Yang's Model as Triply Special Relativity and the Snyder's Model--de Sitter Special Relativity Duality

We show that if Yang's model as the earliest TSR is completed, it should contain both Snyder's quantized space-time model as the earliest DSR, the de Sitter SR and their duality.


I. INTRODUCTION
Some sixty years ago, Snyder [1] proposed a quantized space-time model by means of the projective geometry approach to the de Sitter (dS)-space of momentum with two universal constants: c and 4 dimensions. But, how to realize Yang's model in 4 dimensions completely? In his very short paper, Yang did not answer the question. Recall that there are three 4-d maximally symmetric spacetimes with maximum symmetries of ten generators, which are just the Mink/dS/AdS-space with ISO(1, 3)/SO(1, 4)/SO (2,3) invariance, respectively. Thus, it is impossible to realize Yang's so (1,5) algebra with fifteen generators on one space of 4-dimensions in the sense of Riemann geometry and Lie symmetry. The TSR realization of Yang's so (1,5) algebra gives a tentative 4-d realization. But, it is in terms of a deformed algebra with non-commutative geometry. The fact that there are two so (1,4) subalgebras with a common homogeneous Lorentz algebra so(1, 3) in Yang's so (1,5) suggests another kind of realization: A pair of dS-spaces of 4-dimensions with a dual relation.
In this paper we show that if Yang's model can be completed with such a kind of 4-d realizations at both classical and quantum level, this complete Yang model should contain both Snyder's quantized space-time model, the dS special relativity and their duality.
This paper is arranged as follows. In section II, we first recall and complete Yang's model with a UV-IR dual invariance in a 6-d dimensionless Mink-space at both classical and quantum level. Then, in section III, we show that the two so(1, 4) subalgebras in the complete Yang model relevant to the space-time coordinate operatorsx µ and the momentum operatorsp µ are the same as Snyder's so(1, 4) algebra of quantized space-time and the algebra for 'quantized' energy, momentum, and angular momentum in a dS-space of spacetime, respectively. We also present a way to get Snyder's model, the dS special relativity and their duality from the Yang model. Finally, we end with some concluding remarks.

II. A COMPLETE YANG MODEL AND A UV-IR DUALITY
Under Yang's so(1, 5) algebra, there is an invariant quadratic form of signature −4 [2] in a 6-d dimensionless Mink-space M 1,5 . Then, the metric in M 1,5 reads Thus, there is a 12-d phase space (M, Ω) with a symplectic structure Ω and the non-vanishing basic Poisson bracket in (ζ A , N A ): Obviously, the dimensionless 6-'angular momentum' L AB := ζ A N B − ζ B N A as the classical counterpart of Yang's operators (see below) form an so(1, 5) algebra under Poisson bracket: Under canonical quantization, in the 'coordinate' picture withN A = i ∂ ∂ζ A , [ζ A ,N B ] = −iδ A B , they become operatorsL AB forming the algebra under Lie bracket. Now, the following operators are just the operators in Yang's model [2] up to some redefined coefficientŝ with ǫ 123 = ǫ 23 1 = 1 and ζ j = η jA ζ A . They form Yang's so(1, 5) algebra as follows: together with an so(1, 3) for the 4-d angular momentum operators.
It is clear that there are two so(1, 4) for coordinate operatorsx µ and momentum operatorsp ν , respectively, with a common so(1, 3) forl µν . It is also clear that in Yang's algebra with respect to the 6-d 'angular momentum' there is a Z 2 = {e, r|r 2 = e} dual transformation with Since a is near or equal to the Planck length ℓ P and R is the radius of a dS universe, the invariance under the Z 2 dual transformation is a UV-IR duality.

YANG MODEL
A. Snyder's model from the Yang model

Snyder considered a homogeneous quadratic form
It is a model via homogeneous (projective) coordinates of a 4-d momentum space of constant curvature, a dS-space of momentum. In fact, it can also be started from a dShyperboloid H a in a 5-d Mink-space of momentum with radius 1/a Snyder's inhomogeneous projective momentum is almost the same as the momentum in Beltrami coordinates. In order to preserve orientation, the antipodal identification should not be taken so that the Beltrami atlas should contain at least eight patches to cover the hyperboloid (see, e.g. [9]). In the patch U 4+ , η 4 > 0, Snyder's Beltrami momentum read Now the metric in the patch becomes ds 2 a = g µν a dq µ dq ν with where q µ = η µν q ν . Along geodesic that is the great 'circle' on H a , the spacetime 'coordinates' and angular momentum are conserved Importantly, from these conserved Killing observables and q 0 = E as energy 1 , it follows an important identity It would mean that there is some 'wave packet' moving with constant 'group velocity'. Namely, a law of inertia-like in space of momentum hidden in Snyder's model [19].
Regarding such a 'wave packet' as an object in the space of momentum, a 8-d phase space (M a , ω a ) for can be constructed and locally there are Snyder's momentum q µ as canonical momentum and the conjugate variables X µ as canonical coordinates (q µ , X µ = Rg µν a dq ν /ds a ) with a symplectic structure ω a and basic Poisson brackets {q µ , X ν } a = −δ ν µ , {q µ , q ν } a = 0, {X µ , X ν } a = 0. Then Snyder's space-time 'coordinates' x µ a and angular momentum l µν a can be expressed in terms of these canonical variables (q µ , X µ ). And it is straightforward to show that they form an so(1, 4) under Poisson bracket.
In a momentum picture of the canonical quantization, the operators of Snyder's 'coordinates' and angular momentum are just ten Killing vectors of the model, up to some coefficients, Together with 'boost'M ai =x i a q 0 +x 0 a q i =:l 0i a =L 0i a and '3-angular momentum'L ai = − 1 2 ǫ ijk (x j a q k − x k a q j ) =: 1 2 ǫ ijkl jk a = 1 2 ǫ ijkL jk a , they are the components of 5-d angular momentumL AB a and form an so(1, 4) algebra: Obviously, Snyder's quantized space-time 'coordinates' so(1, 4) algebra is the same as the coordinate so(1, 4) subalgebra of Yang's so (1,5). But, the operators of canonical coordinatesX µ are still commutative.
In order to get Snyder's model from the complete Yang model, we consider a dimensionless dS 5 ∼ = H ⊂ M 1,5 : Take a subspace I 1 of H ⊂ M 1,5 as an intersection where P| ζ 4 =0 is a hyperplane defined by ζ 4 = 0. Introduce dimensional coordinates a free particle with mass /a may move uniformly along a great 'circle' defined by a conserved 5-d angular momentum with an Einstein-like formula for mass /a The conserved momentum and angular momentum of the particle can be defined as And the Einstein-like formula becomes where p Rµ = η µν p ν R and l Rµν = η µρ η νσ l ρσ R . In a Beltrami atlas of the BdS-spacetime [9], the Beltrami coordinates read in the patch U 4+ , ξ 4 > 0, and the metric becomes ds 2 R = g Rµν dy µ dy ν with g Rµν = σ −1 (y)η µν + R −2 y µ y ν σ −2 (y), σ(y) = 1 − R −2 y ν y ν > 0, (3.20) where y µ = η µν y ν . For the free particle with mass /a along geodesic that is the great 'circle' on H R , its motion becomes uniform motion with constant coordinate velocity. In fact, its momentum and angular momentum are constants. This leads to the law of inertia for the particle: For the particle, there is an associated phase space (M R , ω R ) and locally there are Beltrami coordinates as the canonical coordinates and the covariant 4-momentum as canonical momentum (y µ , P µ = a g µν dy ν /ds R ) with a symplectic structure and basic Poisson brackets in the patch {y µ , P ν } R = −δ µ ν , {y µ , y ν } R = 0, {P µ , P ν } R = 0. Now, the conserved Killing momentum and angular momentum of the particle can be expressed in terms of the canonical variables and form an so(1, 4) algebra under the Poisson bracket.
In a canonical coordinate picture of the canonical quantization, the operators of these conserved Killing Beltrami momentum and angular momentum are just ten Killing vectors of the model up to some coefficients forming an so (1, 4) under Lie bracket together with an so(1, 3) for angular momentum operatorsl µν R . This is the same as the momentum subalgebra of the Yang model.
It is remarkable that the conserved Killing Beltrami momentum lead to the law of inertia in the patch and it holds globally in the atlas patch by patch. In fact, the dS special relativity can be set up based on the principle of inertia and the postulate of universal constants, the speed of light c and the dS-radius R [9].
In order to show that there is indeed the BdS-model of the dS special relativity from the complete Yang model, let us consider another subspace I 2 of H ⊂ M 1,5 (3.9) of the Yang model as an intersection where P| ζ 5 =0 is a hyperplane defined by ζ 5 = 0. Introduce dimensional coordinates It is also straightforward now to find that the A, B = 5 components of the 6-d 'angular momentum' operatorsL AB in the Yang model consist of a 5-d angular momentum, which is just the angular momentum operatorsL AB R in the dS special relativity. And Yang's momentum, angular momentum operatorsp µ ,l µν in (2.3) and their subalgebra are just the Killing Beltrami momentum, angular momentum operatorsp µ R ,l µν R in the Beltrami model of the dS special relativity. Thus, the complete Yang model really contain the BdS-model of dS special relativity as a sub-model.

C. The Snyder's model-dS special relativity duality in the Yang model
It is important to see [19] that between Snyder's model and the dS special relativity, there is also a Z 2 = {e, s|s 2 = e} dual exchange with This is also a UV-IR exchange. And it is isomorphic to the Z 2 duality in Yang's model (2.4). The Snyder's model-dS special relativity duality contains some other contents. One is that the cosmological constant Λ should be a fundamental constant in the Nature like c, G and . This is already indicated in Yang's model as long as R = (3/Λ) 1/2 is taken.
Thus, not only both Snyder's model and the dS special relativity are sub-models in the complete Yang model, but their Z 2 duality transformations are contained in that of the Z 2 duality in the Yang model as well.

IV. CONCLUDING REMARKS
The above 'surgery' for two subspaces I 2 and I 2 of a dimensionless H ⊂ M 1,5 (3.9) in the Yang model shows that both Snyder's model and the BdS-model of dS special relativity are really sub-physics of the complete Yang model. And the UV-IR duality in the Yang model is just the one-to-one exchange of the Snyder model-dS special relativity duality.
It is quite possible that there are some other physical implications and/or relations with other dualities, such as the T-duality and S-duality, if a and R may have other identifications.
It should be mentioned that a Yang-like model with an so(2, 4) symmetry on a dimensionless (2, 4)-d flat space M 2,4 can be set up and all similar issues for Snyder's model and the dS special relativity or for an anti-Snyder's model on an AdS-space of momentum and the AdS special relativity can also be realized started with a dimensionless AdS 5 ∼ = H or its boundary ∂(AdS 5 ) ∼ = N ⊂ M 2,4 .
Since Yang's algebra is just the Lie algebra form of the deformed algebra in TSR, which is a generalization of DSR, it should be explored from the point of view in our approach what are the relations with the other DSR models and the TSR. It seems that the duality exists between other DSR models and dS-spacetime since some DSR models can be realized as Snyder's model in different coordinates on the dS-space of momentum, the corresponding coordinates on the dS-spacetime may also be taken. But, this may cause some issues due to the non-inertial effects from the viewpoint of the dS special relativity.
Finally, we would like to emphasize that the complete Yang model should be regarded as a theory of the special relativity based on the principle of inertia in the both spacetime and space of momentum as well as the postulate on three universal constants c, ℓ P and R.
All above issues and related topics should be further studied.