Hermitian realizations of the Yang model

The Yang model is an example of noncommutative geometry on a background spacetime of constant curvature. We discuss the Hermitian realizations of its associated algebra on phase space in a perturbative expansion up to sixth order. We also discuss its realizations on extended phase spaces, that include additional tensorial and/or vectorial degrees or freedom.


Introduction
In recent years noncommutative models in curved spacetime have been extensively investigated, either from a formal perspective [1][2][3][4][5][6][7][8][9][10][11][12], also in connection with quantum field theory [13][14][15], or in view of their application in the study of phenomenological effects in cosmology [16][17].However, the first example of noncommutativity on a curved spacetime background was proposed by C.N. Yang [18] already in 1947, soon after Snyder had introduced the idea of a noncommutative spacetime [19].Yang's proposal was based on an algebra which included phase space and Lorentz generators, where the commutation relations between the components of the position operators, as well as those of the momentum operators were not trivial, giving rise to a spacetime displaying both noncommutativity and curvature.
The noncommutative Yang algebra is a 15-parameter algebra, isomorphic to so (1,5), defined by the relations where α and β are real parameters and η µν the flat metric and we use natural units, h = c = 1.We interpret the operators xµ and pµ as coordinates of the quantum phase space, M µν as generators of the Lorentz transformations and h as a further scalar generator, necessary to close the algebra.The algebra (1) is invariant under Born duality [20] It contains as subalgebras both the de Sitter and the Snyder algebras, to which it reduces in the limit β → 0 and α → 0, respectively.
We have investigated the Yang model in previous papers.In particular, in [10][11] we have considered noncommutative models in a spacetime of constant curvature and discussed their realizations on a quantum phase space.These models preserve the Lorentz invariance and, besides Yang proposal, include some generalizations [1][2].Later, in [12], we have discussed the possibility of obtaining Yang model by symmetry breaking of an algebra defined in an extended quantum phase space that includes also tensorial generators, of the kind introduced in [21][22][23][24][25] for the Snyder model.
Following [12], in this paper we shall investigate the realizations of the Yang algebra in terms of a restricted number of operators of a Hilbert space, the simplest case being realizations in terms of phase space variables x µ and p µ [10][11].However, we shall use a more efficient procedure than in [12] for going to higher orders.Moreover, several possibilities arise depending on how many operators are introduced to generate the Hilbert space, as we discuss in the following sections.For example, one may consider the Lorentz generators as independent from the phase space ones, as proposed in [21][22][23][24] in the case of the Snyder model.Some choices may be useful to obtain Hopf algebra structures, which are not possible in a phase space realization.In general, we shall only obtain perturbative realizations of the algebra, since analytic results seem to be out of reach.

Realizations of Yang model on quantum phase space
In this section, we look for Hermitian realizations in quantum phase space, with where xµ , pµ , M µν and h are functions of phase space operators x µ and p µ that satisfy the Heisenberg algebra with and p µ ⊲ 1 = 0, M µν ⊲ 1 = 0.
In the limit α = 0, a Hermitian realization of the algebra (1) is given by [11] xµ Analogously, when β = 0, a realization is However, when both α = 0 and β = 0, we get where with This result is different from iη µν h(x, p) and therefore xµ (β) and pµ (α) are not a realization of the Yang algebra.In order to construct a true realization of the Yang algebra, we fix pµ = pµ (α) and define xµ = e iG xµ (β)e −iG , choosing G such that [x µ , pν ] = iη µν h.In general, we can expand G as where g 2m,2n are functions of x2 , p 2 and D. From xµ = e iG xµ (β)e −iG , it follows Then, up to sixth order in α and β, we get Hence, Substituting in (7), it follows Requiring that only terms proportional to η µν survive, one obtains (see Appendix A) Hence, at this order, and then and [x µ , pν ] = iη µν h, with We point out that infinitely many realizations can be obtained from xµ and pµ in ( 16) by similarity transformations, defined by acting simultaneously with e iG(D ,x 2 ,p 2 ) on all generators xµ , pµ , M µν and h obtained above.Note that M µν is invariant under these transformations because G is a function of Lorentzinvariant operators, but h is not invariant since [G, h] = 0.
Special classes of realizations are with G given in (15) and At sixth order in α in β we have In particular, for

Realizations of extended Yang model on quantum phase space
A different realization of the Yang algebra can be obtained introducing additional tensorial generators xµν = −x νµ , similarly to what has been done in [21][22][23][24][25] for the Snyder model or in [26] for a more general setting.They are assumed to satisfy In this case, we consider realizations of Lorentz generators of the form and M µν ⊲ 1 = xµν ⊲ 1 = x µν , where x µν are commuting variables.
From the expansion (9), we get at fourth order in α, β, Hence, There are infinitely many realizations obtained from xµ and pµ with arbitrary similarity transformations that are invariant under Lorentz transformations and act on all generators xµ , pµ and h simultaneously.Note that M µν is invariant under these transformations but h is not.

Realizations of Yang model on double quantum phase space
A different class of realizations can be obtained by adding to the generators x µ , p µ of the Heisenberg algebra new generators q µ and k µ satisfying a second Heisenberg algebra, and These realizations are more symmetric in the phase space variables and might permit the definition of a Hopf structure.We shall call the phase space obtained by the addition of q µ and k µ double quantum phase space.
In the limit α → 0, a realization of the Yang model in this space is given by with nonvanishing parameters b and b, with b − b = 1.Analogously, when β = 0, As usual, if both α = 0 and β = 0, xµ (β) and pµ (α) are not a realization of the Yang algebra.
In order to construct realizations of the Yang model in this space, as in sections 2 and 3, we set pµ = pµ (α) and xµ = e iG xµ (β)e −iG , and construct the operator G such that [x µ , pν ] = iη µν h.In general, G can be expanded as in (9).
Proceeding as usual, we get at fourth order in α and β, where D = 1 2 (k and Again, infinitely many realizations can be obtained by acting on (35), (36) with similarity transformations.

Realizations of extended Yang model on double quantum phase space
Let us finally consider the Yang model with both additional phase space generators and additional Lorentz generators.Realizations of this kind have been considered in [12] in a slightly different formalism, see sect.6.The additional Lorentz generators xµν are introduced as in sect.3, such that and M µν ⊲ 1 = xµν ⊲ 1 = x µν , where x µν are commutative parameters.
Proceeding as usual, one can show that realizations up to second order in α 2 , β 2 are in this case with Also in the present case infinitely many realizations can be obtained by similarity transformations.

Concluding remarks
In this paper we have assumed that in the limit α = 0, β = 0, the Yang algebra (1) reduces to the ordinary Heisenberg algebra with Lorentz algebra action and h → 1. Realizations are obtained in terms of quantum phase space and double quantum phase space with or without tensorial coordinates.
An approach to the Yang algebra alternative to the one we have considered here is to view it as a Lie algebra with 15 generators xµ , pµ , M µν and ĥ.When all structure constants go to zero, it reduces to a commutative space with coordinates x µ , q µ , x µν and h with relations xµ ⊲ 1 = x µ , pµ ⊲ 1 = q µ , M µν ⊲ 1 = x µν and h = 0. Realizations of this Yang algebra can be found using the method of realizations of Lie algebras described in [9,26].These realizations are linear in the position coordinates, but are given by power series in the momenta.Such approach was used in [22][23][24] for the extended Snyder model and in [12] for the Yang model.
Finally, let us notice that the Yang model can be obtained from the so(1, 5) algebra with 15 generators M AB (A, B = 1, . . .5), through the relations xµ = βM µ4 , pµ = αM µ5 and ĥ = αβM 45 .A realization of so (1,5) in symmetric ordering has been presented in [26] and can be used for the Yang model as well.
As future prospects of our investigations we may envisage the possibility of constructing a star product and a twist using the double quantum phase space.Also the definition of a field theory on a spacetime based on the Yang model can be pursued from the present results and would be of great interest.