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Generalized Quantum Relativistic Kinematics: a Stability Point of View

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincare-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.


Introduction
A prevailing theme of the last decade or so in physics has been the search for an algebraic signature of quantum gravity. Lorentz symmetry violation, spacetime non-commutativity and modified dispersion relations, among other novelties, have been proposed as signals our antennas should be tuned for, in the search for a scheme where quantum objects could be heavy too. Even before that, physicists exasperated by the darker side of quantum field theory, sought their way out of the maze of infinities in the form of a spacetime granularity that would exorcise, the hope was, their ultraviolet nemeses.
The usual suspect in many of these endeavors has been the nature of spacetime, typically codified in a kinematical algebra. Accordingly, the above search has often focused on the possible deformations of the Lie algebra G PH , i.e., the Poincaré algebra, extended by the inclusion of the position operators and the Heisenberg commutation relations, or one of its subalgebras. These attempts can be roughly divided into three categories, based on the mathematical framework of their approach (or the absence thereof). The first category comprises deformations of Lie type, where the commutators of the generators are linear functions of the same. There exists a well-developed mathematical formalism to deal systematically with such deformations, which has not always been used by physicists. We comment more extensively on this type of deformations below. In category number two enter quantum group type deformations, which are generalizations of the classical group concept and form particular examples of Hopf algebras with a universal R-matrix R, that solves the (universal) quantum Yang-Baxter equation 2 . The linearity of the Lie case is lost 3 , but the construction of the algebra is canonical, given an R with suitable properties. Extensive work in the eighties and nineties has provided a solid mathematical background for these deformations, with applications overflowing to an impressive list of fields. Finally, recent years have seen a plethora of articles loosely classified under the generic misnomer (see [1]) "Doubly Special Relativity" (see, e.g., [2]), which form the third category. A common feature among them, and in some sense the defining one, is that the commutators of the algebra are given by general analytic functions of the generators 4 . To our knowledge, there is no well-defined mathematical framework guaranteeing the self-consistency of these deformations, not to mention their physical applicability, partly because they are not complete, e.g., the fate of the spacetime sector is often left unclear. Subsequent work [18] showed that endowing the above deformations with considerable more (Hopf) structure, results in their identification with particular forms of the κ-Poincaré Hopf algebra, proposed about a decade ago [22] (see also [23]). Both of the last two categories suffer from serious physical problems in the many-particle sector, e.g., in a consistent definition of such basic quantities like the total momentum of a system of particles.
In this paper we deal with Lie-type deformations of standard quantum relativistic kinematics. We undertook this project with three main goals in mind: 1. Emphasize the Lie algebra stability point of view and present in an accessible form the relevant mathematical apparatus, along the lines of Ref. [25], which motivated the present work.
2. Apply the formalism to the problem at hand to obtain a complete, detailed map of the deformation territory in the vicinity of G PH .
3. Interpret physically the generators of the algebra and investigate the nature of the deformations.
The structure of the paper was conceived accordingly, with each of the subsequent three sections dealing with one of the above goals. In Sect. 2 we give a self-contained review of the standard Lie algebra deformation theory and explain why stable structures are more likely to prove useful in physical applications than unstable ones. The section ends with a relatively detailed example, the passage from Galilean to relativistic kinematics, illustrating the use of the formalism, as well as the (alas, a posteriori) predictive power of the stability point of view. Sect. 3 contains a detailed analysis of the options available in deforming G PH . We take as our starting point classical ( = 0) relativistic kinematics (an unstable algebra) and, with the help of some computer code we wrote, explore the various paths that lead to stable algebras. We find that there is essentially one path, its first stop introducing Heisenberg's relations. Thus, given Galileo's conception of spacetime as input, our program zeroes in on the Poincaré-plus-Heisenberg algebra G PH in about a minute. We find this motivating enough to inquire about what lies further ahead. Following this path to its end, we find ourselves in a three-dimensional deformation space of stable Lie algebras, with a double instability cone through its origin. The section ends with a description of relations between our work and earlier treatments in the literature. This concludes the mathematical part of the paper -inferences about physics will have to wait the physical identification of the generators, which we undertake in Sect. 4. There, we argue that the position operators do not have the right properties to serve as Lie algebra generators. In doing so, we are in disagreement with all previous works. Retracing the steps that lead to the definition of the relativistic center-of-momentum concept for a system of particles, we come to the conclusion that the appropriate generators, in the case of a massive, spinless particle, are the moment operators, given essentially by the positions rescaled by the mass operator for the particle. This shows that the algebra G QR of standard quantum relativistic kinematics, differs from G PH and, in the above case, lies on the instability cone. Furthermore, from the algebraic point of view, there is a single deformation direction introducing non-commutativity among the momenta. Sect. 5 comments on the findings and outlines directions for future work.

Lie Algebra Deformations and the Concept of Stability
In this section we summarize the elements of standard Lie algebra deformation theory that will be of use in the rest of the paper. Our exposition follows mostly the original source for this material [27,28], as well as [25]. Sect. 2.5 follows Ref. [6], echoing ideas originating in the Batalin-Vilkovisky quantization (see, e.g., [35]). Background information on Lie algebra cohomology can be extracted (not without some effort) from [5,14]. The foundations of deformation theory are laid out in the classic [10], while plenty of newer material is contained in the book-length [11]. An elegant generalization to bialgebra deformations was given in [12], with still further generalizations to Drinfeld algebras, and much more, appearing in [32] -this latter reference also contains a rather comprehensive bibliography. An exposition of related material, with physical applications in mind, can be found in [9].
This is usually written as a cyclic sum, a form that, in the case at hand, obscures its content. To clarify the latter, take as an example the case where x is a Lorentz group generator, J µν , and y, z are other generators carrying Lorentz indices, say, Y ρ , Z σ respectively. Suppose µ(y, z) = µ(Y ρ , Z σ ) = W . Substituting this in the l.h.s. above, one finds that the Jacobi identity requires that the transformation properties of W under the Lorentz group are derived solely from those of Y ρ , Z σ , i.e., in this case, W ought to transform as a second-rank covariant tensor. Another way of saying this is that µ itself is a Lorentz scalar, an observation that we use later on. Given a basis {T A }, A = 1, . . . , n of V , the product µ is specified by giving all vectors µ(T A , T B ), 1 ≤ A < B ≤ n. The coordinates of these vectors in the basis are, up to a factor of i, the structure constants of the algebra, which are antisymmetric in the lower two indices (a sum over repeated indices is implied). In the above equation we follow the standard physics practice of expressing the (non-associative) Lie product as the commutator [·, ·] w.r.t. an associative operator product, as well as the inclusion of an imaginary unit, related to the hermiticity of the generators. In terms of the structure constants, the Jacobi identity becomes Relaxing for the moment this latter constraint, i.e., taking into account only the antisymmetry in the lower two indices, one is left with N (n) = n 2 (n − 1)/2 arbitrary constants f AB C , A < B. Consider now the space R N , with each of the f 's ranging along an axis. For each value of (A, B, C, S), (3) describes a quadratic hypersurface in this space. The intersection of these hypersurfaces is the space L n of all possible n-dimensional Lie algebras -we sketch it as a surface in Fig. 1. Referring to this figure, consider the point P of L n -it corresponds to the Lie algebra G P , whose structure constants The space L n of n-dimensional Lie algebras (sketch). P is surrounded by equivalent points and hence, G P ∼ G PM , for all P M sufficiently close to P . In contrast, in the tangent space of L n at Q, there are directions that lead outside of the GL(n, R) orbit Orb(Q). Q will move along these directions when ψ 1 in (6) is a non-trivial element of H 2 (G Q ). Notice that, for all n, the surface passes through the origin, which corresponds to the n-generator abelian algebra.
are given by the coordinates of P . Under a linear redefinition of the generators via a GL(n) matrix M , the structure constants transform as and P moves to P M . Clearly, no new physics is to be expected from such a redefinition, G P and G PM being isomorphic. What we are really interested in then, from a physical point of view, is not L n itself, but, rather, the space of equivalence classes into which L n splits under the above action of GL(n), each class being the GL(n) orbit Orb(P ) of any point P in the class 5 . The crucial observation to be made here is that there exist two types of points in L n : those that are completely surrounded by equivalent points (corresponding to isomorphic algebras) and those whose neighborhoods 6 include non-equivalent points, sketched as P and Q respectively in Fig. 1. Any infinitesimal perturbation of the structure constants of G P will necessarily lead to an isomorphic Lie algebra -the orbit Orb(P ) is open in L n . We call G P stable or rigid. On the other hand, there exist infinitesimal perturbations of G Q that lead outside of Orb(Q) and, hence, to non-isomorphic algebras -we call G Q (infinitesimally) unstable.
In physical applications, structure constants are often given by experimentally determined fundamental constants of the theory. The experimental errors involved render the position of the corre-sponding algebra in L n uncertain. If the algebra employed is unstable, the physical predictions of the theory become ill defined, as they depend critically on the exact value of the structure constants, which is not known. Additionally, new measurements or improved data analysis, may move the algebra to a new position. If the algebra is stable, the physical theory based on it will maintain its qualitative validity. We conclude that stable algebras give rise to robust physics.

Deformations and H 2
Given a Lie algebra G 0 = (V, µ 0 ), i.e., the Lie product of X, Y ∈ V is supplied by where t is a formal parameter. The corresponding t-dependent structure constants, define a curve P t in L n , which passes through P 0 (corresponding to G 0 ) at t = 0. The l.h.s. of (6) is bilinear and antisymmetric, hence the ψ m on the r.h.s. are G-valued, bilinear antisymmetric maps We will call such maps 2-cochains (over V ), extending the definition in the natural way (i.e., via plinearity and total antisymmetry) to p-cochains ψ (p) , which accept p arguments 7 . The vector space of p-cochains over V will be denoted by C p (V ). Notice that the 1-cochains are simply linear maps from V to V , the antisymmetry requirement being meaningless in this case. Also, the space of 0-cochains is V itself. Next, for an arbitrary Lie product µ, we define a coboundary operator s µ , which maps p-cochains to (p + 1)-cochains, s µ : C p → C p+1 , according to (−1) r µ T Ar , ψ (p) (T A1 , . . . ,T Ar , . . . , T Ap ) + r<s (−1) r+s ψ (p) µ(T Ar , T As ), T A0 , . . . ,T Ar , . . . ,T As , . . . , T Ap (hats denote omitted terms). For example, for φ ∈ C 1 and ψ ∈ C 2 , where µ(X, Y ) = [X, Y ]. It can be shown that s 2 µ = 0, a result that relies on the Jacobi identity that µ satisfies -a compact proof is given in Sect. 2.3. The relevance of s µ to the problem at hand becomes evident when one imposes the Jacobi identity on the deformed commutator in (6). Writing out this identity and evaluating its t-derivative at t = 0, one finds that ψ 1 must satisfy s µ0 ⊲ ψ 1 = 0. We call a p-cochain ψ (p) annihilated by s µ , s µ ⊲ ψ (p) = 0, a p-cocycle, and denote the vector space of p-cocycles by Z p (V, s µ ). What we have found above is that infinitesimal deformations of Lie algebras are generated by 2-cocycles, the converse being also true.
It remains to determine which of the above infinitesimal deformations lead to isomorphic Lie algebras. As mentioned already, isomorphic Lie algebras result from a linear redefinition of the generators with some invertible matrix M ∈ GL(n, R) (see (4)). In the case of a deformation, M = M t is t-dependent, with M 0 = I n (the unit n × n matrix). Then the deformed, t-dependent commutator, is given by for any X, Y in G 0 . Taking M t in a neighborhood of the identity, M t = I n + tQ, with t small, one readily computes the corresponding first-order (in t) change to the commutator, But the linear map Q : X = X A T A → QX = X R Q R S T S is, as mentioned earlier, a 1-cochain. Comparing the O(t)-term in the r.h.s. of (12) with the first of the examples in (10) shows that the O(t)-change in the commutator, i.e., the 2-cochain ψ 1 in (6), is given by We call a p-cochain ψ (p) that is in the image of s µ , ψ (p) = s µ ⊲ φ (p−1) , a trivial p-cocycle, or, a p-coboundary. The vector space of p-coboundaries will be denoted by B p (V, s µ ). Since s 2 µ = 0, all coboundaries are cocycles, B p ⊆ Z p . What the above result shows is that infinitesimal deformations of G 0 towards isomorphic Lie algebras are generated by 2-coboundaries.
Conversely, assume that all 2-cocycles are trivial. Given a deformed commutator as in (6), there exists a linear map (1-cochain) φ 1 : G 0 → G 0 , such that the ψ 1 appearing in the r.h.s. of that equation is given by ψ 1 = s µ0 ⊲ φ 1 . Consider now a linear redefinition of the generators by the matrix M 1 = e −tφ1 and compute the new t-commutator [X, Y ] ′ t . The result is given by (12), with the substitutions Using (6) to expand the r.h.s. in powers of t, we see that the term linear in t in [X, Y ] ′ t cancels. Repeating the procedure one may eliminate one by one all powers of t, thus bringing the original tcommutator in coincidence with the undeformed one [X, Y ] 0 , using nothing more than successive linear redefinitions of the generators. We conclude that the two commutators define isomorphic Lie algebras, the matrix giving the isomorphism being M = . . . M 2 M 1 , with M m = e −tφm and s µ0 ⊲ φ m = ψ m .
We may summarize the contents of this section in the following geometrical picture: the tangent space 8 T P0 L n to L n at P 0 is (isomorphic to) Z 2 , the space of 2-cocycles. The subspace of T P0 L n leading to isomorphic Lie algebras, i.e., the tangent space to the GL(n)-orbit Orb(P ) is B 2 , the space of 2-coboundaries. To close the familiar circle of definitions, we define the quotient space H p ≡ Z p /B p , in which two cocycles are identified if they differ by a coboundary, as the p-th cohomology group 9 of G 0 . The non-trivial elements of H 2 (if any) correspond to directions in T P0 L n that lead to Lie algebras infinitesimally close to G 0 but non-isomorphic to it. A sufficient condition then for the stability of G 0 is the vanishing of its second cohomology group H 2 (G 0 ). Whitehead's lemma states that this condition is satisfied by all semisimple Lie algebras [16] -we conclude that semisimple Lie algebras are stable. It is worth pointing out that the above is not a necessary condition. As explained in Sect. 2.4, although a non-trivial 2-cocycle may exist, obstructions originating in H 3 (G) can render it non-integrable, in which case the corresponding finite non-trivial deformation does not exist. Concrete examples of stable Lie algebras with non-trivial H 2 have been constructed, typically as semidirect products. For example (see [29]), denote by S the simple 3-dimensional Lie algebra over C and by ρ n the irreducible representation of weight n of S on W ≡ C 2n+1 . The semidirect product L n = W ⋊ ρn S, for n > 5 and odd, is a stable Lie algebra, while its second cohomology group is non-trivial. To deal with such cases, non-cohomological approaches have been developed, relying on techniques of non-standard analysis. A classification algorithm for stable Lie algebras exists, relying on a theorem that such algebras posses a standard non-zero generator whose adjoint representation is diagonalizable. Although tedious, the algorithm permits, in principle, the classification of all stable Lie algebras, in any dimension -for more details we refer the reader to [3,13].

The ⊼ product
It turns out that calculations involving expressions like (1), or (10), simplify considerably when a particular product, the subject of this section, is introduced among p-cochains [28].
where σ ranges over all permutations such that σ(0) < . . . < σ(n) and σ(n + 1) < . . . < σ(m + n) (these are known as riffle shuffles with cut at n + 1). When both α and β are 2-cochains, as will often be the case, the above formula reduces to where, for a p-cochain ψ (p) , Notice that ⊼ is non-associative, but satisfies instead (the commutative-associative law). The (graded) commutator of α, β is defined as Consider now a Lie algebra G = (V, µ), µ ∈ C 2 (V ) = Alt 1 (V ). It is easy to see that the Jacobi identity for µ, Eq. (1), can be put in the form (the first equality is an immediate consequence of (19)). Furthermore, the action of the coboundary operator s µ on an arbitrary (p + 1)-cochain ψ ∈ Alt p (V ) is given by i.e., s µ is equal, up to a sign depending on the order of the cochain it acts on, to the operator D µ ≡ µ, · . Thus, all operations introduced in earlier sections can be expressed in terms of the ⊼ product.
It can be shown that the graded commutator of (19) satisfies a graded Jacobi identity, where α ∈ Alt m (V ), β ∈ Alt n (V ) and γ ∈ Alt p (V ). This property, together with bilinearity and graded antisymmetry, implies that Alt(V ) ≡ n Alt n (V ) is a graded Lie algebra.
We derive now a number of interesting results, illustrating along the way the efficiency afforded by the formalism introduced in this section. First, the proof of s µ • s µ = 0 may be given in a simplified form. Up to an irrelevant sign, it translates into D µ • D µ = 0 and, for an arbitrary α ∈ Alt(V ), where the second equality follows from the graded Jacobi identity for ·, · , Eq. (22), and the last one from the Jacobi identity for µ, Eq. (20). Second, the equation for finite deformations may be derived easily. If µ is a Lie product, µ ′ = µ+ φ will also be one if µ ′ , µ ′ = 0, from which one gets immediately the deformation equation which reduces to the cocycle condition for infinitesimal φ. Third, Eq. (22) implies that D µ is a graded derivation in Alt(V ), i.e., where α ∈ Alt m (V ) and β ∈ Alt(V ). One may then conclude that if α, β are cocycles, α, β ∈ Z(Alt(V ), D µ ), then so is α, β , and that if, additionaly, γ is a coboundary, γ ∈ B(Alt(V ), D µ ), then so is α, γ . These two facts, in turn, imply that the quotient space H(Alt(V ), D µ ) is itself a graded Lie algebra.

Obstructions and H 3
Given a Lie algebra G = (V, µ) and a deformation µ t , Then the deformation equation for φ t , Eq. (24), implies an infinite sequence of equations for the φ n , one for each power of t. The equations corresponding to t, t 2 and t 3 , are 10 If φ 1 is a 2-cocycle, as (27) demands, then φ 1 , φ 1 is a 3-cocycle, since D µ is a (graded) derivation w.r.t. the ·, · product. But then, (28) demands that this 3-cocycle be a coboundary, which may not be the case if H 3 (V, D µ ) is non-trivial. We see then that the existence of non-trivial 3-cocycles may render infinitesimal deformations (φ 1 above) non-integrable. If φ 1 , φ 1 is indeed a trivial 3-cocycle, so that (28) admits a solution, an obstruction may occur in the next step, i.e., in (29), and so on. It can be shown that all of these obstructions lie in H 3 , so that, if H 3 is trivial, every non-trivial 2-cocycle is the first order term of some finite deformation [28].
The following remarks will prove useful: 1. If a non-trivial 2-cocycle φ also satisfies φ, φ = 0, then it satisfies the deformation equation (24). In that case, the truncated deformation µ t = µ + tφ is a Lie product for every t, if µ is one, regardless of the structure of H 3 .
2. If there are several nontrivial 2-cocycles φ i and all their anticommutators are zero, φ i , φ j = 0, ∀i, j, then an arbitrary linear combination of them also satisfies the finite deformation equation, and the space of finite deformations becomes a vector space, spanned by the φ i 's.
3. In both of the above cases, infinitesimal deformations along non-trivial 2-cocycles are guaranteed to lead, as we saw earlier, to non-isomorphic algebras. This is not necessarily the case for finite deformations: the algebra µ + tφ may become isomorphic, for particular finite values of t, to the algebra µ. Notice also that, in general, the algebras µ + tφ, for various finite values of t, may not be isomorphic among themselves. The infinitesimal version of this is that the algebras µ + tφ, for various (infinitesimal) values of t, are all isomorphic, as long as t does not change sign. µ + tφ might well be non-isomorphic to µ − tφ, even for t infinitesimal.
Interestingly enough, these scenarios are realized in the stability analysis of the Galilean algebra, in Sect. 2.6, as well as in that of the PH algebra, in Sect. 3.

Coboundary operator as exterior covariant derivative
It is obvious from the definition given above, that a p-cochain can be realized as a G-valued left invariant (LI) p-form on the group manifold G corresponding to G, with the generators T A now extended to LI vector fields. Denoting by {Π A } the LI 1-forms on G dual to the generators {T B }, we write ψ (p) as Then the action of s given in (9) coincides with that of an exterior covariant derivative ∇, with the connection 1-form Ω given by The nilpotency of s follows now from the vanishing of the curvature 2-form Θ = dΩ + Ω 2 , due to the Jacobi identity, while 2-cocycles are covariantly constant G-valued LI 2-forms (see, e.g., [6]). Notice that the requirement that s ⊲ ψ (2) = 0, with ψ (2) as in (31), reduces to which is, as expected, the first-order term, in t, of the Jacobi identity for the structure constants f + tψ. The use of the differential forms language permits writing out cochains as geometrical objects, as in (31), rather than listing their components, a practice we adher to in the following. The point of view sketched here has been further developed, in the case of compact Lie algebras, in [36], the motivation there being the study of BRST cohomology. The appropriate coboundary operator 11 , called there the BRST operator, is realized in terms of fermionic coordinates and their dual derivatives. An involution of the algebra of the latter, made possible by the invertibility of the Killing form, gives rise to a dual object, the anti-BRST operator, and a grade-preserving Laplacian. Further generalizations, involving higher order invariant tensors of the algebra, have been explored in [7]. We have developed similar techniques to deal with the non-compact case, reinstating the connection term, and used them in one of our programming approaches -we defer further details to a future publication.

An example: the shortest path from Galileo to Einstein
Consider the Galilean algebra G G of non-relativistic kinematics, where J a , K a , a = 1, 2, 3, are the generators of rotations and boosts, respectively, and indices are raised and lowered with the unit metric. The 2-cochain µ that corresponds to this Lie product is, in the language of the preceding section, We adopt here the convention that, in 1-forms, unbarred indices refer to rotations, while barred ones to boosts, so that, e.g., Πā, K b = δ a b (notice that bars are important in forms but make no difference in Kronecker deltas or in the summation convention). By an argument based on the observation made after Eq. (1), we conclude that only scalar (under rotations) cochains need be considered. We simplify further the notation taking advantage of the fact that, due to the limited number of generators and invariant tensors, a simple listing of the nature of the 1-forms and generators that enter in any given cochain, of up to second degree, is sufficient to reconstruct it (there is only one way to contract the indices). For example, µ above is given by (a factor of 1/p! is included whenever p 1-forms of the same type are multiplied).
We inquire now about the stability of this algebra. The most general scalar 1-cochain is given by with φ JJ = Π a ⊗ J a etc.. Applying ∇ to obtain the most general 2-coboundary we get On the other hand, the most general scalar 2-cochain is given by We set ∇χ = 0 to obtain We conclude that β 1 = β 4 and β 3 = 0, so that the most general 2-cocycle is given bỹ with arbitrary c i . Comparison ofχ with ∇φ shows that only χ KKJ is a non-trivial 2-cocycle, giving for the second cohomology group Accordingly, G G is infinitesimally unstable. By noting that χ KKJ , χ KKJ = 0, we conclude that µ t = µ + tχ KKJ yields a one-parameter deformation of the algebra for finite t (see the comment at the end of Sect. 2.4). A look at the form of χ KKJ shows that the deformation only adds a rotation generator in the r.h.s. of the K-K commutator, leaving the rest of the commutators intact. The Lorentz algebra, describing relativistic kinematics, sits at t = − 1 c 2 , where c is the velocity of light and, being semisimple, it is stable.

Stable Quantum Relativistic Kinematics
Hopefully, the above example will have aroused the interest of the reader enough to follow us as we embark on the search for a stable Lie algebra, encompassing relativistic and quantum effects. Our starting point is the fourteen generator Poincaré-plus-positions algebra augmented by a central generator M , to appear later in the r.h.s. of the Heisenberg commutator. We follow the practice of omiting all zero commutators, the metric used is g = diag(1, −1, −1, −1) and c, the speed of light, is taken equal to 1. The resulting fifteen generator algebra, describing classical ( = 0) relativistic kinematics (plus the extra generator M ) we call G CR ("Classical Relativity"). The reader might want to identify J µν with the Lorentz algebra generators and P µ with the momenta (and, even, Z µ with the positions) but we will focus initially on the strictly algebraic problem of stability, and only digress on interpretational aspects, which hold some surprises, in Sect. 4. The 2-cochain µ CR , corresponding to G CR , is given by A straightforward calculation shows that µ CR , µ CR = 0, confirming that the Jacobi identity is satisfied in G CR .

Calculation of H 2 (G CR )
We computed the second cohomology group H 2 (G CR ) with the help of MATHEMATICA. We did this in two independent ways. In the first one, the components of cochains were calculated explicitly, one-by-one, while in the second a symbolic approach was followed, dealing, e.g., with sums of the form Π µ Π ν ⊗ J µν without expanding them further. The first approach has the advantage of generality, as it can deal, practically without further fine-tuning, with any Lie algebra -details of the calculation are given in the appendix (see Sect. A). The second approach is generally faster, at the price of adjustments needed every time a new object (e.g., an invariant tensor) is introduced. In both approaches, the remark made after Eq. (1) shows that we may consider only Lorentz scalars, drastically reducing the workload. We have, nevertheless, implemented this only in our second approach, to keep the first as general as possible. The result of both calculations is i.e., there are five nontrivial generators, with representatives given by As in Ex. 2.6, we adopt a compact notation where undotted indices in forms refer to P 's, dotted ones to Z's, so that, e.g., Πμ, Z ν = δ µ ν . As before, dots make no difference in Kronecker delta's or epsilon tensors. With a slight abuse of notation, Π M denotes the 1-form that detects the generator M .

Finite deformations of G CR
Each of the cocycles in Eqs. (51)-(55) represents a direction of a possible infinitesimal deformation. For example, the first of these, ψ H , when added to µ CR , adds the Heisenberg commutator to G CR , while each of the other four renders M noncentral. There are two questions that arise now: 1. Are these infinitesimal deformations integrable? 2. Are linear combinations of these infinitesimal deformations integrable?
To this end, we compute the commutators among the ψ's and find that the only nonzero ones are those between ψ H and the rest of the ψ's -it will prove convenient in what follows to use the linear combinations ψ − = ψ ZMZ − ψ PMP and ψ + = ψ ZMZ + ψ PMP , Regarding the first question above, the fact that the diagonal commutators are all zero implies that µ CR + tψ A , for t finite, gives a deformation of G CR , where ψ A is any of the five generators of H 2 (G CR ) given above, Eqs. (51)-(55). For the second question, the fact that the commutators among the last four ψ's are all zero implies that any linear combination of these ψ's gives rise to a finite deformation as above. The case of deformations that mix ψ H with the other four generators needs special treatment.
We consider an infinitesimal deformation along the 2-cocycle φ 1 , One easily checks that φ 1 , φ 1 = 0 (the relevant anticommutators are given in Eqs. (56)-(59) above), so that (28) is not trivially satisfied. For the above mentioned anticommutators we find that the first three are trivial, where On the other hand, ψ H , ψ + turns out to be non-trivial. Accordingly, the infinitesimal deformation generated by φ 1 is integrable if, and only if, β + = 0. In that case, Eq. (28) is satisfied with With an eye on (29), we compute φ 1 , φ 2 and find that it vanishes, so that φ 3 = 0 (see (29)). Also, φ 2 , φ 2 = 0, implying that φ 4 , and all higher order φ n 's vanish, and the finite deformation truncates at second order, Without loss of generality, we may put t = 1 and write the result as a form that will prove useful shortly.

Heisenberg's route: the algebra G PH (q)
We want to explore here what happens if one follows, along with Heisenberg, the historical route and only deforms G CR along ψ H . We consider, accordingly, the stability of the algebra G PH (q) ("Poincaré plus Heisenberg"), with corresponding 2-cochain µ PH (q) given by (we assume henceforth that q = 0). The commutators defining it are given by Eqs. (46)-(48), plus the Heisenberg commutator -for the sake of locality we collect them all here, We will have more to say about (71) in Sect. 4 -for the moment, we ask the reader to accept it as a reasonable (i.e., covariant and of Lie-type) form of the familiar Heisenberg relation. As in the previous case, of G CR , we first tackle the purely algebraic problem of stability, and leave questions of physical interpretation for Sect. 4. Meanwhile, the temptation should be resisted to consider G PH as the algebra of "quantum relativistic kinematics" -it is argued later on that it is not. We find that H 2 (G PH (q)) is nontrivial, where ψ H itself is still a cocycle, albeit a trivial one now (for all nonzero q). This means that, starting at G PH (q), and moving along ψ H , one arrives at isomorphic algebras. But moving along ψ H amounts to changing the value of q, without changing its sign. We conclude that the algebras G PH (q), for all nonzero values of q, of the same sign, are isomorphic. On the other hand, one can easily change the sign of q by a redefinition of the generators, e.g., by rescaling all Z's by some negative number, or exchanging P and Z (notice that in both examples, the corresponding matrix that effects the redefinition has positive determinant, i.e., it lies in the connected component of GL (15, R)). The upshot of all this is that all G PH (q), for nonzero q, are isomorphic.

Finite deformations of G PH (q)
Reasoning as in Sect. 3.2, we compute the commutators among the ζ's, and find that they all vanish. Accordingly, every linear combination of the above cocycles, added to µ PH , provides a finite deformation of G PH . For a generic combination ζ( α), the 2-cochain µ(q, α) = µ PH (q) + ζ( α), is clearly identical to the one found before, Eq. (69), with the identifications (β 1 , β 2 , β − ) → (α 1 , −α 2 , α 3 ). We have arrived then at the same result, either deforming G CR along a direction that truncates to second order in the deformation parameter, or by performing two succesive deformations (with intermediate stop at G PH (q)), each truncating to first order. The corresponding deformed algebra is given by the commutators of G CR , Eqs. (46)-(48) (notice that the Heisenberg commutator is not included), plus the following We denote the resulting algebra by G PH (q, α). We see that, for a generic deformation, the P 's cease to commute among themselves, and so do the Z's, M is no longer central, while the Heisenberg commutator receives an additional term, proportional to J µν .

The instability cone
Relevant questions that emerge now are: 1. Are the above deformations G PH (q, α), for various values of α, (finally) stable?
2. Are there deformations that are isomorphic among themselves?
To answer the first question, we compute, as always, the second cohomology group and find where χ = ζ 1 + ζ 2 satisfies χ, χ = 0. G PH (q, α) is, accordingly, stable everywhere outside the instability surface α 2 3 = α 1 α 2 in α-space. The latter represents a double cone with the apex at the origin and its axis along the first diagonal in the α 1 -α 2 plane, parallel to χ (see Fig. 2). We will refer to the various regions of α-space with their relativistic nicknames ("future", "past", etc.), with the future including the positive α 1 -α 2 quadrant. Notice that, off the cone, χ is a trivial cocycle, χ = ∇ξ, and ξ has a pole on the cone. Regarding the second question above, from the fact that each algebra outside the light cone is isomorphic to all algebras in its neighborhood, we conclude that all algebras in, say, the future, are isomorphic among themselves (similarly for the past and the elsewhere). A slight refinement of the argument, using the fact that tangent vectors to the cone are trivial cocycles, leads to the conclusion that all algebras in, say, the future cone, are isomorphic among themselves (similarly for the past cone), with the apex, i.e., G PH (q, 0) ≡ G PH (q), in a class by itself. In conclusion, there are six equivalence classes of algebras, given by the various regions the α-space is divided into by the double light cone.
A glance at Fig. 2 helps visualize several aspects of the stability analysis that were mentioned earlier (see Sect. 2.4). For example, starting at some (unstable) point of, say, the future cone, it is clear that infinitesimal deformations along the cocycle χ given above lead either to the future or to the elsewhere, both being non-isomorphic to the original algebra but also between themselves. Since χ, χ = 0, one may also consider finite deformations, In this case, it is clear that there exists a negative value of t (t 0 = −α 1 − α 2 ) such that the resulting algebra µ t0 = µ PH (q, −α 2 , −α 1 , √ α 1 α 2 ) lies on the past light cone and is, therefore, non-isomorphic Elsewhere Figure 2: The (α 1 , α 2 , α 3 ) deformation space of G PH (q), with a representative of each of the six equivalence classes drawn (these are, in relativistic parlance, the future and past light-cones, the apex, the future, the past and the elsewhere). The two cones and the apex at the origin correspond to unstable algebras -the rest of the space to stable ones. For all classes, a representative exists with α 3 = 0 (the little spheres denote such representatives). With the identification of Z µ with the moment operator (see Sect. 4), the origin corresponds to G PH (q), while G QR , the Lie algebra of standard quantum relativistic kinematics, lies on the future cone, at α = (0, q, 0). Stabilizing deformations of G QR , generated, e.g., by χ, may lead either towards the future (isomorphic to so(1, 5)) or towards the elsewhere (isomorphic to so(2, 4)). Both choices introduce non-commutativity of the P 's, differing in the sign of the associated curvature.
to µ t , for generic t. Finally, had we chosen instead a non-trivial cocycle orthogonal to the axis of the cone, rather than parallel to it 12 , there would exist a finite deformation along it isomorphic to the original algebra, given by the "antipodal" point on the future cone, (α 2 , α 1 , − √ α 1 α 2 ).

Gauging away the α's
The above conclusion makes it evident that, for each of the above classes, a representative exists with α 3 = 0. The deformation space then, from the algebraic point of view, is essentially the α 1 -α 2 plane. To find explicitly a linear redefinition of the generators that moves an arbitrary point in α-space to the α 1 -α 2 plane, we notice that the transformation (P, Z) → (P ′ , Z ′ ), given by leaves the value of q = 1 in the Heisenberg commutator invariant, while, in α-space, it induces the It can be checked that, in the particular case where the P -Z transformation is a rotation by an angle θ, a b c d = cos θ sin θ − sin θ cos θ , the matrix M that results from (85) describes a rotation in α-space, around the axis of the cone, by an angle of 2θ, counterclockwise as seen from the future. Such a rotation clearly leaves all isomorphism classes invariant and can be chosen so as to move any particular point to the α 1 -α 2 plane. Similarly, for algebras in the elsewhere, isomorphic algebras exist with either α 1 or α 2 equal to zero -this is not true for algebras in the past or the future (the orbits of the latter under the above rotation do not intersect the α 1 -α 3 or α 2 -α 3 planes). Having said that, due consideration should be given to the fact that physicists can be single-minded in what regards their preferred set of generators, e.g., working with arbitrary linear combinations of momenta and positions could be frowned on. If such preferences are given priority, one might be forced to work with a version of the deformed algebra with α 3 = 0.

Isomorphisms
This brings our stability analysis to a conclusion. The algebra G PH (q, α) we have arrived at is given by Eqs. (46)-(48) and (77)-(81), with corresponding 2-cochain The novelties are that the P 's don't commute, the Z's don't commute, M is no longer central, and an extra term appears in the Heisenberg commutator. The nature of the three deformation parameters α i is discussed in Sect. 4.5, after the physical identification of the generators has been carried out. As it has been pointed out in [17,25], the above algebra, off the instability cone, is isomorphic to some so(m, 6 − m), where m depends on the signs of α 1 , α 2 (taking α 3 = 0). The isomorphism, given in [25], is as follows: denote the generators of so(m, 6−m) by {J µν , J µ4 , J µ5 , J 45 } -their commutation relations are analogous to those of the Lorentz group, Eq. (46), with a metricḡ that is taken diagonal, with entries ±1, and coinciding with g in the Lorentz sector. Assuming the identifications one finds the commutators Comparing the first of these with the Heisenberg commutator gives ρ = −qστ . Substituting this in the other two, and comparing with (80), (81), respectively, shows that i.e., assuming q > 0, (see Fig. 2). On the light-cone, the above semisimple (and, hence, stable) algebras go over to the corresponding semidirect product [17].

Relation with other algebras
As mentioned in the Introduction, several non-commutative spacetime Lie algebras have been proposed over the years, but, as a rule, they fail to provide a complete set of generators. We provide below an account of these earlier attempts, emphasizing from the outset that our list of references does not pretend to be complete. The first, to our knowledge, publication regarding a non-commutative, Lorentz covariant spacetime is due to Snyder, dating from 1947. Apparently, as J. Wess has documented and publicized, the idea can be traced back to Heisenberg, who, in a letter to Peierls, suggested that the ultraviolet infinities of quantum field theory could be tamed by assuming noncommuting spacetime coordinates. Peierls soon found an altogether different application in the calculation of the lowest Landau level of a system of electrons in a magnetic field with impurities, using noncommuting coordinates in the potential-like function describing them. He also passed on the idea to Pauli, who described it to Oppenheimer, who shared it with Snyder, who published Ref. [34] (our source is [15]). Snyder's position operators fail to commute among themselves, exactly as in (79). His momenta, however, commute, and the positionmomenta relations contain non-linear terms. An early attempt at formulating electrodynamics in this non-commutative spacetime followed shortly after [33]. Later in that year, Yang [37], pointed out that by introducing what we have called M , one can render the algebra linear. Additionally, he proposed non-commuting momenta, exactly as in (78) and the accompanying P -M relations, Eq. (80) (with α 3 = 0). The so isomorphism was also given, although with a particular choice for the signs ǫ 4 , ǫ 5 (both equal to −1). In fact, getting to some known, preferably semisimple, algebra (like so), seems to have been his guiding principle in completing the set of commutators.
Several years later, Khruschev and Leznov [17] provided a further deformation, essentially the one given by our ζ 3 , i.e., by the terms proportional to α 3 in (77)-(81). Although their article cited above appeared in 2002, it quotes this result (or, at least, significant parts of it) from an earlier work of theirs, dating from 1973, which we have not had access to. Their approach is via a straightforward solution of the Jacobi identities, and does not include the stability point of view. Apart from the deformation itself, they also provide information on the Casimirs of the deformed algebras, and mention the α 1 α 2 − α 2 3 = 0 relation, with α 1 α 2 = 0, as a semisimplicity criterion for the deformed algebra. The so identifications are given (they actually use o(n, 6 − n)) but the possibility of gauging away α 3 by a redefinition of the generators is not pointed out. Some steps towards constructing field theories on these quantized spaces were taken in [21].
The work of Vilela Mendes [25], which motivated ours, appeared in 1994. There, for the first time, the stability criterion for the non-commutative spacetime algebra is invoked and its relevance, more generally, for the algebraic structures employed in physical theories is convincingly advocated. The approach taken in determining the stable form of the algebra is a minimalistic one: the so algebras are proposed ab initio, being obviously deformations, and their semisimplicity is invoked to guarantee their stability. Economical as it may be this approach, it leaves neverheless pending the question of uniqueness, prompting us to undertake the present systematic search. The instability double cone is not mentioned in the above work. Also, although Snyder's work was known to the author, it seems he did not come across Yang's contribution. Various applications have been considered by Vilela Mendes and co-workers in [4,24,26].
In recent years, as mentioned in the Introduction, several "Doubly Special Relativity" algebras have been proposed. They all ignore the (initially) central generator M . In [18] it was shown that all commutation relations of the deformed algebras, except the P -Z ones, can be brought into a Lie form by appropriate non-linear redefinitions of the generators. The Lie form found coincides with that provided by the α 2 deformation above. Furthermore, it was pointed out in [8], that by taking M into account, the "Triply Special Relativity" of [19] is linearized, and the resulting, Lie-type, deformation is the one provided by α 1 above (this was essentially a repeat of Yang's observation on Snyder's proposal, applied to the momentum sector). Thus, it seems that when M is taken into account, non-linear redefinitions bring the "Multi-Special Relativity" algebras into one of the forms found above 13 . These observations suggest that, before leaving the tried and tested Lie algebra framework, Lie deformations introducing new invariant scales, like the ones proposed earlier, should be studied carefully, and the need for non-linearity should be critically examined.
Finally, the following obvious fact should be emphasized: isomorphic, or even identical, algebras may correspond to radically different physics if the generators that enter in them are interpreted in different ways. In this respect, all of the above mentioned works coincide in the physical identification of the generators, in particular, in the fact that the Z µ 's should be interpreted as position operators. As we explain in the section that follows, our view differs.

Some Physical Considerations
We deal, finally, with a number of interpretational issues. We would like to warn the reader that the material in this section is still in its formative stage, and several aspects of what follows are still under investigation. Nevertheless, we feel it is worthwhile pointing out alternative possibilities in the physical identification of the generators we have been studying. The content here is mostly qualitative and the tone, accordingly, informal. We keep complexity to a minimum by treating the case of a massive, spinless particle only -a more complete analysis will have to wait, like so many other things, a future work.

The coproduct of Lie algebra generators
We wish to discuss the physical meaning of the coproduct of Lie algebra generators. We will use the Poincaré algebra as an example, but the discussion applies to general Lie algebras.
Consider applying a translation T a to a particle, located at x. As a result, the particle shifts to x + a. Imagine now that, under closer inspection, the particle is seen to be a bound state of two other particles. To translate by a what is now known to be a two-particle system, one applies the translation T a to each of the constituent particles in the system. The n-particle case, n > 2, is handled by further subdivision of either of the two particles above. Similar considerations hold for rotations or boosts. This observation is formalized in the following manner. The state of the system under study is represented by a state vector |ψ in some Hilbert space H. To a possible transformation of the system, e.g., a rotation R αβγ parametrized by Euler's angles, one associates an operator D(R αβγ ), acting on H. When the system is revealed to consist of, say, particles 1 and 2, the state space becomes H 1 ⊗ H 2 , where H i is the state space of particle i. The observation made above then implies that the operator representing where D i is the representation of rotations in H i . This is true for all representations D i -we may accordingly conclude that the abstract rotation operator R αβγ acts on tensor products as R αβγ ⊗R αβγ and call this latter operator the coproduct ∆(R αβγ ) of R αβγ . Particular cases then are handled by taking the appropriate representation of this universal formula, e.g., D 1 ⊗ D 2 above. The fact that rotations should compose in the same way, whether applied to a simple or to a composite system, is expressed algebraically by the requirement that 14 ∆(R 1 R 2 ) = ∆(R 1 )∆(R 2 ). Our formalism respects this for, if R 1 R 2 = R 3 , then the product law in the tensor product being (A ⊗ B)(C ⊗ D) = AC ⊗ BD. We summarize: for all transformations T in the Poincaré group • the coproduct ∆(T ) is grouplike, • ∆ is an algebra homomorphism, Now write T = e A , with A in the Poincaré algebra G P and define ∆ to be linear in the entire U (G P ), the universal enveloping algebra 15 of G P -a simple calculation then shows that ∆(A) = A ⊗ 1 + 1 ⊗ A (this is a logarithm turning a product into a sum, as usual). We conclude that • The generators of grouplike transformations are primitive, with J tot = J 1 + J 2 as the archetypical example from quantum mechanics. In other words, the physical quantities corresponding to generators of grouplike transformations are additive under system composition (or extensive, in thermodynamics parlance). All Lie algebra generators are of this nature. From a geometric point of view, the definition of pointwise multiplication of functions on the group manifold, (f h)(g) = f (g)h(g), is what fixes the coproduct of the point (transformation) g to be grouplike. At the infinitesimal level, this becomes the Leibniz rule satisfied by the generators (this is another way of interpreting (94)), consistent with their representation as first-order differential operators on the group manifold. The above considerations prompt us to only allow primitive operators as Lie algebra generators.
In [25], it is argued that the 1-dimensional Heisenberg commutator, [p, x] = −i, can be interpreted as defining a stable Lie algebra. The justification for this claim is made through the observation that one could equally well choose a function of x as a coordinate, in particular, y = e ix . Then, the Heisenberg relation takes the form [p, y] = y, which indeed defines a stable 2-dimensional algebra. Apart from the unsuitability of e ix as coordinate over the entire x-axis, it should be clear from our earlier discussion that we cannot agree with this argument, since y is no more primitive than x.

The Lie form of the Heisenberg algebra
As mentioned after our first reference to the Heisenberg commutator, Eq.(71), there are a number of remarks that we would like to make regarding its proposed form. One usually first encounters the Heisenberg commutator in the form which is unsatisfactory for (at least) two reasons. The first has to do with Lorentz covariance -the obvious remedy is to consider instead the form leaving for a future brainstorm the elucidation of its physical implications (notice that time is promoted to an operator). The second reason is of a technical nature: dealing with a Lie algebra, the r.h.s. of (96) ought to be linear in the generators. The usual solution followed in the literature is to introduce a new, central generator M , with [P µ , X ν ] = i qg µν M .
The resulting three-generator Lie algebra is referred to as the Heisenberg algebra -the physical interpretation of M is generally left obscure. It might at first seem that there is little to be gained from writing out M explicitly, since it commutes with everything, but when deformations of the algebra are considered, it will be essential to do so since, as a result of the deformation, M might cease to be central (for an example of the type of problems that may arise by suppressing M , see [8]). Is (97), at last, an acceptable form of the Heisenberg algebra? That the answer should still be negative follows easily from our remarks about the primitiveness of Lie algebra generators. First, if M in the r.h.s. of (97) were primitive (and hence extensive), the effective Planck's constant for a composite system would be the sum of those for its constituent parts, providing for several concrete examples of fuzzy spheres (e.g., the earth, with q Earth ∼ = 10 14 Kgr m 2 /sec). Second, X µ is not primitive. There are various ways to see this. To begin with, it is rather obvious that position is not an extensive quantity: if two particles are glued together at x µ , their composite system is also located at x µ , not at 2x µ . Another way is to look at the corresponding finite transformation. X µ may be considered, up to a sign, the generator of translations in momentum space. But the apparent symmetry (via duality) between momenta and positions should be treated with care. In particular, although translations in spacetime are grouplike, those in momentum space are not. If a particle of 4-momentum p is translated, in momentum space, by k, it ends up with 4-momentum p+k. If now it is discovered that it is actually made up of two other particles and each of them is translated by k, then the composite particle would be translated by 2k. This latter example reveals something about the nature of the grouplike operator that should replace e X , the logarithm of which would be acceptable as a Lie algebra generator. Roughly speaking, it should somehow detect the mass of the particle and translate in momentum space by a quantity proportional to it. Notice that, despite the elementary nature of the considerations in this section, there seems to exist a consensus in the literature that X µ is primitive 16 .

The coproduct of the position operator
So, if X µ is not primitive, what is its coproduct ∆(X µ )? The answer is that, in general, ∆(X µ ) does not exist. To see why this is so, let us first specify what exactly is it that we want the position operator to do for us. For a single localized particle it is clear that X µ should return its position, but what should X µ (via its coproduct ∆(X µ )) do on a two-particle system? Clearly, if the two particles are glued together and the composite system is localized, we should get the same answer whether we operate with X µ on the composite particle or with ∆(X µ ) on the two-particle system. When the two particles are far apart and/or have different velocities, the natural requirement would be that ∆(X i ) (i.e., the spatial part of ∆(X µ )) should return the position of their center-of-momentum (or center-of-inertia), i.e., the relativistic refinement of the newtonian center-of-mass concept, which is the natural "effective position" of a relativistic composite system 17 . The problem is that the centerof-momentum 3-vector is not the spatial part of any 4-vector, in other words, the "effective position" of a composite relativistic system does not behave as a 4-vector. As a result, different observers locate the center-of-momentum of a system at different points. This, in turn, implies that ∆(X µ ) does not satisfy the same commutation relations with ∆(L ρσ ) as X µ does with L ρσ (see footnote 14), in other words, ∆ fails to be a homomorphism of the algebra, which proves our assertion.
The above conclusion might well be correct, but, certainly, there are composite systems the "effective position" of which, for all practical purposes, behaves like a 4-vector (e.g., an α-particle). This implies that although, strictly speaking, ∆(X µ ) does not exist, one might nevertheless define an approximate coproduct that works provided it is applied on a restricted class of systems -intuitively, systems that can fool the observer into thinking they are a single, localized particle. To make this statement precise, we note that the center-of-momentum spatial coordinates of a (non-interacting) two-particle system are given by where E ≡ E 1 + E 2 is the total energy of the system (this formula makes it clear that R is not the spatial part of any 4-vector). Assume now 18 that the system under study is such that in its centerof-momentum frame all energies E i are nearly equal to the corresponding rest masses, E i ∼ = m i -we will call such a system psychron, from the greek ÝÙÕÖÓÒ for "cold". Then, in the above frame, (98) reduces to the Newtonian formula for the center-of-mass. Moreover, when boosting to an arbitrary frame, all energies in the r.h.s. of (98) transform by the same γ-factor, which cancels, so that the l.h.s. transforms as a spatial vector. We conclude that, for psychron 2-particle systems, the relation where m 12 ≡ m 1 + m 2 , defines the effective position x 12 of the system as a 4-vector. Denoting by M the mass operator, M 2 = P µ P µ , and brushing aside ordering ambiguities, we conclude from (99) that the moment operator Z µ ≡ X µ M is primitive, when applied to psychron systems (M is also primitive on such systems). We note furthermore that Z µ is of exactly the form anticipated by the argument at the end of Sec

The algebra of standard quantum relativistic kinematics
We investigate the repercussions of the above interpretation of Z µ , M , in identifying the algebra of standard quantum relativistic kinematics. In the latter, the momenta commute and so do the positions, while their cross-relations are given by the Heisenberg commutator, Eq. (96). But then the Z's, in terms of which the algebra should be expressed, do not commute, and neither do the Z's with M , where [X µ , f (P )] = −i q∂f (P )/∂P µ was used 19 . Notice that the Z-Z non-commutativity is a purely quantum (q = 0) phenomenon and has no connection to spacetime non-commutativity. We recognize the r.h.s. of (100) as (a multiple of) the covariant form of the orbital angular momentum generator, L µν = q −1 (X µ P ν − X ν P µ ). For a massive, spinless particle then, we have A look at (79), (81), shows that the above relations, Eqs. (101) and (102), are of exactly the form furnished by the α 2 deformation, with α 2 = q. We conclude that, for a massive, spinless particle, the algebra G QR of standard quantum relativistic kinematics is given by i.e., in α-space, it lies on the surface of the future cone, along the α 2 -axis, at α 2 = q (see Fig. 2). It is worth noting that, with the above interpretation of the Z's, the Heisenberg algebra by itself does not close, the Z-Z commutator generating the Lorentz group.

The nature of the deformations
The deformation corresponding to ζ 1 introduces non-commutativity among the momenta and renders M non-central. Its origins lie in the instability of the Poincaré algebra, which stabilizes to the simple De Sitter algebras so (1,4) or so (2,3). The corresponding parameter, α 1 , has dimensions [L] −1 [M ], so that R ≡ /α 1 is a length, the radius of curvature of the manifold on which the various so algebras of Sect. 3.7 act. It has been suggested in [25] that, as long as one is interested in the kinematics in the tangent space to the manifold, rather than the group of motions of the manifold itself, one may take the R → ∞ limit, i.e., one may essentially disregard the above deformation. On the other hand, in [19], the suggestion has been made that R 2 may set the scale for the cosmological constant Λ. In any case, this deformation is a familiar and thoroughly studied one.
When the Z µ are identified with the position operators, the deformation generated by ζ 2 turns on spacetime non-commutativity. α 2 in that case has dimensions [L][M ] −1 , so that ℓ ≡ √ α 2 is a length, the inverse of which has, in the past, been conjectured to set the scale for the masses of the elementary particles. However, if that were the case, the effects of the deformed commutators would by now have been measured, so this proposal had to be abandoned. A more recent tendency is to regard ℓ as the Planck length, and attribute the non-commutativity to quantum gravity effects (see, e.g., [19] and references therein). Whatever the interpretation of the new length scale may be, the above identification of the Z's seems to us to suffer from a somewhat incredulous prediction: the extent to which the coordinates of a particle do not commute, i.e., the local "fuzzines" in spacetime due to, e.g., quantum gravity effects, depends, in general, on the position of the origin (since J µν , the particle's angular momentum, does). In particular, the coordinates of a particle at the origin commute. We think it improbable that such a state of affairs can be succesfully incorporated in a consistent physical scheme, and invite workers pursuing this direction to address what, to us, seems like a neglected pathology. In conclusion, then, we think it fair to say that interpreting the Z's as spacetime coordinate operators of a particle makes it improbable for the α 2 deformation to have the physical applications proposed in the literature. On the other hand, our identification of the Z's with the moment operators leads to the conclusion that G QR , the standard, experimentally tested, quantum relativistic algebra in which, in particular, the spacetime coordinates commute, is given, in the case of a massive spinless particle, by α 2 = q, with the experiment fixing the value q = . If the interpretation advocated above is correct, then, a look at Fig. 2 shows that the only deformations left to explore are those generated by ±χ, leading to the future or the elsewhere, respectively, both introducing non-commutativity of the momenta.
The α 3 deformation signals a more radical departure from G QR , so much so that, in Ref. [25], it is practically discarded as unphysical. Ref. [17], on the other hand, treats it on an equal footing and observes that, with the Z's as positions, α 3 is dimensionless, so that α 3 is a new fundamental constant with dimensions of action. When the Z's are taken as moments, α 3 acquires dimensions of mass. In either case, the physical implications of the deformation are somewhat obscure and deserve further study.

Concluding Remarks
We have pursued in this paper the stability point of view to its ultimate consequences. Our systematic algebraic analysis has recovered previous results, establishing their uniqueness, and shedding light along the way on various technical issues, in particular, the interrelations among the deformations found. A fundamental departure from the established lore has been our identification of the Z µ generators with the moment operators of a (massive, spinless) particle, having concluded that the position operators lack the essential property of primitiveness, necessary for all Lie algebra generators.
We think that a number of questions raised here deserve further study. First, we would like to generalize the concept of the moment operators to the case of particles with spin, and/or zero mass. Second, representation theoretical aspects of the problem should be examined, in particular, a Wigner-type classification should be carried through. It would also be of interest to develop some degree of intuition regarding the deformed kinematics, e.g., by clarifying the coexistence of the Lorentz contraction with an invariant length scale.