Spinorial Snyder and Yang Models From Superalgebras And Noncommutative Quantum Superspaces

The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors.


Introduction
In order to obtain quantum gravity models which reconcile two basic theories in physics, namely general relativity (GR) and quantum mechanics (QM), it appears desirable to introduce noncommutative (NC) quantum space-times (see e.g. [1]- [5]) and look for also the quantum deformations of quantum-mechanical relativistic phase space algebra with quantum NC fourmomenta (see e.g. [6]- [9]). Further, because the spinorial variables are even more fundamental that the vectorial ones (see e.g. quark model of hadrons or Penrose twistor theory) it is interesting to look for the algebraic systems providing quantum-deformed spinorial variables.
In this short paper we consider firstly various NC Lorentz-covariant models of quantum space-times and quantum-deformed relativistic phase spaces, which originate from the ones proposed firstly in 1947 by Snyder [1] and Yang [6].
These historically first NC structures of quantum space-times and quantum relativistic phase spaces are directly associated with the Lie algebras describing D=4 space-time symmetries which contain D = 4 Lorentz algebra as its subalgebra. Further, these models will be naturally extended to supersymmetric ones, with supercharges promoted to fundamental quantum-deformed fermionic spinorial variables.
whereĝ =k ⊕ĥ and 4 The sub(super)algebraĥ enters as the covariance (super)algebra, andô(3, 1) ⊂ h 5 ; the generatorsk via Snyderization procedure are defining quantum (super) space coordinates, and to quantum phase (super)space if we are able to embedd ink some quantum-deformation of relativistic Heisenberg algebra. We observe that in our Snyderization procedure of superalgebraĝ all supercharges belonging tok will be promoted to quantum-deformed spinors, which will transform under the spinorial covering groups (see e.g. (1.1)-(1.2)). In the case of semisimple superalgebrasĝ the fermionic odd spinorial generators form an algebraic basis ofĝ because for such Lie superalgebras all the bosonic generators can be described as bilinear products of supercharges. Subsequently, in our quantum-deformed covariant superspaces the spinorial quantum coordinates are primary, i.e. they describe the algebraic basis of U(ĝ). That property inclined us to use the names "spinorial Snyder" or "spinorial Yang" models for those obtained by the Snyderization of supercharges in semisimple Lie superalgebras.
We add that the quantum (super) spaces obtained via described above Snyderization procedure have two important properties: i) The algebraic construction of quantum (super)spaces inherits from Lie (super)algebras the validity of Jacobi identities i.e. the respective quantum (super)spaces are described by the associative (super)algebras.
ii) Because underlying classical Lie (super)algebras are endowed with Hopfalgebraic structure, the quantum (super)spaces inherit as well the coalgebraic structure, described for classical Lie (super)algebra generators by primitive coproducts.
The plan of this paper is the following. Firstly in Sect. 2 we recall briefly the algebraic construction of quantum D=4 Snyder space-time and quantum D=4 Yang relativistic phase spaces. In Sect. 3 we consider two D = 4 supersymmetric Snyder models of quantum dS and quantum AdS superspaces. Further, in Sect. 4 we describe three types of quantum Yang phase superspaces (one in D = 3, two in D = 4), which can be linked with quantum-deformed supersymmetric Heisenberg algebras. In last Section we provide outlook and final remarks.
It should be mentioned that the idea of using Lie (super)algebra relations for description of NC quantum superspaces was already considered earlier in the literature (see e.g. [23]- [24]), but these examples studied in the literature were not providing physically the most interesting physically cases of D = 4 Lorentzcovariant quantum superspaces with AdS and dS quantum fermionic spinors 4 Because superalgebraĝ is split into bosonic and fermionic sectorĝ =ĝ (0) ⊕ĝ (1) , by [·, ·} we denote graded commutator (i, j = 0, 1 mod 2) describing their odd sectors. One can add however that interest in Snyder and Yang type models is increasing; in particular recently Zoupanos et al [9], [25] applied the ideas based on Yang type models in order to describe the dynamics of D = 4 fundamental interactions, with included quantum gravity sector.
2 Snyder quantum space-times and Yang quantum phase spaces 2.1 Snyder quantum space-times D=4 dS and AdS algebras are described by the following five-dimensional orthogonal algebras (A=0,1,2,3,4) with signature η AB = diag(−1, 1, 1, 1, ǫ) and ǫ = η 44 = ±1, (ǫ = 1 for dS algebra and ǫ = −1 for AdS algebra). If following the Snyderization procedure we postulate taht M µ4 = 1 λx µ , where λ is an elementary length in quantum physics given usually by the Planck length λ p = G c 3 ≃ 1.6 · 10 −33 cm, we obtain besides the Lorentz algebra generators M µν (µ = 0, 1, 2, 3) the relations describing NC Snyder space-times 6 The difference between dS and AdS Snyder space-times consists only in difference of sign on rhs of relation (2.3). The algebra with the basis (M µν ,x ρ ) introduces algebraically an elementary relativistic quantum system, with D=4 quantum (A)dS space-timesx µ as Lorentz algebra module and Lorentz transformations providing the D=4 relativistic covariance of Snyder equations (2.2-2.3). Snyder models (see (2.1-2.3) are Born-dual (x µ ↔p µ , M µν unchanged, λ → 1 R ) to the momentum space realizations (M µν ,p µ ) ofô(4, 1) orô(3, 2) algebras with generators describing the automorphisms of five-dimensional pseudospheres where M µ4 = Rp µ describe the generators of curved translations on the pseu- In fact Snyder constructed his model as aimed at the description of NC geometry at ultra short (Planckian) distances, in order to regularize geometrically the ultraviolet divergencies in renormalization procedure of quantized fields. Born duality formalizes a physical as well as some philosophical concept that one can relate the micro and macro world phenomena -the first ones of quantum nature described by NC geometry, and the second linked with classical de-Sitter dynamics of general relativity at very large cosmological distances.
In the outlook we would like to comment on some possible directions of future studies, namely: i) Our method of passing from Snyder models describing quantum spacetimes to Yang models providing quantum-deformed Heisenberg algebra relations is quite general, what also permits to extend supersymmetrically Yang models in order to obtain quantum-deformed supersymmetric extension of Heisenberg algebra. It should be recalled however the old Snyder idea of adding "by hand" to quantum space-time coordinatesx µ the commuting fourmomenta p µ . In such an axiomatic method we postulate further the general covariant formula for the quantum-deformed basic commutator (for β see (2.3)) [22] [x µ , p ν ] = iη νν F (βp 2 ) + βp µ p ν G(βp 2 ) (5.4) and impose all required Jacobi identities. Such a problem posed for D = 4 AdS Snyder models has a general solution with one of the functions F ,G remaining arbitrary. Such axiomatic approach can be proposed also for supersymmetric extension of Snyder model in order to specify the anticommutator relations for fermionic counterparts (ξ α ,π α ) of the quantumdeformed Heisenberg algebra basis (x µ ,p µ ) 9 .
ii) If one uses superalgebras for obtaining via Snyderization the spinorial degrees of freedom, the spinors will appear necessarily as Grasmannian, what is desirable in the framework of QFT. One can however employ the "bosonic" coset of matrix groups with employment of spinorial covering groups -in such a case one gets curved bosonic spinors. A good example is provided by the case of conformal Penrose twistors (t A ∈ C; A = 1 · · · 4), which are the fundamental representations of D = 4 conformal "bosonic" group. Describing twistors by bosonic cosets or as N = 1 superconformal odd cosets one gets the following two different choices (see e.g. [34]) Penrose twistors t iii) It is known since eighties [36], [37] that local gauging of D = 4 and D = 5 quaternionic dS superalgebras leads to the appearance of gauge ghost fields. Recently however it appeared proposals (see e.g. [39]) that D = 4 dS supergravity without ghost fields can be obtained by suitably spontaneously broken D = 4 superconformal gravity. Further algebraic understanding of this idea is still desired.