Generalized Yang Poisson Models on Canonical Phase Space

. We discuss the generalized Yang Poisson models. We construct generalizations of the Yang Poisson algebra related to o (1 , 5) algebra discussed by Meljanac and Mignemi (2023). The exact realizations of this generalized algebra on canonical phase space are presented and the corresponding differential equations are solved in simple cases. Furthermore, we discuss the Poisson algebras related to o (3 , 3) and o (2 , 4) algebras


Yang Poisson model
The Yang model of noncommutative geometry on a curved background spacetime was proposed in [18] and is a generalization of the noncommutative geometry first introduced by Snyder [17].This model is defined in terms of a 15-parameter algebra isomorphic to o(1, 5) containing the generators of the Lorentz algebra with the coordinates of phase spaces, and was generalized in [4].Later, Snyder-de Sitter or triply special relativity model, where this symmetry is realized nonlinearly, was proposed in [1,3,5,14].Realizations of the Yang model have been discussed in [6,8,10,11,13].They cannot be obtained in closed analytic form, but only as power series in coupling constants.In the classical limit, where quantum commutators are replaced by Poisson brackets, the algebra has a simpler structure and no ordering problems arise.The corresponding realizations of this Yang Poisson model can be obtained in an exact form [12].
We look for realizations of xµ and pµ on a phase space with coordinates x µ and momenta p µ satisfying the canonical algebra Realizations of the Yang Poisson algebra on a canonical phase space of coordinates x µ and p µ , with the Lorentz algebra generators realized as M µν = x µ p ν − x ν p µ were analyzed and discussed in [12].Special solution for realizations of xµ and pµ are given by xµ where Generally, it holds that in terms of original variables xµ and pµ , This result for h is universal and generally valid in the Yang Poisson model [12].The most general realizations are obtained using the group of automorphisms applied to the special solution defined with ϕ 2 (z) = 0 [12].
In the present paper, we shall consider generalized Yang Poisson models and their related models, as well as their exact realizations.In Section 2, we define generalizations of the Yang Poisson algebra (1.1)-(1.6)and construct realizations of this algebra solving the corresponding differential equations, presenting the general results in Section 3. Furthermore, we discuss the related Poisson algebras in Section 4.

Generalized Yang Poisson models
The most general new generators linear in xµ , pµ and M µν are where M µν = x µ p ν − x ν p µ is unchanged and A, B, φ, ψ, a µ , b µ are dimensionless parameters with AB ̸ = 0.
3 General solution Proof .First, we have that By induction on n and using (1.5), we prove the relations h, . . ., h, h, xµ . . .
For the case when n = 2k, it is easy to see, using (1.5) that for k = 1, we get Assume that the relation (3.4) holds for some k > 1.Then by the induction assumption and using (1.5), we have h, . . ., h, h, xµ . . .
Similarly, by induction on n and using (1.5), we prove that relation (3.5) holds.Now, inserting relations (3.4) and (3.5) in (3.3), we finally prove that (3.1) holds.Also, in a similar way, we prove (3.2). ■ Let us now define an operator L G acting on a canonical phase space spanned by x µ and p µ as where G(x, p) and f (x, p) are functions on classical phase space.Furthermore, we define and then construct an operator O such that where h is given in (1.9).The general structure of an operator O acting on x(0) Solving perturbatively (3.6), a unique solution was found for G in [12], where ) .
The summation of the equation (3.7) for G gives an exact result, with the property The composition of mappings e φ L • e L G can be calculated perturbatively applying the BCH formula.where which implies that in order to prove (3.8) is sufficient to prove that the relation holds.We prove (3.9) by induction on n.For n = 1, using the Jacobi identity, we get Let us assume that the relation (3.9) holds for some n > 1.Then by the induction assumption and using the Jacobi identity, we have Then in the first order we get The most general realizations of Xµ , Pµ and h are obtained using the group of automorphisms applied to the special solution X(1) µ , P (0) µ and h (1,0) , namely Xµ = O F X(1) µ and h = O F h (1,0) , where O F = e L F and F is arbitrary function of α 2 x 2 , β 2 p 2 and z.Alternatively, we can write These solutions satisfy the above seven differential equations (2.17)-(2.23).
Remark 3.3.Note that the Poisson brackets are covariant under the action of e L F i.e. they satisfy for any function F , f , g.Also e L F M µν = M µν if F, M µν = 0 and e L F η µν = η µν .
Remark 3.4.Using realizations of the Snyder model [7], we can write the corresponding realizations of the Snyder Poisson model.From the results that were found in [7], we get , where

Related Poisson models
In this section, we introduce related Poisson algebras generalizing (1.3), (1.4) and (1.5): where ϵ 2 1 = ϵ 2 2 = 1.The Yang Poisson model in Section 1 corresponds to the ϵ 1 = ϵ 2 = 1 case which is related to o(1, 5) algebra.Now, let us consider the case when ϵ 1 = ϵ 2 = −1.A special solution for realizations of xµ and pµ in this case is given as where The In the case when ϵ 1 = 1 and ϵ 2 = −1, a special solution is given by xµ where The new generators Xµ and Pµ are given by where Finally, in the last case when ϵ 1 = −1 and ϵ 2 = 1 a special solution is given by xµ where The Note that the algebras generated with cases when ϵ 1 = 1, ϵ 2 = −1 and ϵ 1 = −1, ϵ 2 = 1 are related to o(2, 4) algebra.

Discussion
The generalized Yang models are examples of noncommutative geometry on a background spacetime of constant curvature that display a duality between position and momentum manifolds.
In this paper, we have obtained the exact realizations of a generalized Yang Poisson algebra on a canonical phase space related to the o(1, 5) algebra.These realizations are simpler than in the quantum case.The results we have obtained can be considered as a limit of the quantum formalism for ℏ → 0, presenting a classical approximation of the quantum realizations.
Moreover, we have discussed the Poisson algebras related to o(3, 3) and o(2, 4) algebras.These models correspond to different physical settings, namely, the case ϵ 2 = −1 is related to the symmetries of anti-de Sitter spacetime, while ϵ 1 = −1 corresponds to the so-called anti-Snyder algebra [15].We recall that the anti-Snyder algebra enjoys rather different properties from the Snyder algebra, in particular concerning the existence of a maximum allowable momentum.
Possible applications of our results are in cosmology, since the present model could be useful in describing effects of noncommutativity in early stages of inflation, and in the investigation of modifications of the dynamics of simple mechanical systems caused by the deformed symplectic structure.The most elementary example is the harmonic oscillator, that has been studied in [12] in the o(1, 5) case, exhibiting a modification with respect to the canonical solution, with the period that becomes energy dependent.In the o(3, 3) and o(2, 4) cases, one expect similar modifications, analogous to those found in [16] for the related TSR setting.
In the present paper, a canonical phase space is defined with coordinates x µ , momenta p µ and Lorentz generators M µν = x µ p ν − x ν p µ .However, it is possible to define the extended coordinates x µν with corresponding momenta p µν and to interpret the Lorentz generators M µν as extended coordinates x µν and h as an additional scalar coordinate.In this framework, the algebra (2.3)-(2.12) is generated by 15 coordinates.The corresponding realizations can be obtained from the quantum realizations of the generalized Yang models, presented in [6,9].

Proposition 3 . 2 .
Application of the BCH formula to a Poisson algebra with the classical Poisson brackets gives e L A • e L B = e L C , (3.8) new generators Xµ and Pµ generate a new class of Poisson algebras isomorphic to the initial Yang Poisson algebra.The new algebra generated by Xµ , Pµ , M µν and h is given by Xµ new generators Xµ and Pµ , linear in xµ , pµ and M µν , are given in (2.1) and (2.2).Also, a new algebra, related to o(3, 3) algebra, generated by Xµ , Pµ , M µν and h is given in (2.3)-(2.11),where ρ A, B, φ, ψ, a µ , b µ are dimensionless parameters with AB ̸ = 0. New generators Xµ and Pµ together with M µν and h generate a new algebra, isomorphic to the initial Yang Poisson algebra, which is given in (2.3)-(2.10),where h new generators Xµ and Pµ are given in (4.1) and (4.2) and a new algebra generated by them together with M µν and h is given in (2.3)-(2.10),where h