REVIEW
Covariant realizations of kappa-deformed space
High Energy Physics - Theory · Submitted 27 Feb 2007 · 46 references
Pith's verdict: A clean classification of covariant kappa-realizations with a useful coproduct/star-product dictionary, but the abstract's 'all realizations are physically equivalent' is only demonstrated for type I.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
The noncommutative κ-deformed space admits infinitely many rotation-covariant rewritings, and the authors argue they all describe the same physics.
The reading
What carries the argument
The load-bearing device is the shift operator $Z$, defined by $[Z,�hat{x}_��]=ia_��Z$ and $[Z,D_��]=0$, which compresses the deformed Heisenberg algebra into the compact form $[D_��,�hat{x}_��]=��_{�岒}Z^{-1}+ia_��D_��$ and yields the rotation generators as $M_{�岒}=(�hat{x}_��D_��-�hat{x}_��D_��)Z$. The second device is the free parameter function $ϕ(A,B)$: it labels the infinite family of realizations, and the coefficients $c_k$ of the similarity exponent $U=(x∂)Σ_k c_k A^k$ are fixed recursively by the derivatives of $ϕ$ at the origin, which is what makes the equivalence of type-I star products explicit rather than assumed.
Extended reading notes
Core claim
The paper's claim is that the κ-deformed Euclidean algebra generated by noncommuting coordinates, undeformed rotation generators $M_{�岒}$, and mutually commuting vector-like Dirac derivatives $D_��$ can be realized in infinitely many ways on ordinary commuting coordinates $x_��$ and derivatives $��_��$, and that all of these realizations are physically equivalent. The realizations are obtained by solving the coupled consistency equations that follow from the defining commutators, and they fall into two families. In type I, $�hat{x}_��=x_��ϕ+i(x∂)(a_��γ_1+ia^2∂_��γ_2)$ with $γ_1,γ_2$ fixed by $ϕ$ and its derivatives; in type II an extra term $i(ax)∂_��$ is present. Within each family $ϕ$ is free, subject to $ϕ(0,0)=1$, so the solution set is infinite. Choosing $ϕ_L=1-A$, $ϕ_R=1$, $ϕ_S=A/(e^A-1)$ reproduces the left, right and Weyl-symmetric orderings, and $ϕ_N=-A+√(1-B)$ gives a natural realization in which the Dirac derivative is simply $D_��=∂_��$ and the rotation generators take their undeformed form. For every realization the paper gives the coproduct of $∂_��$ and the corresponding star product, and shows that type-I star products are all conjugate to the right-ordering star product by an explicit similarity operator $S=e^{U}$.
Load-bearing premise
The claim that all covariant realizations are physically equivalent rests on showing each one is reached from the right realization by a similarity transformation, and the paper constructs that transformation only for the type-I family with $ϕ$ depending on $A$ alone, leaving the natural type-II realization's membership in the same orbit asserted rather than built.
Editorial extensions
If this is right
- Any physical quantity computed in one covariant realization can be transported to another by the similarity operator, so the ordering ambiguity in κ-space is not an ambiguity of predictions.
- The star product of any type-I realization follows from the right-ordering star product by conjugation with $S=e^{U}$, with the coefficients determined recursively from the derivatives of $ϕ$ at zero.
- In the natural realization, where $D_��=∂_��$ and $M_{�岒}=x_��∂_��-x_��∂_��$, scalar fields, invariants and an invariant integral can be written in ordinary flat-space form, with the star product replacing pointwise multiplication in the action.
- Quantum-gravity models in 2+1 dimensions built on $SU(2)$ or $SU(1,1)$ algebras become candidates for the same covariant-realization treatment.
Reading between the lines
- The equivalence argument is a statement about star products of functions, not about the full algebra of quantum observables; whether it survives a quantization in which $�hat{x}$ and $D$ are ordered inside interacting actions is a separate question the paper does not test.
- Because the free function $ϕ$ acts like a choice of frame, one could look for an invariant that distinguishes realizations only when the deformation vector $a$ is allowed to vary from point to point.
- If the equivalence holds, the natural realization's simple Laplacian and $D_��=∂_��$ could make loop calculations in a κ-deformed field theory no harder than ordinary ones, with deformation effects confined to the star-product vertices.
- The same similarity-orbit technique should apply to Snyder space, which the authors note is of the same extended-Lie-algebra type, and would give a concrete test of whether the equivalence statement generalizes beyond κ-deformation.
Original abstract
We study a Lie algebra type $\kappa$-deformed space with undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. Space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study a Lie algebra type $\kappa$-deformed space with undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. Space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study a Lie algebra type $\kappa$-deformed space with undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. | contribution 0.99 | 0.93 | 0.15 | asserted 0.99 |
| s1 | Space deformation depends on an arbitrary vector. | result 0.39 | 0.39 | 0.06 | asserted 1.00 |
| s2 | Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. | contribution 0.62 | 0.95 | 0.20 | asserted 1.00 |
| s3 | The corresponding coproducts and star products are found and related in a new way. | result 0.65 | 0.94 | 0.12 | asserted 1.00 |
| s4 | All covariant realizations are physically equivalent. | result 0.78 | 0.78 | 0.48 | asserted 1.00 |
| s5 | Specially, a few simple realizations are found and discussed. | contribution 0.72 | 0.95 | 0.05 | asserted 1.00 |
| s6 | The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization. | contribution 0.95 | 0.87 | 0.07 | asserted 0.97 |
What was examined, and what was not
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References
How the literature uses it
38 arXiv papers cite this in 146 sentences
Papers citing it · 95 sentences without a confident label
Builds on it 3 papers
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In this paper we are establishing numerous results valuable from both, mathematical and physical point of view, generalizing previous results on realizations [ref] .
Kappa-Minkowski spacetime, Kappa-Poincar\'{e} Hopf algebra and realizations · Introduction · 0.98
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The investigations carried out in this paper are along the track of developing general techiques of calculations, applicable for a widest possible class of NC spaces and as such are a continuation of the work done in a series of previous papers [ref] .
Kappa-deformed Snyder spacetime · Introduction · 0.84
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The investigations carried out in this paper are along the track of developing general techiques of calculations, applicable for a widest possible class of NC spaces and as such are a continuation of the work done in a series of previous papers [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] .
Lie algebraic deformations of Minkowski space with Poincare algebra · Introduction · 0.87
Uses its method or tool 5 papers
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The approach we take here is to express the $κ$-deformed Laplacian in terms of the derivatives with respect to the commutative coordinates and deformation parameter, developed in [ref] .
Spectral Dimension of kappa-deformed space-time · Kappa-deformed Laplacian in Euclidean space-time · 0.99
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To this aim we shall use explicit construction inspired by covariant Heisenberg realizations proposed firstly in [ref] : [equation] in terms of undeformed Weyl–Poincaré algebra ([ref])–([ref]) satisfying the canonical commutation relations: [equation] The transformation $(x^μ,M_μν,p_μ)⟶(^μ,M_μν,P_μ)$ is invertible.
$\kappa$-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems · $κ$-Minkowski spacetime and $κ$-Poincaré Hopf algebra with “$h$-adic” topology · 0.98
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For this, we consider the deformed coproduct for a fully covariant realization of the $κ$-Minkowski algebra [ref] .
Deformed Oscillator Algebras and QFT in $\kappa$-Minkowski Spacetime · Introduction · 0.80
Compares with it 2 papers
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In papers [ref] , [ref] , [ref] the symmetric realization corresponding to symmetric Weyl ordering was used, but therein $(D_μ)$ and $(M_μν)$ are given by Eqs.([ref]),([ref]) with $φ(A) = A/e^A -1,$ whereas in our approach, which uses the notion of natural realization, $D_μ$ and $M_μν$ have particularly simple form [ref] , namely $D_μ = ∂_μ$ and $M_μν = x_μ ∂_ν - x_ν ∂_μ .$ Hence, both realizations are covariant and equivalent.
Kappa-Minkowski space-time and the star product realizations · Free scalar field theory on $κ$-deformed Minkowski space · 1.00
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Substituting the ansatz of $ϕ(P̂)$ and $χ_μν$ in ([ref]), we get [equation] Thus $χ_μν$ is now exactly solved and the ansatz for $ϕ(P̂)$ reduces to: [equation] Finally, substituting $ϕ(P̂)$ and $χ_μν$ ([ref]), in ([ref]), we get [equation] So eventually we have determined the deformed brackets which will help us to derive the deformed transformation under Poincare generators, as [equation] With these we reproduce the results which are already existing in the literature [ref] etc.
Symmetries of $\kappa$ Minkowski space-time: A possibility of exotic momentum space geometry? · Deformed symmetries of $κ$ Minkowski spacetime · 0.80
Cites it as support 3 papers
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Our motivation is to unify -Minkowski spacetime and Lorentz algebra in the unique Lie algebra [ref] .
Kappa-Minkowski spacetime, Kappa-Poincar\'{e} Hopf algebra and realizations · Introduction · 0.97
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Namely, in the natural realization, where $D_μ = ∂_μ,$ the Dirac derivative $D_μ$ is a vector under Lorentz algebra and the coproduct $_N ( D_μ)$ transforms in a covariant way [ref] .
Kappa-Minkowski space-time and the star product realizations · Free scalar field theory on $κ$-deformed Minkowski space · 0.96
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Let us mention that the relation ([ref]) is the unique way to obtain a Lie algebra generated by and $x̂_μ$ (see [ref] ).
Generalized Poincare algebras, Hopf algebras and kappa-Minkowski spacetime · Lorentz algebra and -Poincaré-Hopf algebra · 0.96
Cites it as background 23 papers
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In this section we briefly summarise the results of [ref] relevant for the present study.
Newton's Equation on the kappa space-time and the Kepler problem · $κ$-space-time, symmetries and dispersion relation · 0.94
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Generalized Klein-Gordon equation using the Casimir on $κ$-space is written as $□(1-a^2/4□)Φ(x)-m^2Φ(x)=0$ [ref] .
$\kappa$-deformed Dirac Equation · $κ$-Poincare Algebra · 0.85
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The coalgebraic part, which includes coproducts for translation ($P_μ= -i_μ$), rotation and boost generators [ref] , [ref] , is given by the following relations, [equation] In the above expressions $Z$ is the shift operator, Eq.([ref]), whose copruduct is simply given by [equation] The operator $□$ is a deformed d'Alambertian operator [ref] , [ref] , [ref] , [equation] with the property $□→^2$ as $a →0$.
Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance · Hopf algebra structure · 0.81
Discussion
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