N ) and the corresponding q - Euclidean lattice

We present the Euclidean Hopf algebra Uq(e N ) dual of Fun(Rq >⊳SOq−1(N)) and describe its fundamental Hilbert space representations [6], which turn out to be rather simple “ lattice-regularized ” versions of the classical ones, in the sense that the spectra of squared momentum components are discrete and the corresponding eigenfunctions normalizable. A suitable notion of classical limit is introduced, so that we recover the classical continuous spectra and generalized (non-normalizable) eigenfunctions in that limit. Introduction Since their birth quantum groups [2] have found a number of different applications to physics and mathematics. In particular they can be used to generalize the ordinary notion of space(time) symmetry. This generalization is tightly coupled to a radical modification of the ordinary notion of space(time) itself. From this viewpoint inhomogenous group symmetries such as Poincaré’s and the Euclidean one yield physically relevant candidates for quantum group generalizations; Minkowski space M and Euclidean R one are then the corresponding space(time) manifolds. One can generalize the latter by the N -dimensional (N ≥ 3) Euclidean space Rq [3], its symmetry by the q-Euclidean one carried by the Hopf-algebra E q := R N q >⊳SOq(N) [9, 13, 10] or equivalently by its dual [9, 6], which here we will call Uq(e ). In Ref [6] we classified the fundamental Hilbert space representations of Uq(e ); here we represent the latter results in a more pedagogical and explicit way and add some new ones. A major physical motivations for such generalizations is the desire to discretize space(time) (or momentum space) in a “ wise ” way for QFT regularization purposes. Nowadays such a discretization is usually performed by approximating the points of the space(time) (or Alexander-von-Humboldt fellow momentum space) continuum by the points of a lattice. In the case of the cubic Euclidean lattice, for instance, the coordinates x (i = 1, 2..., N) can assume only the values an, where a is the lattice spacing and n ∈ Z; one chooses as a basis of the Hilbert space H of physical states the set {|n, ..., n >}ni∈Z of eigenvectors of the N commuting observables x with eigenvalues an. On the other hand, it is known that standard lattices used in regularizing QFT do not carry representations of discretized versions (in the form of discrete subgroups) of the associated inhomogenous groups; actually, the notion of a group is too tight for this scope. For instance, the Euclidean cubic lattice is invariant only under a discretized version of the translation subgroup of the Euclidean group, but not of the rotation one; in other words, we are able only to represent the latter subgroup on H. On the contrary, the notion of symmetry provided by quantum groups is broad enough to allow the existence of lattices whose points are mapped into each other under the action of the whole inhomogeneous q-groups. The main purpose of this paper is to describe how this occurs in the case of the q-Euclidean symmetry and how in the limit q → 1 one recovers the ordinary representation spaces. One concludes that the q-Euclidean lattice introduced in ref. [6] seems very appealing in view of full covariant regularizations of Euclidean QFT; actually, a q-deformed version of the ε-tensor on Rq is also available [4, 6, 11], so allowing the construction of the pseudo-tensors which are needed for chiral field theories. The main difference w.r.t the cubic lattice stems from the following fact. The N configuration-space coordinates x (as well as the momenta p) don’t commute with each other; therefore we can use a complete set of commuting observables consisting only partially (i.e. for about one half) of functions of the p (or, alternatively, of the x) and, as for the rest, by angular momentum components. Their spectra are discrete. The lattice in the present situation has namely [ 2 ] dimensions in p-space and [ 2 ] dimensions ([a] denotes the integer part of a) in angular momentum space; to each point of the lattice there corresponds a unique eigenvector belonging to a basis of H and labeled by N integers. Notably, under the action of the generators of the q-Euclidean algebra each vector is mapped simply into a new one with labels differing at most by ±1. In the sequel we will consider as algebra of observables the one generated by p’s and the angular momentum components, since they generate the physically relevant q-deformed Euclidean algebra (qtranslations + q-rotations); but both the commutation relations and the representation theory would be exactly the same (under the replacement x → p) if we considered the x instead. In section 2 we briefly introduce the q-deformation Uq(e N ) of the universal enveloping algebra of the Euclidean Lie algebra e which we are going to adopt as quantum symmetry. We will be quite explicit in the case N = 3, 4, for which we also write down the analog of the Pauli-Lubanski casimirs. Uq(e N ) is the Euclidean analogue of the q-deformed Poincare’ Hopf algebra (of u.e.a. type) Ref. [15, 9]. In both cases the inhomogeneous Hopf algebra contains the homogeneous one as a Hopf subalgebra which can be obtained from it by setting p = 0,Λ = 1 (Λ is the “dilaton”), and all commutation relations are homogeneous in p, contrary to what happens for inhomogenous Hopf algebras obtained through contractions [1, 7, 8]. Representation theory is also developed in a similar way as in ref. [15]. Section 3 is devoted to a detailed description of fundamental (i.e. irreducible oneparticle) Hilbert space ∗-representations of Uq(e N ) (we will call them “ irreps ” in the sequel). The case N = 3 is analysed first, as an introduction to the general case. We choose a Cartan subalgebra (i.e. a complete set of commuting observables) consisting basically of two parts, [ 2 ] squared momentum components and [ 2 ] angular momentum components ([a] denotes the integer part of a). The points of the spectra make up a q-lattice. One important fact is that the irreps turn out to be of highest weight type. Moreover, they can be obtained from tensor products of the singlet one (i.e. the one describing a particle with zero Uq(so(N))-highest weight) and some representation of Uq(so(N)); for instance, the irreps with N = 3 are obtained from the tensor product of the q-boson (i.e. zero spin) representation of Uq(e ) with a representation of some spin j ∈ N of Uq(so(3)) ≈ Uq(su(2)), in analogy with the undeformed case. The spectra of all observables are discrete, in particular the spectra of squared momentum components, as expected. The corresponding eigenvectors are normalizable and make up an orthogonal basis of the Hilbert space of each irrep. A cumbersome “ kinematical PT (parity + timeinversion) asymmetry ” appears in the structure of the spectra of the angular momentum observables; it disappears in the limit q → 1. In section 4 we clarify in which sense the Euclidean algebra/representations go to the classical ones in the limit q → 1. In the classical representation we know that the eigenvectors of operators which are only functions of the momenta are distributions, tipically they are delta-functions in momentum space. We show how to construct q-dependent integer labels ni(q) and coefficients α(q) such that α(q)|ni(q) >q (eigenvectors belonging to the q-representation) are delta-convergent functions in the limit q → 1. We can think of the irreps studied in section 3 as describing the (time-independent) dynamics of a free nonrelativistic particle with arbitrary “ generalized ” Uq(so(N))-spin on Rq . The subalgebra Ûq(e N ) := Uq(e N )/(Λ − 1) can be considered as the quantum group symmetry of the hamiltonian H := (p · p) 2M . (0.1) of the system; therefore all states with a given energy should be obtained from each other by the action of Ûq(e N ), as in the classical case, different eigenspaces of the energy should be obtained from each other by the action of the dilatation operators Λ. Some notational remarks are necessary before the beginning. For representation purposes we will assume in section 3 that q ∈ R, and we will limit ourselves to the case 0 < q ≤ 1; the case q > 1 can be treated in an analogous way. We set h = h(N) = { 0 if N = 2n+ 1 1 if N = 2n to allow a compact way of writing relations valid both for even and odd N . Unless stated differently, in our notation a space index i can take all the integer values between −n and n including/excluding i = 0 if N = 2n+1, 2n respectively. When N = 2n there is a complete invariance of the validity of all the results under the exchange of indices i = −1 ↔ i = 1, so that we will normally omit writing down explicitly the results that can be obtaind by such an exchange. We will often use the shorthand notation [A,B]a := AB − aBA (⇒ [·, ·]1 = [·, ·]). Indices are raised and lowered through the q-deformed metric matrix C := ||Cij||, for instance ai = Cija , a = Caj , Cij := q δi,−j, (0.2) where (ρi) := { (n− 1 2 , n− 3 2 , ..., 1 2 , 0,− 2 ..., 1 2 − n) if N = 2n+ 1 (n− 1, n− 2, ..., 0, 0, ..., 1− n) if N = 2n. (0.3) C is not symmetric and coincides with its inverse: C = C. 1 The Euclidean ∗-algebra Uq(e ) The Hopf algebra which we are going to use, Uq(e N ), was constructed in Ref. [9] and in equivalent form in ref. [6] by an inhomogeneous extension of the Hopf algebra Uq(so(N)) of “infinitesimal q-rotations” (in analogy with the undeformed construction). Uq(e ) is the Hopf dual of Fun(Rq >⊳SOq−1(N)) [13, 9]. In ref. [6] work, we added to the DrinfeldJimbo generators of the latter first the q-derivatives on Rq as infinitesimal generators p i of q-translations and then one more generator Λ, generating dilatations; the coalgebra and antipode for Uq(e ) were derived from the Leibnitz-rule of q-differential operators (the role of the coalgebra in representation theory is to allow the construction of many-particle representations starting from one-particle ones). On the algebra Uq(e N ) there exists a notion of complex conjugation ∗, (which will play the role of hermitean conjugation of operators). However, since the coalgebra is uncompatible, at least in the usual sense, with the ∗-structure, here we focus the attention on the algebra structure of Uq(e ) which we need to develop the theory of one-particle representations. 1.1 A Chevalley basis of Uq(so(N)) A Cartan-Weyl basis of Uq(so(N)) is the set {L ij , (k) 1 2} (i < j, 6= −j; n ≥ l ≥ 1) with commutation relations given below. Its elements were realized in Ref. [5] as q-differential operators on Rq ; this is the q-deformed analogue of realizing the generators of so(N)) as “angular momentum components”. To help the reader in the identification of the corresponding classical angular momentum components, we give here their classical limits L q→1 −→ x∂ − x∂, k − 1 q2 − 1 q→1 −→ x∂ − x∂, (1.1) where x, ∂ denote the classical coordinates/derivatives, ∂x = δ + x∂; the latter are chosen not to be real, but complex combinations such that (x) = x, (∂) = −∂. According to this construction, Uq(so(N)) is realized as a subalgebra of the differential algebra on Rq . The k’s generate a Cartan subalgebra of Uq(so(N)). The elements L ,L,k(k) (together with L,L,kk in the case N = 2n) for i = h, h+ 1, ..., n are “ Chevalley generators ” (i.e. algebraically independent generators) of Uq(so(N)) coinciding [5] with the Drinfeld-Jimbo ones, up to some rescaling of the roots L by suitable functions of k (k ≡ 0). The correspondence between the Chevalley generators L corresponding to positive roots and the spots of the Dynkin diagram of so(N) is shown in fig. 1. All the other generators L can be constructed starting from them as follows: [L,L]q = q lL [L,L]q = q lL, n ≥ k > l > j ≥ −h(N) (1.2) [L,L]q−1 = q lL [L,L]q−1 = q lL 2 ≤ l < k ≤ n (1.3) [L,L] = qL [L,L] = L 1 < k ≤ n if N = 2n+1; (1.4) these relations can be easily verified by the reader in the limit q = 1 using the limits (1.1). Once introduced the basis {L ,k} (i < j, 6= −j; n ≥ l ≥ 1), then the commutation relations satisfied by the Chevalley generators can be summarized in the following way. • Commutation relations between the generators of the Cartan subalgebra and the simple roots: