REVIEW
Canonical and Lie-algebraic twist deformations of Galilei algebra
High Energy Physics - Theory · Submitted 8 Jan 2008 · 51 references
Pith's verdict: Two genuinely new Galilei space-times worth having, but model (iii) has a scaling inconsistency in Section 4 that the paper misses.
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Referee report
The paper
The paper claims
By twisting the Poincaré symmetry of relativity and then letting the speed of light go to infinity, the paper derives five quantum Galilean symmetries, each with its own noncommutative version of space and time — two of them new.
The reading
What carries the argument
The Drinfeld twist and its contraction. A twist element $F$ satisfying the classical cocycle condition $F_{12}(Δ_0⊗1)F = F_{23}(1⊗Δ_0)F$ deforms only the coproduct, $Δ_0(a) ↦ F Δ_0(a) F^{-1}$, and the contracted twist factors $K = exp(-i/2 θ^{kl}∂_k∧∂_l)$, $exp(-i/2 ζ ∂_γ∧(x^k∂_l - x^l∂_k))$, $exp(-i θ̂^{0k}/κ̂ ∂_t∧∂_k)$, $exp(-i/2 κ̂ ∂_t∧(x^k∂_l - x^l∂_k))$ and $exp(-i/2 κ ∂_l∧ t∂_k)$ are what generate the star products. The star product of functions on the Hopf module, $f⋆g = ω∘(K^{-1}⊲f ⊗ g)$, is the mechanism that converts each twist factor into a specific set of coordinate commutators, so the twist factor and the space-time geometry are two faces of the same object.
Extended reading notes
Core claim
Starting from the canonical twist $F_θ = exp( i/2 θ^{μν}P_μ∧P_ν)$ and the Lie-algebraic twist $F_ζ = exp( i/2 ζ^λP_λ∧M_{αβ})$, the paper rescales $P_0 = Π_0 c$, $M_{i0} = cV_i$ and takes $c→∞$. With twist parameters held fixed this yields the canonically deformed Galilei Hopf algebra $U_θ(G)$ and the Lie-algebraic $U_ζ(G)$, whose space-time commutators are the magnetic (space-space) form $[x^i,x^j] = 2iθ^{ij}$ with $[t,x^i]=0$, and a new Lie-algebraic space-time with $[x^i,x^j]$ linear in $x$ and classical time. With twist parameters scaled by $c$ the paper obtains three more: an electric (time-space) canonical model with $[t,x^i] = 2iθ̂^{0i}/κ̂$ and commuting spatial coordinates; a nonrelativistic $κ$-Minkowski model $[t,x^i] = iκ̂(δ^{l i}x^k - δ^{k i}x^l)$, $[x^i,x^j]=0$, with quantum time and classical three-space; and a final model with $[x^i,x^j] = iκ t(δ^{l i}δ^{k j} - δ^{k i}δ^{l j})$, $[t,x^i]=0$, with quantum space and classical time. The paper also reports where the programme fails: for $θ^{ij}≠0$ the contracted coproducts diverge, and for boost-like carriers $[M_{k0},P_l]$ the rotation coproduct has no $c→∞$ limit.
Load-bearing premise
The paper assumes that taking $c→∞$ of the twisted coproducts, and of the corresponding twist factors, always lands on a genuine Hopf algebra, but it never displays the coassociativity, counit, antipode or twist-cocycle checks that would establish this for the five contracted structures.
Editorial extensions
If this is right
- If the contractions produce genuine twists, the star products (57) and (59) are associative and the five Galilean quantum groups can be used as symmetry algebras for deformed nonrelativistic field theories and phase spaces.
- The model with $[t,x^i] = iκ̂(δ^{l i}x^k - δ^{k i}x^l)$ and $[x^i,x^j]=0$ gives a nonrelativistic analogue of $κ$-Minkowski space in which only time fails to commute with space, isolating time-ordering effects from spatial noncommutativity.
- The model with $[x^i,x^j] = iκ t(δ^{l i}δ^{k j} - δ^{k i}δ^{l j})$ has noncommutativity proportional to $t$, so the quantum geometry switches off at $t=0$ and grows with elapsed time, a qualitatively different behaviour from constant-$θ$ models.
- The reported divergences — $θ^{ij}≠0$ in the canonical case and boost-like carriers in the Lie-algebraic case — delimit which relativistic twists admit a nonrelativistic limit, and hence which Planck-scale kinematics have a slow-particle regime at all.
Reading between the lines
- Associativity of the star products (57) and (59) is equivalent to the classical cocycle condition for the contracted twist factors (41) and (45); a direct check of that condition would settle the consistency of both new models in one short computation, and it is the natural next thing to do with this catalogue.
- Because the commutator in (59) scales with $t$, the deformation defines a running noncommutativity scale rather than a fixed one, which invites comparison with settings where the quantum-gravity scale is set by proper time or by cosmological evolution instead of a constant mass parameter.
- The two new models look dual under exchange of which generator carries the deformation (rotation versus boost), so one might expect a third, mixed case in which both a rotation and a boost appear in the twist; the paper's carrier analysis does not cover it.
- If only the $c$-independent canonical twist survives cleanly, then space-space (magnetic) noncommutativity is the robust nonrelativistic limit, while time-space (electric) noncommutativity requires the $c$-dependent scaling — a distinction that constrains how noncommutative effects could show up in slow-particle phenomenology.
Original abstract
We describe various nonrelativistic contractions of two classes of twisted Poincare algebra: canonical one ($\theta_{\mu\nu}$-deformation) and the one leading to Lie-algebraic models of noncommutative space-times. The cases of contraction-independent and contraction-dependent twist parameters are considered. We obtain five models of noncommutative nonrelativistic space-times, in particular, two new Lie-algebraic nonrelativistic deformations of space-time, respectively, with quantum time/classical space and with quantum space/classical time.
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 describe various nonrelativistic contractions of two classes of twisted Poincare algebra: canonical one ($\theta_{\mu\nu}$-deformation) and the one leading to Lie-algebraic models of noncommutative space-times. The cases of contraction-independent and contraction-dependent twist parameters are considered. We obtain five models of noncommutative nonrelativistic space-times, in particular, two new Lie-algebraic nonrelativistic deformations of space-time, respectively, with quantum time/classical space and with quantum space/classical time.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We describe various nonrelativistic contractions of two classes of twisted Poincare algebra: canonical one ($\theta_{\mu\nu}$-deformation) and the one leading to Lie-algebraic models of noncommutative space-times. | contribution 0.98 | 0.85 | 0.11 | asserted 0.99 |
| s1 | The cases of contraction-independent and contraction-dependent twist parameters are considered. | method 0.58 | 0.54 | 0.04 | asserted 0.91 |
| s2 | We obtain five models of noncommutative nonrelativistic space-times, in particular, two new Lie-algebraic nonrelativistic deformations of space-time, respectively, with quantum time/classical space and with quantum space/classical time. | result 0.95 | 0.95 | 0.77 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
36 arXiv papers cite this in 103 sentences
Papers citing it · 54 sentences without a confident label
Builds on it 1 paper
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In this article we propose canonical and Lie-algebraic twist deformations of quantum groups dual to the recovered in [ref] Galilei Hopf algebras $𝒰_.(𝒢)$.
Canonical, Lie-algebraic and quadratic twist deformations of Galilei group · Final remarks. · 0.89
Uses its method or tool 1 paper
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Further, in the second step of our investigation, we also perform three nonrelativistic contractions ( [ref] , [ref] ; see also [ref] ) of our generalized Poincare Hopf structures.
Generalized twist deformations of Poincare and Galilei Hopf algebras · Introduction · 0.92
Cites it as background 29 papers
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In second Section we remaind the canonically-deformed Galilei Hopf algebras proposed in [ref] .
The energy-momentum conservation law in two-particle system for twist-deformed Galilei Hopf algebras · Introduction · 0.86
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In Section 3 we recall Lie-algebraic noncommutative quantum Galilei space-time proposed in article [ref] .
The Henon-Heiles system defined on Lie-algebraically deformed Galilei space-time · Introduction · 0.85
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In this section we very shortly recall the basic facts associated with the (twisted) canonically deformed Galilei Hopf algebra $U_θ(G)$ and with the corresponding quantum space-time [ref] .
The Henon-Heiles system defined on canonically deformed space-time · Canonically deformed Galilei space-time · 0.97
What it cites
Cites as background 1 paper
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In particular, the $r$-matrix ([ref]) for the special choices ${M_αβ= M_03,ϵ=1,2}$ and ${M_αβ=M_12,ϵ=0,3}$ has been discussed some time ago in [ref] and also considered very recently in [ref] .
Noncommutative fields and actions of twisted Poincare algebra · Canonical and Lie-algebraic twisted Poincaré algebra · 0.87
Discussion
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