Relativistic time-commutative dynamics with $\kappa$-plane noncommutativity

Alessandro Moia; Giovanni Amelino-Camelia; Stefano Stocchetti

REVIEW 2 major objections 4 minor 32 references

Relativistic time-commutative dynamics with $\kappa$-plane noncommutativity

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that deformed relativistic symmetries on a noncommutative spacetime can be represented by ordinary first-quantized observables, and that for the time-commutative κ-plane two distinct, consistent two-particle dynamics exist

desk verdict A substantial, mostly checkable construction of the TCκ first-quantized framework; the interacting sector is the soft spot, and the abstract oversells 'fully relativistic.' read the letter →

arxiv 2607.15261 v1 pith:TMAF7YJH submitted 2026-07-16 hep-th gr-qcmath-phmath.MP

Alessandro Moia , Stefano Stocchetti , Giovanni Amelino-Camelia This is my paper · ORCID

classification hep-thgr-qcmath-phmath.MP MSC 81R5081R60

keywords time-commutativeκ-planedeformedPoincarésymmetryHopfalgebranoncommutativespacetimecovariantquantummechanicsGalileanlimitharmonicpotentialtwo-particlecompositionlaw

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that deformed relativistic symmetries on a noncommutative spacetime can be described by ordinary first-quantized observables, bypassing the usual Noether-theorem obstruction. Specializing to the 2+1D time-commutative κ-plane, it constructs the full deformed Poincaré Hopf algebra and a canonical symmetry algebra at first order in the noncommutativity scale ℓ. In the Galilean limit, it identifies deformed position operators and a deformed free Hamiltonian, then analyzes two particles interacting through a harmonic potential. It finds two admissible composition laws for the two-particle symmetry generators, each requiring a specific deformation of the harmonic potential, so that the structure of the generators and the interaction law are interdependent.

What carries the argument

The central object is the canonical symmetry algebra CGκ (with its Poincaré parent CPκ), defined by explicit deformed commutators such as [k̂i,p̂j] = i(δijm + ℓϵijm p̂1 − ℓδi1δj1m p̂2) and [r̂,p̂i] = i(ϵikp̂k + (ℓ/2)δi1p̂₁² − ℓδi1p̂₂²). The invariant constraint (117) fixes the deformed Galilean Hamiltonian, and the deformed position operators x̂j(0) (143) realize the κ-plane noncommutativity on the physical Hilbert space. These objects carry the argument: symmetry covariance is checked by imposing that the two-particle generators satisfy the same CGκ algebra, which determines both the composition laws and the required deformations of the harmonic potential.

What would settle it

Explicitly compute the cross-particle commutator [x̂A_1(0), x̂B_2(0)] in the two-particle Hilbert space using the TCκ coproduct or braiding; any nonzero result at order ℓ would contradict the tensor-product ansatz and invalidate the two potentials. Alternatively, check the Jacobi identities for the proposed two-particle generators; a violation at order ℓ would falsify the claim.

Watch

Extended reading notes

Core claim

For the time-commutative κ-plane, defined by [x0,xj]=0 and [x1,x2]=iℓx1, the physically relevant symmetry algebra is not the abstract Hopf algebra but its canonical counterpart CPκ, generated by pμ, r, and ni with deformed mixed commutators and invariant constraint Cκ = pαpα − m²c² − ℓp₁²p₂. Taking c→∞ yields the Galilean canonical algebra CGκ, in which boost-momentum commutators acquire order-ℓ mass-dependent terms. For a single particle, position observables x̂j(0) satisfying the κ-plane commutator [x̂1(0),x̂2(0)] = iℓx̂1(0) are identified, and the free Hamiltonian becomes ĥ = (p̂kp̂k + ℓp̂₁²p̂₂)/2m. For two particles with a harmonic interaction, ordinary additive composition of generators

Load-bearing premise

The two-particle analysis assumes the composite observable algebra is the ordinary tensor product O_tot = O1 ⊗ O2, so position operators of different particles commute, even though the TCκ coproducts are noncocommutative; if a braided tensor product is required instead, the composition laws and deformed potentials (178) and (189) would be modified by order-ℓ terms and the claimed two consistent models could fail.

Editorial extensions

If this is right

If the construction is correct, deformed spacetime symmetries yield genuine conserved observables, so Noether-like consequences survive without a generalized Noether theorem.
The total momentum of two interacting particles cannot be the naive sum; consistent dynamics force nonlinear composition laws.
The harmonic interaction must be deformed at order ℓ; the usual potential would break deformed Galilean covariance.
The two admissible composition laws tie the momentum-space geometry (one gives a de Sitter momentum space) to the form of the interaction.
The single-particle Hilbert-space realization gives explicit differential operators, enabling concrete spectra, expectation values, and uncertainty-principle computations.

Reading between the lines

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

The main open threat: because the TCκ coproducts are noncocommutative, multi-particle states may require a braided tensor product rather than the ordinary tensor product assumed in Eq. (153); if so, cross-particle position commutators become nonzero at order ℓ and both composition laws and potentials (178) and (189) would need modification.
A systematic classification of admissible composition laws for generic Lie-algebra spacetimes could reveal a physically distinguished law selected by requirements such as cluster decomposition or separability of distant subsystems.
The sign reversal between the abstract Hopf commutators and the canonical commutators warns that phenomenological predictions should be derived from the canonical algebra, not the abstract one; using the wrong algebra can flip the order-ℓ corrections.
The same covariant-quantum-mechanics construction could be applied to other noncommutative spacetimes, generating deformed Hamiltonians and two-particle composition laws with potentially testable low-energy signatures.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear X,

Quick take: this is the real thing, not a sketch. The authors construct a first-order TCκ Poincaré Hopf algebra with boosts and time translations (Sec. 3), the corresponding canonical symmetry algebra CPκ, its Galilean limit CGκ, a free-particle deformed Heisenberg algebra, and two consistent two-particle composition laws with deformed harmonic potentials. I spot-checked (89)-(91) against (96)-(98) and (134) against (117); they pass. The construction is derived, not fitted, and it goes well beyond the heuristics of [14], correcting a sign error along the way. The resolution of the 'no-pure-boost' puzzle—separating abstract Hopf generators from canonical observables—is the most convincing part of the paper and makes physical sense.

Soft spots, in rough order. The abstract says 'fully relativistic' two-particle model; the body only ever delivers Galilean-relativistic dynamics for interacting particles. That mismatch should be fixed before publication. The general Noether-like claim is conditional on the ansatz (38)-(39); the paper itself defers the domain question to a footnote and the conclusion. It is an honest limitation, but it means the broad framing is stronger than the proven result. The two-particle section is algebraic only: no physical Hilbert space for the composite system is constructed, and I did not find a check that the total generators preserve the defining relations of O_tot, in particular the cross-particle commutators [x^A,x^B]=0. The ordinary tensor product in (153) is an assumption, not a consequence. That said, the stress-test worry that a braided tensor product is forced by the noncocommutative coproduct does not land here: the authors are representing the canonical Lie algebra CGκ by derivations on an algebra, not building an H-module algebra, so ordinary tensor product is not automatically wrong. But it needs justification or at least an explicit physical argument. Minor: coassociativity/compatibility (60) is asserted in one sentence; probably routine, but a reader shouldn't have to take it on faith.

Who is this for? Anyone working on κ-Minkowski/DSR phenomenology or first-quantized noncommutative models. It deserves a serious referee and, after the abstract/overclaim and two-particle consistency checks are dealt with, could be a useful reference. I would send it to review.

Referee Report

2 major / 4 minor

Summary. The paper develops a first-order-in-ℓ deformation of the 2+1D Poincaré and Galilei symmetries for the time-commutative κ-plane, and uses it to build first-quantized particle models. It first constructs a TCκ Poincaré Hopf algebra with coproducts and antipodes (Section 3), then introduces a 'canonical' symmetry algebra CPκ represented by operators in covariant quantum mechanics (Section 4), and takes the Galilean limit to obtain the canonical algebra CGκ (Section 5). A single-particle model is then formulated as an irreducible representation of CGκ, with deformed position operators and a deformed free Hamiltonian (Section 6). The final section studies two particles interacting through a deformed harmonic potential and claims two admissible composition laws for the deformed Galilei generators, each tied to a specific deformation of the interaction potential.

Significance. If the two-particle construction is accepted, this is a valuable proof-of-concept: it shows that a deformed relativity principle can be realized in first-quantized models, that symmetry generators can be connected to observables in a Noether-like way, and that interaction potentials and composition laws can be consistently deformed at first order in ℓ. The algebraic core is largely self-contained, the free-particle Hamiltonian (134) follows from the invariant constraint (117), and the checkable commutators, e.g. (89)-(91) versus (96)-(98), are consistent. The paper is explicit that results are first-order in ℓ, and the main limitation is the assumption that the two-particle observable algebra is the ordinary tensor product (153). The central claims are accordingly conditional, but the general construction is a useful step beyond the heuristic treatment of Ref. [14].

major comments (2)

[§7, Eq. (153)] The composite algebra is taken as the ordinary tensor product O_tot = O^1 ⊗ O^2, so cross-particle position operators commute. This is a physical assumption that is not derived from the TCκ Hopf structure. The coproducts (46)-(48) are non-cocommutative, and κ-deformed multi-particle systems are often formulated with a braided tensor product, for which [x_i^A, x_j^B] can be of order ℓ. The potentials (178) and (189) and the composition laws (158)-(163), (168)-(169), (179)-(181), (184) are derived under the ordinary-product assumption. If the correct composite algebra is braided, these cross-particle commutators change at order ℓ and the claimed consistency with CGκ could fail. Please justify the ordinary tensor product as the physically appropriate choice for TCκ, or redo the two-particle analysis with the braided product and show that the results are unaffected at order ℓ.
[§3, Eqs. (45)-(60); §7, Eqs. (178)-(189)] Several load-bearing algebraic claims are asserted without proof. The coassociativity of the coproducts (45)-(48) is stated as 'easy to verify', the homomorphism compatibility (60) is stated as a 'tedious but straightforward computation', and the two-particle potentials (178) and (189) are said to follow from 'enforcing conditions' (175)-(177) and (186)-(188) without showing the computation. These are central to the paper: the Hopf algebra Pκ supports the whole construction, and the two-particle admissibility is the main new result. The authors should supply the explicit verifications in an appendix or as ancillary material.

minor comments (4)

[§2 and Conclusions] The paper correctly acknowledges that the general framework is conditional on the linear ansatz (38)-(39), whose full domain of validity is deferred to future work. This caveat should be stated prominently in the abstract or introduction, since the general Noether-like claim is presented rather broadly.
[§7, Eqs. (170)-(173)] The index contractions in the two-particle commutators, e.g. δ_{1i} δ_{j2} δ_{1k} P_j P_k and δ_{ij} δ_{1j} P_2, are hard to parse. Please spell out the summation conventions or rewrite them with explicit sums.
[Title/Abstract] The words 'fully consistent' and 'full characterization' should be qualified by 'at first order in ℓ', since all constructions are at leading order in the noncommutativity parameter.
[§7] Minor typos: 'coalgebric' should be 'coalgebraic'; 'no more' should be 'no longer' in places. Ref. [29] is cited for homogeneous spaces but not for braided tensor products; if braiding is to be dismissed, give a specific reference or argument.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; core derivations are self-contained and conditional ansätze are stated, not disguised.

full rationale

The derivation chain is self-contained at the level claimed. The core objects—the TCκ Hopf algebra Pκ (Eqs. 45–48), the canonical algebra CPκ (Eqs. 92–98), the constraint (99), the Galilean limit CGκ (Eqs. 110–116), the free-particle observables (Eqs. 144–148), and the two-particle potentials (178)/(189)—are obtained by explicit computation from the stated commutation rules and ansätze, not by importing an input as a prediction. The general Noether-like step is explicitly conditional on the representation ansatz (38)–(39), which is then verified for TCκ; any residual conditionality is admitted rather than hidden. Reliance on prior work (Refs. [14,24]) is motivational or heuristic: Ref. [14] is described as heuristic and is corrected (wrong sign), and Ref. [24] supplies the CQM framework but is re-derived in Section 2. The two-particle analysis makes an explicit structural assumption, the ordinary tensor product O_tot = O1 ⊗ O2 (153); if the correct multiparticle algebra for the noncocommutative TCκ coproduct were braided, cross-particle commutators could change, but this is a stated assumption and a possible correctness gap, not a circular reduction. The potentials (178) and (189) are solved from the invariance conditions [P_i,V]=[R,V]=[K_i,V]=0, so they are consistency requirements, not fitted data. No step reduces by construction to its own input. Score 1 reflects the presence of self-citations and explicit ansätze, none load-bearing.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the CQM framework, the linear-ansatz representation of transformation parameters, first-order truncation in ℓ, and the ordinary tensor product for two particles. No new physical entities are introduced; the deformed position operators and canonical generators are constructed, not postulated.

assumptions (5)

domain assumption Covariant quantum mechanics framework: kinematical Hilbert space, Hamiltonian constraint, physical observables commute with the constraint.
Adopted from Refs. [19-23]; the entire construction of canonical generators in Secs. 2-4 lives in this framework.
ad hoc to paper The linear ansatz (38)-(39) for noncommutative transformation parameters as functions of real parameters.
The general Noether-like result is conditional on this representation; the paper states its full domain 'will be investigated elsewhere' (footnote 5). It is verified explicitly for TCκ in Sec. 4.
domain assumption First-order truncation in the noncommutativity scale ℓ.
All deformed algebras, constraints, and potentials are computed only at first order in ℓ; no proof is given that the structure lifts to all orders.
ad hoc to paper Ordinary tensor product for the two-particle algebra (Eq. 153), with no braiding.
Assumed without discussion despite noncocommutative coproducts (46)-(48); load-bearing for the two-particle composition laws and potentials.
standard math Stone-von Neumann uniqueness of the irreducible representation of the Heisenberg algebra.
Used in Sec. 6 to represent the auxiliary canonical algebra on L2(R2, d2p) via the momentum representation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Relativistic time-commutative dynamics with $\kappa$-plane noncommutativity." pith.science (2026). https://pith.science/paper/TMAF7YJH

@misc{pith2026260715261,
  author       = {Pith},
  title        = {Pith review of: Relativistic time-commutative dynamics with $\kappa$-plane noncommutativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMAF7YJH}},
  note         = {Machine review of arXiv:2607.15261}
}

abstract

In the last decades, spacetime noncommutativity and the associated deformations of relativistic symmetries have attracted a lot of interest, as several phenomenological windows into quantum gravity are approaching genuine Planck-scale sensitivity. However, the physical significance of the mathematical structures introduced to deal with spacetime noncommutativity is still debated, and some crucial pieces are missing or poorly understood. From time to time it has been suggested that valuable insight into these conceptual challenges could come from the analysis of appropriately designed first-quantized toy models, in which major technical and interpretive issues can be effectively managed or sidestepped altogether. In this paper we take seriously this suggestion and develop the first fully consistent and fully relativistic first-quantized model of two particles propagating and interacting on a noncommutative spacetime. The specific spacetime noncommutativity implemented in our model, which we call 'time-commutative $\kappa$-plane', had already been proposed as a suitable arena for a first-quantized analysis, but previous studies were mostly heuristic and failed to provide a full description of the deformed relativistic symmetries. We here go much beyond those pioneering attempts: we find a full characterization of the appropriate deformed Poincar\'e symmetry algebra as well as its Galilean limit; we build a single-particle quantum model carrying an irreducible representation of the deformed Galilei algebra; we exhibit two consistent, Galilean-relativistic descriptions of a system of two quantum particles interacting via a deformed harmonic potential, finding that the structure of the two-particle symmetry generators is intimately connected with the deformation of the interaction law.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references

[14]

Amelino-Camelia, G

G. Amelino-Camelia, G. Fabiano and D. Frattulillo,Total momentum and other noether charges for particles interacting in a quantum spacetime, Symmetry17(2025), 227

2025
[1]

Amelino-Camelia,Quantum-spacetime phenomenology, Living Rev

G. Amelino-Camelia,Quantum-spacetime phenomenology, Living Rev. Relativ.16 (2013), 5

2013
[2]

Majid,Hopf algebras for physics at the Planck scale, Class

S. Majid,Hopf algebras for physics at the Planck scale, Class. Quantum Grav.5(1988), 1587

1988
[3]

V . G. Drinfeld,Quantum groups, J. Math. Sci.41(1988), 898

1988
[4]

Majid,Foundations of quantum group theory, Cambridge University Press (2000)

S. Majid,Foundations of quantum group theory, Cambridge University Press (2000)

2000
[5]

Majid,Quantum groups and noncommutative geometry, J

S. Majid,Quantum groups and noncommutative geometry, J. Math. Phys41(2000), 3892

2000
[6]

Amelino-Camelia and S

G. Amelino-Camelia and S. Majid,Waves on noncommutative space–time and gamma- ray bursts, Int. J. Mod. Phys. A15(2000), 4301

2000
[7]

Szabo,Quantum field theory on noncommutative spaces, Phys

R. Szabo,Quantum field theory on noncommutative spaces, Phys. Rept.378(2003), 207

2003

Show all 32 references

[8]

Agostini, G

A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marcianó and R. Altair Tacchi,Gener- alizing the Noether theorem for Hopf-algebra spacetime symmetries, Mod. Phys. Lett22 (2007), 1779

2007
[9]

Amelino-Camelia, G

G. Amelino-Camelia, G. Gubitosi, A. Marcianò, P. Martinetti and F. Mercati,A no-pure- boost uncertainty principle from spacetime noncommutativity, Phys.Lett. B671(2009), 298

2009
[10]

Freidel, J

L. Freidel, J. Kowalski-Glikman and S. Nowak,Field theory onκ-minkowski space revis- ited: Noether charges and breaking of lorentz symmetry, Int. J. Mod. Phys. A23(2008), 2687. 31

2008
[11]

Amelino-Camelia, F

G. Amelino-Camelia, F. Briscese, G. Gubitosi, A. Marcianò, P. Martinetti and F. Mercati, Twisted Hopf symmetries of canonical noncommutative spacetimes and the no-pure-boost principle, Phys. Rev. D78(2008), 025005

2008
[12]

Moia,Noncommutative spacetime symmetries from covariant quantum mechanics Adv

A. Moia,Noncommutative spacetime symmetries from covariant quantum mechanics Adv. High Energy Phys.2017(2017), 4042314

2017
[13]

Amelino-Camelia,Planck-scale soccer-ball problem: A case of mistaken identity, En- tropy19(2017), 400

G. Amelino-Camelia,Planck-scale soccer-ball problem: A case of mistaken identity, En- tropy19(2017), 400

2017
[15]

Majid and H

S. Majid and H. Ruegg,Bicrossproduct structure of the k-Poincare group and non- commutative geometry, Phys. Lett. B.334(1994), 348

1994
[16]

Lukierski, H

J. Lukierski, H. Ruegg and W.J. Zakrzewski,Classical and Quantum Mechanics of Free Relativistic Systems, Ann. Phys.243(1995), 90

1995
[17]

Lukierski, A

J. Lukierski, A. Nowicki and H. Ruegg,New quantum Poincaré algebra andκ-deformed field theory, Phys. Lett. B293(1992), 344

1992
[18]

Ballentine,Quantum mechanics: a modern development, World Scientific (1998)

L.E. Ballentine,Quantum mechanics: a modern development, World Scientific (1998)

1998
[19]

Halliwell,Trajectories for the wave function of the universe from a simple detector model, Phys

J. Halliwell,Trajectories for the wave function of the universe from a simple detector model, Phys. Rev. D64(2001), 044008

2001
[20]

Gambini and R

R. Gambini and R. A. Porto,Relational time in generally covariant quantum systems: Four models, Phys. Rev. D63(2001), 105014

2001
[21]

Reisenberger and C

M. Reisenberger and C. Rovelli,Spacetime states and covariant quantum theory, Phys. Rev. D65(2002), 125016

2002
[22]

Teitelboim,Quantization of Gauge Systems, Princeton University Press (1992)

M.Henneaux and C. Teitelboim,Quantization of Gauge Systems, Princeton University Press (1992)

1992
[23]

Marolf,Refined algebraic quantization: Systems with a single constraint, Banach Cen- ter Publications39(1997), 331

D. Marolf,Refined algebraic quantization: Systems with a single constraint, Banach Cen- ter Publications39(1997), 331

1997
[24]

Amelino-Camelia, V

G. Amelino-Camelia, V . Astuti and G. Rosati,Relative locality in a quantum spacetime and the pregeometry ofκ-minkowski, Eur. Phys. J. C73(2013), 2521

2013
[25]

Amelino-Camelia, Valerio Astuti and G

G. Amelino-Camelia, Valerio Astuti and G. Rosati,Predictive description of planck-scale- induced spacetime fuzziness, Phys. Rev. D87(2013), 084023

2013
[26]

Woronowicz,Differential calculus on compact matrix pseudogroups (quantum groups), Commun.Math

S.L. Woronowicz,Differential calculus on compact matrix pseudogroups (quantum groups), Commun.Math. Phys.122(1989), 125

1989
[27]

Currie, T.F

D.G. Currie, T.F. Jordan, and E.C.G. Sudarshan,Relativistic invariance and hamiltonian theories of interacting particles, Rev. Mod. Phys.35(1963), 350

1963
[28]

Majid,A Quantum Groups Primer, Cambridge University Press, 2002

S. Majid,A Quantum Groups Primer, Cambridge University Press, 2002. 32

2002
[29]

Mercati,T-minkowski noncommutative spacetimes I: Poincaré groups, differential cal- culi, and braiding, Prog

F. Mercati,T-minkowski noncommutative spacetimes I: Poincaré groups, differential cal- culi, and braiding, Prog. Theor. Exp. Phys.2024(2024), 073B06

2024
[30]

Strocchi,An Introduction to Non-Perturbative Foundations of Quantum Field Theory, International Series of Monographs on Physics, Oxford University Press (2013)

F. Strocchi,An Introduction to Non-Perturbative Foundations of Quantum Field Theory, International Series of Monographs on Physics, Oxford University Press (2013)

2013
[31]

Strocchi,An Introduction to the Mathematical Foundation of Quantum Mechanics, World Scientific (2005)

F. Strocchi,An Introduction to the Mathematical Foundation of Quantum Mechanics, World Scientific (2005)

2005
[32]

Amelino-Camelia, G

G. Amelino-Camelia, G. Gubitosi and G. Palmisano,Pathways to relativistic curved mo- mentum spaces: de Sitter case study, Int. J. Mod. Phys. D25(2016), 1650027. 33

2016

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [14]

Amelino-Camelia, G

G. Amelino-Camelia, G. Fabiano and D. Frattulillo,Total momentum and other noether charges for particles interacting in a quantum spacetime, Symmetry17(2025), 227

2025

[2] [1]

Amelino-Camelia,Quantum-spacetime phenomenology, Living Rev

G. Amelino-Camelia,Quantum-spacetime phenomenology, Living Rev. Relativ.16 (2013), 5

2013

[3] [2]

Majid,Hopf algebras for physics at the Planck scale, Class

S. Majid,Hopf algebras for physics at the Planck scale, Class. Quantum Grav.5(1988), 1587

1988

[4] [3]

V . G. Drinfeld,Quantum groups, J. Math. Sci.41(1988), 898

1988

[5] [4]

Majid,Foundations of quantum group theory, Cambridge University Press (2000)

S. Majid,Foundations of quantum group theory, Cambridge University Press (2000)

2000

[6] [5]

Majid,Quantum groups and noncommutative geometry, J

S. Majid,Quantum groups and noncommutative geometry, J. Math. Phys41(2000), 3892

2000

[7] [6]

Amelino-Camelia and S

G. Amelino-Camelia and S. Majid,Waves on noncommutative space–time and gamma- ray bursts, Int. J. Mod. Phys. A15(2000), 4301

2000

[8] [7]

Szabo,Quantum field theory on noncommutative spaces, Phys

R. Szabo,Quantum field theory on noncommutative spaces, Phys. Rept.378(2003), 207

2003