith review arXiv 2001

[8] [7]

J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. B264, 331 (1991)

View on Pith 1991

[9] [8]

S. Majid and H. Ruegg, Phys. Lett. B334, 348 (1994)

View on Pith 1994

[10] [9]

H.S. Snyder, Phys. Rev. 71, 38 (1947)

View on Pith 1947

[11] [10]

H. Ruegg in Integrable Systems, Quantum Groups, and Quantum Field Theories, L.A. Ibort and M.A. Rodr ´iguez eds., NATO ASI series, vol. 409, Kluwer Academic Pub- lishers, Dordrecht, 1993; J. Lukierski, H. Ruegg and V.N. Tolstoy, in Quantum Groups: Formalism and Ap- plications, Polish Scientiﬁc Publishers, (1995), p. 259

View on Pith 1995

[12] [11]

P. Kosi´ nski, J. Lukierski, P. Maslanka, J. Sobczyk, Mod. Phys. Lett. A10, 2599 (1995)

View on Pith 1995

[13] [12]

J. Magueijo and L. Smolin, hep-th/0112090

View on Pith Pith review arXiv

[14] [13]

J. Kowalski-Glikman and S. Nowak, hep-th/0203040

View on Pith Pith review arXiv

[15] [14]

J. Lukierski and A. Nowicki, hep-th/0203065

View on Pith Pith review arXiv

[16] [15]

J. Lukierski and A. Nowicki, Proceedings of Quantum Group Symposium at Group 21, (July 1996, Goslar) Eds. H.-D. Doebner and V.K. Dobrev, Heron Press, Soﬁa, 1997, p. 186

View on Pith 1996

[17] [16]

A. Nowicki, in Proceedings of IX Max Born Sympo- sium New Symmetries in the Theories of Fundamental Interactions, Polish Scientiﬁc Publishers, (1997), p. 43 [math.QA/9803064]

View on Pith arXiv 1997

[18] [17]

P. Ma´ slanka, J. Math. Phys. 34, 6025 (1993)

View on Pith 1993

[19] [18]

J. Lukierski, Proceedings of Quantum Group Symposium at Group 21, (July 1996, Goslar) Eds. H.-D. Doebner and V.K. Dobrev, Heron Press, Soﬁa, 1997, p. 173

View on Pith 1996

[20] [19]

J. Lukierski, H. Ruegg and W.J. Zakrzewski, Ann. Phys. 243, 90 (1995)

View on Pith 1995

[21] [20]

J. Lukierski, Proceedings of the III International Wor k- shop on Classical and Quantum Integrable Systems, JINR (1998), [ hep-th/9812063]

View on Pith Pith review arXiv 1998

[22] [22]

M.R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2002) [ hep/th-0106048]

View on Pith 2002

[23] [23]

H.S. Snyder, Phys. Rev. 71, 68 (1947)

View on Pith 1947