PHYSICAL REVIEW D 99, 106005 (2019)

Noncommutative Fourier transform for the via the Duflo map

Daniele Oriti Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany and Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstrasse 37, D-80333 München, Germany, EU

Giacomo Rosati INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy

(Received 11 February 2019; published 13 May 2019)

We defined a noncommutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the noncommutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map.

DOI: 10.1103/PhysRevD.99.106005

I. INTRODUCTION of (noncommutative) Fourier transform to establish the (unitary) equivalence between this new representation and Physical systems whose configuration or phase space is the one based on L2 functions on the group configura- endowed with a curved geometry include, for example, tion space. point particles on curved spacetimes and rotor models in The standard way of dealing with this issue is indeed to condensed matter theory, and they present specific math- renounce to a representation that makes direct use ematical challenges in addition to their physical interest. of the noncommutative variables, and to use In particular, many important results have been obtained in instead a representation in terms of irreducible representa- the case in which their domain spaces can be identified tions of the , i.e., to resort to spectral decom- with (Abelian) group manifolds or their associated homo- position as the proper analogue of the traditional Fourier geneous spaces. transform in flat space. In other words, instead of using Focusing on the case in which the nontrivial geometry is some necessarily generalized eigenbasis of the noncom- that of configuration space, identified with a Lie group, mutative momentum (Lie algebra) variables, one uses a this is then reflected in the noncommutativity of the basis of eigenstates of a maximal set of commuting conjugate momentum space, whose basic variables corre- operators that are functions of the same. This strategy spond to the Lie derivatives acting on the configuration has of course many advantages, but it prevents the more manifold, and that can be thus identified with the Lie geometrically transparent picture, more closely connected algebra of the same Lie group. At the quantum level, this to the underlying classical system, that dealing directly noncommutativity enters heavily in the treatment of the with the noncommuting Lie algebra variables would system. For instance, it prevents a standard L2 representa- provide. tion of the Hilbert state space of the system in momentum In recent times, most work on these issues has been picture, and may lead us to question whether a represen- motivated by quantum gravity, where they turned out to tation in terms of the (noncommutative) momentum play an important role. This happened in two domains. The variables exists at all. Further, if such a representation first is effective (field theory) models of quantum gravity could be defined, one would also need a generalized notion based (or inspired) by noncommutative geometry [1]. Such models have attracted a considerable interest because they offer a framework for much quantum gravity phe- Published by the American Physical Society under the terms of nomenology [2]. From the physical point of view, these the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to models are based on the hypothesis that some form of the author(s) and the published article’s title, journal citation, spacetime noncommutativity [1] constitutes a key remnant, and DOI. Funded by SCOAP3. at a macroscopic, continuum level, of more fundamental

2470-0010=2019=99(10)=106005(27) 106005-1 Published by the American Physical Society DANIELE ORITI and GIACOMO ROSATI PHYS. REV. D 99, 106005 (2019) quantum gravity structures, and that one key way in which because it plays in particular a key role in quantum gravity, this manifests itself is by relaxing [3] (or making relative in the context of spin foam models, group field theories as [4,5]) the usual notion of locality on which effective field well as effective noncommutative spacetime models. The theory (and most spacetime physics) is based. At the quantization map we use, which is the key technical input mathematical level, spacetime noncommutativity in these of the whole construction, is the Duflo map [16]. This has models is the conjugate manifestation of curvature in already found some applications in quantum gravity momentum space (and thus in phase space) [6]. In most [17–22], and it had been used in [15] for the compact of them, moreover, this curved momentum space is SUð2Þ case. For our present purposes, the mathematical modeled by a Abelian Lie group (the phase space is features of this quantization map that makes it a preferred therefore its cotangent bundle) or homogeneous spaces choice are its generality (it also can be applied to any semi- constructed from it, hence the relevance of the noncom- simple Lie algebra) and its characterizing property of mutative Lie algebra representation and of the noncommu- mapping the subalgebra of invariant functions (under the tative Fourier transform for Lie groups (see e.g., [7–9]). The ) on the classical phase space faithfully to a second domain is that of nonperturbative quantum gravity subalgebra of invariant operators in the quantum theory. proper, more specifically the broad area comprising group This is mathematically desirable, but also physically field theory [10], spin foam models [11] and loop quantum crucial, in our opinion. gravity [12]. Here again the cotangent bundle of a Lie The plan of the paper is as follows. In the next section, we group plays a prominent role: it is the classical phase space summarize the results of [15], for general Lie groups, which of the variables associated to the (edges of the) lattices, form the basis for our subsequent analysis and construction which replace continuum spacetime manifolds and re- for the Lorentz group. In Sec. III we use the Duflo map to present the basic interaction processes of the fundamental construct the Lie algebra representation of a quantum degrees of freedom of all these formalisms. The interpre- system whose classical phase space is the cotangent bundle tation of these algebraic variables have, in turn, a discrete of the Lorentz group, and then, in Sec. IV, using the same geometric interpretation, as discrete gravity connection (the quantization map, we obtain the noncommutative Fourier group variables) and conjugate discrete metric variables transform that ensure its equivalence with the more tradi- (the Lie algebra elements). Once again, the Lie algebra tional group representation, constructing explicitly the representation and the noncommutative Fourier transform corresponding noncommutative plane waves. These are allow a representation of the formalisms in terms of more the key new results of this paper. We also offer a quick geometrically transparent data [13,14]; this, in turn, facil- comparison with related results in the literature, concerning itates model building as well as the extraction of effective the definition and application of noncommutative methods macroscopic physics. for similar systems, both at the mathematical level and in In this article we are not going to deal with these physical quantum gravity contexts. Moreover, we show how our aspects, although we will discuss them briefly. We focus results generalize to the case of homogeneous spaces of the instead on the formal aspects of the noncommutative Lorentz group. Finally, in the last section, we show the Fourier transform. Among the many issues motivated by application of our tools to the computation of the propagator its intrinsic mathematical interest, it is important to clarify of a single nonrelativistic point particle on the hyperboloid, its general properties for arbitrary Lie groups and its as an example. explicit construction in the challenging case of noncompact ones. The mathematical foundations of the Lie algebra representation and of the noncommutative Fourier trans- II. BACKGROUND: LIE ALGEBRA form have been explored in very general terms, e.g., in [15]. REPRESENTATION AND A key result of that work has been to show how both the NONCOMMUTATIVE FOURIER star product and the noncommutative plane waves that TRANSFORM characterize fully such algebra representation and its The mathematical foundations of the noncommutative equivalence with the other representations of the underlying Fourier transform have been clarified in [15], and we quantum system, are uniquely determined by the initial base our present work on that analysis. We summarize its choice of quantization map that defines the same quantum results here. system starting from its classical phase space and algebra of observables. Moreover, this has been shown for arbitrary semisimple locally compact Lie groups. Thus, while the A. Classical phase space and its quantization explicit constructions reported in the same paper only dealt We consider physical systems whose configuration with the SUð2Þ, the same procedure can be space is a Lie group G, and whose momentum space straightforwardly extended to the noncompact case. This is coincides with the dual g of the Lie algebra g of G, giving what we do in this paper, focusing on the Lorentz group the classical phase space as the cotangent bundle SLð2; CÞ because of its general physical interest and TG ≅ G × g. The classical algebra of observables is

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∞ thus the Poisson algebra PG ¼ðC ðT GÞ; f·; ·g; ·Þ, such the logarithm map, kðgÞ¼−i lnðgÞ, setting for the corre- 1 ˆi ˆi ˆi that sponding operators ϵGðk Þ¼0, SGðk Þ¼−k and ∂ ∂ ∂ ∂ X∞ X f g k f g ≡ L − L ˆi ˆp1 ˆp ˆq1 ˆq ff; gg ig if þ cij Xk; ð1Þ Δ ðk Þ¼ B k k k ⊗ k k k ; ð4Þ ∂Xi ∂Xi ∂Xi ∂Xj G p1pkq1ql n¼1 k;l∈N kþl¼n ∞ for functions f; g ∈ C ðT GÞ, Euclidean coordinates Xi on ≅ Rd ≔ L where B ∈ R are the coefficient of the Taylor g (d dimðGÞ); i are Lie derivatives on G (with p1pkq1ql respect an orthonormal basis of right-invariant vector expansion for the Baker-Campbell-Hausdorff formula k i B i ∈ A fields), and cij are the structure constants of the Lie k ðghÞ¼ ðkðgÞ;kðhÞÞ . Defining the lift of f G onto ≡ ik algebra g ð≅ gÞ with i; j; k ¼ 1; …;d (summation over the Lie algebra as fkðkÞ fðe Þ, the coproduct can be ˆ ˆ repeated indexes is assumed). extended to the corresponding f ∈ UG as ΔGðfÞ ≡ A P ˆi The quantization of a maximal subalgebra of G (for fkðΔGðk ÞÞ. These definitions satisfy the requirements which the quantization is consistent) amounts to the for the coproduct to be induced by the group multiplication definition of an abstract operator -algebra U, obtained as ΔðfÞðg; hÞ¼fðghÞ, as well as the group unit and from the classical Poisson algebra. The quantization is inverse to induce counit and antipode, ϵðfÞ¼fðeÞ, defined through a quantization map Q∶A → U. The SðfÞðgÞ¼fðg−1Þ. required properties of the quantization map are as follows. First, QðfÞ ≕ fˆ for all f ∈ A ⊂ C∞ðGÞ, A being the G G B. The group and algebra representations subalgebra of A ⊂ C∞ðG × gÞ constant in the second ˆ In order to get a concrete realization of the abstract argument, and QðXiÞ ≕ Xi, satisfying observable algebra U, one has to construct its representa- d tions, in which the elements of the abstract algebra become ½f;ˆ gˆ¼0; ½Xˆ ; fˆ¼iL f ∈ U ; ½Xˆ ; Xˆ ¼ic kXˆ ; i i G i j ij k operators acting on specific Hilbert spaces. ð2Þ For the system at hand, one can immediately define a “group representation” π of U on the Hilbert space L2ðGÞ ˆ G for all f; gˆ ∈ UG with UG ¼ QðAGÞ. Second, when of square-integrable functions on G with respect to the Haar ∞ restricted to functions Ag ⊂ C ðG × g Þ constant in the measure dg. This is done by diagonalizing all the operators Q ˆ ˆ first factor, maps to a completion of the universal f ∈ UG and setting the action of the Xi to be given by the enveloping algebra of g, Ug ≔ QðAg Þ ≅ UðgÞ.Ifwe Lie derivatives, so that restrict to the space of polynomials in g, then Q encodes π ˆ ψ ≡ ψ the operator ambiguity coming from the noncommutativity ð GðfÞ ÞðgÞ fðgÞ ðgÞ; ˆ ˆ ˆ of the elements Xi. The algebra U generated by f and ðπ ðX ÞψÞðgÞ ≡ iL ψðgÞ; ð5Þ ˆ G i i Xi is then called the “quantum (observable) algebra” ∞ 3 for T G. for all ψ ∈ Cc ðGÞ where we restrict the domain of πG to Ug is endowed with a natural Hopf algebra structure the sbspace of AG of smooth compactly supported func- 2 inherited from the universal enveloping algebra U, with tions CcðGÞ [dense in L ðGÞ]onG. The inner product for Δ ϵ 2 coproduct g , counit g and antipode Sg ψ; ψ 0 ∈ L ðGÞ is defined as usual as Z Δ ˆ ˆ ⊗ 1 1 ⊗ ˆ g ðXiÞ¼Xi þ Xi; ψ ψ 0 ≡ ψ ψ 0 h ; iG dg ðgÞ ðgÞ: ð6Þ ˆ ˆ ˆ G ϵg ðXiÞ¼0;Sg ðXiÞ¼−Xi: ð3Þ That (5) is a representation of the quantum algebra (2) is An Hopf algebra structure in U can be obtained2 for G ensured by the fact that the Lie derivative satisfies the exponential Lie groups by considering canonical coordi- Leibniz rule respect to the pointwise product of functions. nates (of the first kind) k∶G → g ≅ Rd defined through Such property can be expressed as a compatibility con- dition between the coproduct Δg of Ug and the pointwise 1 0 0 0 ∞ See, e.g., [23] for a general discussion of Poisson structures product mG∶f ⊗ f → f · f for f; f ∈ C ðGÞ. More in on cotangent bundles. While there are in general other possible Poisson structures for TG, we here stick to the form (1), which is 3 compatible with the description of the phase space T G comply- This is necessary to ensure that fψ lies in CcðGÞ for all ∞ ing with the on G × g implementing the ψ ∈ Cc ðGÞ. In general, since we are dealing with unbounded (coadjoint) action of G on g. The explicit expression (1) for the operators, for their representation on a Hilbert space H, their semidirect product TG ≅ G × g can be found in [24]. domain of definition must be restricted to some dense subspaces 2This construction of the coproduct for exponential Lie groups of H such that their images under the action of the operators are was introduced in [15]. contained in H.

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π ∞ ˆ general if is a representation of U on a space Fm with for all f⋆ ∈ Ag ⊂ C ðg Þ such that fðXiÞ¼Qðf⋆Þ ∈ Ug . ∶ ⊗ 0 → 0 multiplication m f f f ·m f , the compatibility with With this definition the ⋆-product and quantization map are ˆ the coproduct Δ can be expressed, for an operator T in U, related, as in the deformation quantization framework, by by the commutative diagram 0 −1 0 f⋆ ⋆ f⋆ ¼ Q ðQðf⋆ÞQðf⋆ÞÞ: ð12Þ

ð7Þ As anticipated, the choice of quantization map determines uniquely the ⋆-product to be used in constructing the algebra representation. The faithfulness of the representation (8) is then guar- anteed by noticing that the ⋆-product is such that the expressing the identity πðTˆ Þ ∘ m ¼ m ∘ ðπ ⊗ πÞðΔðTˆ ÞÞ. observables depending on Xˆ are represented through an Besides the group representation π , a representation i G → π ˆ based on functions ϕðXÞ of coordinates on g (thus algebra isomorphism, while f g ðfÞ is a homomor- analogous of standard functions on momentum space) phism due to commutativity of ∂i on g. Finally, one can “ ” ˆ j dj can also be defined: the algebra representation . The show (see [15]) that the commutator ½Xi; zˆ ¼iLiz is underlying noncommutativity of the Lie algebra has trans- reproduced if the ⋆-product satisfies the property encoded late into an appropriately defined noncommutativity of in the commutative diagram (7), i.e., if the functions ϕðXÞ. This is done by the introduction of ⋆ i i a -product compatible, in the sense of diagram (7), with πg ðzˆ Þ ∘ mg ¼ mg ∘ ðπg ⊗ πg ÞðΔGðzˆ ÞÞ: ð13Þ i the coproduct ΔG of the algebra of operators zˆ correspond- ing to a specific parametrization of G. As we will see, the Thus, the compatibility condition (7) can also be under- required ⋆-product can be defined directly from the choice stood as expressing the compatibility between the choice of of quantization map Q. quantization map, i.e., of ⋆-product, and the choice of More precisely, one takes coordinates z∶G → g ≅ Rd on coordinates on the group used in defining the quantum i 0⃗ 0 ∂ i 0⃗ δi algebra U. For more details on the algebra representations, G satisfying zkð Þ¼ , ∂kj zkð Þ¼ j, where k are canoni- cal coordinates. The coordinates operators zˆi are defined by see [15]. ! ˆ i ∞ imposing f ¼ fzðzˆ Þ, where fz ∘ z⃗≡ f for all f ∈ C ðGÞ. The algebra representation πg then is defined by the actions C. The noncommutative Fourier transform The noncommutative Fourier transform F∶L2ðGÞ → ˆ 2 F −1∶ 2 → 2 ðπg ðXiÞϕÞðXÞ ≡ Xi ⋆ ϕðXÞ; L⋆ðg Þ [and its inverse L⋆ðg Þ L ðGÞ] is the i i intertwiner between the group and algebra representation ðπ ðzˆ ÞϕÞðXÞ ≡ −i∂ ϕðXÞ; ð8Þ g (and vice versa), ensuring their unitary equivalence. It is expressed as the integral transform: where the second equation means that Z 2 ˆ ⃗ ψ˜ ðXÞ ≔ FðψÞðXÞ¼ dgEgðXÞψðgÞ ∈ L⋆ðg Þ; ðπ ðfÞϕÞðXÞ¼f ð−i∂ÞϕðXÞ; ð9Þ g k GZ d −1 d X 2 ik ∞ ∞ ψ ≔ F ψ˜ ⋆ ψ˜ ∈ where f ðkÞ ≔ fðe Þ ∈ C ðgÞ for all f ∈ C ðGÞ. The ðgÞ ð ÞðgÞ¼ d EgðXÞ ðXÞ L ðGÞ: k ð2πÞ algebra representation is thus defined on the function space g 2 ∞ ð14Þ L⋆ðg Þ ∋ ϕðXÞ, which denotes the quotient Cc ðg Þ=N of the space of smooth compactly supported functions (see footnote 3) on g by the subspace N , such that the inner The kernel of the transformation EgðXÞ is called the product respect to the Lebesgue measure ddX on g noncommutative plane wave. Its defining equations are identified by requiring that the intertwined function spaces Z d define a representation of the same quantum algebra, and 0 d X ¯ 0 ϕ ϕ ≔ ϕ ⋆ ϕ applying the action of U in the different representation. The h ; ig d ð ÞðXÞð10Þ g ð2πÞ explicit form of EgðXÞ, and its existence, depends then again on the choice of a quantization map, and thus, in turn, N ≔ ϕ∈ ∞ ∶ ϕ ϕ0 0 is non degenerate, i.e., with f C ðg Þ h ; ig ¼ g. on the choice of a deformation quantization ⋆-product. The first equation in (8) is actually a consequence of a stronger The intertwining property can be expressed generally as condition that one imposes on the ⋆-product: ˆ ˆ ˆ F ∘ πGðTÞ¼πg ðTÞ ∘ F, where T ∈ U. Applying it to the ˆ ζˆi ˆ operators Xi and respectively, one obtains the following ðπ ðfðXiÞÞϕðXÞÞ ≡ f⋆ðXÞ ⋆ ϕðXÞ; ð11Þ g conditions to be satisfied by the Kernel EgðXÞ:

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− −iLiEgðXÞ¼Xi ⋆ EgðXÞ; group G, one can define the coordinates kðgÞ¼ i lnðgÞ to lie in the principal branch, and amend the ⋆-product by a −i∂iE ðXÞ¼ziðgÞE ðXÞ; ð15Þ g g projection onto the principal branch of the logarithm, ⋆→⋆p. With this definition the set of plane waves con- which can be integrated (modulo the nontrivial global ⋆ properties of G)as stitutes a representation of G respect to the p-product, since5 L⃗ ikðhÞ·X Eðhg; XÞ¼ekðhÞ· Eðg; XÞ¼e⋆ ⋆ Eðg; XÞ; E ðXÞ ⋆ E ðXÞ¼E ðXÞ: ð21Þ ⃗ g p h gh Eðg; X þ YÞ¼eY·∂ Eðg; XÞ¼eiζðgÞ·YEðg; XÞ; ð16Þ We will omit in the following the suffix p and implicitly where kðhÞ¼−i lnðhÞ ∈ g are canonical coordinates, and assume that the projection on the principal branch is implemented. X∞ 1 fðXÞ k g e⋆ ¼ f ⋆ ⋆ fðXÞð17Þ Thus, in terms of canonical coordinates ð Þ on the n! |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n¼0 group, the noncommutative plane waves takes the form of a n times ⋆-exponential with respect to the ⋆-product corresponding Combining the two expressions one can show that the to the chosen quantization map. plane wave is given by the ⋆-exponential in terms of the However, from the second of Eqs. (16), it follows also i canonical coordinates on the group manifold that there exists a choice of coordinates z ðgÞ such that the same star-exponentials take the form of classical ikðgÞ·X exponentials EgðXÞ¼e⋆ X∞ n i A izðgÞ·X ¼ kj1 ðgÞkjn ðgÞX ⋆ ⋆ X : ð18Þ EgðXÞ¼ ðgÞe : ð22Þ n! j1 jn n¼0 with a multiplicative prefactor AðgÞ¼Egð0Þ. The choice The existence and explicit form of the plane wave (18) is of such coordinates ziðgÞ and the corresponding prefactor subject to the possible nontrivial global properties of G.For AðgÞ are also determined by the quantization map. For this the definition of the Fourier transform, since Eðg; XÞ has to 4 second form the multivaluedness issue affects the choice of be considered only under integration, weak exponentiality coordinates z∶G → g. of G is a sufficient condition. Moreover, in case G has Finally, it is possible to show (see [15]) that when the ⋆- compact , the logarithm map is multivalued, and product is verified to lead to a noncommutative plane wave the definition in terms of canonical coordinates has to refer ikðgÞ·X of the form (22), i.e., when E ðXÞ¼e⋆ ¼ AðgÞeizðgÞ:X, only to one of the charts covering the group manifold, and g then the z-coordinates satisfy the compatibility equa- extended appropriately beyond it (this can be done in ⋆ several ways, see [15] and references cited there). Let us tion (13), i.e., they are compatible with the -product in discuss this point a little further, briefly. the sense of the commutative diagram (7). This means From the relation (12) it follows that the product of plane that the algebra representation is guaranteed to exist, waves is given by and to be obtained by noncommutative Fourier transform from the group representation, only if such coordinates can −1 ikðgÞ·Xˆ ikðhÞ·Xˆ be found. EgðXÞ ⋆ EhðXÞ¼Q ðe e Þ iBðkðgÞ;kðhÞÞ·X e⋆ ; 19 ¼ ð Þ D. Summary where BðkðgÞ;kðhÞÞ is given by the application of the Baker- Let us summarize schematically the main elements Campbell-Hausdorff (BCH) formula to kðgÞ;kðhÞ ∈ g, and introduced so far. One starts from the phase space of a system whose configuration space is a Lie group G,given ik·X ik·Xˆ Qðe⋆ Þ¼e : ð20Þ by the cotangent bundle TG ≅ G × g, and its Poisson algebra PG. A quantization of (a maximal subalgebra A of) In order to have a one-to-one relation between elements of PG can be obtained through a quantization map Q, E ≔ ik:Xˆ ∶ ∈ ⊂ the set of plane waves fe k gg Ug and the leading to the quantum algebra U ≈ UG × Ug . The latter is endowed, for each of its subspaces UG and Ug , 4A group G is said to be exponential if its exponential map with a natural Hopf-algebra structure characterized by ∶ → ∀ ∈ expG ðgÞ G is onto (surjective), in the sense that g G, ðΔ ; ϵ ;S Þ and ðΔ ; ϵ ;S Þ. ∃ ∈ “ ” G G G g g g x g such that expGðxÞ¼g. A group G is said to be weakly exponential [25,26] if expG is dense in G, in the sense that the ∶ ∈ 5 closure expGg of expG g ¼fexpGðxÞ x gg is equal to G. See Without the projection on the principal branch, the coordinate also the discussion at the beginning of Sec. III A. BiðkðgÞ;kðhÞÞ in (19) may lie in any branch of the logarithm.

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Two different representation of U can be defined, the exponential. Exponentiability indeed ensures that the 2 ∶ → group representation πG on L ðGÞ, and the algebra repre- image of the exponential map exp g G is onto, i.e., 2 sentation πg on L⋆ðg Þ. The latter is characterized by the that all the points of G are obtained through exponentiation definition of a suitable ⋆-product compatible with the of its Lie algebra. Also, if a group is weakly exponential quantization map, and determined by it. Both representa- [25,26] the image of exp ∶g → G is dense in G. Since the tions satisfy the compatibility condition between their noncommutative Fourier transform is defined only under natives product and coproduct, as depicted in the diagram integration, weak exponentiability is a sufficient condition (7), which can then be understood as a condition for the for its uniqueness, as the canonical coordinates kðgÞ¼ existence of the representation. −i lnðgÞ defining the plane waves will cover the whole The noncommutative Fourier transform F∶L2ðGÞ → group except for a set of points of zero measure. So, in 2 L⋆ðg Þ is the intertwiner between the group and algebra order to define the noncommutative Fourier transform for representation, ensuring their unitary equivalence. Its kernel the Lorentz group we must first discuss the relation is the noncommutative plane wave EgðXÞ, whose existence between its parametrization and the exponential map (we and precise expression depend again on the chosen quan- follow mainly the discussion in [29]). tization map, and thus of ⋆-product. The explicit form of the The (homogeneous) Lorentz group is usually defined 6 plane wave can indeed be written in terms of ⋆-exponentials by two basic matrix representations: as the group SO(3,1) 4 4 EgðXÞ¼exp⋆ ðkðgÞ · XÞ, where kðgÞ¼−i lnðgÞ are of real orthogonal × matrices with unit determinant, canonical coordinates, or as ordinary exponentials (with a preserving the bilinear form of Lorentzian signature (3,1), or as the group SL 2; C of complex unimodular 2 × 2 prefactor) EgðXÞ¼AðzðgÞÞ exp ðzðgÞ · XÞ in terms of ð Þ ∂ j j 2; C coordinates z such that z 0 0 and z 0 δ matrices. As topological groups, SLð Þ is the double kð Þ¼ ∂ki kð Þ¼ i 2 C ik covering of SO(3,1). Moreover since SLð ; Þ is simply [zkðkÞ¼zðe Þ]. The existence of these kind of coordinates connected it is also the universal covering of SO(3,1), while ensures that the compatibility condition (7) is fulfilled, tying SO(3,1) is isomorphic to SLð2; CÞ=fI;−Ig, where I and −I together the existence of the noncommutative Fourier trans- are the central elements of SLð2; CÞ, and thus SO(3,1) is form with the one of the algebra representation. doublyconnected. While for a compact connected (matrix) group one can III. THE DUFLO MAP AND LIE ALGEBRA prove that the exponential map is surjective [30], when the REPRESENTATION FOR THE group is noncompact, as for SO(3,1) or SLð2; CÞ, surjec- LORENTZ GROUP tiveness is not guaranteed. In particular one can show that We will now proceed to constructing the structures SO(3,1) is exponential, while SLð2; CÞ is not [31], i.e., not introduced in the previous section for the case in which all the points of SLð2; CÞ are in the image of the the configuration space of our system is the Lorentz group. exponential map applied to its Lie algebra slð2; CÞ. We will first look for suitable coordinates in which to define However, a complex connected Lie group, as is the case the noncommutative plane waves. As explained, the def- of SLð2; CÞ, is always weakly exponential (see for instance inition of the plane wave depends on the quantization map, [32]), so that the image of exp ∶slð2; CÞ → SLð2; CÞ is which in turn specifies the ⋆-product structure. We will dense. As we stressed, this is all that is needed for a well- work the Duflo quantization map [16,27,28], which has defined noncommutative Fourier transform and algebra already found a number of applications in the quantum representation. gravity context [17–22]. After defining the properties of the An element of SLð2; CÞ can be parametrized as Duflo map, we will derive the explicit expression of the Duflo plane wave for the Lorentz group, as a star a ¼ a012 þ a⃗· σ⃗; ð23Þ exponential, and as the ordinary exponential with prefactor in terms of suitable coordinates on the group. We will then where 12 is the identity matrix in 2 dimensions, a0, aj, with discuss the properties of the algebra representation and j ¼ 1, 2, 3, are complex numbers, and σj are Pauli matrices show explicitly the structure of the star product for the (Hermitian traceless 2 × 2 matrices) lowest order polynomials.       These structures, in particular the plane waves, will 01 0 −i 10 σ1 ¼ ; σ2 ¼ ; σ3 ¼ ; ð24Þ finally allow us to define the noncommutative Fourier 10 0 0 −1 transform for the Lorentz group for the Duflo quantization i map, in the following Sec. IV. satisfying the Lie brackets A. Parametrization of the Lorentz group 6We restrict in this manuscript to the so-called proper orthoc- As discussed, the definition of the noncommutative ronous Lorentz group, the identity component of O(3,1), con- plane wave, and thus of the noncommutative Fourier sisting of the set of Lorentz transformations that preserve the transform, requires the group to be at least weakly orientation of “spatial” and “temporal” dimensions.

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½σi; σj¼2iϵijkσk: ð25Þ ½Mab;Mcd¼iðηadMbc þ ηbcMad − ηacMbd − ηbdMacÞ; ð34Þ The conjugacy classes of SLð2; CÞ are instead represented by the matrices [31,32] so that an element of SO(3,1) is (repeated indexes are       α 0 11 −11 summed, throughout the manuscript) ; ; : ð26Þ 0 α−1 01 0 −1   Λ α i α ð Þ¼exp 2 ijMij : ð35Þ Canonical coordinates (of the first kind) on SLð2; CÞ are defined [30,31] by applying the exponential map to its Lie algebra slð2;CÞ. One way of realizing this is as the Defining the generators complexification of SU(2) by means of a complex rotation 1 vector [29] ϵ 1 2 3 Ji ¼ 2 ijkMjk;Ni ¼ M0i;i;j¼ ; ; ; ð36Þ ζ ¼ ρ þ iβ ;i¼ 1; 2; 3 ð27Þ i i i corresponding to rotation and boost transformations, sat- isfying the commutation rules where ρi, βi are real parameters, so that, choosing the basis of sl 2;C represented by Pauli matrices, an element of ð Þ ϵ ϵ SLð2; CÞ is written as ½Ji;Jj¼i ijkJk; ½Ji;Nj¼i ijkNk;   ½Ni;Nj¼−iϵijkJk; ð37Þ i a ¼ exp ζ⃗· σ⃗ : ð28Þ 2 we can express an element of the group as

With this definition one gets the relations Λðρ⃗; β⃗Þ¼exp fiρ⃗· J⃗þ iβ⃗· N⃗g; ð38Þ   1 1 sin ð2 ðϕ þ iηÞÞ a0 ¼ cos ðϕ þ iηÞ a ¼ i ζ ; ð29Þ where 2 j ϕ þ iη j 1 ϕ η ρ ϵ α β α where we define the complex rotation angle þ i , with i ¼ 2 ijk jk; i ¼ 0i: ð39Þ η ≥ 0, related to ζ⃗by (denoting x2 ¼ x⃗· x⃗) The explicit expression of Λðρ⃗; β⃗Þ is given in the 2 2 2 2 2 2 ⃗ ðϕ þ iηÞ ¼ ζ ; ϕ − η ¼ ρ − β ; ϕη ¼ ρ⃗· β: ð30Þ Appendix A1, where the relation between SO(3,1) and SLð2; CÞ is also discussed. It is important to notice that The meaning of ϕ and η can be better understood the parameters domain is different for the two groups: considering the action of a SLð2; CÞ matrix (23) on a point the multivaluedness of the logarithmic map (the inverse of in Minkowski spacetime x ¼ðx0;x1;x2;x3Þ defined as the exponential map) results from the periodicity of the   compact of rotations. As shown in Appendix A1, x0 x3 x1 − ix2 x 01 ⃗ σ⃗ þ analogously to the relation between SO(3) and SU(2), ¼ x þ x · ¼ 1 2 0 3 ; ð31Þ 2 C “ ” x þ ix x − x SLð ; Þ covers twice SO(3,1), with the isomor- phism SOð3; 1Þ ≃ SLð2; CÞ=f1; −1g. so that In order to show the weak exponentiability of the Lorentz group, one can study the behavior of the logarithm map by x0 ¼ axa† → x0 ¼ ΛðaÞx: ð32Þ inverting relation (28). The study of the branch points for the logarithmic map for the Lorentz group is performed in The transformation matrices ΛðaÞ form the elements Appendix A2. Restricting it to its principal values, the of the group SO(3,1). This is identical to the group of complex rotation vector is holomorphic in a0 except for a linear transformations leaving invariant the metric branch cut extending on the real axis of a0 from −1 to −∞. η ¼ diagf1; −1; −1; −1g, described by 4 × 4 unimodular Thus, except for the branch cut, the canonical coordinates orthogonal matrices generated by the set of (purely imagi- provided by the exponential map, represented by the nary traceless) matrices defined by complex rotation vector ζ⃗¼ ρ⃗þ iβ⃗, are holomorphic functions parametrizing the whole SLð2; CÞ group. η δ − η δ 0 1 2 3 4 ϕ ðMabÞkl ¼ ið ak bl al bkÞ; a;b;k;l¼ ; ; ; ; ð33Þ Accordingly, the corresponding range of values of for which the complex rotation vector is single-valued is with commutation rules ϕ ∈ ð−2π;2π for SLð2; CÞ and ϕ ∈ ð−π; π for SOð3; 1Þ.

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This defines the principal branch respectively for the of the “symmetrization map” S, SLð2; CÞ and SOð3; 1Þ parametrizations. 1 X Thus, in the following, we consider a generic element S ˆ ˆ ðXi1 Xi Þ¼ Xiσ Xiσ ; ð42Þ n ! 1 n of the Lorentz group in its exponential representation (38) n σ∈S or (28) n with S is the symmetric group of order n, with the function ⃗ ⃗ ⃗ n g ¼ exp fiρ⃗· J þ iβ · Ng; ð40Þ on the Lie algebra g   where the generators J and N are the ones defined in this sinh 1 ad i i 2 x ∈ section, and, depending on the chosen group representa- jðxÞ¼det 1 ;xg; ð43Þ 2 adx tion, can be considered to be the generators (36) of SO(3,1), 1 1 J ¼ ϵ M and N ¼ M0 , or the generators J ≡ σ i i i 2 ijk jk i i i 2 i where in (41) one substitutes x with ∂ ¼ ∂=∂Xi where ≡ 1 σ 7 2 C and Ni 2 i i of the real Lie algebra slð ; ÞR, satisfying X ∈ g . Here adx is in the of g. in both cases the Lie brackets (37), the distinction between Notice also that with this definition the Duflo function (43) the two groups being given by the domain of the “rotation” coincides with the Jacobian of the exponential map [34,35], parameter ϕ as discussed above. An element of the Lie i.e., for g ¼ expðxÞ i algebra is then x ¼ x ei ∈ g, with ei the basis of generators i ei ≡ ðJi;NiÞ, and x the set of canonical coordinates dg ¼ jðxÞdx: ð44Þ ki ≡ ðρi; βiÞ. As we have explained in Sec. II, once we have chosen the Duflo quantization map, the compatibility between the B. The Duflo quantization map for the Lorentz associated ⋆-product, given by relation (12) as group and the noncommutative plane waves 0 −1 0 The Duflo map was introduced in [16] as a generaliza- f⋆ ⋆ f⋆ ¼ D ðDðf⋆ÞDðf⋆ÞÞ; ð45Þ tion to arbitrary finite-dimensional Lie algebras of the Harish-Chandra isomorphism between invariant polyno- and the coproduct on the algebra of functions on the group, mials on the dual of a Lie algebra and the center of the and thus the existence of the algebra representation, is corresponding universal enveloping algebra. The isomor- guaranteed by the construction of the noncommutative phism was proved in a different form within the formalism plane wave, the kernel of the transformation between the of deformation quantization of general Poisson manifolds group and algebra representations. Our next goal is then to [27] (see also [28]), and used then in different contexts in derive the explicit expression of this noncommutative plane the quantum gravity literature [17–22,33]. An important wave, and in particular its expression as a standard property of the Duflo map is that it realizes the isomor- exponential (with a certain prefactor) in terms of specific phism between the set SymðgÞg of invariant polynomials coordinates on the group. under the (adjoint) action of G, and the set UðgÞg of We need first to calculate the function jðxÞ for the G-invariant differential operators on the enveloping alge- Lorentz group. We consider a generic element of the bra, corresponding to the center of the enveloping algebra, Lorentz group written in its exponential representation UðgÞg ¼ ZðUðgÞÞ. Thus, invariant polynomials SymðgÞg (38) or (28), map to Casimirs of the quantum algebra, and the map ⃗ preserves the algebra structure of such polynomials, so the g ¼ exp fiρ⃗· J⃗þ iβ · N⃗g; ð46Þ Duflo quantization map D has the remarkable property that g for two elements α and β of UðgÞ , DðαÞDðβÞ¼DðαβÞ. where the generators Ji and Ni are the ones defined in the We point out that while the Duflo map is uniquely previous section, and can be considered to correspond to determined by the isomorphism property only on the the Ji,Ni matrices generating SO(3,1) or to the generators g ≡ 1 σ ≡ 1 σ 9 2 C invariant subspace SymðgÞ of SymðgÞ, we will extend it Ji 2 i, Ni 2 i i of the real Lie algebra slð ; ÞR,in to all of SymðgÞ in a natural way.8 both cases satisfying the Lie brackets (37), the distinction Concretely, the Duflo map is defined by the composition between the two groups being given by the domain of the rotation parameter ϕ as discussed in the previous section. 1 D ¼ S ∘ j2ð∂Þð41Þ The calculation of the Duflo function jðxÞ on the element (40) is performed in Appendix B1, and, for an element of i i the Lie algebra x ¼ xJJi þ xNNi,gives 7slð2; CÞ can be considered a real Lie algebra generated of dimension 6 with basis vectors fσig; fiσig, not to be confused with its real forms suð2Þ or slð2; RÞ. 9slð2; CÞ can be considered to be a real Lie algebra generated 8 We refer to [15] for the definition of the symmetric algebra of dimension 6 with basis vectors fσig; fiσig, not to be confused SymðgÞ and the universal enveloping algebra UðgÞ. with one of its real forms suð2Þ or slð2; RÞ.

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1 2 ˆ J ⃗ ˆ N ρ⃗ ⃗J β⃗ ⃗N 1 j sinh ð2 xζÞj D−1 iρ⃗ðgÞ·X þiβðgÞ·X i ðgÞ·X þi ðgÞ·X 2 4 EgðXÞ¼ ðe Þ¼e⋆ j ðxÞ¼ 2 ; ð47Þ jxζj ⃗J ⃗ ⃗N ¼ Aðρ⃗ðgÞ; β⃗ðgÞÞeiρ⃗ðgÞ·X þiβðgÞ·X ; ð52Þ pffiffiffiffiffiffiffiffiffiffiffiffi ⃗ ⃗ i i i where xζ ¼ xζ · xζ and xζ ¼ xJ þ ixN. We can now 1 with apply the Duflo function j2ð∂Þ on exponentials i ρi J βi N i ≡ ⃗ exp ðik XiÞ¼exp ði Xi þ i Xi Þ, with coordinates k Aðρ⃗ðgÞ;βðgÞÞ ðρi; βiÞ corresponding to canonical coordinates on the ϕ 2 η 2 group k g −i ln g , and coordinates on g X ≡ ðgÞ þ ðgÞ ð Þ¼ ð Þ i ¼ 2 1 2 1 2 1 2 1 ; J N 4ðcosh ð2 ηðgÞÞsin ð2 ϕðgÞÞ þ sinh ð2 ηðgÞÞcos ð2 ϕðgÞÞÞ ðXi ;Xi Þ associated respectively to the Ji and Ni gener- 10 i → ∂i i → ∂i ators. Thus, after the substitution xJ J;xN N (i.e., ð53Þ i → ∂i ∂i ∂i xζ ζ ¼ J þ i N), we obtain where ðϕ; ηÞ ≡ ðϕðρ⃗; β⃗Þ; ηðρ⃗; β⃗ÞÞ. Expression (52) shows 1 ⃗J ⃗ ⃗N ðj2ð∂Þ expÞðiρ⃗· X þ iβ · X Þ that within the Duflo quantization map the coordinates for 1 2 which the noncommutative plane wave takes the form of j sinð ζÞj ⃗J ⃗⃗N ¼ 4 2 eiρ⃗·X þiβ·X the standard exponential are still the canonical coordinates jζ2j zðgÞ¼kðgÞ¼−i lnðgÞ ≡ ðρ⃗; β⃗Þ, but with a prefactor 2 1 η 2 1 ϕ 2 1 η 2 1 ϕ cosh ð2 Þsin ð2 Þþsinh ð2 Þcos ð2 Þ ⃗J ⃗⃗N A ρ⃗β⃗ ¼ 4 eiρ⃗·X þiβ·X ; ð ; Þ, amounting, as expected from Eq. (44), to the ϕ2 þ η2 inverse of the square root of the Jacobian of the exponential ð48Þ map [see Eq. (A19)]. Notice that this is exactly what happens, for example, in the simpler SUð2Þ case, studied in [15]. We have thus obtained the explicit form of the plane where ζi ¼ ρi þ iβi, and we have considered also the parametrization (30), ðϕ þ iηÞ2 ¼ ζ2, ϕ2 − η2 ¼ ρ2 − β2, wave as a standard exponential, ensuring the existence of the algebra representation, which we are now going to ϕη ρ⃗· β⃗. ¼ discuss. Notice now that the exponential keeps the same form under the symmetrization map (42): C. The noncommutative algebra

i i ˆ representation from the Duflo map Sðeik Xi Þ¼eik Xi : ð49Þ We mentioned in Sec. II, and it was proven in [15], that the Using the above results, we finally obtain the action of the coordinates for which the noncommutative plane wave can Duflo map on the exponential function be written as a standard exponential (plus prefactor), satisfy the compatibility relation (13) (or (7) )withthe⋆-product, 2 1 2 1 2 1 2 1 which, in turn, ensures the existence of the algebra repre- ⃗J ⃗⃗N cosh ð ηÞsin ð ϕÞþsinh ð ηÞcos ð ϕÞ D eiρ⃗·X þiβ·X 4 2 2 2 2 ð Þ¼ ϕ2 η2 sentation for this choice of quantization map. In other words, þ having derived relation (52)–(53), we have ensured that, for ˆ J ⃗ˆ N × eiρ⃗·X þiβ·X : ð50Þ the ⋆-product associated to the Duflo map, the action (8) ϕ ∈ ∞ ≡ J N ≡ ρ β on functions ðXÞ Cc ðg⋆Þ (Xi ðXi ;Xi Þzi ð i; iÞ, i The last equation allows us to define the noncommuta- ∂ ¼ ∂=∂Xi) tive plane wave corresponding to the Duflo quantization ˆ map for the Lorentz group. This can be given in its ðπg ðXiÞϕÞðXÞ ≡ Xi ⋆ ϕðXÞ; ⋆ expression in terms of -exponential (18), which in turn i i ðπ ðzˆ ÞϕÞðXÞ ≡ −i∂ ϕðXÞ; ð54Þ encodes the property (45) [as in (20)], and thus defines the g ⋆-product for the Duflo map. We have indeed is a representation of the quantum algebra (2)

J ⃗ N iρ⃗ðgÞ·X⃗ þiβðgÞ·X⃗ ρ⃗ ˆ J β⃗ ˆ N DðE ðXÞÞ ¼ Dðe⋆ Þ¼ei ðgÞ·X þi ðgÞ·X : ð51Þ ˆ J ˆ J ϵk ˆ J ˆ J ˆ N ϵk ˆ N g ½Xi ; Xj ¼i ijXk; ½Xi ; Xj ¼i ijXk ; ˆ N ˆ N − ϵk ˆ N At the same time the last equation can be inverted, and ½Xi ; Xj ¼ i ijXk ; ð55Þ using Eq. (50) we get the relation between the plane wave ˆ ˆ ˆ as a ⋆-exponential and as a standard exponential with ½ρˆi; ρˆj¼½βi; βj¼½ρˆi; βj¼0; ð56Þ prefactor (22): ˆ ˆ Ld ½Xi; zj¼i Xi zj: ð57Þ 10 More precisely Xi are the coordinates on the Lie algebra g defined by the basis fe˜ig dual to feig ≡ fJi;Nig with the While the above statement can be proved in general (see [15]) ˜ ˜ canonical pairing hei;eji¼δij, X ¼ Xiei ∈ g . using the properties of the ⋆-product, we can still show more

106005-9 DANIELE ORITI and GIACOMO ROSATI PHYS. REV. D 99, 106005 (2019) explicitly that this is indeed the case for the structures derived d ite L fðgÞ¼ fðe i gÞj 0 ð63Þ in the previous section. Notice first that relations (56) are Xi dt t¼ trivially satisfied by the representation (54) due to the commutativity of ordinary derivatives. Let us then evaluate can be expressed in terms of coordinates zi for which ⋆ the -product between the lowest order monomials. EgðXÞ¼AðzðgÞÞ exp ðizðgÞ · XÞ as Considering expressions (52)11 the ⋆-product between n coordinates of g can be evaluated through the formula L j ∂ Xi EgðXÞ¼Li ðzðgÞÞ zj EgðXÞ

n j izðgÞ·X ∂ ¼ L ðzðgÞÞð∂z AðzðgÞÞ þ iXjAðzðgÞÞÞe X ⋆ X ⋆ X −i n i j i1 i2 in ¼ð Þ i1 j2 i ∂ ∂ ∂ n 0 k1 k2 k3 k1¼k2¼¼kn¼ ð64Þ −1 B⃗ … ˆ × D ðei ðk1;k2; ;knÞ·XÞ: ð58Þ so that From this formula one obtains (the explicit calculation is ∂j L j A izðgÞ·X reported in Appendix B2, where third order monomials are ð Xi EgðXÞÞ ¼ iLi ðzðgÞÞ ðzðgÞÞe also reported for completeness): j ∂ A A þ Li ðzðgÞÞð zj ðzðgÞÞ þ iXj ðzðgÞÞÞ i 1 × ∂jeizðgÞ·X: ð65Þ XJ ⋆ XJ XJXJ ϵk XJ − δ ; i j ¼ i j þ 2 ij k 6 ij i We now use the second of relations (15), and the relation XJ ⋆ XN XJXN ϵk XN; i j ¼ i j þ 2 ij k L j Xi zjðgÞ¼Li ðzðgÞÞ to rewrite the last expression as i 1 XN ⋆ XN XNXN − ϵk XJ δ : 59 i j ¼ i j 2 ij k þ 6 ij ð Þ L L ið Xi zjðgÞÞEgðXÞþizjðgÞ Xi EgðXÞ; ð66Þ

These relations show immediately that Eqs. (55) are fulfilled which, substituted in (62), and using again the second of by the representation (54), i.e., that (15), returns (61). Thus, we confirm the relations (54) define a representation of (57). Moreover, as mentioned ⋆ ⋆ ϕ − ⋆ ⋆ ϕ ϵk J ⋆ ϕ Xi Xj ðXÞ Xj Xi ðXÞ¼i ijXk ðXÞ: ð60Þ already, one can show in general (see [15]) that this amounts to the compatibility between (the coproduct for) ⋆ Finally we must check the commutators (57).Itis the coordinates ziðgÞ and the -product in the sense of the enough to consider the action (54) on plane waves (52), commutative diagram (7). since, if the noncommutative Fourier transform can be defined, as we will do in the next section, any function can IV. THE NONCOMMUTATIVE FOURIER be decomposed in terms of its plane-wave basis. We need to TRANSFORM FOR THE LORENTZ GROUP show then that The noncommutative plane wave (52) is the kernel of the noncommutative Fourier transform, the transform between − iX ⋆∂jE ðXÞþi∂jðX ⋆ E ðXÞÞ i g i g the group and algebra representation. We have now all the i L z g j E X : 61 ¼ ð Xi jð ÞÞjzjðgÞ¼−i∂ gð Þ ð Þ material to show the form of such noncommutative Fourier transform and to discuss some of its properties. We will From relations (15) defining the plane wave we have that discuss also its relation with other known constructions for the left-hand side (l.h.s.) can be rewritten as a Fourier transform on the group, as well as with the Plancherel decomposition into irreducible representations. − L ∂j L izjðgÞ Xi EgðXÞþ ð Xi EgðXÞÞ: ð62Þ A. The NC Fourier transform for SLð2;CÞ The Lie derivative In order to write the expression of the noncommutative Fourier transform we first need to evaluate the μ 11Obviously, knowing the definition of the Duflo map (41) and ðgÞ for the Lorentz group in canonical coordinates the general relation between the quantization map and the kiðgÞ ≡ ðρiðgÞ; βiðgÞÞ, such that dg¼μðρ⃗ðgÞ;β⃗ðgÞÞd3ρ⃗d3β⃗. associated ⋆-product (12), the action of any operator in the The calculation is performed in Appendix A3.Fromthe ⋆ algebra representation and the explicit expression of any - definition (14), considering the expression of the plane wave monomial can be computed directly. Having at hand the formula for the exponential elements, however, allows a more immediate (52) and the Haar measure (A19) the Fourier transform 2 2 −1 calculation. F∶L ðgÞ → L⋆ðg Þ and its inverse F are

106005-10 NONCOMMUTATIVE FOURIER TRANSFORM FOR THE LORENTZ … PHYS. REV. D 99, 106005 (2019) Z Z d 3 3 ⃗ −1 ⃗ ρ⃗⃗J β⃗⃗N d X ψ˜ ðXÞ¼FðψÞðXÞ¼ d ρ⃗d βA ðρ⃗; βÞei ·X þi ·X ψðgÞ; ψ˜ ψ˜ 0 ≔ ψ˜ ⋆ ψ˜ 0 h ; ig d ð ðXÞ ðXÞÞ G ð2πÞ Zg ψðgÞ¼F −1ðψ˜ ÞðgÞ Z dgψ g ψ 0 g ψ; ψ 0 ; 71 1 ¼ ð Þ ð Þ¼h iG ð Þ 3 J 3 NA ρ⃗β⃗ −iρ⃗·X⃗J−iβ⃗·X⃗N ⋆ ψ˜ G ¼ 6 d X d X ð ; Þe ðXÞ; ð2πÞ g 2 2 so that we may identify L⋆ðg Þ¼FðL ðGÞÞ. ð67Þ (iv) The ⋆-product is dual to the product on G under the noncommutative Fourier transform: ⃗ ⃗ where ðρ⃗; βÞ¼ðρ⃗ðgÞ; βðgÞÞ, and the integration measure g ψ˜ ⋆ ψ˜ 0 ¼ ψ ψ 0 d3ρ⃗d3β⃗is the standard Lebesgue measure, with the param- Z eters restricted as explained in Sec. III A depending if G is where ψ ψ 0ðgÞ¼ dhψðgh−1Þψ 0ðhÞ: ð72Þ SLð2; CÞ or SOð3; 1Þ: η ∈ ½0; ∞Þ and ϕ ∈ ð−2π; 2π for G SLð2; CÞ while ϕ ∈ ð−π; π for SOð3; 1Þ,withϕ2 − η2 ¼ ρ2 − β2 ϕη ρ⃗ β⃗ 3 J 3 N Moreover one can see that, in canonical coordinates , ¼ · . The integration measure d X d X is i i i 3 3 3 3 k ðgÞ ≡ ðρ ðgÞ; β ðgÞÞ, the Haar measure coincides with the Lebesgue measure d XJd XN with XJ ∈ R , XN ∈ R . the inverse square of the factor (53) coming from the Duflo As explained in Sec. II, the Fourier transform so defined acts map: as intertwiner between the group and algebra representation, where the intertwining property can be expressed as ˆ ˆ ˆ μðρ⃗ðgÞ; β⃗ðgÞÞ ¼ A−2ðρ⃗ðgÞ; β⃗ðgÞÞ: ð73Þ F ∘ πGðTÞ¼πg ðTÞ ∘ F, where T ∈ U. From the definition of the Fourier transform (67), we get the general following properties (we refer to [15] for the This remarkable feature reflects the property (44) of the general proofs): Duflo map, and from it some interesting consequences arise (i) The group multiplication from the left is dually in the characterization of the Fourier transform: represented on FðψÞðXÞ as ⋆-multiplication by (i) notice first that the property (73) implies that the g E −1 ðXÞ: inner product of two plane waves respect to , g which is the group delta function, becomes the Z pointwise product between plane waves. Indeed from (69), for g; h ∈ G, FðLgψÞðXÞ¼ dhEhðXÞψðghÞ G Z Z

−1 ⋆ F ψ d ⋆ d ¼ Eg ðXÞ ð ÞðXÞ: ð68Þ d XEgðXÞ EhðXÞ¼ d XEghðXÞ g g ¼ δðghÞ: ð74Þ −1 − (ii) Noticing that EgðXÞ¼Eg ðXÞ¼Egð XÞ we get 2 the delta function on L ðgÞ Using the transformation law (70) for canonical coordinates zðgÞ¼ðρ⃗ðgÞ; β⃗ðgÞÞ, Z ddX δ δd −1 d ðgÞ¼ ðzðgÞÞ¼ d EgðXÞ δðghÞ¼μ ðzðgÞÞδ ðzðgÞþzðhÞÞ ð2πÞ Z g Z −1 1 ⃗J ⃗⃗N μ d iðzðgÞþzðhÞÞ·X d3XJd3XNA ρ⃗;β⃗e−iρ⃗·X −iβ·X ¼ ðzðgÞÞ d Xe ; ð75Þ ¼ 6 ð Þ ð69Þ g ð2πÞ g where we used the ordinary representation of the 0 L j δj delta function δdðzðgÞþzðhÞÞ with respect to the for coordinates zðgÞ such that zðeÞ¼ , iz ðeÞ¼ i , which has the property (see A5) Lebesgue measure ddX. Noticing that the last ex- pression is nonzero only for zðhÞ¼−zðgÞ, and that Að−zðgÞÞ ¼ AðzðgÞÞ, we can rewrite it as δðghÞ¼μðzðhÞÞ−1δdðzðgÞþzðhÞÞ; ð70Þ μ−1ðzðgÞÞjA−2ðzðgÞÞj Z μ ðzðgÞÞ being the Haar measure factor dg ¼ d A izðgÞ·XA izðhÞ·X d × d X ðzðgÞÞe ðzðhÞÞe : ð76Þ μðzðgÞÞd zðgÞ. g 2 2 (iii) F is an isometry from L ðGÞ to L⋆ðg Þ in that it preserves the L2 norms (6) and (10): But from (73), for the Duflo map, this implies

106005-11 DANIELE ORITI and GIACOMO ROSATI PHYS. REV. D 99, 106005 (2019) Z 6 product, δ⋆ðX; YÞ¼δðX; YÞ, with the only impor- δðghÞ¼ d XEgðXÞ ⋆ EhðXÞ g tant caveat that the parameters relative to the Z compact subgroup have compact range. Specifically, 6 ϕ ϕ ∈ −2π 2π ¼ d XEgðXÞEhðXÞ: ð77Þ the restriction on the domain of , ð ; g (or ϕ ∈ ð−π; π for SO(3,1)), implies the domain of ⃗ 2 ðρ⃗; βÞ to be restricted by the condition This implies that the Duflo L⋆ inner product coincides with the usual L2 inner product, and 2 ⊆ 2 2 therefore L⋆ðg Þ L ðg Þ as an L norm-complete rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1   vector space for the Duflo map. Indeed, considering ρ2 − β2 ρ2 − β2 2 4 ρ⃗β⃗2 the representation property (54) it follows by lin- 2 ð Þþ ð Þ þ ð : Þ earity from (76) that for a generic quantization map ∈ ð−2π; 2πðorð−π; π for SOð3; 1ÞÞ: ð83Þ Z Z ⃗ ddXϕ˜ ðXÞ ⋆ ψ˜ ðXÞ¼ ddXðσði∂Þϕ˜ ðXÞÞψ˜ ðXÞ; g⋆ g⋆ Notice that for the special case in which the boost ð78Þ and rotation parameters are collinear, i.e., ρ⃗:β⃗¼ ρβ, −1 2 this expression reduces to a condition on the where σðzÞ ¼ μðzÞjAðzÞj . Using again (73), for modulus of the (canonical) rotation parameter the Duflo map last relation reduces to Z Z d ϕ˜ ⋆ ψ˜ d ϕ˜ ψ˜ d X ðXÞ ðXÞ¼ d X ðXÞ ðXÞ; ð79Þ ρ ∈ 0 2π 0 π 3 1 g⋆ g⋆ ½ ; Þ; ðor ½ ; Þ for SOð ; ÞÞ: ð84Þ

showing that under integration, the Duflo ⋆-product coincides with the pointwise product. Thus, (82) tells us that the Duflo noncommutative (ii) A similar feature characterizes the inner product of delta behaves as a standard delta distribution when two plane waves respect to G. It is easy to see from considering the commutative space of variables X, the definition of the Fourier transform that it where however due to the restriction on the param- corresponds to the noncommutative Dirac delta in eter range associated to the compact subgroup, some 2 L⋆ðg Þ of the values of the X spaces are restricted to take Z 1 discrete values. This is the expected result, due to the dgE X E Y δ⋆ X; Y ; compactness of the corresponding sections of the 2π 6 gð Þ gð Þ¼ ð Þ ð80Þ ð Þ SLð2;CÞ conjugate space. In fact, the same feature was pointed out in previous works concerning noncom- acting as a standard delta distribution with respect to mutative harmonic analysis on compact groups [18]. the ⋆-product: Z Z B. Relation between the noncommutative Fourier d d d Xδ⋆ðXÞ ⋆ ψ˜ ðXÞ¼ d Xψ˜ ðXÞ ⋆ δ⋆ðXÞ transform and the Fourier expansion in g g group unitary irreducible representations ψ˜ 0 ; 81 ¼ ð Þ ð Þ The noncommutative Fourier transform allows to switch between the group representation and the algebra repre- where δ⋆ðXÞ¼δ⋆ðX; 0Þ. But in particular for the sentation of the quantum algebra U. In standard harmonic Duflo map it follows from (73) that analysis, however, functions on the group are expanded in Z 1 terms of unitary irreducible representations. In that case a 6 dgEgðXÞEgðYÞ different generalization of the Fourier transform is consid- ð2πÞ 2 C SLZð ; Þ ered, consisting in a unitary map from square-integrable 1 2 2 ˆ 3ρ⃗3β⃗iρ⃗·ðX⃗J−Y⃗JÞþiβ⃗·ðX⃗N −Y⃗N Þ functions L ðGÞ to square-integrable functions L ðGÞ on ¼ 6 d d e ; ð82Þ ð2πÞ SLð2;CÞ the Pontryagin dual. The harmonic analysis on the Lorentz group is devel- i.e., the inner product of two plane waves reduces to oped for instance in [29]. It is shown that a function ψðgÞ ∈ the standard orthogonality expression in canonical L2ðSLð2; CÞÞ can be expanded in terms of irreducible coordinates with respect to the Lebesgue measure unitary (infinite dimensional) representations of the prin- d3ρ⃗d3β⃗, and the noncommutative delta function cipal series, by means of the Plancherel decomposition (and reduces to the standard delta respect to the pointwise its inverse)

106005-12 NONCOMMUTATIVE FOURIER TRANSFORM FOR THE LORENTZ … PHYS. REV. D 99, 106005 (2019) Z χ −χ ψˆ χ χ ψ and denote, for any representation and of the principal j1j2q1q2 ¼ dgDj1j2q1q2 ðgÞ ðgÞ; 2 j 2 j SLð2;CÞ series, LmðU1Þq and LmðU2Þq the Hilbert spaces of, Z 12 1 Xþ∞ respectively, the measurable functions ψðgÞ¼ dr ðm2 þ r2Þ Z 2 R m¼−∞ j j ϕψ ðu1jχÞq ¼ dμðu2ÞKψ ðu1;u2jχÞϕqðu2Þ; X∞ Xj1 Xj2 Z χ ψˆ χ × Dj1j2q1q2 ðgÞ j1j2q1q2 ; j j ϕψ u2 χ dμ u1 ϕ u2 Kψ u1;u2 − χ : j1;j2¼jð1=2Þmj q1¼−j1 q1¼−j2 ð j Þq ¼ ð Þ qð Þ ð j Þ ð89Þ ð85Þ One can show that the image of L2ðSLð2; CÞÞ by the where χ ¼ðm; rÞ labels the representation and, in the Fourier transform (85)Rcan be mapped isometrically into a ⊕ ∞ j 2 j Hilbert space H dχ⨁ 1 ⨁ L U1 (or principal series, r is a real continuous parameter, while m ¼ r≥0 j¼j mj q¼−j mð Þq 2 2 R 2 takes discrete values. dχ ¼ drðm þ r Þ is the Plancherel H ⊕ χ⨁∞ ⨁j 2 j χ ¼ r≥0 d 1 q¼−jLmðU2Þq) such that a left (or ψˆ j¼j2mj measure, and the Fourier coefficients j1j2q1q2 are matrix 2 χ right) translation in L ðSLð2; CÞÞ generates irreducible elements of functions L2ðGˆ Þ. D ðgÞ are matrix j1j2q1q2 unitary representations χ of the principal series in elements of irreducible unitary representations in the so- 2 j 2 j L ðU1Þ (L ðU2Þ respectively). “ ” ϕj m q m q called canonical basis, spanned by the set qðuÞ¼ By means of Eqs. (67) and (85) one can obtain the 2 1 1=2 j 1 0 1 2 … 2 2 C ð j þ Þ Dð1=2Þm;qðuÞ, j ¼j2 mjþn, n ¼ ; ; ; , relation between the expansion of a function L ðSLð ; ÞÞ 2 in terms of noncommutative plane waves and the one in −j ≤ q ≤ j, complete in LmðuÞ, the space of measurable functions ϕðuÞ on SU(2) covariant on the right cosets terms of irreducible group representations (Plancherel of U(1) as ϕðγuÞ¼eimωϕðuÞ,withu ∈ SUð2Þ and γ ¼ modes): exp ðiωÞ 0 j1 Z ð Þ.HereDq qðu1Þ define the SU(2) 1 0 exp ð−iωÞ 1 ψˆ χ 3 J 3 N χ ⋆ ψ˜ j1j2q1q2 ¼ 6 d X d X Dj1j2q1q2 ðXÞ ðXÞ; unitary irreducible representations (Wigner matrices), and ð2πÞ g χ Z Xþ∞ the matrices Dj1j2q1q2 ðgÞ can be decomposed as 1 ψ˜ ðXÞ¼ dr ðm2 þ r2Þ X 2 R χ j1 j2 χ η m¼−∞ Dj1j2q q ðgÞ¼ Dq1qðu1ÞDqq2 ðu2Þdj1j2qð Þ; ð86Þ 1 2 ∞ q X Xj1 Xj2 χ ψˆ χ × Dj1j2q1q2 ðXÞ j1j2q1q2 1 2 − − where u1, u2 ∈ SUð2Þ and η is the “boost” parameter, and j1;j2¼jð = Þmj q1¼ j1 q1¼ j2 χ η dj1j2qð Þ can be written in the integral representation ð90Þ Z Z 1 χ 1=2 1=2 j1 χ χ d ðηÞ¼ð2j1 þ 1Þ ð2j2 þ 1Þ dtd ð2t − 1Þ j1j2q ð1=2Þm;q Dj1j2q1q2 ðXÞ¼ dgDj1j2q1q2 ðgÞEgðXÞ: ð91Þ 0 SLð2;CÞ j2 2 − 1 −η 1 − η ði=2Þr−1 × dð1=2Þm;qð td Þ½te þð tÞe : ð87Þ C. Comparison with existing results on noncommutative Fourier transform Thus, the above makes use of the fact that the space for noncompact groups 2 LmðuÞ decomposes into a direct orthogonal sum of As we discussed in the Introduction, the noncommutative 2 1 H ( j þ )-dimensional Hilbert spaces j carrying each an Fourier transform has played an important role in the context 2 ∞ irreducible representation of SU(2): LmðuÞ¼⨁ 1 Hj. j¼j2mj of quantum gravity, and more specifically in effective The Fourier transform (85) realizes the decomposition of models based on the idea of spacetime noncommutativity. the regular (right and left) representation, carried by the In particular, when the noncommutativity of spacetime Hilbert space L2ðSLð2; CÞÞ, into irreducible unitary rep- coordinate operators is of Lie algebra type, the associated resentations of the principal series. More precisely, con- momentum space can be described as a curved manifold sider the kernel Kψ ðu1;u2jχÞ of the Fourier transform (85) with a Lie group structure. The idea of a curved momentum defined by [here dμðuÞ is the Haar measure for SU(2)] space dates back to M. Born [36], as a way to make more Z symmetric the role of configuration space (generically ψˆ χ μ μ ϕj1 χ ϕj2 j1j2q1q2 ¼ d ðu1Þd ðu2Þ q1 ðu1ÞKψ ðu1;u2j Þ q2 ðu2Þ; 12 2 j 2 j The Hilbert spaces LmðU1Þq and LmðU2Þq are equivalent, as it exists an intertwining operator between the χ and −χ ð88Þ representations.

106005-13 DANIELE ORITI and GIACOMO ROSATI PHYS. REV. D 99, 106005 (2019) curved, in a GR setting) and momentum space, as a possible algebra type, and the associated momentum space is the key to quantum gravity. The same idea was later formalized group manifold AN3, corresponding to half of de Sitter more rigorously in the language of Hopf-algebras (quantum space [43]. The properties of the noncommutative Fourier groups) [37], through the introduction of a suitable notion of transform for κ-Minkowski have been considered in several integration. It has been then further investigated and devel- studies. The Hopf-algebra point of view, where a “time oped for several examples of noncommutative spaces (and ordering” prescription is used to define the noncommuta- spacetimes). tive plane wave is taken in [7–9], and the corresponding Examples of noncommutativity with a noncompact quantization map discussed in [44]. The point of view of group manifold for the momenta have been also consid- the group structure of the momentum manifold is instead ered13. Particularly relevant for our analysis is a work central in [6], and in [45] its formulation in terms of group investigating a Snyder [38] kind of noncommutativity in field theory has been analyzed. In our language we can three dimensions [39]. From a group theoretical point of understand the noncommutative Fourier transform dis- view, it is possible indeed to consider Snyder noncommu- cussed in these works, which is the one associated to the tative spacetime to be described by the quotient so-called “bicrossproduct” basis of κ-Poincar´e [42],to soð4; 1Þ=soð3; 1Þ, so that the associated momentum space correspond to a quantization map consisting in a specific consists in the homogeneous space SOð4; 1Þ=SOð3; 1Þ, i.e., time-ordering prescription for the noncommutative oper- de Sitter space. In [39] a “Euclidean” version of three- ators (see again [44], where alternative choices of orderings dimensional Snyder spacetime is considered, where are also considered). Once more, while the physical idea momentum space consists in the homogeneous space given behind the time-ordering is transparent, it is not clear if this 3 1 3 by the coset SOð ; Þ=SOð Þ, the three-dimensional hyper- leads to any advantage from the mathematical perspective H boloid 3. The case treated fits well with the context of the of the algebra of quantum observables, compared to the ⋆ work we have presented here, since the -product intro- Duflo map. duced in Snyder configuration space and the noncommu- tative Fourier transform relating it to the curved momentum space are induced by the group and algebra structure of the D. Aside the noncommutative Fourier transform Lorentz group. They have not been directly derived from a for the homogeneous space SLð2;CÞ=SUð2Þ choice of quantization map, but rather postulated at the Using the results obtained for the Lorentz group, we can onset of the analysis. Using our framework, however, this describe as well the properties of the noncommutative quantization map can be in principle read out of the Fourier transform for its homogeneous spaces. We discuss ⋆ postulated -product. In particular, the authors of [39] here only the homogeneous space H3 ≅ SLð2; CÞ=SUð2Þ consider two different ⋆-products (a nonassociative and an (or SOð3; 1Þ=SOð3Þ). This is relevant for the physical associative one), corresponding to two different paramet- applications mentioned in the previous subsections, and 3 1 3 rizations of the SO(3) and SOð ; Þ=SOð Þ sectors, fol- one more physical application of our formalism using the lowed by the application of the symmetric map (42) to the same homogeneous space will be given in the next section. two sectors separately, i.e., to a ordering prescription for the In fact, it is also an interesting domain for spin foam models noncommutative operators corresponding to the Cartan and group field theory constructions for Lorentzian quan- decomposition. It is not clear what are the properties of tum gravity in 4d [11,46]. the chosen quantization maps nor why they would be Consider the decomposition of an element of preferable to the Duflo map, from a mathematical per- SLð2; CÞ spective. The more detailed relation with our construction will be studied in future work. We will discuss in the next ⃗ ⃗ ⃗ section the implementation of our noncommutative Fourier g ¼ kh ¼ exp ðib · NÞ exp ðir⃗· JÞð92Þ transform, based on the Duflo quantization map, for a 3 1 of SOð ; Þ, and for the homo- where h¼expðir⃗·J⃗Þ∈SUð2Þ, corresponding to the H ∼ 3 1 3 geneous space 3 SOð ; Þ=SOð Þ. This may facilitate Cartan decomposition. Parametrizing a generic point on a future detailed comparison of the construction presented H3 as in this paper and the one of [39]. Another much studied example of spacetime noncom- q q 1 − q⃗ σ⃗≡ q ; q⃗; mutativity associated to a curved noncompact momentum ¼ 0 · ð 0 Þ ð93Þ space related to the Lorentz group, is that of κ-Minkowski [40–42] (with its associated Hopf-algebra of symmetries, q is defined by the SLð2; CÞ action on the origin ⃗ κ-Poincar´e). In this case, the noncommutativity is of Lie qa ¼ð1; 0Þ ≡ 1:

† † 13 1 − ˆ σ⃗ We mention also the work [48], where the non-commutative q ¼ gqag ¼ kqak ¼ coshðbÞ sinhðbÞb · 2 Fourier transform for the SLð ; RÞ group is discussed, in ≡ ˆ connection with some known results for ð2 þ 1ÞD gravity. ðcoshðbÞ; sinhðbÞbÞ: ð94Þ

106005-14 NONCOMMUTATIVE FOURIER TRANSFORM FOR THE LORENTZ … PHYS. REV. D 99, 106005 (2019)

2 C 2 2 Thus, the quotient space SLð ; Þ=SUð Þ can be identified b ib⃗·X⃗N with the coset14 gH, H ≡ SUð2Þ. Accordingly, in the EkðXÞ¼ 2 1 e ; 4sinh ð2 bÞ splitting (92) the boost element k is a representative 2 of the set of equivalence classes ½k≔fkh; ∀ h∈SUð2Þg r ir⃗·X⃗J EhðXÞ¼ 2 1 e ; ð101Þ defining gH. 4sin ð2 rÞ H So, we can identify functions fH3 ðqÞ on 3 with functions fðgHÞ on the coset SLð2; CÞ=SUð2Þ, constant and, from the property (21) of ⋆-product, on the orbits of SU(2), through the projection Z EkhðXÞ¼EkðXÞ ⋆ EhðXÞ: ð102Þ ≡ ∈ H fH3 ðqÞ fðgHÞ¼ dhfðghÞ;q3; SUð2Þ We can now use Eq. (95) and the expression for the g ∈ SLð2; CÞ;h∈ H ≡ SUð2Þð95Þ (inverse) Fourier transform (67) to write Z where, for integrable functions on H3, the Haar measure dg ≡ induces a SL 2; C -invariant Haar measure dq on H3 such fH3 ðqÞ dhfðghÞ ð Þ 2 that [47] SUð Þ Z Z 1 Z Z Z dh ddXE X ⋆ f˜ X ; ¼ 2π 6 ghð Þ ð Þ ð103Þ dgfðgÞ¼ dq dhfðghÞ: ð96Þ ð Þ SUð2Þ g SLð2;CÞ H3 SUð2Þ which can be rewritten, using (notice that E ðXÞ¼ The Haar measure for the decomposition (92) factorizes gh E −1 −1 ðXÞ¼E ðXÞ ⋆ E ðXÞÞ,as (see A4) into the product of the measure on H3 and the h g h g measure on the subgroup SU(2) Z Z 1 d ˜ fH ðqÞ ≡ d X dhE ðXÞ ⋆ E ðXÞ ⋆ fðXÞ: ∈ H3 ∈ 2 3 6 h g dg ¼ dqdh; q ;hSUð Þ; with ð2πÞ g SUð2Þ 2 2 1 sinh ðbÞ 3 ⃗ sin ð2 rÞ 3 ð104Þ dq ¼ d b;dh¼ 4 d r⃗: ð97Þ b2 r2 The plane wave can be written in Cartan coordinates as Using (101) and (97) we notice that the first term reduces to the ordinary delta function (with compact range) on the ⃗ J ⃗ N “ ” ⃗⃗ ⃗⃗⃗ iρ⃗ðb;r⃗Þ·X⃗ þiβ⃗ðb;r⃗Þ·X⃗ rotation sector of the algebra EkhðXÞ¼Aðρ⃗ðb; rÞ; βðb; rÞÞe ; ð98Þ Z Z ⃗ 1 1 ⃗⃗J where ðρ⃗ðb; rÞ; βðb; rÞÞ are given by the BCH dhE X d3⃗e−ir·X δ3 X⃗J 2π 3 hð Þ¼ 2π 3 r ¼ ð Þ formula (A20). Notice now that from (A20) it follows that ð Þ SUð2Þ ð Þ SUð2Þ ð105Þ for b ¼ 0; ρ⃗¼ r⃗; ϕ ¼ r; β ¼ η ¼ 0; ⃗ for r ¼ 0; β⃗¼ b; η ¼ b; ϕ ¼ ρ ¼ 0; ð99Þ with jr⃗j ≤ 2π [or jr⃗j ≤ π for SO(3,1)]. We can use now the property of the Duflo ⋆-product under integral (79) to so that, from the expression of the Duflo factor (53), eliminate the first ⋆-product in (104) to obtain r2 Z A ρ⃗b 0;r⃗; β⃗b 0;r⃗ A r⃗;0 ; 1 ð ð ¼ Þ ð ¼ ÞÞ¼ ð Þ¼4 2 1 ≡ d δ3 ⃗J ⋆ ˜ sin ð2rÞ fH3 ðqÞ 3 d X ðX ÞEgðXÞ fðXÞ: ð106Þ ð2πÞ g 2 A ρ⃗⃗ 0 β⃗⃗ 0 A 0 ⃗ b ð ðb;r¼ Þ; ðb;r¼ ÞÞ¼ ð ;bÞ¼ 2 1 : ð100Þ 4sinh ð2bÞ Notice that in this formula, as in the previous ones leading to it, the ⋆-product used remains the one of SLð2; CÞ, and ⃗ Considering that if h ¼ 1 r⃗¼ 0 and if k ¼ 1 b ¼ 0, we can all functions are treated as functions on the TðSLð2; CÞÞ write the plane wave for the particular group elements phase space, only appropriately restricted in their depend- 15 khjh¼1 and khjk¼1 as ence on the domain.

14 15 The decomposition works similarly for H3 ≅SOð3;1Þ=SOð3Þ, Notice also the slight abuse of notation: the pointwise the difference being again in the range of the rotation parameter. In product is between the ⋆-function defined by (105) (correspond- 3 this case (see also Sec. III A) a point in the space H3 is defined by ing, as a function on R , to the usual delta function), and the ⃗ ⋆ the 4-vector ðq0; q⃗Þ, while the action on the origin q0 ¼ð1; 0Þ is function obtained by -multiplying the noncommutative plane ∈ 3 1 ˜ given by q ¼ gqa for g SOð ; Þ. We omit in the following the wave with the function f; this pointwise product is then evaluated distinction between the two cases unless needed. on the point X.

106005-15 DANIELE ORITI and GIACOMO ROSATI PHYS. REV. D 99, 106005 (2019) Z Z V. AN EXAMPLE: PARTICLE 00 0 −1 Kðq ;q; tÞ¼ lim dg1 dgN−1Kϵðg0 g1Þ ON THE HYPERBOLOID N→∞ −1 We will consider now the motion of a free particle on a × KϵðgN−1gNÞ; three-dimensional (spatial) hyperboloid, as a very simple † 00 0 ðq ¼ g q g ;q ¼ q ;q0 ¼ q Þð109Þ example of application of our formalism in a well-under- j j a j N stood context. The quantization of the system can be described in terms of a path-integral formulation. We will of a spherical function KϵðgÞ (the short time propagator), show that, by means of the noncommutative Fourier satisfying transform, the quantum propagator can be easily formu- −1 lated in the algebra representation, as its expression KϵðgÞ¼Kϵðh1 gh2Þ; ∀ h1;h2 ∈ SUð2Þ: ð110Þ assumes, in this simple case, the form one would expect from the classical action by identifying the (noncommu- This is our object of interest. We will give its explicit tative) classical momenta with the algebra elements. In this expression in the following, as well as its decomposition in sense, the main virtue of the noncommutative Fourier irreducible representation of the group, the analogue of the transform and of the algebra representation is to allow Peter-Weyl decomposition for compact groups. This will for a description of the quantum system in which the have to be compared to the one obtained via noncommutative underlying classical theory is manifest. Fourier transform. In order to do so, we have to introduce The path-integral formulation of quantum mechanics in some more harmonic analysis on the hyperboloid. terms of noncommutative momenta was described in [49] Functions on the group, satisfying the relation (110), i.e., for the case of SO(3), and the corresponding propagator constant in the two-sided coset HgH [for H ¼ SUð2Þ], can for a free particle on a sphere was derived. In dealing be expanded in terms of zonal spherical functions Dl ðgÞ H ≅ 2 C 2 00 with the homogeneous space 3 SLð ; Þ=SUð Þ [or [50–52]. Spherical representations (or class 1 representa- 3 1 3 SOð ; Þ=SOð Þ], we will refer to [49] for the details on the tions) Dl of G are unitary irreducible representations which general characterization of quantum mechanics in non- have null vectors jψ αi invariant under transformations of a commutative momentum basis. subgroup of G. SU(2) is a massive subgroup of SLð2; CÞ, A remark on notation: we characterize the spatial hyper- l meaning that there is only one vector jψ 0i in any D boloid introducing a length scale l, corresponding to the invariant under SU(2). Then one can define spherical radius of curvature, as y lq lq0; lq⃗, so that the ¼ ¼ð Þ functions (unit) hyperboloid defined by (94) parametrizes the surface μ 2 2 2 yμy ¼ y − y⃗ ¼ l . Similarly we define the dimensional 0 ψ l ψ Dl ψ ⃗ ðgÞ¼h j ðgÞj 0ið111Þ coordinates on H3 x⃗¼ lb. such that ψ lðghÞ¼ψ lðgÞ, i.e., constant in the left coset gH, A. The propagator in the group representation so that ψ lðgÞ can be considered as functions on the Considering the characterization of the homogeneous homogeneous space H3. Choosing a basis fjuiig such l space H3 ≅ SLð2; CÞ=SUð2Þ of Sec. IV D, the finite time that ju0i¼jψ 0i, the matrix elements of D ðgÞ given by propagator is defined as the expectation value in the Hilbert 2 space L ðH3Þ Dl Dl m0ðgÞ¼humj ðgÞju0ið112Þ

K q00;q0; t00 − t0 q00 U t00 − t0 q0 : 107 are called associated spherical functions. A zonal spherical ð Þ¼h j ð Þj i ð Þ l function D00ðgÞ is thus an associated spherical function constant on the two-sided coset HgH, i.e., such that for any 2 0 l 0 l where jqi are vectors in L ðH3Þ, and the time evolution h; h ∈ SUð2Þ D00ðhgh Þ¼D00ðgÞ. operator is defined in terms of the particle momentum p as For the spherical principal series the parameter l labeling the representation is continuous and takes the value l ¼   −1 þ ir (r ≥ 0). In terms of zonal spherical functions the 2 p t Fourier expansion of a function fðgÞ invariant on the two- UðtÞ¼exp −i : ð108Þ 2mℏ sided coset HgH takes the simpler expression Z 1 ∞ ˆ lDl The propagator can be described in the group representa- fðgÞ¼ 2 drdlf 00ðgÞ; ð2πÞ 0 tion by noticing [50,51] that the finite time propagator Z (107) can be expressed as the convolution in the group ˆ l l −1 f ¼ dgfðgÞD00ðg Þ; ð113Þ manifold

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Dl Dl −1 “ ” ℏ where 00ðgÞ¼ 00ðg Þ, and dl is a dimension factor Pˆ ¼ Xˆ N; ð118Þ defined by the relation l Z 1 ˆ ℏ ˆ J “ ” l l0 −1 0 while we denote R ¼ l X the momenta relative to the D00ðgÞD00ðg Þ¼ δðl − l Þ: ð114Þ SLð2;CÞ dl rotation sector (which are trivial when we restrict to the hyperboloid). The spherical functions are eigenfunctions of the Following the construction of [49], we define a set of ⃗ ⃗ ⃗ ⃗ 6 Laplace-Beltrami operator on H3, that can be identified states fjP; RijP; R ∈ R⋆g in the noncommutative momen- with the operator p2 in the time evolution (108), and tum basis, by their inner product with the group basis corresponds to a Casimir of the Lorentz algebra, restricted   to its boost components. l ⃗ ⃗ ≡ ⃗ ⃗ β⃗ ⃗ ρ⃗ ⃗ The zonal spherical function coincides with the matrix hgjP; Ri EgðP; RÞ¼exp⋆ i ð · P þ · RÞ : ð119Þ ð0;2rÞ ℏ element d000 ðηÞ of the SLð2; CÞ representations defined in (87) and takes the explicit form By virtue of properties (69) and (80) they form a complete η and orthonormal basis respect to the ⋆-product: l ð0;2rÞ sin ðr Þ D00ðbðgÞÞ ¼ d000 ðηðgÞÞ ¼ : ð115Þ r sinhðηÞ   2πℏ 6 ⃗ ⃗⃗0 ⃗0 ⃗ ⃗0 ⃗ ⃗0 hP; RjP ; R i¼ δ⋆ðP − P ; R − R Þ; The dimension factor can be evaluated to d ¼ r2. l l Z In the group representation (i.e., configuration space), 3 ⃗ 3 ⃗ d Pd R ⃗ ⃗ ⋆ ⃗ ⃗ 1ˆ and using polar coordinates, the short time propagator takes 6 jP; Ri hP; Rj¼ : ð120Þ R6 ð2πℏ=lÞ the form [50,51] ⋆

Kðq;q0;ϵÞ¼hq00jUðϵÞjq0i On this Hilbert space, corresponding to the algebra repre-      3 2 2 sentation of the system, we can define the time evolution ml 2 i ml ℏ ϵ ¼ exp ðcoshΘ−1Þþ ; operator on the Lorentz group (108) to be 2πiℏϵ ℏ ϵ 8ml2 ð116Þ it 2 2 UðtÞ¼− ðPˆ − Rˆ Þ; ð121Þ 2mℏ where Θ is the angle of the hyperbolic rotation trans- 0 0 ˆ 2 2 forming q into q: q ¼ gq . As shown in Appendix C1the where C ¼ Pˆ − Rˆ is the quadratic Casimir of the quantum angle Θ coincides with the modulus of the hyperbolic Lorentz algebra. The full propagator is rotation b in (94) (and then with η, when restricting to the hyperboloid). 00 0 00 0 KGðg ;g ;tÞ¼hg jUðtÞjg i The expansion of the propagator in terms of zonal Z spherical representations (the analogue of the Peter-Weyl d3Pd⃗ 3R⃗ 00 ⋆ 0 ¼ 6 hg jP;Ri hP;RjUðtÞjg i decomposition for compact Lie groups) takes instead the 6 2πℏ l R⋆ ð = Þ expression [50,51] Z P 3 3 − i t ðP ⋆P −R ⋆R Þ d Pd⃗ R⃗ 2mℏ i i i i Z   ⃗ ⃗⋆ i ∞ 2 ¼ 6 Eg00 ðP;RÞ e⋆ 1 ℏr R6 2πℏ l 00 0; − ⋆ ð = Þ Kðq ;q tÞ¼ 2 dr exp i 2 t 2π 0 2ml ⃗ ⃗ ⋆Eg0 ðP;RÞ: ð122Þ sin ðxðg−1g Þr=lÞ r 0 N ; × −1 l ð117Þ P sinh ðxðg0 gNÞ= Þ ⋆ − ⋆ The expression iðPi Pi Ri RiÞ is the quadratic Casimir in the algebra representation C⋆, obtained with ⃗ with x ¼jxj. the inverse Duflo map, and reduces to [see (B31)] This is the expression we would have to compare with the one obtained via noncommutative Fourier transform. X 2 2 2 ℏ C D−1 Cˆ P ⋆ P − R ⋆ R P⃗ − R⃗ : ⋆ ¼ ð Þ¼ ð i i i iÞ¼ þ l2 B. The propagator in the noncommutative i momentum representation ð123Þ We identify the particle momenta in the algebra repre- ˆ N sentation with the operators X ∈ QðAg Þ relative to the Notice that C⋆ ⋆-commutes with functions of X and we can “boost” sector, through the rescaling rewrite the propagator as

106005-17 DANIELE ORITI and GIACOMO ROSATI PHYS. REV. D 99, 106005 (2019) Z Z 3 ⃗ 3 ⃗ 3 ⃗ 3 ⃗  2  − i C i ⃗2 ⃗2 ℏ d Pd R 2 ℏt ⋆ d Pd R 3 −2 ℏtðP −R þ 2Þ ⃗⃗ ℏ 00 0; m ⋆ −1 00 0; δ ⃗ m l ⋆ ix·P= KGðg ;g tÞ¼ 6 e⋆ Eg00g0 ðP; RÞ Kðq ;q tÞ¼ 6 ðRÞ e e 6 2πℏ l 6 2πℏ l R⋆ ð = Þ R⋆ ð = Þ ¼ K ðg ¼ g00g0−1; tÞð124Þ ðx=2lÞ2 G × : ð130Þ sinh2ðx=2lÞ where we used that EgðXÞ¼Eg−1 ðXÞ. Now we exploit relation (96) to project the propagator The expression for the propagator we have obtained, on the homogeneous space H3 (see also Appendix C2and in its form (129) or (130), is the direct analogue of the [53]): commutative one, and presents the noncommutative alge- Z bra variables in the precise role we expect for the classical 00 0 00 0−1 momenta in a path integral expression (here within a Kðq ;q; tÞ¼ dhKGðg ¼ g g ; tÞð125Þ SUð2Þ Fourier transform to configuration basis). This is the main result we wanted to show for this simple application of our 00 00 00† 0 0 0† where q ¼ g qag and q ¼ g qag . Following the dis- formalism. cussion of Sec. IV D, we rewrite the plane wave exploiting As a further remark, we point out that the propagator the splitting (92) and notice that the “boost” part of the could be expressed in terms of only commutative momenta plane wave now takes the form (100), in physical coor- by calculating explicitly the ⋆-product in (130). We leave ⃗ dinates x⃗ðgÞ¼lbðgÞ, the calculation to a future work. We notice however for the interested reader that the calculation will necessarily result 2l 2 in some quantum correction to the classical expression. ⃗ ⃗ ðx= Þ ⃗⃗ ℏ E ðP; RÞ¼ eix·P= : ð126Þ Considering that the action of the Casimir on a plane wave k sinh2ðx=2lÞ can be rephrased as the Laplacian in terms of Lie Plugging these relations into the propagator, we obtain derivatives [see relations (8)], we expect that the quantum corrections arising from the ⋆-product can be encoded in an P 2 2 2 Z − i C K ℏ l l 3 ⃗ 3 ⃗ 2 ℏt ⋆ extra multiplicative term ðt =ðm Þ;x = Þ consisting is d Pd R m 2 00 0; i ⋆ ⃗ ⃗ ℏ l Kðq ;q tÞ¼ 6 e⋆ EkðP; RÞ some combination of the dimensionless terms t =ðm Þ R6 ð2πℏ=lÞ 2 2 Z⋆ and x =l , reducing to identity in the classical limit (which ℏ → 0 l → ∞ ⋆ ⃗ ⃗ we take to be followed by ). In this case we dhEhðP; RÞ; ð127Þ δ3 SUð2Þ can apply the delta function ðRÞ, and we are left with

−1 −1 Z ⃗⃗ ⃗ 00 0 ⃗ 00 0 3 ⃗ 2 2 where ðx; rÞ¼ðxðg g Þ; rðg g ÞÞ. Again, the last term d P − i tðP⃗ þℏ Þ ⃗⃗ ℏ 00 0; ≃ 2mℏ l2 ix·P= 3 ⃗ Kðq ;q tÞ 3 e e is just the delta function δ R on the rotation part (105), 3 2πℏ l ð Þ R⋆ ð = Þ and we get, using the property (79) to eliminate the last   tℏ x2 ðx=2lÞ2 ⋆-product, × K ; : ð131Þ ml2 l2 sinh2ðx=2lÞ Z 3 3   ⃗ ⃗ i d Pd R − tC⋆ 00 0; δ3 ⃗ 2mℏ ⋆ ⃗ ⃗ Kðq ;q tÞ¼ 6 ðRÞ e⋆ EkðP; RÞ : We can separate the radial and angular part of momenta R6 ð2πℏ=lÞ ⋆ P⃗¼ Pnˆ (P ¼jP⃗j) and, considering that the Casimir does ð128Þ not depend on the direction of P⃗, integrate the plane wave on the unit sphere to get Notice further that due to the ⋆ commutativity of the Z   Casimir, the propagator can be rewritten in terms of a ∞ 2 2 1 l − i tðP2þℏ Þ tℏ x ⋆ 00 0; ≃ 2mℏ l2 K single -exponential as Kðq ;q tÞ 2 dPe 2 ; 2 2π 0 ℏ ml l Kðq00;q0; tÞ l xðg00g0−1Þ sin ðxðg00g0−1ÞP=ℏÞ Z  P  × P 2 −1 ; ð132Þ 3 ⃗ 3 ⃗ ℏ 2lsinh ðxðg00g0 Þ=lÞ d Pd R − i t ðP ⋆P −R ⋆R Þþ ix⃗·P⃗ δ3 ⃗ 2mℏ i i i i i ℏ ¼ 6 ðRÞ e⋆ : R6 2πℏ=l ⋆ ð Þ where we also wrote explicitly the dependence on the ð129Þ group element. Comparing this result with the one for the propagator in Expression (128) can be simplified noticing that from the the representation basis (117), we notice the similarity of l properties of the Duflo map to preserve the algebra of the results, provided one uses the identification r ¼ ℏ P. invariant polynomials on g, the star-exponential involving While the “classical” appearance of the propagator in the the Casimir reduces to the standard exponential [see (B29)], algebra representation would be maintained for a more and using also (126) we can rewrite the propagator as complicated quantum system, e.g., a particle in an external

106005-18 NONCOMMUTATIVE FOURIER TRANSFORM FOR THE LORENTZ … PHYS. REV. D 99, 106005 (2019) Z  Z  potential and, more generally, with a more complicated i t0 00 0; D D _a ⃗ action, in particular one depending on more than the Kðq ;q tÞ¼ q P exp ℏ ds x ðgðsÞÞPa  t  quadratic invariant of the momenta, the similarity of the 1 ℏ2 resulting expression with the expression in representation − 2 1 − 6K 2 P þ l2 ð 0Þ : ð137Þ basis would be lost. This is to be expected, since harmonic m functions are only eigenstates of the quadratic Casimir, while the states (120) are generalized eigenstates of all momentum operators. Finally, we can ascribe the correction The propagator takes the explicit form of a path-integral terms in the exponential to the choice of (Duflo) quantiza- whose action appearing at the exponential coincides with tion map (see the discussion of a similar term in the the classical (Hamiltonian) action, confirming what we had compact case in [49]). anticipated looking at the infinitesimal propagator in the To conclude our remark, we consider again expression algebra representation. One can notice the quantum cor- (131)and take the infinitesimal propagator, for t → ϵ, rection in the action [whose explicit form should be 00 0; ϵ K ϵℏ x2 ⋆ Kðq ;q Þ. Expanding the factor ðml2 ; l2Þ in powers calculated solving the -product in (130)], as a shift in of ϵ and x2=l2, we rewrite it as the energy, which characterizes the Duflo quantization map we have chosen. Similar corrections had been found     ϵℏ 2 2 4 in the Euclidean case, with a different quantization map, K x ≃ 1 K x ϵ x 2 ; 2 þ 0 2 þ Oð ÞþO 4 ; ð133Þ in [49]. ml l l l where K0 is some constant factor. Notice that, as expected VI. CONCLUSIONS in the algebra representation (54), we can rewrite functions We have defined a noncommutative algebra representa- of x=⃗ l as differential operators acting on the plane wave tion for quantum systems whose phase space is the through the substitution xi → −iℏ∂=∂P . We can thus i cotangent bundle of the Lorentz group T SLð2; CÞ, and integrate by part the functions involving those factors the noncommutative Fourier transform ensuring the unitary and, noticing also that the prefactor Að0; x=⃗ lÞ is at least equivalence with the standard group representation. Our fourth order in x=⃗ l, we obtain construction, following the general template presented in [15],isfrom first principles in the sense that all the Kðq00;q0; ϵÞ Z structures are derived from the single initial input of a 3 ⃗ 2 i 2 ℏ choice of quantization map for the classical system. Our d P ⃗⃗ ℏ − ϵðP þ ð1−6K0ÞÞ 2 ix·P= 2mℏ l2 ϵ ¼ 3 e e þ Oð Þ: specific construction corresponds to the choice of the Duflo R3 ð2πℏ=lÞ ⋆ quantization map, a choice motivated by the special ð134Þ mathematical properties of this map as well as by the interesting physical applications of the same, as we dis- Making explicit its dependence on the group elements we cussed in the text. can reexpress the infinitesimal propagator as We have left all possible physical applications (beside Z   the simple case of a point particle) of our construction d3P i x⃗g00g0−1 00 0; ϵ ≃ ϵ ð Þ ⃗ aside, in this paper. However, we believe that our results Kðq ;q Þ 3 exp · P R3 ð2πℏ=lÞ ℏ ϵ could be of considerable impact in this direction. We have   1 ℏ2 in mind in particular the application to quantum gravity, − 2 1 − 6K 2 P þ l2 ð 0Þ : ð135Þ which can take two parallel paths. The first is in the context m of model building for Lorentzian 4d quantum gravity, within the group field theory and spin foam formalisms, In the continuum limit, for the infinitesimal propagator we so far quite limited (model building based on the Duflo map 0 00 ϵ ⃗ −1 0 can rewrite g ¼ gðtÞ, g ¼ gðt þ Þ, so that (xðgg Þ¼ ) in the Riemannian context is discussed in [54]). In this   context, Lorentzian model building should proceed along- x⃗ðg0−1g00Þ dgðtÞ ¼ x⃗ g−1ðtÞ ; ð136Þ side a more careful investigation of causal (or, better, ϵ dt “precausal”) properties of the resulting models, possibly inspired by the early work [55]. The second is in the context which corresponds to the boost part of the canonical of noncommutative spacetime field theories, where on the coordinates associated to the time derivative of the one hand our construction can offer a more mathematically −1 Maurer-Cartan form λ P¼ g dg, inducing the curved metric solid ground for the construction of effective quantum λcλc in the manifold gab ¼ c a b. Denoting these coordinates gravity models, while on the other hand the possible a iλa as x ðgÞ¼x i ðgÞ we thus finally get the finite time phenomenological implications of the mathematical pecu- propagator liarities of the Duflo map could be identified.

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APPENDIX A: SOME PROPERTIES OF THE While for the SU(2) subgroup of SLð2; CÞ we can see by LORENTZ GROUP PARAMETRIZATION setting η ¼ β⃗¼ 0 in (29) that 1. Explicit expression for the SO(3,1) matrices and the ρ 0 β 0 ρ 4π β 0 1 relation between SO(3,1) and SL(2,C) representations að ¼ ; ¼ Þ¼að ¼ ; ¼ Þ¼ 2; ⃗ The explicit expression of Λðρ⃗; βÞ can be obtained aðρ ¼ 2π; β ¼ 0Þ¼−12: ðA5Þ directly from (32) or evaluating the matrix exponential (38), and it is given by This shows that, analogously to the relation between SO(3) 2; C “ ”  and SU(2), SLð Þ covers twice SO(3,1), manifesting 1 1 the isomorphism SOð3; 1Þ ≃ SLð2; CÞ=f1; −1g. Λ0 ϕ2 η2 ρ⃗2 β⃗2 η 0 ¼ 2 2 ð þ þ þ Þ cosh When ϕ ¼ 0 þ 2nπ and η ≠ 0 (ρ ¼ 0, β ¼ η), the matrix ϕ þ η 2  Λ represents a pure boost 1 2 þ ðϕ2 þ η2 − ρ⃗2 − β⃗Þ cos ϕ ;   2 B00 B0i Λ ρ⃗ 0; β⃗ A6 1 ð ¼ Þ¼ B B ð Þ Λ0 ¼ ð−ρ ðϕ sinh η − η sin ϕÞ 0i ij i ϕ2 þ η2 i where − βiðη sinh η þ ϕ sin ϕÞþϵijkρjβkðcosh η − cos ϕÞÞ; 1 β i i Λ0 ¼ ð−ρ ðϕ sinh η − η sin ϕÞ B00 ¼ cosh β; B0 ¼ − sinh β; ϕ2 þ η2 i i β − β η η ϕ ϕ − ϵ ρ β η − ϕ β β ið sinh þ sin Þ ijk j kðcosh cos ÞÞ B δ i j β − 1   ij ¼ ij þ 2 ðcosh Þ: ðA7Þ 1 1 β⃗ Λi ¼ δ ðϕ2 þ η2 − ρ2 − β2Þ cosh η j ϕ2 þ η2 ij 2  1 ϕ2 η2 ρ2 β2 ϕ 2. Branch cuts for the canonical coordinates þ 2 ð þ þ þ Þ cos When ϕ ≠ 0 þ 2nπ or η ≠ 0, a can be diagonalized and β β ρ ρ η − ϕ þð i j þ i jÞðcosh cos Þ Λ can be put in a normal form:

þ ϵijkðβkðϕ sinh η − η sin ϕÞ    eðϕþiηÞ=2 0 a ¼ aδa−1; δ ¼ ; ðA8Þ − ρ η η ϕ ϕ − ϕ iη =2 kð sinh þ sin ÞÞ ; ðA1Þ 0 e ð þ Þ

ΛðaÞ¼ΛðaÞΛðδÞΛða−1Þ; where the relation between ϕ, η, ρ⃗, β⃗, are given by (30). 0 1 η 00 η This shows the relation between canonical coordinates on cosh sinh B C SLð2; CÞ and SO(3,1). As mentioned in the main text, the B 0 cos ϕ sin ϕ 0 C ΛðδÞ¼B C: ðA9Þ parameters domain is different for the two groups: the @ 0 − sin ϕ cos ϕ 0 A multivaluedness of the logarithmic map (the inverse of η 00 η the exponential map) is determined by the periodicity of the sinh cosh compact subgroup of rotations. When η ¼ 0 (jρ⃗j¼ϕ, Then Λ belongs to an equivalent class of elements γðϕ; ηÞ β⃗ 0 2 C ¼ ), the group SLð ; Þ [see (29)] reduces to SU(2) corresponding to the first of (26). If we restrict 0 ≤ ϕ < 2π, Λ and the matrix represents a pure rotation each of these classes corresponds to a unique complex   rotation angle, while one class γðϕ; ηÞ of elements of 103 Λ ∈ SOð3; 1Þ corresponds to two classes γðϕ; ηÞ and Λðρ⃗; β⃗¼ 0Þ¼ ; ðA2Þ 03 R γðϕ þ 2π; ηÞ of elements of SLð2; CÞ. ϕ 0 2 π η 0 ζ⃗2 0 PIf ¼ þ n and ¼ , so that ¼ , and if also ζ 2 0 ρ⃗ 0 β⃗ 0 γ 0 0 where jj jj ¼ ( ¼ , ¼ ), then a belongs to ð ; Þ. The second class in (26) can be obtained when ϕ ¼ 0, η ¼ 0,so ρ ρ ρ P R δ ρ i j 1 − ρ − ϵ k ρ ζ⃗2 0 ζ 2 0 ρ⃗2 β⃗2 ρ⃗ β⃗ 0 ij ¼ ij cos þ 2 ð cos Þ ijk sin ; ðA3Þ that ¼ ,but jj jj > ( ¼ , · ¼ but ρ ρ 2 ρ⃗2 þ β⃗ > 0).16 The third class in (26) cannot be obtained so that 16For instance the representative of the second class in (26) β −1 ρ 1 Λ ρ⃗ 0 β⃗ 0 Λ ρ⃗ 2π β⃗ 0 1 can be obtained by setting 1 ¼ , 2 ¼ and all the other ðj j¼ ; ¼ Þ¼ ðj j¼ ; ¼ Þ¼ 4: ðA4Þ components to zero.

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8 [31,32] with the exponential map. Indeed the group 3 ⃗ 3 ⃗ 2 C dg ¼ 2 d ad a ðA15Þ SLð ; Þ is not exponential. But one can show that it is ja0ða⃗Þj weakly exponential. In order to do so consider the inverse map where

2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 log a0 a − 1 2 ζ ð þð 0 Þ Þ ⃗ 1 ⃗2 j ¼ 2 1 aj: ðA10Þ a0ðaÞ¼ þ a : ðA16Þ i ða0 − 1Þ2 ζ ζ The study of the branch points of this function shows that, The Jacobian to complex vector ð ; Þ (28) can be restricting it to its principal values, the complex rotation evaluated to vector is holomorphic in a0 except for a branch cut −1 −∞ sin ð1 ζÞ sinðζÞ 2 extending on the real axis of a0 from to . Indeed J ⃗ ⃗ → ζ⃗ζ⃗ 2 ðða; a Þ ð ; ÞÞ ¼ 2 ; ðA17Þ the function in (A10) is single valued for all complex a0. 4ζ For a0 real with −1 ≤ a0 ≤ 1 the numeratorpffiffiffiffiffiffiffiffiffiffiffiffiffi in (A10) is 1 1 − 2 nothing but arccosða0Þ¼i logða0 þ i a0Þ, which, from which extended to complex numbers, has branch cuts on the real axis for a0 < −1 and a0 > 1. The interval −1 ≤ a0 ≤ 1 jsin4ð1 ζÞj 2 2 3ζ 3ζ (and the imaginary axis of a ) is realized by η 0, dg ¼ 4 d d : ðA18Þ j ¼ jζ j ϕ ∈ ð−2π; 2π, i.e., by the SU(2) subgroup of rotations. The half-line a0 real with a0 > 1 is taken by pure boosts Finally, in terms of the real canonical coordinates ðρ; βÞ (27) ϕ ¼ 0, η ≠ 0, and we can write the numerator as pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi the Haar measure takes the form (J ððζ; ζÞ → ðρ; βÞÞ ¼ 8) arccoshða0Þ¼logða0 þ a0 þ1 a0 −1Þ, which, extended to complex a0, has a branch cut on the real axis for a0 < 1.   cosh2ð1 ηÞsin2ð1 ϕÞþsinh2ð1 ηÞcos2ð1 ϕÞ 2 The branch cut a0 < −1 is given by the composition of 4 2 2 2 2 3ρ⃗ 3β⃗ dg ¼ 2 2 d d : these two functions. The branch point a0 ¼ −1 corresponds ϕ þ η to the third class in (26). Thus, except for the branch cut, the ðA19Þ canonical coordinates provided by the exponential map, ζ⃗ ρ⃗ β⃗ represented by the complex rotation vector ¼ þ i , Notice that the Haar measure in terms of canonical coor- are holomorphic functions parametrizing the whole dinates manifests the property (44) of the Duflo map. SLð2; CÞ group. 4. Haar measure in Cartan decomposition 3. Haar measure for SLð2;CÞ Considering the representation (27) and (92) we obtain Considering an element the relations   αβ       ∈ 2 C 1 1 1 a ¼ SLð ; Þ; ðA11Þ ζ ⃗ ⃗ γδ cos 2 ¼ cosh 2 jbj cos 2 jrj     1 1 the Haar measure is defined [29,47] as (the factor 2 is − ⃗ ⃗ ˆ ˆ i sinh 2 jbj sin 2 jrj b · r; arbitrary)       1 1 ⃗ 1 2 sin ζ ζˆ ¼ cosh jbj sin jr⃗j rˆ β γ δ β γ δ 2 2 2 dg ¼ δ 2 d d d d d d ; ðA12Þ     j j 1 1 ⃗ ⃗ ˆ þ i sinh 2 jbj cos 2 jrj b and has the invariance property     1 1 −1 − ⃗ ⃗ ˆ ∧ ˆ dg ¼ dðhgÞ¼dðghÞ¼dg : ðA13Þ i sinh 2 jbj sin 2 jrj b r; ðA20Þ

In the parametrization (23) evaluating the Jacobian 2 2 2 where ζ⃗¼ρ⃗þiβ⃗and ζ ¼ ζ⃗ ¼ðϕ þ iηÞ . Equation (A20)

2 is nothing but the Baker-Campbell-Hausdorff (BCH) for- a3 − a0 J ððβ; γ; δ; β; γ; δÞ → ða;⃗a⃗ÞÞ ¼ 4 ðA14Þ mula for the Cartan group element (92): a0 ρ⃗ ⃗ β⃗ ⃗ ⃗ ⃗ ⃗ ⃗ one gets the measure g ¼ expði · J þ i · NÞ¼expðib · NÞexpðir · JÞ: ðA21Þ

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Defining the quantities 5. Delta on the group in canonical coordinates   Consider that for canonical coordinates in G 1 ζ⃗ ⃗ ζ Z ¼ sin 2 ζ ; zðghÞ¼BðzðgÞ;zðhÞÞ; ðA30Þ   1 ⃗ ⃗ ˆ where BðzðgÞ;zðhÞÞ is the given by the BCH formula B ¼ sinh 2 jbj b;   respect to the Lie algebra g. Indeed 1 ⃗ ⃗ ˆ B R ¼ sin 2 jrj r; ðA22Þ gh ¼ eizðgÞ·xeizðhÞ·x ¼ ei ðzðgÞ;zðhÞÞ·x; g; h ∈ G; x ∈ g: ðA31Þ we get Since zðg−1Þ¼−zðgÞ, zðeÞ¼0, pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi ⃗2 ⃗2 ⃗2 ⃗ ⃗ 1 − Z ¼ 1 þ B 1 − R − iB · R ðA23Þ BðzðgÞ;zðhÞÞ ¼ 0 for zðhÞ¼−zðgÞ: ðA32Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi The delta transforms with the inverse of the Jacobian of ⃗2 ⃗ ⃗2 ⃗ ⃗ ⃗ Z⃗¼ 1 þ B R þ i 1 − R B þ iR ∧ B ðA24Þ transformation

⃗ δðghÞ¼δdðBðzðgÞ;zðhÞÞÞ In terms of the Z coordinates the measure is [see (A15) and ⃗ ⃗ ddB z g ;z h −1 consider that a ¼ iZ] ð ð Þ ð ÞÞ δd ¼ d ðzðgÞþzðhÞÞ: ðA33Þ d zðgÞ zðhÞ¼−zðgÞ 8 dg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi d3Zd⃗ 3Z⃗: ðA25Þ 2 2 But from the invariance of the Haar measure dðghÞ¼dg, j 1 − Z⃗j so that ⃗ ⃗ ⃗ The Jacobian of transformation from Z to ðB; RÞ is dg ¼ ddzðgÞμðzðgÞÞ ¼ ddzðghÞμðzðghÞÞ ðA34Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ⃗ 3 ⃗ 1 ⃗2 it follows that the Jacobian is nothing but d Zd Z 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ B 1 − ⃗2 1 ⃗2 ⃗ ⃗ 2 3 3 ¼ 2 ðð R Þð þ B ÞþðR · BÞ Þ: ⃗ ⃗ ⃗ d d Rd B 1 − R d BðzðgÞ;zðhÞÞ μðzðgÞÞ ¼ ; ðA35Þ ðA26Þ ddzðgÞ μðBðzðgÞ;zðhÞÞÞ μ 0 μ 1 The measure then becomes and we find, since ðzðgÞ¼ Þ¼ ðeÞ¼ , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δðghÞ¼μ−1ðzðgÞÞδdðzðgÞþzðhÞÞ: ðA36Þ 1 ⃗2 64 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ B 3 ⃗ 3 ⃗ dg ¼ 2 d Rd B: ðA27Þ 1 − R⃗ APPENDIX B: SOME CALCULATION FOR THE ⃗ ⃗ DUFLO QUANTIZATION MAP AND THE We can now rewrite it in terms of b and r coordinates. ASSOCIATED ⋆-PRODUCT Considering the Jacobian of transformation (apart from numerical factors) 1. Calculation of the Duflo factor The Duflo function (43) can be evaluated explicitly 1 ⃗ ⃗ 3 ⃗ sin ð2 jrjÞ sinðjrjÞ 3 making use of the identity d R ¼ d r⃗; 4 ⃗2   jrj 1 X sinh ð xÞ B2 1 ⃗ ⃗ 2 ¼ exp n x2n ; ðB1Þ 3 ⃗ sinh ð2 jbjÞ sinhðjbjÞ 3 ⃗ 1 x 2nð2nÞ! d B ¼ ⃗2 d b; ðA28Þ 2 n≥1 4jbj with B2n Bernoulli numbers, so that, from the property of the measure becomes the determinant

2 ⃗sin2 1 r⃗ det ðexpðAÞÞ ¼ exp ðTrAÞ; ðB2Þ 4 sinh jbj ð2 j jÞ 3⃗ 3 ⃗ dg ¼ ⃗2 ⃗2 d rd b: ðA29Þ jbj jrj we get

106005-22 NONCOMMUTATIVE FOURIER TRANSFORM FOR THE LORENTZ … PHYS. REV. D 99, 106005 (2019)   X i i i i B2 the Lie algebra is x ¼ x L þ x R ¼ x J þ x N , jðxÞ¼exp n Trðad Þ2n : ðB3Þ L i R i J i N i 2 2 ! x we get n≥1 nð nÞ i i i i i − i In order to simplify the calculations we adopt the xL ¼ xJ þ ixN;xR ¼ xJ ixN; ðB11Þ following standard redefinition of the Lorentz generators: and 1 1 ⃗ ⃗− ⃗ ⃗ ⃗ ⃗ L ¼ 2 ðJ iNÞ; R ¼ 2 ðJ þ iNÞ; ðB4Þ 1 2 1 j sinh ð2 xζÞj 2 4 j ðxÞ¼ 2 ; ðB12Þ satisfying the brackets jxζj ϵ ϵ 0 pffiffiffiffiffiffiffiffiffiffiffiffi ½Li;Lj¼i ijkLk; ½Ri;Rj¼i ijkRk; ½Li;Rj¼ : ⃗ ⃗ i i i where xζ ¼ xζ · xζ and xζ ¼ xJ þ ixN. ðB5Þ Notice that if we rewrite the exponential function on g as We have thus split the algebra in two mutually commuting sets of suð2Þ generators L and R , in terms of which a i i exp ðik XIÞ¼exp ðiζ⃗· X⃗ Þ exp ðiζ⃗ · X⃗ Þ XI ∈ g; group element (40) takes the form I L R ðB13Þ g ¼ exp ðiζ⃗· L⃗Þ exp ðiζ⃗ · R⃗ÞðB6Þ where in this basis the coordinates on g are17 where ζi ¼ ρi þ iβi as above. Defining a generic element I i i of g in this basis as x ¼ x EI ¼ xLLi þ xRRi, for 1 1 1 … 6 1 XL ¼ ðXJ − iXNÞ;XR ¼ ðXJ þ iXNÞ; ðB14Þ I ¼ ; ; , with EI ¼ LI for I ¼ ,2,3,EI ¼ RI−3 for i 2 i i i 2 i i I ¼ 4, 5, 6, the adjoint representation is given by the matrix K I K K ðadXÞJ ¼ x cIJ , where cIJ are the structure constant of g we can also apply the Duflo function directly on the form K ϵ K 1 1 given by (B5) as cIJ ¼ i IJ for I, J, K ¼ , 2, 3, while (B10), i.e., apply the function j2ð∂Þ to the exponential K ϵ K−3 4 cIJ ¼ i I−3J−3 for I, J, K ¼ , 5, 6. It follows by direct (B13) to get computation that 1 2 1 j sin ð2 ζÞj I K ðj2ð∂Þ expÞðikIX Þ¼4 exp ðikIX Þ; ðB15Þ adXadY ¼fðadxÞ JðadyÞI g I ζ2 I   j j δk ⃗ ⃗ − k 0 f j xL · yL xLjyLg 3×3 ¼ ; 0 δk ⃗ ⃗ − k which coincides with (48). 3×3 f j xR · yR xRjyRg

ðB7Þ 2. Explicit calculation of the ffiffiffiffiffiffiffiffiffi p ⋆-product on monomials so that, with the notation x ¼ x⃗· x⃗(¼jx⃗j if x⃗is real) Considering expressions (18) and (19), the ⋆-product 2 2 2 2 2 2 TrðadxadxÞ¼ x ¼ xL þ xR: ðB8Þ between n coordinates of g can be evaluated through the formula With a similar calculation one finds that

2 2 2 ⋆ ⋆ n 2 n 2 n Xi1 Xi2 Xi Trðadx Þ¼ xL þ xR ; ðB9Þ n ∂n −i n and finally ¼ð Þ i1 j2 i ∂k ∂k ∂k n 0   1 2 3 k1¼k2¼¼kn¼ X ⃗ ˆ B −1 iBðk1;k2;…;k Þ·X 2 2n 2n 2n × D ðe n Þ: ðB16Þ jðxÞ¼exp 2 2 ! ðxL þ xR Þ n≥1 nð nÞ  X   X  The term to differentiate in the last expression can be B2 B2 2 n 2n 2 n 2n rewritten as ¼ exp 2 2 ! xL exp 2 2 ! xR n≥1 nð nÞ n≥1 nð nÞ ⃗ ˆ ⃗ ˆ ⃗ ˆ 2 1 2 1 −1 ik1·X ik2·X ik1·X sinh x sinh x AðBðk1;k2; …;k ÞÞS ðe e e Þ: ðB17Þ 16 ð2 LÞ ð2 RÞ n ¼ 2 2 : ðB10Þ xL xR For the lowest order powers for the ⋆-product we get We can now rewrite the Duflo factor in the canonical basis ≡ 17 ˜ generated by ei ðJi;NiÞ [for which we have canonical This can be seen using the duality relations hei; eji¼δij i ≡ ρi βi i J i N i L i R coordinates k ð ; Þ]. Considering that an element of from which hx; Xi¼xJXi þ xNXi ¼ xLXi þ xRXi .

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2 2 −i ∂ 2 ðiÞ ð Þ ð Þ 2 ⃗i ⃗j −1 ˆ ˆ where we indicated with A … ðka;kb; …;kcÞ the X ⋆ X A k1;k2 i k1k S X X ; kðk1;k2; ;;knÞ i j ¼ j ð kðk1;k2Þð Þþ 2 ð i jÞÞ ∂ i ∂ 0 k1 k2 k¼ (i)-order mixed term in ka;kb; …;kc of the n-ple Duflo factor. ðB18Þ ⃗ ⃗ In terms of the complex vector ζ ¼ ρ⃗þ iβ, defining − 3∂3 ⃗ ⃗ ð iÞ ð3Þ Bðk1;k2; …;k Þ¼ζðζ1; ζ2; …; ζ Þ, we can calculate the X ⋆ X ⋆ X ¼ ðA ðk1;k2;k3Þ n n i j k i j k kðk1;k2;k3Þ ∂k1∂k2∂k3 k¼0 BCH formula from the SL(2,C) representation (28) 3 m n l S−1 ˆ ˆ ˆ þ i k1 k2k3 ðXmXnXlÞ     1 1 2 iζ⃗σ ð Þ l 2 · ζ⃗ σ 1 ζ⃗ σ ζˆ σ þ iðA ðk1;k2Þk3 e ¼ cos · þ i sin · · ; ðB20Þ kðk1;k2;k3Þ 2 2 ð2Þ l A k1;k3 k þ kðk1;k2;k3Þð Þ 2 ð2Þ l from which follows for the double BCH (the BCH coming þ A ðk2;k3Þk1ÞX Þ; ðB19Þ kðk1;k2;k3Þ l from the product of two exponentials)   1 1 ζ 1 ζ ζˆ 1 ζ 1 ζ ζˆ − 1 ζ 1 ζ ζˆ ∧ ζˆ ˆ sin ð2 1Þ cos ð2 2Þ 1 þ cos ð2 1Þ sin ð2 2Þ 2 sin ð2 1Þ sin ð2 2Þ 1 2 ζ ζ1; ζ2 ζ ζ1; ζ2 : tan 2 ð Þ ð Þ¼ 1 1 1 1 ˆ ˆ ðB21Þ cos ð2 ζ1Þ cos ð2 ζ2Þ − sin ð2 ζ1Þ sin ð2 ζ2Þζ1 · ζ2

The Duflo factor (52) [or inverse (48)] can be expanded as 1 S−1ðXˆ Xˆ Þ¼X X þ S−1ð½Xˆ ; Xˆ Þ; ðB26Þ (fkg¼fζg or fρ; βg) i j i j 2 i j 1 A ¼ 1 þ ðζ2 þ ζ2ÞþOðk4Þ 1 24 −1 ˆ ˆ ˆ S−1 ˆ ˆ S ðXiXjXkÞ¼XiXjXk þ 2 ðXi ð½Xj; XkÞ 1 2 2 4 ¼ 1 þ ðρ − β ÞþOðk Þ: ðB22Þ −1 ˆ ˆ −1 ˆ ˆ 12 þ XjS ð½Xi; XkÞ þ XkS ð½Xi; XjÞÞ ζ⃗ ζ ζ 1 Since the BCH (B21) for is at least linear in 1 and 2,itis þ S−1ð3½Xˆ ; ½Xˆ ; Xˆ enough to consider the expansion of the BCH up to 2nd 12 i j k k ˆ ˆ ˆ ˆ ˆ ˆ order in the coordinates f g − ½Xj; ½Xi; Xk − ½Xk; ½Xi; XjÞ: ðB27Þ ⃗ ⃗ ⃗ 1 ⃗ ⃗ ζðζ1; ζ2Þ ≃ ζ1 þ ζ2 − ζ1 ∧ ζ2; ðB23Þ 2 Substituting (B24), (B25) in (B22), and using the commu- or tation relations (55) we can finally calculate from (B18) and 1 (B19). We here report the explicit expression of third order ⃗ ⃗ ⋆ ρ⃗ðρ1; ρ2Þ ≃ ρ⃗1 þ ρ⃗2 − ðρ⃗1 ∧ ρ⃗2 − β1 ∧ β2Þ; monomials for the Duflo-map -product (repeated indexes 2 are summed): 1 β⃗β β ≃ β⃗ β⃗ − ρ⃗ ∧ β⃗ β⃗ ∧ ρ⃗ ð 1; 2Þ 1 þ 2 2 ð 1 2 þ 1 2Þ: ðB24Þ i XJ ⋆ XJ ⋆ XJ ¼ XJXJXJ þ ðϵl XJXJ þ ϵl XJXJ For the triple BCH we can use the associativity18 property i j k i j k 2 ij k l ik j l Bðζ1; ζ2; ζ3Þ¼Bðζ1; Bðζ2; ζ3ÞÞ, from which we get, up to 1 i þ ϵl XJXJÞ − δ XJ − ϵ ; 2nd order jk i l 2 ik j 12 ijk i 1 J ⋆ J ⋆ N J J N ϵl N J ϵl J N ρ⃗ρ ρ ρ ≃ ρ⃗ ρ⃗ ρ⃗ − ρ⃗ ∧ ρ⃗ ρ⃗ ∧ ρ⃗ Xi Xj Xk ¼ Xi Xj Xk þ ð ijXk Xl þ ikXj Xl ð 1; 2; 2Þ 1 þ 2 þ 3 2 ð 1 2 þ 1 3 2 1 ρ⃗ ∧ ρ⃗ − β⃗ ∧ β⃗ − β⃗ ∧ β⃗ − β⃗ ∧ β⃗ ϵl XJXN δ XN − 2δ XN ; þ 2 3 1 2 1 3 2 3Þ; þ jk i l Þþ6ð jk i ik j Þ ⃗ ⃗ ⃗ ⃗ 1 ⃗ ⃗ i βðβ1; β2; β2Þ ≃ β1 þ β2 þ β3 − ðρ⃗1 ∧ β2 þ ρ⃗1 ∧ β3 XN ⋆ XN ⋆ XN XNXNXN − ϵl XNXJ ϵl XNXJ 2 i j k ¼ i j k 2ð ij k l þ ik j l ⃗ ⃗ ⃗ ⃗ þ ρ⃗2 ∧ β3 þ β1 ∧ ρ⃗2 þ β1 ∧ ρ⃗3 þ β2 ∧ ρ⃗3Þ: 1 ϵl XNXJ δ XN ; þ jk i l Þþ2ð ik j Þ ðB25Þ i XN ⋆ XN ⋆ XJ XNXNXJ − ϵl XJXJ − ϵl XNXN The symmetrization map (42) on 2nd and 3rd order i j k ¼ i j k 2ð ij k l ik j l monomials is such that 1 i − ϵl XNXN 2δ XJ − δ XJ ϵ : jk i l Þþ6ð ik j jk i Þþ12 ijk 18 The associativity of the BCH comes from the associativity of ðB28Þ the group product.

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From the properties of the Duflo map it is also easy to q0 ¼ coshðΘÞ; prove that the Duflo star product preserves the algebra of q1 ¼ sinhðΘÞ sinðα3Þ sinðα1Þ; SymðgÞg. One can check explicitly for instance that given Cˆ ˆ 2 − ˆ 2 Θ α α the quadratic Casimir ¼ XJ XN of the quantum alge- q2 ¼ sinhð Þ sinð 3Þ cosð 1Þ; bra, such that q3 ¼ sinhðΘÞ cosðα3Þ: ðC4Þ

C D−1 Cˆ J ⋆ J − N ⋆ N ⃗2 − ⃗2 − 1 ⋆ ¼ ð Þ¼Xi Xi Xi Xi ¼ XJ XN ; By comparison with Eq. (94) we find that ðB29Þ Θ ¼ b; ˆ ˆ ˆ 2 b ¼ðsinðα3Þ sinðα1Þ; sinðα3Þ cosðα1Þ; cosðα3ÞÞ: ðC5Þ C · C ¼ DðC⋆Þ · DðC⋆Þ¼DðC⋆Þ; ðB30Þ Moreover, considering the Jacobian i.e., 3 ⃗ 2 d b ¼ Θ sinðα1ÞdΘdα1dα3; ðC6Þ 2 C⋆ ⋆ C⋆ ¼ C⋆ ðB31Þ the measure (97) becomes

2 dg ¼ sinh ðΘÞ sinðα1ÞdΘdα1dα3dh: ðC7Þ APPENDIX C: SOME PROPERTIES OF THE HOMOGENEOUS SPACE H ≈SL 2;C =SU 2 3 ð Þ ð Þ 2. Projection to H3 of the propagator 1. Hyperbolic rotation angle The propagator in H3 can be rewritten as We show in this subsection the relation between the Kðq00;q0; tÞ¼hq00jUðtÞjq0i Cartan parametrization of Sec. IV D and the parametriza- Z tion in terms of “hyperbolic rotations” used in [50,51]. ¼ dg0 These can be considered as the generalization to hyperbolic 2 C SLZð ; Þ space of Euler angles rotations. The only change respect to the Cartan splitting of Sec. IV D is in the “boost” sector × dg00hq00jg00ihg00jUðtÞjg0ihg0jq0i: SLð2; CÞ=SUð2Þ [or SOð3; 1Þ=SOð3Þ]. In this case an SLð2;CÞ element of SLð2; CÞ [or SO(3,1)] is split as ðC8Þ

g ¼ Kh; ðC1Þ Defining hg0jq0ihq00jg00i¼δðq0k0−1Þδðq00k00−1Þδðh0h00−1ÞðC9Þ where h ∈ SUð2Þ [or SO(3)], and K ∈ SLð2; CÞ=SUð2Þ [or SOð3; 1Þ=SOð3Þ] such that for the Cartan splitting (92) g0 ¼ k0h0, g00 ¼ k00h00, we get Z Z Z K ¼ h1ðα1Þh3ðα3ÞK3ðΘÞ; ðC2Þ Kðq00;q0; tÞ¼ dk0 dh0 dk00 H 2 H Z3 SUð Þ 3 where h1ðα1Þ and h3ðα3Þ are Euclidean rotations around 00 0 0−1 00 00−1 respectively the 1 and 3 axes, while K3ðΘÞ is an hyperbolic × dh δðq k Þδðq k Þ rotation (a boost) along the 3 axis: SUð2Þ × δðh0h00−1Þhg00jUðtÞjg0i 8 1 1 Z coshð2ΘÞ12 −sinhð2ΘÞσ3 in SLð2;CÞ=SUð2Þ; > 00 0 >0 1 ¼ dhhq hjUðtÞjq hi; ðC10Þ <> coshðΘÞ 00sinhðΘÞ SUð2Þ B C K3ðΘÞ¼ B 0100C >B C in SOð3;1Þ=SOð3Þ: i.e., >@ 0010A :> Z 00 0 00 0 sinhðΘÞ 00coshðΘÞ Kðq ;q; tÞ¼ dhKGðq h; q h; tÞ SUð2Þ ðC3Þ Z 00 0−1 ¼ dhKGðg ¼ g g ; tÞðC11Þ K 1 0⃗ H SUð2Þ The action of on the origin qa ¼ð ; Þ of 3 gives a † generic point of q ¼ KqaK (q ¼ Kqa for SOð3; 1Þ=SOð3Þ) where we also used the invariance of the SU(2) Haar 0 of H3 as measure dðhh Þ¼dh.

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