Deformed Quantum Phase Spaces, Realizations, Star Products and Twists

We review deformed quantum phase spaces and their realizations in terms of the undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta, and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Meljanac, Stjepan, Štrajn, Rina
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Language:	English
Published:	Інститут математики НАН України 2022
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Cite this:	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists. Stjepan Meljanac and Rina Štrajn. SIGMA 18 (2022), 022, 20 pages

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author	Meljanac, Stjepan Štrajn, Rina
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description	We review deformed quantum phase spaces and their realizations in terms of the undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta, and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed quantum phase spaces, and Snyder-type spaces are considered. Examples of linear realizations of the κ-Minkowski spacetime are elaborated. Finally, some new results on quadratic deformations of quantum phase spaces and a generalization of the Yang and triply special relativity models are presented.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 022, 20 pages Deformed Quantum Phase Spaces, Realizations, Star Products and Twists Stjepan MELJANAC a and Rina ŠTRAJN b a) Division of Theoretical Physics, Ruder Bošković Institute, Bijenička cesta 54, 10002 Zagreb, Croatia E-mail: meljanac@irb.hr b) Department of Electrical Engineering and Computing, University of Dubrovnik, Ćira Carića 4, 20000 Dubrovnik, Croatia E-mail: rina.strajn@unidu.hr Received December 27, 2021, in final form March 14, 2022; Published online March 23, 2022 https://doi.org/10.3842/SIGMA.2022.022 Abstract. We review deformed quantum phase spaces and their realizations in terms of undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed quantum phase spaces and Snyder type spaces are considered. Examples of linear realizations of the κ-Minkowski spacetime are elaborated. Finally, some new results on quadratic deformations of quantum phase spaces and a generalization of Yang and triply special relativity models are presented. Key words: deformed quantum phase spaces; realizations; star products; twists 2020 Mathematics Subject Classification: 81R60 1 Introduction Noncommutative NC spaces appeared in theoretical physics in the efforts to understand and model Planck scale phenomena. The first proposed model of NC geometry was that of the Snyder spacetime [116]. In order to obtain quantum gravity models that reconcile general relativity (GR) and quantum mechanics (QM), one of the main ideas is to introduce noncommutative quantum spacetimes [32, 33, 37] and also look for quantum deformations of the quantum-mechanical relativistic phase space algebra [118]. For example, a widely studied model is the κ-Minkowski spacetime, where the parameter κ is usually interpreted as the Planck mass or the quantum gravity scale and the coordinates themselves close a Lie algebra [7, 30, 35, 65, 66, 72]. A successful approach to noncommutative geometry is based on the formalism of Hopf al- gebras [28, 71], describing the relativistic symmetries of the quantum spacetime [11, 14]. The κ-Poincaré quantum group [38, 73, 119], as a possible quantum symmetry of the κ-Minkowski spacetime, allows for the study of deformed relativistic spacetime symmetries and the corre- sponding dispersion relations [5, 8, 12, 13, 77]. It is an example of a Hopf algebra, where the algebra sector is the same as that of the Poincaré algebra, but the coalgebra sector is deformed. In general, in the Hopf algebra framework, it is possible to deform the Hopf algebra using a twist element which satisfies the 2-cocycle condition, which again produces a Hopf algebra, with the algebra sector unaltered and the coalgebra sector deformed. Deformations of relativistic sym- metries play an important role in the study of phenomenologically relevant effects of quantum gravity [2, 3, 4, 9, 54, 55, 70]. The interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries was presented in [15]. mailto:meljanac@irb.hr mailto:rina.strajn@unidu.hr https://doi.org/10.3842/SIGMA.2022.022 2 S. Meljanac and R. Štrajn A powerful tool in the study of NC spaces is that of realizations in terms of the Weyl– Heisenberg algebra [20, 22, 34, 41, 42, 46, 77, 79, 83, 84, 95, 97, 115]. Namely, the NC coordinates are expressed in terms of the commutative coordinates and the corresponding momenta, which allows one to simplify the methods of calculation on the deformed spacetime. Every realization corresponds to a specific ordering and one special example is the Weyl realization, related to the symmetric ordering. Exponential formulas [21, 88, 89, 90, 99] are related to the deformed coproduct of momenta in NC spaces and also appear in the computations of star products, which are both needed for the definition of a field theory, the notion of differential calculus and other calculations in a NC spacetime [6, 29, 43, 49, 69]. Exponential formulae, which were used to obtain the coproducts and star products were also presented in [23, 87, 101]. A Lie deformed phase space obtained from a twists in the Hopf algebroid approach was considered in [47, 51, 63, 64]. A class of deformed quantum phase spaces, i.e., a deformed Heisenberg algebra, is generated with NC coordinates x̂µ and commutative momenta pµ defined as [x̂µ, x̂ν ] = iΘµν(l, x̂, p), [pµ, x̂ν ] = −iφµν(l, p), [pµ, pν ] = 0, (1.1) with all Jacobi identities satisfied and l a real parameter of order of the Planck length. There are also deformed quantum phase space models with NC momenta p̂µ including the cosmological radius R [55, 118]. The undeformed phase space is defined as [xµ, xν ] = 0, [pµ, pν ] = 0, [xµ, pν ] = −iηµν , µ, ν = −1, . . . , n− 1, where ηµν = diag(−1, 1, . . . , 1), xµ are coordinates and pµ are momenta. Examples of NC spaces where Θµν (1.1) do not depend on the momenta pµ are the canonical theta space [33, 110], with Θµν = const, Lie algebra type spaces [6, 7, 19, 29, 30, 31, 35, 36, 40, 63, 65, 66, 67, 72, 79, 102], for which Θµν = iCµνλx̂λ, and quadratic deformations of Minkowski space, with Θµν = Θµνρσx̂ρx̂σ [67, 117]. In the Snyder space [16, 17, 27, 39, 44, 61, 62, 86, 103, 104, 105, 106, 107, 108, 109, 113, 114, 116], one has [x̂µ, x̂ν ] = il2Mµν , where Mµν are Lorentz generators, NC coordinates x̂µ do not close an algebra between themselves, but x̂µ and Mµν close a Lie algebra. The Snyder space and the κ deformed Snyder space [89, 90] lead to a non associative star product and non coassociative coproduct [21, 39]. Recently, the extended Snyder model and the κ deformed extended Snyder model with additional tensorial coordinates were proposed [91, 92, 93]. These models are of Lie algebra type NC spaces in which star products are associative and coproducts are coassociative [92]. Some new developements in the applications of NC geometry to physics can be found in [24, 25, 26, 57, 59, 60, 74, 75, 111, 112]. In this review we survey the above mentioned types of deformed quantum phase spaces, their properties and the relations between realizations of NC coordinates in terms of undeformed phase space, star products and twists. Many technical results important for this review have appeared previously in the literature and are cited appropriately. However this list of references is not exhaustive. We also present some new results in Sections 3, 4, 6 and 7. The plan of paper is as follows. In Section 2, the Lie deformed quantum phase space and realizations are presented. In Section 3, a star product from realizations is constructed. In Section 3.1, we present the Snyder space and its extension with tensorial coordinates. In Sec- tion 4, the coproduct of momenta and twist from star product and realizations are obtained. In Section 5, some examples of linear realizations of the κ deformed Minkowski spacetime, spe- cially the right covariant, left covariant and light like realizations, are revisited. In Section 6, some aspects of quadratic deformations of quantum phase spaces are elaborated, specially in Section 6.1, quadratic deformations of the Minkowski space from dilatation. A generalization of Yang and triply special relativity models is given in Section 7. Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 3 2 Lie deformed quantum phase space and realizations The undeformed quantum phase space is defined with coordinates xµ and momenta pµ [xµ, xν ] = 0, [pµ, pν ] = 0, [pµ, xν ] = −iηµν , µ, ν = 0, 1, . . . , n− 1, where ηµν = diag(−1, 1, . . . , 1). The generalization of ηµν to a metric with arbitrary Lorentz signature is straightforward. We consider the deformed quantum phase space defined with noncommutative (NC) coordi- nates x̂µ and momenta pµ of the type [x̂µ, x̂ν ] = ix̂αCµνα(l, p) + idµν(l, p), [pµ, x̂ν ] = −iφµν(l, p), [pµ, pν ] = 0, (2.1) where l is a parameter of order of the Planck length and summation over repeated indices is assumed. For l = 0, Cµνα(0, p) = 0 and dµν(0, p) = 0. Functions Cµνα and φµν depend on momenta pα, where Cµνα are a generalization of the structure constants. For example, the original Snyder space is defined with [x̂µ, x̂ν ] = il2Mµν , (2.2) where Mµν are Lorentz generators. The realization Snyder proposed [116] is given by x̂µ = xµ + l2(x · p)pµ, Mµν = xµpν − xνpµ = x̂µpν − x̂νpµ. Hence Cµνα(l, p) = l2(ηµαpν − ηναpµ). If Cµνλ are structure constants, then NC coordinates x̂µ close a Lie algebra. A perturbative construction of φµν(l, p) corresponding to the symmetric ordering can be found in [34]. From [pµ, x̂ν ] = −iφµν(l, p), (2.1), it follows that the realization of NC coordinates x̂µ can be written as x̂µ = xαφαµ(l, p) + χµ(l, p), (2.3) and the inverse is given by xν = (x̂µ − χµ(l, p)) ( φ−1 ) µν , and pµ = −i ∂ ∂xµ . (2.4) If l = 0, φµν(0, p) = ηµν . All Jacobi relations for the class of deformed quantum phase spaces/deformed Weyl–Heisenberg algebras (2.1) (obtained from (2.3)) are satisfied. Realizations of the type given in equation (2.3), used in physical applications, were studied for example for κ-Minkowski spaces in [41, 42, 49, 77, 79, 84, 95, 97], for Snyder spaces in [20, 21, 86], and for the extended Snyder model with tensorial coordinates in [91, 92, 93]. A special class of deformed quantum phase spaces/deformed Weyl–Heisenberg algebras for which φµν(l, p) is at most linear in pα is given by x̂µ = xµ + lKβµαxαpβ + χµ(l, p), Kβµα ∈ R, 4 S. Meljanac and R. Štrajn where φαµ(l, p) = ηαµ + lKβµαpβ. For χµ(l, p) = 0, it holds [x̂µ, x̂ν ] = il(Kµνα −Kνµα)xα + il2(KβµαKανσ −KβναKαµσ)xσpβ. For example, the linear realization for the case of the κ Snyder deformation was presented in [90]. The Lie algebra is closed in NC coordinates x̂µ if KβµλKλνα −KβνλKλµα = (Kµνλ −Kνµλ)Kβλα, and the structure constants are Cµνα = Kµνα −Kνµα, see for example [48, 63, 64, 78, 79, 100]. 3 Star product from realizations Let us define the ▷ action xµ ▷ f(x) = xµf(x), pµ ▷ f(x) = −i ∂f(x) ∂xµ . For the realization given in equation (2.3), using ▷, it follows pµ ▷ e iqx = qµe iqx, eikαx̂α ▷ eiqβxβ = eiJα(k,l,q)xα+ih(k,l,q), kα, qα ∈Mn, where x̂µ = xαφαµ(l, p) + χµ(l, p) and Mn denotes the Minkowski space, for some J , h [101]. If kα = 0, Jµ(0, l, q) = qµ, h(0, l, q) = 0. If qα = 0, Jµ(k, l, 0) = Kµ(k, l). If l = 0, Jµ(k, 0, q) = kµ+qµ. Jµ(k, l, q) and h(k, l, q) can be constructed perturbatively using [101, Theorems 1 and 2]. Compact results for Jµ(tk, l, p) and h(tk, l, p), where pα is the momentum operator (2.4), are given by Jµ(tk, l, p) = ( etkαOα ) (pµ), h(tk, l, p) = ( etO − 1 O ) (kβχβ(l, p)), where O = kαOα, Oα = ad−ixβφβα(l,p) . Jµ(tk, l, p) and h(tk, l, p) are unique solutions of the partial differential equations ∂Jµ(tk, l, p) ∂t = kβφµβ(J(tk, l, p)), ∂h(tk, l, p) ∂t = kβχβ(J(tk, l, p)), (3.1) with boundary conditions Jµ(0, l, p) = pµ, h(0, l, p) = 0. From Jµ(k, l, q) and h(k, l, q) we can obtain the star product eikx ∗ eiqx = eiK −1(k)x̂−ih(K−1(k),0) ▷ eiqx = eixD(k,l,q)+iG(k,l,q), (3.2) Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 5 where Dµ(k, l, q) = Jµ ( K−1 µ (k, l), l, q ) = ( exp ( K−1 β (k, l)φαβ(l, q) ∂ ∂qα )) (qµ), (3.3) G(k, l, q) = h ( K−1(k, l), l, q ) − h ( K−1(k, l), l, 0 ) . (3.4) K−1 is the inverse map K−1 µ (K(k, l)) = Kµ ( K−1(k, l) ) = kµ. The star product, (3.2), and the method of calculation are a generalization of the method first proposed in [88, 89]. The method from [88, 89] was applied in [21, 23, 78, 85, 86, 87, 90, 91, 92, 93, 99, 100]. The star product, (3.2), can be associative or nonassociative. For example, the star product for the Snyder model is nonassociative [21, 39]. If NC coordinates x̂µ, (2.3), close a Lie algebra, then the corresponding star product is associative [83]. According to the PBW theorem, if coordinates x̂µ, µ = 0, 1, . . . , n−1 generate a Lie algebra ĝ and xµ = x̂µ ▷ 1, µ = 0, 1, . . . , n− 1 generate a commutative algebra g, then i) enveloping algebras U(ĝ) and U(g) are isomorphic, ii) if f̂ ▷ 1 = f , U(ĝ) ▷ 1 = U(g), then we define the inverse map ▶ f ▶ 1 = f̂ , g ▶ 1 = ĝ, and the star product f ∗ g = f̂ ĝ ▷ 1, (f ∗ g) ▶ 1 = f̂ ĝ, where f̂ , ĝ ∈ U(ĝ) and f, g ∈ U(g). This star product is associative f ∗ (g ∗ h) = (f ∗ g) ∗ h, f, g, h ∈ U(g). Namely f ∗ (g ∗ h) ▶ 1 = f̂ ( ĝĥ ) = f̂ ĝĥ, (f ∗ g) ∗ h ▶ 1 = ( f̂ ĝ ) ĥ = f̂ ĝĥ. Generally, if the star product, (3.2), is associative, i.e.,( eik1x ∗ eik2x ) ∗ eik3x = eik1x ∗ ( eik2x ∗ eik3x ) , then it holds Dµ(D(k1, l, k2), l, k3) = Dµ(k1, l,D(k2, l, k3)), (3.5) and G(k1, l, k2) + G(D(k1, l, k2), l, k3) = G(k2, l, k3) + G(k1, l,D(k2, l, k3)). Addition of momenta is defined with kµ ⊕ qµ = Dµ(k, l, q). Note that Dµ(k, l, q) ∈ R. For example, let the NC coordinates x̂i, i = 1, 2, 3 close the su(2) algebra [x̂i, x̂j ] = 2ilϵijkx̂k, 6 S. Meljanac and R. Štrajn and the realization of x̂i is given by x̂i = xi √ 1− l2p2 + lϵijkxjpk, p2 = p21 + p22 + p23. Then [x̂i, pj ] = i ( δij √ 1− l2p2 + lϵijkpk ) , and ki ⊕ qi = Di(k, q) = ki √ 1− l2q2 + √ 1− l2k2qi + lϵijkkjqk. This example of the su(2) NC space appeared in connection with the 3 dimensional quantum gravity model [36], see also [58, 99]. Note that the universal formula for Lie algebra generators as formal power series of the corresponding structure constants with coefficients in Bernoulli numbers was given in [34, 45]. The explicit star product of the cotangent bundle of a Lie group [45] corresponds to the star product (3.2), where x̂ is expressed in terms of the universal formula, i.e., φ(l, p) is the generating function for Bernoulli numbers, related to the symmetric ordering [34]. 3.1 Snyder space and an extension with tensorial coordinates Examples of non associative star products are related to the following realizations of coordi- nates x̂µ x̂µ = xµφ1 ( l2p2 ) + l2(x · p)pµφ2 ( l2p2 ) . Then the commutation relations [x̂µ, x̂ν ] lead to a generalized Snyder algebra [86] [x̂µ, x̂ν ] = il2Mµνψ ( l2p2 ) , where Mµν = xµpν − xνpµ are Lorentz generators. Specially, for ψ ( l2p2 ) = 1, it becomes the Snyder algebra originally proposed in [116]. In the original Snyder model NC coordinates x̂µ do not close a Lie algebra and the corre- sponding star products are non associative. For example, the star product corresponding to the realization x̂µ = xµ + l2(x · p)pµ (3.6) leads to xµ ∗ xν = xµxν , and xµ ∗ (xν ∗ xρ) = xµxνxρ − l2(ηµνxρ + ηµρxν), (xµ ∗ xν) ∗ xρ = xµxνxρ − l2 2 (ηµρxν + ηνρxµ + 2ηµνxρ). Hence, the star product is non associative (xµ ∗ xν) ∗ xρ − xµ ∗ (xν ∗ xρ) = l2 2 (ηµρxν − ηνρxµ). Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 7 Using the realization in equation (3.6), and equations (3.1), (3.3), the function Dµ(k, l, q), defining the star product (3.2), was obtained in [21] Dµ(k, l, q) = 1 1− l2(k · q) ( kµ − l2 1 + √ 1 + l2k2 kµ(k · q) + √ 1 + l2k2qµ ) , for which equation (3.5) is not satisfied, implying that the star product is non associative. The corresponding coproduct ∆pµ = 1 1− l2pα ⊗ pα ( pµ ⊗ 1− l2 1 + √ 1 + l2p2 pµpα ⊗ pα + √ 1 + l2p2 ⊗ pµ ) is non coassociative. Note that in the original Snyder space NC coordinates x̂µ and Lorentz generators Mµν close a Lie algebra, but Mµν ▷ 1 = 0. Since NC coordinates x̂µ do not close a Lie algebra between themselves, additional tensorial coordinates x̂µν are introduced [39, 91] instead of Mµν and they satisfy x̂µν ▷ 1 = xµν , where xµν are commutative tensorial coordinates. Both the coordinates x̂µν and xµν and the canonical momenta pµν transform as Lorentz generators Mµν . Consequently, the new star prod- uct is associative and the coproduct is coassociative [39, 91, 94]. Using the above extended Snyder space, a unification with the κ-Minkowski space is also proposed in the form of associative realizations of the κ deformed extended Snyder model [92, 93]. A similar extension with additional commutative tensorial coordinates θµν was proposed in the context of the DFR NC space, changing the constant θµν to tensorial coordinates θµν [1, 10]. 4 Coproduct of momenta and twist from star product and realizations The relation between the star product and the twist operator is given by (f ∗ g)(x) = mF−1(▷⊗ 1)(f(x)⊗ g(x)), where m is the multiplication map, A⊗B → AB. Using the star product (3.2) and the above relation for f ∗ g, a family of twist operators can be writen as [85, 86, 87] F−1 = : exp((i(1− u)xα ⊗ 1 + u1⊗ xα)(∆−∆0)pα): exp(iG(p⊗ 1, 1⊗ p)), (4.1) where : : denotes normal ordering, in which the xs are to the left of the p’s, u is a real parameter and ∆pα = Dα(p⊗ 1, 1⊗ p), ∆0pα = pα ⊗ 1 + 1⊗ pα, where Dα(l, k, q) is given in (3.3) and G in (3.4). Applying [101, Theorem 1], we can find the twist in the form without normal ordering, see for example [100, Section 4]. It is important to note that ∆pα is the coproduct of momenta and it holds ∆pα = F∆0pαF−1. 8 S. Meljanac and R. Štrajn This can be proved using the identity n∑ k,l=0 (−1)n xkpxl k!l! = 0, for n ≥ 2. Alternatively, we can obtain the twist in the following way. The coproduct ∆pµ is obtained from ∆pα = Dα(p⊗ 1, 1⊗ p) and (3.3) ∆pµ = ( exp ( K−1 β (p)⊗ φαβ(p) ∂ ∂pα )) (1⊗ pµ). The coproduct is coassociative if and only if the star product is associative, i.e., if x̂µ close a Lie algebra. We can define new momenta pWµ corresponding to the Weyl realization pWµ = K−1 µ (p), with the property pWµ ▷ eiK(k)·x = kµe iK(k)·x, and [ pWµ , e ik·x̂] = kµe ik·x̂. Hence, the coproduct of momenta is ∆pµ = e−ipWα ⊗x̂α(1⊗ pµ)e ipWβ ⊗x̂β . Then we obtain the relation between ∆pµ and ∆0pµ ∆pµ = F∆0pµF−1 = e−ipWα ⊗x̂αeipβ⊗xβ (∆0pµ)e −ipα⊗xαeip W β ⊗x̂β . If χµ(l, p) = 0, then the twist operator for the realization x̂µ = xαφαβ(l, p) is given by F−1 = e−ipα⊗xαeip W β ⊗xγφγβ(l,p). (4.2) If χµ(l, p) ̸= 0, then the twist F−1 is F−1 = e−ipα⊗xαeip W β ⊗xγφγβ(l,p)eiG(p⊗1,1⊗p), (4.3) and the consistency check is x̂µ = mF−1(▷⊗ 1)(xµ ⊗ 1) = xαφαµ(l, p) + χµ(l, p). Note that the following identity holds :ei(1⊗xα)(∆−∆0)pα : = e−ipα⊗xαeip W β ⊗xγφγβ(l,p). The twists in this section, specially (4.1) and (4.3), are given in the Hopf algebroid approach [47, 51, 63, 64] and generally do not satisfy the cocycle condition in the Hopf algebra sense [96]. However, in the Hopf algebroid approach, these twists satisfy a generalized cocycle condition if and only if the star product is associative, i.e., x̂µ close a Lie algebra [98]. In the next section we give examples for the case when these twists can be transformed into Drinfeld twists satisfying the cocycle condition in the Hopf algebra sense. Note that another construction of the twist for the original Snyder model (2.2) was presented in [76]. Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 9 5 Linear realizations For a linear realization x̂µ = xµ +Kβµαxαpβ, (5.1) the differential equation for Jµ(tk, q) is given by dJµ(tk, q) dt = kµ + kαKβαµJβ(tk, q), Jµ(0, q) = qµ. The solution for Jµ(tk, q) is then Jµ(tk, q) = kα ( eK(tk) − η K(k) ) αµ + qα ( eK(tk) ) αµ , (5.2) where Kµν(k) = Kµανkα. (5.3) For t = 1 Jµ(k, q) = kα ( eK(k) − η K(k) ) αµ + qα ( eK(k) ) αµ . For qµ = 0 Jµ(k, 0) = kα ( eK(k) − η K(k) ) αµ = Kµ(k). (5.4) The inverse of Kµ(k) is given by K−1 µ (k) = kWµ . (5.5) The expansion of kWµ in terms of kα was given in the appendix of [78]. Then Dµ(k, q) = Jµ ( K−1(k), q ) = kµ + qα ( eK(kW ) ) αµ , (5.6) and the coproduct for pµ is ∆pµ = pµ ⊗ 1 + ( eK(pW ) ) αµ ⊗ pα. (5.7) The twist in the Hopf algebroid approach corresponding to the linear realization x̂µ = xµ + Kβµαxαpβ is F = exp ( −ipWµ ⊗Kβµαxαpβ ) = exp ( −iKβα ( pW ) ⊗ xαpβ ) , (5.8) where pWµ = K−1 µ (p). The coproduct for pµ, ∆pµ = F∆0pµF−1 is identical to the equation above, (5.7). Note that Kµ(p) = ( eK(p) − η K(p) ) αµ pα and pµ = ( eK(pW ) − η K ( pW ) ) αµ pWα . The expression for pWµ was given in the appendix of [78]. For κ-Minkowski and the corresponding linear realizations an explicit proof of the cocycle codition is given in [48] and more generally in [78]. The conditions under which linear realiza- tions (5.1) generate a Lie algebra are given in Section 2. Other examples of linear realizations were presented in [63, 64, 79, 90, 100]. 10 S. Meljanac and R. Štrajn 5.1 Right covariant realization of κ-Minkowski space time The right covariant realization of κ-Minkowski space is given by x̂µ = xµ − aµx · p, [x̂µ, x̂ν ] = i(aµx̂ν − aν x̂µ). Jµ(k, q) for the right covariant realization satisfies the following differential equation dJµ(tk, q) dt = kµ − a · kJµ(tk, q), with initial condition Jµ(0, q) = qµ. J (0) µ (tk, q) = tkµ + qµ for aµ = 0. The solution of the above differential equation is Jµ(tk, q) = kµ 1− e−ta·k a · k + qµe −ta·k. For t = 1, Jµ(k, q) = kµ 1− e−a·k a · k + qµe −a·k. For kµ = 0, Jµ(0, q) = qµ. For qµ = 0, Jµ(k, 0) = kµ 1− e−a·k a · k ≡ Kµ(k). From here it follows that a ·K = 1− e−a·k, a · k = − ln(1− a ·K(k)), e−a·k = 1− a ·K. Hence, the inverse of Kµ(k) is given by K−1 µ (k) = −kµ ln(1− a · k) a · k = kWµ . Furthermore, Dµ(k, q) = Jµ ( K−1(k), q ) = kµ + qµ(1− a · k), ∆pµ = pµ ⊗ 1 + (1− a · p)⊗ pµ. In the Hopf algebroid approach, the twist corresponding to the right covariant realization is given by F = e−ipWβ ⊗x̂βeipα⊗xα = eAeB, where A = −ipWα ⊗ xα + ia · pW ⊗ x · p, B = ipα ⊗ xα, [A,B] = ( a · pW ⊗ 1 ) B. Using a special case of the BCH formula, we get (see [48, Appendix C]) F = exp ( A+ B ( a · pW ⊗ 1 1⊗ 1− e−a·pW⊗1 )) Using the relation pWµ = −pµ ln(1− a · p) a · p , we get F = e−i ln(1−a·p)⊗D, D = xαpα, and ∆pµ = pµ ⊗ 1 + (1− a · p)⊗ pµ. (5.9) We have shown that the twist in the Hopf algebroid approach corresponding to the right covariant realization of κ-Minkowski is identical to the Jordanian twist leading to the same right covariant realization and satisfying the cocycle condition in the Hopf algebra sense. Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 11 5.2 Left covariant realization of κ-Minkowski space time The left covariant realization of κ-Minkowski spacetime is given by x̂µ = xµ(1 + a · p), [x̂µ, x̂ν ] = i(aµx̂ν − aν x̂µ). Jµ(k, q) for the left covariant realization satisfies the following differential equation dJµ(tk, q) dt = kµ(1 + a · J(tk, q)), with initial condition Jµ(0, q) = qµ. J (0) µ (tk, q) = tkµ + qµ for aµ = 0. The solution of the above differential equation is Jµ(tk, q) = kµ eta·k − 1 a · k (1 + a · q) + qµ. For t = 1, Jµ(k, q) = kµ ea·k − 1 a · k (1 + a · q) + qµ. For qµ = 0, Jµ(k, 0) = kµ ea·k − 1 a · k ≡ Kµ(k). It follows that a ·K = ea·k − 1, a · k = ln(1 + a ·K(k)), and the inverse of Kµ(k) is given by K−1 µ (k) = kµ ln(1 + a · k) a · k = kWµ . Furthermore, Dµ(k, q) = Jµ ( K−1(k), q ) = kµ(1 + a · q) + qµ, ∆pµ = pµ ⊗ (1 + a · p) + 1⊗ pµ. The twist in the Hopf algebroid approach corresponding to the left covariant realization is F = e−ipWβ ⊗x̂βeipα⊗xα = eAeB, where A = −ipWβ ⊗ xβ − ipWβ ⊗ xβa · p, B = ipα ⊗ xα, [A,B] = −(ln(1 + a · p)⊗ 1)B = − ( a · pW ⊗ 1 ) B. Using a special case of the BCH formula we get F (see [48, Appendix C]) F = exp ( A+ B ( −a · pW ⊗ 1 1⊗ 1− ea·pW⊗1 )) . Using pWµ = pµ ln(1 + a · p) a · p , a · pW = ln(1 + a · p), we get F = e−ipWβ ⊗xβa·p = e−iaαpWβ ⊗Lβα , Lβα = xβpα, ∆pµ = pµ ⊗ (1 + a · p) + 1⊗ pµ. We have shown that the twist in the Hopf algebroid approach corresponding to the left covariant realization of the κ-Minkowski spacetime is different from the Jordanian twist leading to the same left covariant realization which satisfies the cocycle condition in the Hopf algebra sense, F = exp(−iD ⊗ ln(1 + a · p)). Although these twists are different, they give the same deformed Hopf algebra. Drinfeld twists F = exp(−i ln(1−a·p)⊗D), (5.9), and F = exp(−iD⊗ln(1+a·p)) belong to extended Jordanian twists for Lie algebras [56]. Interpolations between Jordanian twists, right and left covariant realizations of the κ-Minkowski spacetime were presented in refs. [23, 80, 81, 82]. 12 S. Meljanac and R. Štrajn 5.3 Light like realization of κ-Minkowski spacetime The light like realization of the κ-Minkowski space is defined with a2 = 0 and x̂µ = xµ(1 + a · p)− a · xpµ = xµ + aαMµα, satisfying the κ-Minkowski algebra [x̂µ, x̂ν ] = i(aµx̂ν − aν x̂µ), [Mµν , x̂λ] = −i(x̂µηνλ − x̂νηµλ)− i(aµMνλ − aνMµλ), [Mµν , pλ] = −i(pµηνλ − pνηµλ). Using (5.2),(5.3), (5.4), (5.5), (5.6), one obtains the following expression for Dµ(k, q) Dµ(k, q) = kµ(1 + a · q)qµ − aµ k · q 1 + a · k − 1 2 aµ(a · q) p2 1 + a · k , and for the coproduct ∆pµ = Dµ(p⊗ 1, 1⊗ p) = ∆0pµ + ( pµaα − aµ pα + 1 2aαp 2 1 + a · p ) ⊗ pα. (5.10) From the expression for the twist given in equation (5.8), it follows that the Drinfeld twist is given by [48, 49, 50] F = exp ( iaαpβ ln(1 + a · p) a · p ⊗Mαβ ) , which satisfies the cocycle condition in the Hopf algebra sense and the coproduct ∆pµ = F∆0pµF−1 is coassociative and coincides with ∆pµ above, (5.10). Remark 5.1. If a2 ̸= 0, the above realization x̂µ = xµ(1 + a · p)− a · xpµ = xµ + iaαMµα, leads to the κ-Snyder algebra [89, 90] [x̂µ, x̂ν ] = i(aµx̂ν − aν x̂µ) + ia2Mµν . The corresponding twist (4.2) does not satisfy the cocycle condition, the star product is non associative and the coproduct is non coassociative. 6 Quadratic deformations of quantum phase space In [117], deformed Heisenberg algebras were constructed as examples of NC structures and the framework for higher dimensional NC spaces based on quantum groups was studied. Further- more, quadratic deformations of the Minkowski space from twisted Poincaré symmetries were constructed in [67]. The construction was based on the twist F = exp ( i 2 ΘαβγδMαβ ∧Mγδ ) , where Mαβ are Lorentz generators and the r-matrix is given with r = 1 2ΘµνρσMµν ∧ Mρσ. Quadratic deformations of the Galilei group and the Newton equation for classical space were considered in [31]. Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 13 Generally, quadratic algebras, i.e., quadratic deformations, can be defined with the following commutation relations [x̂µ, x̂ν ] = Θµνγδx̂γ x̂δ, where all Jacobi relations have to be satisfied and multiplication in the enveloping algebra U(x̂) is associative. In this case the general realization of the NC coordinates is x̂µ = xαφαµ(iLγδ), where Lγδ = xγpδ generate the gl(n) algebra [Lµν , Lρσ] = i(ηµσLρν − ηρνLµσ). In the lowest order in Kµνγδ we get x̂µ = xµ + ixαLβγKµγβα +O ( K2 ) = xµ + iKµγβαxαxβpγ +O ( K2 ) , with Kµνγδ −Kνµγδ = Θµνγδ. We point out that Sections 3 and 4 cannot be applied to these quadratic deformations. The construction of such quadratic algebras can be performed using twist operators that, besides Lorentz generators, include dilatation operators Dµ and D D = ∑ µ Dµ = ∑ µ xµpµ, Dµ = xµpµ (no summation), and more generally Lµν = xµpν generating the gl(n) algebra. Here we consider a simple case. 6.1 Quadratic deformations of Minkowski space from dilatation We consider the twist F = exp (∑ α,β aαβDα ⊗Dβ ) , aβα = −aαβ, (6.1) where [Dα, Dβ] = 0, [Dα, pβ] = ipαηαβ (no summation), [Dα, xβ] = −ixαηαβ, and pα are momenta. The action of Dα in undeformed quantum phase space is defined as Dα ▷ f(x) = −ixα ∂f(x) ∂xα . The deformed quantum phase space is defined with x̂αx̂β = qαβx̂βx̂α, qαβ = exp(aαβ − aβα) = exp(2aαβ), [Dα, x̂β] = −ix̂αηαβ, pαx̂β − eaβα x̂βpα = −iηαβ exp (∑ γ iaβγDγ ) , cαβ = eaαβ . The realization for x̂α is given by x̂α = xα exp (∑ β iaαβDβ ) = xαϕα, ϕα = exp (∑ β iaαβDβ ) , ϕα ▷ 1 = 1, x̂α ▷ 1 = xα. 14 S. Meljanac and R. Štrajn Note that xαϕα = exp(−aααϕβxα), xαϕβ = exp(−aβαϕβxα), xαf(iDα) = f(iDα − 1)xα, pαf(iDα) = f(iDα + 1)pα. The coproducts are given by ∆Dα = F∆0DαF−1 = Dα ⊗ 1 + 1⊗Dα = ∆0Dα, ∆ϕα = ϕα ⊗ ϕα (no summation), ∆pα = F∆0pαF−1 = pα ⊗ ϕα + ϕ̃α ⊗ pα, ϕ̃α = exp (∑ α iaβαDβ ) , ϕα = exp (∑ β iaαβDβ ) , ∆ϕ̃α = ϕ̃α ⊗ ϕ̃α, ϕαx̂β = eaαβ x̂βϕα, [ϕα, x̂β] = ( eaαβ − 1 ) x̂βϕα. The twist F given in equation (6.1) is abelian and satisfies the cocycle condition. The star product is associative xα ∗ xβ = x̂αx̂β ▷ 1 = x̂α ▷ xβ = eaαβxαxβ, xβ ∗ xα = eaβαxαxβ, xα ∗ xβ = qαβxβ ∗ xα, x̂α = mF−1(▷⊗ 1)(xα ⊗ 1) = xα exp (∑ β aαβDβ ) , (f ∗ g)(x) = mF−1(▷⊗ ▷)(f(x)⊗ g(x)). For aβα = −aαβ, qαβ = exp(2aαβ) = (cαβ) 2, cαα = qαα = 1, ϕ̃α = (ϕα) −1. The ▶ action is defined with f̂ ▶ ĝ = f̂ ĝ, x̂αf̂ ▶ 1 = x̂αf̂ , x̂αf̂ = ( Oα ▶ f̂ ) x̂α, f̂ x̂α = x̂α ( O−1 α ▶ f̂ ) , Oα = ϕα ( ϕ̃α )−1 , Oα ▶ 1 = 1, ϕα ▶ 1 = 1, pα ▶ 1 = 0, pα ▶ x̂β = −iηαβ. Let us define ŷα = mF̃−1(▷⊗ 1)(xα ⊗ 1) = xαϕ̃α = x̂α(ϕα) −1ϕ̃α = x̂α(Oα) −1, where F̃ = Fop, Fop = exp (∑ α,β aαβDβ ⊗Dα ) , ŷα ▷ 1 = xα, ŷα ▶ f̂ = f̂ x̂α. For aβα = −aαβ, Oα = (ϕα) 2. [x̂α, ŷβ] = 0 ∀α, β, ŷαŷβ = exp(−2aαβ)ŷβ ŷα. The special case where qαβ = q for α > β and qαβ = q−1 for α < β was studied in [52, 53]. Remark 6.1. For aβα = aαβ it follows that F = Fop, x̂µ = ŷµ, [x̂µ, x̂ν ] = 0, ϕ̃µ = ϕµ and (f ∗ g)(x) = (g ∗ f)(x), but cαα ̸= 1 and pαx̂β − eaαβ x̂βpα = −iηαβ exp (∑ γ iaβγDγ ) . For the case of one dimension, n = 1, see [18, 117]. Applications to the Fock space representation and the Calogero model in one dimension were considered in [18]. Deformed Quantum Phase Spaces, Realizations, Star Products and Twists 15 7 Generalization of Yang and triply special relativity models In Sections 2–5, we have considered quantum deformed phase spaces (for example the Θ canon- ical space, Lie algebra type spaces, the Snyder space) in which defomation parameters are proportional to the minimal length l. In Section 6, related to quadratic deformations of quan- tum phase space, deformation parameters are dimensionless. There are also deformed quantum phase spaces in which deformation parameters depend on two physical quantities, the minimal length and the cosmological radius R. These models are generated with NC coordinates x̂µ and NC momenta p̂µ. A large class of such deformed quantum phase spaces can be described with algebras containing 2 Snyder algebras as subalgebras, with the same Lorentz algebra generated with Mµν . They are defined as [x̂µ, x̂ν ] = iβ2Mµν , (7.1) [p̂µ, p̂ν ] = iα2Mµν , (7.2) [Mµν , x̂λ] = i(ηµλx̂ν − ηνλx̂µ), (7.3) [Mµν , p̂λ] = i(ηµλp̂ν − ηνλp̂µ), (7.4) [Mµν ,Mρσ] = i(ηµρMνσ − ηµσMνρ − ηνρMµσ + ηνσMµρ), (7.5) [x̂µ, p̂ν ] = igµν , (7.6) [Mµν , gρσ] = i(ηµσgρν − ηνσgρµ + ηµρgνσ − ηνρgµσ), (7.7) [gµν , x̂λ]− [gλν , x̂µ] = iβ2(ηµν p̂λ − ηλν p̂µ), (7.8) [gνµ, p̂λ]− [gνλ, p̂µ] = iα2(ηλν x̂µ − ηµν x̂λ) = −iα2(ηµν x̂λ − ηλν x̂µ), (7.9) [gµν , gρσ] = i ( [[gµν , p̂σ], x̂ρ]− [[gµν , x̂ρ], p̂σ] ) . (7.10) These algebras are Born dual, x̂µ ↔ p̂µ, Mµν ↔Mνµ, gµν ↔ −gνµ, α↔ β. Hermitian realizations of x̂µ, p̂µ, Mµν and gµν can be written as x̂µ = 1 2 ( xµF + F †xµ + pµG+G†pµ ) , p̂µ = 1 2 ( pµH +H†pµ + xµK +K†xµ ) , Mµν = xµpν − xνpµ, gµν = ηµνh0 + α2xµxνh1 + h†1α 2xµxν + β2pµpνh2 + h†2β 2pµpν + αβ(xµpν + pνxµ)h3 + h†3αβ(xµpν + pνxµ) + αβ(xνpµ + pµxν)h4 + h†4αβ(xνpµ + pµxν), gµν − gνµ = 2Mµναβ ( h3 + h†3 − h4 − h†4 ) , where F , G, H, K, h0, h1, h2, h3, h4 are Lorentz invariants depending on x2, x · p, p2. For the Yang model, gµν = ηµνh0. Yang quantum phase spaces [44, 118] and tripy special relativity models [55, 105, 106] are special cases of the above deformed quantum phase spaces. Realizations of these models are more difficult to construct and will be presented elsewhere. Spinorial Snyder and Yang models from super algebras and NC quantum super spaces have recently been constructed [68]. Similarly, generalized spinorial models, super algebras and quantum super spaces can be constructed by extending the above deformed quantum phase spaces (7.1)–(7.10). 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Gen. 27 (1994), 2075–2082. https://doi.org/10.1088/0264-9381/26/24/245020 https://doi.org/10.1002/andp.201000105 https://arxiv.org/abs/0802.1129 https://doi.org/10.1103/physrevd.84.025021 https://arxiv.org/abs/1104.0490 https://doi.org/10.1088/0264-9381/29/21/215019 https://doi.org/10.1088/0264-9381/29/21/215019 https://arxiv.org/abs/1110.0201 https://doi.org/10.1103/physrevd.90.044019 https://arxiv.org/abs/1404.6396 https://doi.org/10.1155/2016/1328284 https://arxiv.org/abs/1501.01447 https://doi.org/10.1016/S0370-2693(01)00339-2 https://arxiv.org/abs/hep-th/0011172 https://doi.org/10.1103/physrevd.104.024017 https://arxiv.org/abs/2105.12573 https://doi.org/10.1088/1361-6382/ac05d7 https://doi.org/10.1088/1361-6382/ac05d7 https://arxiv.org/abs/2010.15734 https://doi.org/10.1103/PhysRevD.70.105006 https://arxiv.org/abs/hep-th/0408193 https://doi.org/10.1016/j.physletb.2008.02.001 https://arxiv.org/abs/0802.1250 https://doi.org/10.1063/1.1369656 https://doi.org/10.1103/PhysRev.71.38 https://arxiv.org/abs/math-ph/9910013 https://doi.org/10.1103/PhysRev.72.874 https://doi.org/10.1088/0305-4470/27/6/030 1 Introduction 2 Lie deformed quantum phase space and realizations 3 Star product from realizations 3.1 Snyder space and an extension with tensorial coordinates 4 Coproduct of momenta and twist from star product and realizations 5 Linear realizations 5.1 Right covariant realization of kappa-Minkowski space time 5.2 Left covariant realization of kappa-Minkowski space time 5.3 Light like realization of kappa-Minkowski spacetime 6 Quadratic deformations of quantum phase space 6.1 Quadratic deformations of Minkowski space from dilatation 7 Generalization of Yang and triply special relativity models References
id	nasplib_isofts_kiev_ua-123456789-211523
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-19T06:23:16Z
publishDate	2022
publisher	Інститут математики НАН України
record_format	dspace
spelling	Meljanac, Stjepan Štrajn, Rina 2026-01-05T12:24:36Z 2022 Deformed Quantum Phase Spaces, Realizations, Star Products and Twists. Stjepan Meljanac and Rina Štrajn. SIGMA 18 (2022), 022, 20 pages 1815-0659 2020 Mathematics Subject Classification: 81R60 arXiv:2112.12038 https://nasplib.isofts.kiev.ua/handle/123456789/211523 https://doi.org/10.3842/SIGMA.2022.022 We review deformed quantum phase spaces and their realizations in terms of the undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta, and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed quantum phase spaces, and Snyder-type spaces are considered. Examples of linear realizations of the κ-Minkowski spacetime are elaborated. Finally, some new results on quadratic deformations of quantum phase spaces and a generalization of the Yang and triply special relativity models are presented. SM thanks S. Mignemi for useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformed Quantum Phase Spaces, Realizations, Star Products and Twists Article published earlier
spellingShingle	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists Meljanac, Stjepan Štrajn, Rina
title	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
title_full	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
title_fullStr	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
title_full_unstemmed	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
title_short	Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
title_sort	deformed quantum phase spaces, realizations, star products and twists
url	https://nasplib.isofts.kiev.ua/handle/123456789/211523
work_keys_str_mv	AT meljanacstjepan deformedquantumphasespacesrealizationsstarproductsandtwists AT strajnrina deformedquantumphasespacesrealizationsstarproductsandtwists

Deformed Quantum Phase Spaces, Realizations, Star Products and Twists

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