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Quantum walks, deformed relativity, and Hopf algebra symmetries Alessandro Bisio, Giacomo Mauro D’Ariano, and Paolo Perinotti ∗ † ‡ QUIT group, Dipartimento di Fisica, Universita` degli Studi di Pavia, via Bassi 6, 27100 Pavia, Italy and Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, 27100 Pavia, Italy We show how the Weyl quantum walk derived from principles in Ref. [1], enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras–the usual Poincaré and theκ-Poincaré algebras. PACSnumbers: 11.10.-z,03.70.+k,03.67.Ac,03.67.-a,04.60.Kz 6 1 0 I. INTRODUCTION ofspacetimeandphasespace. Weconsiderthenonlinear 2 deformation in the two alternative scenarios: the usual y Poincaré and the κ-Poincaré cases. We will see that the a Quantumwalks (QWs)[2–6] andmore generallyquan- construction of spacetime as the dual space to the al- M tum cellular automata (QCA)[7–9] have been recently gebra of translations is left unaffected by any nonlinear considered not only as a tool for quantum simulation of deformationthatrecoversthe linearLorentztransforma- 5 fields[10–12],butalsoforthefoundationofquantumfield 2 theory [1, 13–20]. The QCA framework appears as the tions at the leading order. Whether we obtain the usual spacetime or a noncommutative versionis a feature that natural candidate for the extension of the informational ] is independent on the nonlinear transformation that we h paradigm, which has been crucial in the understanding applytothemomentumoperators. Thisisaslightgener- p offoundationsofQuantumTheory[21–27]),to the foun- alizationoftheresultofRef.[39]whereonlythenonlinear - dation of Quantum Field Theory. t deformations that leave the rotation sector undeformed n The free theory has been derived starting from a de- a were considered. On the other hand, we see that the numerablesetofelementaryquantumsystemsininterac- u construction of the phase space as the left cross-product q tion along with the generalassumptions of homogeneity, algebra between momentum space and spacetime, does [ locality,isotropy,andlinearityofthe interactions[1,18]. depend on the nonlinear deformation. We then derive 2 The whole framework does not require Lorentz covari- the set of deformed Heisenberg commutation relations ance, which results as a subgroupof the dynamical sym- v emerging in our framework both in the usual Poincaré 2 metries of the quantum walk/automaton in the limit of and in the κ-Poincaré cases. Deformed Heisenberg com- 9 small wave-vectors[28, 29]. For general wave-vectors the mutation relationsare an ubiquitous feature of quantum 5 Lorentz transformations are nonlinear, thus realizing a Gravity models, they were first observed in the context 4 model of Doubly Special Relativity (DSR)[30–32]. of String theory [40, 41], then studied on their ownright 0 Inthispaperweconsiderthesimplestcaseofthemen- bymany authors[42–46], andrecently consideredfor ex- . 1 tionedquantumwalkfieldtheoryderivedfromprinciples, perimental verification [47]. 0 namely the one-particle sector of the free Weyl automa- 6 ton of Ref. [1]. We show how the dynamics of this walk 1 enjoys a nonlinearLorentz symmetry,which allowsus to v: introduce Hopf algebras[33–35] for position and momen- II. QUANTUM WALK AND RELATIVITY i X tum of the quantum walk particle, generalizing the role of the Lie algebra of symmetries. We focus on two spe- A quantum walk describes the discrete time evolution r a cialmodels ofHopf algebras: the usualPoincaréandthe of particle on a discrete set Γ. The Hilbert space of the κ-Poincaréalgebras[36]. system is := ℓ2(Γ) Cs where ℓ2(Γ) is the Hilbert space of sHquare summ⊗able function over Γ and Cs is After reviewing the derivation of the Weyl quantum the Hilbert space corresponding to the internal degrees walk in Sect. II along with its symmetries, in Section of freedom of the evolving particle. We introduce the III we analyze the nonlinear relativity symmetry, within ortonormalbasis g ofℓ2(Γ). The physicalinterpreta- thecontextofHopfalgebras—thecanonicalframeworkin {| i} tionisstraightforward: thethestate g ψ correspond whichdeformedrelativitymodelsarestudied[32,37,38]. | i⊗| i to a particle which is localized in g with internal state Weexpoundananalysis,closelyrelatedtotheoneinRef. ψ . The dynamics is described by a unitary operator A [39], where we study how our non linear deformation of | i (A A = AA = I) on . As shown in Ref. [1], the re- theLorentzgroupaffectstheHopfalgebraicconstruction † † H quirements of homogeneity and locality of the dynamics imply that the set Γ is endowed with a graph structure orresponding to the Cayley graph of a group G[50]. The generators of G are represented by a translation opera- ∗[email protected] †[email protected] tor Th acting on ℓ2(Γ) as follows: Th g = gh−1 (T is | i | i ‡[email protected] the right regular representation of G). Then, the homo- 2 geneity and locality assumption imply that the unitary which defines the dispersion relation of the automa- operator corresponding to the quantum walk A can be ton. It is easy to check that, by taking in the limit decomposed as follows: k k =(0,0,0)inEq. (3),thequantumwalkA+ (resp 0 → A )recoversthedynamicsoftheright-handed(respleft- − A= T A (1) handed) Weyl equation. Clearly, taking the same limit h h ⊗ hXS in Eq. (4) gives the usual relativistic dispersion relation ∈ ω2 k2 =0. We noticethatthe same behaviouroccurs where S is the set of generatos and A are operators on −| | Cs. h in the limit k → k2 = √23π(−1,−1,−1) and in the lim- Given a Cayley graph Γ and a fixed dimension s for its k → k1 = √23π(1,1,1), k → k3 = √3π(1,0,0) with the Hilbert space of the internal degrees of freedom, the the chirality exchanged. Because of this reason we refer existence(ornot)ofaquantumwalkonitisahighlynon- to the quantum walks in Eq. (2) as Weyl walks. It is trivial problem. In Ref. [1] some authors of the present a remarkable result that a Lorentz invariantdynamics is manuscript addressed the case in which Γ is the Cay- recoveredfromadynamicalmodelwhichfollowsfromthe ley graph of the Abelian group Z3 and the dimension only assumptions of homogeneity, locality and isotropy, of the internal degree of freedom is s = 2. Moreover, without the relativity principle. they assumed the quantum walk to be isotropic, a con- InthefollowingwewillconsideronlytheA+Weylwalk dition that translates the idea that all the directions on and we will drop the apex in order to simplify the ± the lattice are equivalent. In mathematical terms, there notation. The entire analysis can be straightforwardly mustexistaunitaryrepresentationU overC2 ofagroup applied to the A− case. L of graph automorphisms, transitive over a set of di- In the quantum walk framework space and time are rect generators[51], such that one has T A = not on an equal footing: space is given by the lattice h∈S h ⊗ h structure, while time comes from the discrete steps of Psuml(hp)t∈iSonTsl(,ht)h⊗ereUlisAohnUll†y foonreaaldlml i∈ssiLbl.ePUCnaydleerytghreaspehaos-f the evolution. It is then far from obviuos whether and Z3, which is the one corresponding to the body-centered how it is possible to recover changes of spacetime coor- dinates that mix space and time, like boosts in special cubiclattice,andthereareonlytwoadmissiblequantum relativity. This question was recently addressed and an- walksoverit(uptoalocalchangeofbasis). Theanalytic swerdinRef. [29]wherethenotionofchangeofobserver expressionofthesequantumwalksareeasilygiveninthe Fourier transform basis k = (2π) 3/2 eikx x for quantum walks was defined as as an invertible map (where x clearly denotes a|nielement i−n Z3P) x∈Z3 · | i Lβ over [−π,π]×B, as follows (ω,k) (ω′,k′)= β(ω,k) (5) → L A± :=ZBdk|kihk|⊗A±k where the parameter β labels different changes of A±k :=(2π)−23 eik·yA±y (2) rlaetfetirceencceo-ofrrdaminea.teTshaendidtehaeisdinscortettoetfoimcuessotenpt,hbeutdirsactrheeter yXS to consider (ω,k)–which are constants of motion of the ∈ A±k :=λ±(k)I−in±(k)·σ± quantumwalk–asthe fundamental variables. In this set- ting a symmetry of the dynamics is defined as follows: s c c c s s x y z x y z n±(k):=cxsycz±sxcysz, ∓ cxcysz sxsycz Definition 1 Let A be a quantum walk on Z3. A sym- λ±(k):=(cxcycz ±sxsysz), metryofthedynamicsfor Ais atriple (Lβ,Γβ,Γ˜β), with ∓ defined in Eq (5) and Γ , Γ˜ invertible matrix func- cα :=cos(kα/√3), sα :=sin(kα/√3), α=x,y,z. Ltioβns of (ω,k), such that β β wcuhbeircelBattdiecneoatensdtσhe+B=rillσoudinenzootneeaofvethcteobroodfythceenutesrueadl (sinωI−n(k)·σ)=Γ˜−β1(sinω′I−n(k′)·σ)Γβ. (6) Pauli matrices, while σ− = σT denotes the transposed The set of symmetries SA is a group which we refer to ones. The unitary constraint implies that A±k is unitary as the symmetry group of the quantum walk A. for every k B. Notice that due to the discreteness of the lattice th∈e quantum walk is band-limited in k. The The next step is then to explore whether the symmetry group of the Weyl walk A contains a representation of quantum walk dynamics is determined by the solutions of the eigenvalue equation (A eiω)ψ = 0 that is the Lorentz group which recovers the usual one in the ± − | i regime in which the walk approaches the Weyl equation equivalent to (i.e. near k ,k ,k , and k ). In other words we are 0 1 2 3 (sinωI n (k) σ )ψ(k,ω)=0, (3) asking whether there exists a deformed relativity model ± ± − · which preserves the dynamics of the Weyl walk A. which also implies the identity Deformed(ordoubly)specialrelativityisatheoretical proposal in which one modifies the linear Lorentz trans- sin2ω n (k)2 =0 (4) ± formations in order to have an invariant energy scale −| | 3 in addition to the speed of light. Such a theory has quantum mechanics. The Weyl walk dynamics has been beenproposedbyAmelino-Camelia[30]anddevelopedby singledoutwithoutrequiringLorentzinvariance,whereas otherauthors[31]asakinematicstructurewhichmayun- the Lorentz invarianceis recoveredas a symmetry of the derlie quantum theory of gravity. Indeed, if the Planck dynamics. length were a threshold beyond which quantum gravity effects would become relevant, this length should be the same for all the observers, a statement which clearly III. HOPF ALGEBRA, κ-POINCARE AND disagrees with special relativity. A deformed relativ- NONCOMMUTATIVE SPACETIME ity model consist in replacing the usual (linear) Lorentz transformation Lβ in momentum space as follows: In this section we explore how the deformation of the Lorentz group given by the nonlinear deformation (8) L , β →Lβ manifests itself at the level of the Poincaré algebra. We β = −1 Lβ , (7) will restrict to the (0) case and then drop the (0) apex L D ◦ ◦D D (ω,k) (ω,k), in order to simplify the notation, the generalization for β →L i=1,2,3 is trivial. In order to perform this analysis we where the map is a singular invertible map such that willneedtoconsidertheframeworkofHopfalgebras(for its Jacobian J Dequals the identity in (ω,k) = 0. These a comprehensive introduction to the subject we suggest conditions areDneeded in order to have an invariant en- Ref.[34]). The notion of Hopf algebra generalizes that of ergy,whilerecoveringtheusualphenomenologyatenergy Lie algebra to a less “rigid” object, which is can accom- scales much smaller than the Planck scale. modate a nonlinear version of the Lorentz group, which For a complete derivation where we refer to Ref [29]. is incompatible with a Lie algebra structure. Unfortu- Apart froma null measure set we split the Brillounzone nately, any specific nonlineardeformationof the Lorentz B into four parts Bi, i=0,...3. Each vector ki belongs group, of the kind in Eq. (7), is not sufficient to select tothecorrespondingregionBi. TheregionsBiarechosen a unique Hopf algebra, since there are many compati- such that the compositions (i) = (i) 1 L (i) are ble coproduct structures. Nevertheless it is interesting Lβ D − ◦ β ◦D well defined, with (i) given by to study the role that our deformed Lorentz transforma- D tionplays within the contextofHopf algebras,since this is the canonical context in the specialized literature on ω sinω (i) :Σ Γ , (i) : g(ω,k) , deformed relativity [32, 37, 38]. D i → 0 D (cid:18)k(cid:19)7→ (cid:18)n(i)(k)(cid:19) Σ := (ω,k) s.t. k B ,sin2ω k2 =0 , i i { ∈ −| | } Γ := p R4 s.t. p pµ =0 , A. Classical Poincaré and κ-Poincaré Hopf algebras 0 µ { ∈ } (8) The Lie algebra of the Poincare group is given by the for a suitably defined function g(ω,k) [52]. The maps relations (i) provide a well defined nonlinear representation of Lthβe Lorentz group on each set Σ . [Mi,Mj]=iǫijkMk [Mi,pj]=iǫijkpk i For i=0,2one caneasily checkthat the conditionsof [Mi,Nj]=iǫijkNk [Mi,p0]=0 (9) Definition1aremetifwesetΓ =Λ andΓ˜ =Λ˜ ,pro- [N ,N ]= iǫ M [N ,p ]=iδ p k β k β i j ijk k i j ij 0 − vided that Λ is the right handed spinor representation β [N ,p ]= ip [p ,p ]=0 i 0 0 µ ν oftheLorentzgroup,andΛ˜ istheleft-handedrepresen- − β tation. Fori=1,3thesameholdsprovidedweexchange wherewedenotedwithM thegeneratorsofspatialrota- i the two representations. The four vector (ω,k) ∈ Σi tions, with Ni the generators of boosts, and with pµ the transformsunderthenonlinearrepresentation (i). Since generators of translations—p0 denoting the generator of 3 B =B(apartfromazero-measureset),weLhβavethat time translation. Clearly, if we apply a non-linear map ∪i=0 i to the generators p , the set of commutation relations themaps (i) provideanotionofLorentztransformation µ Lβ (9)isspoiled,andgenerallydoesnotdefineaLiealgebra for any solution of the Weyl QCA dynamics. anymore. However, it is possible to treat such deforma- Wenoticethatthechoiceofthemap(8)isnotunique, tions on formalgrounds,within the more generalsetting since there are many admissible choices for the function ofHopfalgebras. Theuniversalenvelopingalgebraofthe g(ω,k). The symmetry group SA of the Weyl walk A Liealgebra(9)canbeendowedwithaHopfalgebrastruc- containsthenmanydifferentistancesofdeformedrelativ- ture bydefining the primitive co-product∆,antipode S, ity. However, all of them will recover the usual Lorentz and co-unit ǫ as transformations near the points k . The four invariant i regions are interpreted as four different particles (this is ∆(O)=1 O+O 1, the phenomenon of Fermion doubling). ⊗ ⊗ S(O)= O, S(1)=1, (10) Finally,itisworthstressingthereversedperspectiveof − ǫ(O)=0, ǫ(1)=1. thisapproachwithrespecttotheusualoneinrelativistic 4 TheserelationsarejustarephrasingoftheusualPoincaré algebra of translation. Liealgebrastructure(9)inthelanguageofHopfalgebras, If we denote by T the Hopf algebra generated by the wheretheadditionalcoalgebrastructureallowsonetoex- translation generators p one can define the position al- µ press the Leibniz rule for the infinitesimal action of the gebra as the dual hopf algebra T on which T acts co- ∗ group on products of functions through the coproduct. variantly [36]. T is determined by introducing the gen- ∗ This rule can be easily accounted for using the tensor erators x and the pairing µ product structure and the theory of group representations. On the other hand, within the context of Hopf ∂ f(p ),x =f( )[x ](0). (12) µ ν ν algebras any invertible analytical map that transforms h i ∂x µ momenta as p = f (p ) can be treated as a change of ′ν ν µ This way of introducing the pairing follows the classical basis in an infinite dimensional algebra. Even if, from a mathematical perspective, this transformation is just a pairing between the enveloping algebra of R4 with the algebraoffunctionsonR4,i.e.thetranslationgenerators change of basis, it may have significant physical conse- act as derivatives evaluated at the origin. The structure quenceslikee.g.adeformationofthedispersionrelation. of T is then determined by the axioms of Hopf algebra Nonlinear modifications of the translation generators ∗ duality is not the only possible deformation of the classical Poincarésymmetry. It is indeedpossibleto considersce- p,xy = ∆(p),x y narios in which the Hopf-algebraic structure itself is dif- h i h ⊗ i (13) ferent (up to any change of basis) from the classical one pq,x = p q,∆(x) . h i h ⊗ i given by Eqs. (9) and (10). Of particularly interest are Since the momenta commute we have that positions co- thosedeformationsoftheclassicalPoincaréHopfalgebra commute with co-commutators that reduce to the usual one in a suitable limit of values of the deformation parameters. The classification of all ∆x =1 x +x 1. (14) the possibledeformationofPoincaréHopfalgebraisstill µ ⊗ µ µ⊗ an open problem. Thecommutationrelations[x ,x ]aredifferentfrom0 µ ν Up to now the most studied example is the so-called onlyifthecoproductsforthep arenotco-commutative. µ κ-Poincaré Hopf algebra [33, 36], which in the so called Then, if we are dealing with the usual Poincaré alge- “classical basis” [37, 48] takes the following form: bra we will always have a commutative spacetime, independently of the nonlinear mapping we are using to the same algebraic sector define the generators, as their coproduct will still be co- κ ∆(p0)= (K K K−1 K−1)+ commutative. 2 ⊗ − ⊗ The scenario is different in the κ-Poincaré case where 1 + (K 1 p2 K 1) (11) ithasbeenprovedthatthe Hopfalgebradefined byEqs. − − 2κ | | ⊗ (11)leadsto the followingcommutationrelationsfor po- +(K−1pi pi+K−1 K−1 p2) sitions ⊗ ⊗ | | ∆(p )=p K+1 p i i i i ⊗ ⊗ [x ,x ]=0 [x ,x ]= x (15) i j 0 i i where K := κ1(p0+(p20−|p|2+κ2)21) and κ is a real pa- −κ rameter. Onecancheckthatthe usualclassicalPoincaré In this case it could happen that a differrent choice of Hopf algebra is recoveredin the limit κ . the generators p could lead to different commutation µ →∞ Then, starting from the enveloping algebra of the relations. Inthe literature[39]itisprovedthatthe com- Poincaré Lie algebra we have two different roads that mutation relations (15) do not depend on the choice of can be explored: i) assume the coalgebra structure (10) basis as long as it is rotationally invariantand such that andconsidertheclassicalPoincaréHopfalgebra,orii)as- the usual generators are recovered in the limit κ . → ∞ sume Eq. (11) and study the κ- Poincaré Hopf algebra. Itispossibletoslightlygeneralizethisresultbydropping Ononehand, ourscenariosinglesouta setofgenerators the assumption of rotational invariance k that are defined in terms of the classical one p by µ µ Lemma 1 Let : p p = (p) be a transformation the nonlineardeformationp= (k). Ontheotherhand, ′ M 7→ M D of the translation generators such that J (0)=I. Then our model does not prefer any of the different algebric the commutation relations (15) remain uMnchanged. models andit is interesting to consider the consequences of the the nonlinear deformation given by the map in both the classical Poincaréand in the κ- PoincarécaDses. Proof. First we observe that, from the pairing (12) we havethatthe onlytermsinthecocommutators(11)that are relevant for computing the commutators [x ,x ] are µ ν B. From Poincaré Hopf algebra to spacetime the ones that are at most bilinear, i.e. ∆(p )=1 p + 0 0 p 1+1 p p and∆(p )=p 1+1p p +⊗1 p . 0⊗ κ i i⊗ i i i⊗ κ i⊗ 0 ⊗ i Oneofthemostpopularspeculationsconcerntherela- By powerPexpanding M we have p′µ = pµ + κ1mαβpαpβ tion between the algebra of position coordinate and the and by power expanding the inverse function 1 we − M 5 have p = p + 1n p p . It is then easy to verify After some cumbersome but straightforward calcula- µ ′µ κ αβ ′α ′β that, up to the bilinear terms, the coproduct ∆(p ) is tion,wehave,intheclassicalPoincaréHopfalgebracase ′0 co-commutative while the coproducts ∆(p ) are the sum of a co-commutative term and 1p p . S′iince the non- ( 1)δi,2 κ ′i⊗ ′0 [ki,xj]= iδij i − (δi+1,jki+2+δi+2,jki+1) cocommutative term 1p p has the same expression − − κ κ ′i ⊗ ′0 independently ofthe nonlinearmapping ,the commu- [ω,xj]=[ki,t]=0 [ω,t]=i M tation relation for the spacetime variables remains the (21) same.(cid:4) where we used the notation x = 1,y = 2,z = 3 and This result tells us that our nonlinear mapping, which the sums are meant to be modulo 3. Similarly in the satisfies the hypotheses of lemma 1, does not change the κ-PoincaréHopf algebra case we get commutation relations for the spacetime variables. ω [k ,x ]= iδ (1 )+ i j ij − − κ C. From Poincaré Hopf algebra to phase space ( 1)δi,2 i − (δ k +δ k ) i+1,j i+2 i+2,j i+1 − κ (22) We haveseen inthe preceding sectionthat a notionof i 1 [ω,x ]= k x k 2 [k ,t]=0 spacetime can be introduced as the dual T∗ to the Hopf j κ j − 2κ j| | i algebra of translations T. The additional notion of left 1 coregular action [ω,t]=i xj k 2 − 2κ | | p⊲x:= p,x x (16) Differentlyfromthespace-timecommutationrelations, (2) (1) h i the commutation relation between position and momen- allowsto introduceanotionofphasespace[37,49]asthe tum areaffectedby the choice ofthe basis. As onecould leftcrossproduct algebraT∗⋊T wherethemultiplication expect, in both cases we recover the usual commutation is defined as relations between position and momentum as the deformation parameter κ goes to infinity. (x p)(x p)=x(p ⊲x) p p. (17) ′ ′ (1) ′ (2) ′ ⊗ ⊗ ⊗ If we define the isomorphisms IV. CONCLUSION x x 1 p 1 p (18) ∼ ⊗ ∼ ⊗ In this paper we have studied the dynamical sym- it make sense to consider the commutation relation metries of the Weyl quantum walk. As explained in the paper such walk is particularly interesting since [pµ,xν]=xν pµ pµ(1),xν 1 pµ(2) it was derived from general principles without assum- ⊗ −h i ⊗ − (19) + p ,1 x p ing Lorentz covariance, but nevertheless it recovers a µ(1) ν µ(2) h i ⊗ Lorentz-invariant dynamics in the limit of small wave- We will see that the commutation relations (19) will de- vectors. For large wave-vectors the Lorentz group be- pendonthechoiceofthegenerators,i.e. theydependon comesnonlinear,andwe haveamodelofDoubly Special the nonlinear deformation. Relativity. We introducedthe Hopf algebrasfor position We will now compute the commutation relation (19) andmomentumofthequantumwalkparticle,andevalu- for the choice of generators given by the map . Since atedthe structureconstantsofthe algebrasforthe usual D we cannot derive an analytic expression for the inverse Poincaréand the κ-Poincarécases. Generalizinga result map 1 we will consider just the terms up to the first of Ref.[39], we have shown that the spacetime commu- − orderDin 1. We have then tators are left unaffected by any nonlinear deformation κ that recovers the linear Lorentz transformations at the E =ω ω =E leading order. Finally we derived the analytical expres- 1 1 sion up to the first order in the inverse Planck-energy px =kx+ κkykz kx =px− κpypz κ−1 of the deformed Heisenberg commutation relations. 1 1 (20) p =k k k k =p + p p y y x z y y x z − κ κ Acknowledgments 1 1 p =k + k k k =p p p z z x y z z x y κ − κ ThisworkhasbeensupportedbytheTempletonFoun- This result holds the same for any choice of g(ω,k) such dation under the project ID# 43796A Quantum-Digital that g(0)=0. Universe. ∇ 6 [1] G. M. D’Ariano and P. Perinotti, Phys. Rev. A 90, [30] G. 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