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PHYSICAL REVIEW D 102, 086003 (2020)

Spectral and spectra of quantum

† Michał Eckstein 1,2,* and Tomasz Trześniewski 3,2, 1Institute of Theoretical and Astrophysics, National Quantum Information Centre, Faculty of , Physics and Informatics, University of Gdańsk, ulica Wita Stwosza 57, 80-308 Gdańsk, Poland 2Copernicus Center for Interdisciplinary Studies, ulica Szczepańska 1/5, 31-011 Kraków, Poland 3Institute of Theoretical Physics, Jagiellonian University, ulica S. Łojasiewicza 11, 30-348 Kraków, Poland

(Received 11 June 2020; accepted 3 September 2020; published 5 October 2020)

Different approaches to quantum generally predict that the dimension of at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterized by a discrete complex —the dimension spectrum. We show that these two notions complement each other and the dimension spectrum is very useful in unraveling the UV behavior of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum and the κ-Minkowski spacetime, for a few different Laplacians. In particular, we find that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the κ-Minkowski spacetime.

DOI: 10.1103/PhysRevD.102.086003

I. INTRODUCTION preserved only at the intermediate—semiclassical—level, while in the full theory spacetime breaks down into discrete The concept of spacetime, understood as a differentiable elements, determined either by the fundamental , has proven to be extremely fruitful in modeling scale or a regularization cutoff (see [1] for a conceptual gravitational phenomena. However, it is generally sup- discussion). In analogy to systems in condensed matter posed that the smooth breaks down at small physics, configurations of the underlying “atoms of - scales or high energies, due to the quantum effects. ” may form different phases, while (classical) continu- Consequently, many of the familiar notions, such as ous spacetime should emerge in the limit in at least one of causality, distance or dimension, have to be refined within them. Other phases will naturally share some features the adopted new mathematical structure. with the classical phase. Let us stress in this context the An essential property of the hypothetical quantum theory distinction between the continuum limit, which is a of gravity, as well as a useful input for constructing it, is the transition from a discrete protospacetime to the continuous ability to provide meaningful and testable predictions (but still quantum) spacetime, and the classical limit, in concerning deviations of physics from . which we completely recover familiar of general The first step in this direction can be done by characterizing relativity. the structure of (static) quantum spacetime, which replaces In the recent years, calculations of the effective the classical , but can be seen as a of spacetime dimensions have become an ubiquitous certain tangible generalization of the latter. This is possible method to characterize the quantum spacetime. This is only if an unambiguous notion of a spacetime could be one of only a few tools allowing us to find some order in the provided. In some of the approaches, such a notion is diverse landscape of models [2], whose predictions are notoriously difficult to compare. The *[email protected]kstein.pl dimension can be defined in many different ways, some † [email protected] of which are based on mathematical assumptions and some on physical concepts, such as thermodynamics [3]. The Published by the American Physical Society under the terms of most popular notion remains the spectral dimension, which the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to can be seen as determined by the mathematical properties the author(s) and the published article’s title, journal citation, of spectral geometry or by a physical (fictitious) and DOI. Funded by SCOAP3. process. The advantage of the spectral dimension in the

2470-0010=2020=102(8)=086003(20) 086003-1 Published by the American Physical Society MICHAŁ ECKSTEIN and TOMASZ TRZEŚNIEWSKI PHYS. REV. D 102, 086003 (2020) latter context is its dependence on a parameter (auxiliary as determined by noncommutative algebras of diffusion time) that can be identified with the length scale at —ought to be characterized by a discrete subset which the geometry is probed. Therefore, it is naturally of the complex —the dimension spectrum, rather than interpreted as a measure of how the dimension of spacetime a single number. More precisely, in noncommutative geom- changes with scale. etry the dimension spectrum is defined as the set of poles Starting with the seminal paper [4] (updated in [5]), of the spectral zeta functions of geometrical provenance. belonging to the causal dynamical triangulations approach These, on the other hand, are intimately connected with the to quantum gravity, the spectral dimension has been calcu- celebrated heat trace expansion via the Mellin functional lated for Horava-Lifshitzˇ gravity [6], asymptotic safety transform (see [31,32] and also [33]). Both spectral zeta scenario [7], nonlocal quantum gravity [8], spin foam models functions and heat trace expansions are indispensable tools [9] (and kinematical states of loop quantum gravity in general in [34–36], also in its noncommutative [10]), [11,12] and multifractional spacetimes [13]. version [37]. The heat trace can be utilized to compute The almost universal prediction is the the one-loop effective action and allows one to study at the smallest scales (the UV limit) to thevalue of 2, for which the short-distance behavior of propagators, along with general relativity would actually become power-counting quantum anomalies and some nonperturbative effects renormalizable. On the other hand, in some cases values [38]. Consequently, it is justified to expect that the structure different from 2 are obtained in the UV limit, especially in of the entire dimension spectrum of a given noncommutative nonclassical phases of models with a nontrivial phase geometry is relevant for physics. diagram. The situation is similar for quantum gravity in less Among the noncommutative spaces with known dimen- than 3 þ 1 topological dimensions [6,7,13,14] (see also [15]). sion spectra an interesting example is provided by the Quantum spacetime often turns out to be described in Podleś quantum sphere [39]. For a particular choice of terms of broadly understood noncommutative geometry, geometry [40], the dimension spectrum turns out to have which is also sometimes treated as a stand-alone approach to surprising features [41]: First, it exhibits the dimension quantum gravity. A particular example is the κ-Minkowski drop (also called the dimensional reduction) from 2 to 0. spacetime [16], associated with κ-Poincar´e algebra [17,18] Secondly, the associated spectral zeta function contains and widely considered in doubly/deformed poles outside the real axis, suggesting self-. and relative locality. As it was shown in [19], the small-scale Thirdly, these poles are of second order, which is character- behavior of the spectral dimension of κ-- istic for spaces with conical singularities [42]. Finally, time (first calculated in [20]) depends on the Laplacian, the corresponding heat trace expansion turns out to be which can be chosen according to several distinct principles. convergent—in sharp contrast to the case of smooth For 3 þ 1-dimensional κ-Minkowski spacetime, there are at manifolds, where it is only asymptotic. As we will show, least three possibilities in the UV: the dimension decreasing the above features are shared by two other (i.e., to 3, growing to 6 or diverging. However, as we will discuss other Laplacians) on the quantum sphere but the fine details in Sec. IV D, there may be a way to reconcile these of their heat trace expansions are different. The latter fact contrasting results. Let us also note that [21] presents strengthens the observation made in [19] by one of us in the an example of a noncommutative toy model [with Uð1Þ × case of κ-Minkowski space that the spectral dimension SUð2Þ momentum space] that exhibits the dimensional characterizes a given quantum space equipped with a reduction to 2, i.e., the value obtained in the approaches specific Laplacian. to quantum gravity mentioned before. Let us note that the effective dimensionality of spaces Recently, it has also been suggested [22,23], in the with a topological dimension lower than 4 is relevant context of multifractional [24], that the dimension not only from the perspective of toy models of lower- of quantum spacetimes can acquire complex values. These, dimensional (quantum) gravity but also in the context of the on the other hand, result in log-periodic oscillations in problem of entanglement [43]. The reason is that various physical quantities [25] and can possibly affect the the latter can be derived from the heat trace over the cosmic microwave background spectrum [23], introduce a boundary of some region of space [44]. stochastic noise to gravitational waves [26] or modify the The purpose of this work is to revisit the concept of the thermodynamics of photons [27]. More generally, complex spectral dimension from the perspective of the dimension dimensions (or complex critical exponents) and the corre- spectrum. We show that the latter is a valuable rigorous tool sponding log-periodic oscillations can also arise in the to study the UV behavior of the spectral dimension. To this systems with discrete scale invariance, which is observed in end, we first provide, in Sec. II, the definitions of both many contexts, including some particular cases of holog- concepts and highlight the trouble spots. Then, in Sec. III, raphy [28], as well as condensed matter physics, earth- we compute the dimension spectra and spectral dimensions quakes and financial markets, see, e.g., [29]. of three Laplacians on the Podleś quantum sphere [39].We Meanwhile, it has already been recognized by Connes and show that the dimension drop observed by Benedetti [20] Moscovici in 1995 [30] that quantum spaces—understood has a finer structure with the -logarithmic decay and

086003-2 SPECTRAL DIMENSIONS AND DIMENSION SPECTRA OF … PHYS. REV. D 102, 086003 (2020) log-periodic oscillations. For values of the deformation where we integrate over a fiducial V. (It factorizes parameter q close to the classical value 1 the amplitude in the leading term and therefore can be taken to if of these oscillations becomes very small and they are needed). PðσÞ is the probability that after the time σ the invisible in numerical plots. On the other hand, their diffusion will return to the same x ∈ V ⊂ M. The presence is clearly attested for any q by nonreal spectral dimension is now taken to be a function of the scale in the dimension spectrum. Next, in Sec. IV, we study the parameter σ defined as dimension spectra of three different Laplacians on the κ-Minkowski spacetime in 2, 3 and 4 topological dimen- ∂ P σ 2σ ∂P σ σ ≔−2 log ð Þ − ð Þ sions. We utilize these to identify the leading and dSð Þ ¼ : ð5Þ ∂ log σ PðσÞ ∂σ subleading short-scale behavior of the spectral dimensions obtained in [19]. This example uncovers an in In the case of M being a of the definition of the spectral dimension related to the order σ σ “ ” topological dimension d, we have dSð Þ¼d for all . of the Laplacian-like operator. We summarize our find- σ ∈ N σ ings in Sec. V and discuss their consequences for model Therefore, in general, if dSð Þ for a given value of ,it building in quantum gravity. can be interpreted as the effective dimension such that the ordinary diffusion process in dSðσÞ-dimensional Euclidean space would approximately behave as the Δ-governed II. TWO FACES OF DIMENSIONALITY diffusion on M. If we choose the appropriate Laplacian, A. Spectral dimension of a diffusion process small values of σ allow us to probe the ultraviolet structure of M, while large ones correspond to its infrared geometry. The usual perspective in quantum gravity is to introduce However, for sufficiently large σ the function (5) becomes the spectral dimension as a characteristic of the fictitious sensitive to the finite of M and the curvature of g. diffusion (or random walk) process on a given configura- The original definition (5) applies solely in the context of tion space. Let us first consider a diffusion process on a Riemannian manifolds. The departure from smooth geom- Riemannian manifold ðM;gÞ of topological dimension d, etry requires a suitable generalization. which is described by the heat Within the framework of deformed relativistic sym- ∂ metries, the noncommutative geometry of spacetime is ; σ Δ ; σ 0 ∂σ Kðx; x0 Þþ Kðx; x0 Þ¼ ; ð1Þ accompanied by a curved momentum space. Typically, the latter is a non-Abelian Lie , equipped with an with a second-order differential operator Δ in variable x and invariant Haar measure μ. This suggests a natural gener- an auxiliary time variable σ ≥ 0 (playing the role of a scale alization of formula (3) to (cf. [15,19,21]) parameter). In general, the operator Δ does not need to be μν Z the standard Laplacian −g ∇μ∇ν, μ; ν ¼ 1; …;d—it can dμ p μ ð Þ ipμðx−x0Þ −σLðpÞ Kðx; x0; σÞ¼ e e ; ð6Þ be a Laplace-type operator [38] or even a pseudodifferential ð2πÞ4 one (see [45]). In order to solve Eq. (1) one needs to impose appropriate representing the noncommutative Fourier transform initial/boundary conditions. Typically, one chooses the (i.e., inverse of the group Fourier transform) of the function initial condition of the form −σL e . ðdÞ More generally, one can adopt the definition of a heat δ ðx − x0Þ −σT Kðx; x0; σ ¼ 0Þ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð2Þ operator e , which applies for any closed, possibly j det gðxÞj unbounded, operator T acting on a separable H (cf. [45] and Appendix A of [33]). If T is bounded from 4 −σ In particular, in the case of d ¼ and the flat Euclidean below and e T is trace class, one defines the heat trace (or metric, the solution to Eq. (1) can be expressed as an the “return probability”) of an abstract operator T as (inverse) Fourier transform Z X∞ 4 −σ −σλ d p μ P σ ≔ T nðTÞ ; σ ipμðx−x0Þ −σLðpÞ ð Þ TrHe ¼ e ; ð7Þ Kðx; x0 Þ¼ 4 e e ; ð3Þ ð2πÞ n¼0 L Δ where is the momentum space representation of . where λnðTÞ are the eigenvalues of T counted with their To characterize the diffusion process (1) we may use the multiplicities. return probability For a compact Riemannian manifold M and T ¼ Δ, Z the trace can be computed via the standard integral kernel 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi PðσÞ¼ d4x j det gjKðx; x; σÞ; σ > 0; ð4Þ methods and one obtains (from now on we drop the volV V normalization)

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Z pffiffiffiffiffiffiffiffiffiffiffiffiffi −σΔ 4 (4) The direct computation of the spectral dimension 2 ; σ P σ TrL e ¼ d x j det gjKðx; x Þ¼ ð Þ; M from formula (8) requires full knowledge about the spectrum of T, which is seldom granted. One could for any σ > 0. resort to asymptotic formulas for heat traces Formula (7) allows us to extend the notion of the spectral [cf. Eq. (11)] to unveil the small-σ behavior of the dimension (5) beyond the realm of smooth manifolds. For spectral dimension, but the result can be very an abstract operator T one has misleading if one quits the UV sector [48]. (5) A consistent interpretation of d ðσÞ as a scale- P S −σT ∞ −σλ ðTÞ dependent dimension of spacetime requires it to TrHTe λ T e n σ 2σ 2σ nP¼0 nð Þ “ ” dSð Þ¼ −σT ¼ ∞ −σλ : ð8Þ reach the classical value in the IR sector. However, He nðTÞ Tr n¼0 e the latter is equal to 4 only in the very specific 4 instance of R , in which case actually dSðσÞ¼4 Let us now point out a few trouble spots with usage of (7) independently of the value of σ. If the classical and hence the spectral dimension: spacetime has compact topology or nontrivial cur- (1) First, one needs to make sure that formula (7) is well vature, then either d σ tends to 0 or grows to −σ Sð Þ defined. The trace-class property of e T for all infinity as σ → ∞, depending on whether the oper- σ > 0 is guaranteed on general grounds if T is a ator T has a trivial kernel or not (cf. Δsc versus Δsp classical pseudodifferential operator on a compact on Fig. 7). As for the latter, one can use the notion of manifold M [31,32], but it may fail, for instance, on the spectral variance [49] to remove the zero . infinite-dimensional spaces [46]. In either case, in order to recover the correct (2) If the spacetime manifold M is not compact, then dimension in the IR, one has to match the large- −σT e is typically not trace class (actually, not even scale behavior of dSðσÞ of the quantum model with compact) even if T is an honest classical pseudo- the corresponding classical spacetime manifold, as differential operator. Consequently, to define the was done for causal dynamical triangulations heat trace one needs an IR cutoff in the form of a in [14]. trace-class operator F (6) Finally, in order to study the spectral dimension of spacetime, which is characterized by a Lorentzian −σT Pðσ;FÞ ≔ TrHFe : ð9Þ metric, one first has to perform the Wick of it, i.e., an analytic continuation to the Euclidean On a manifold, one can simply take F to be a signature. This is a rather cumbersome procedure function projecting on a compact fiducial volume V, even in the case of curved pseudo-Riemannian as done in formula (4). Then, after restoring the manifolds and it is likely to be even more problem- normalization, V can eventually be taken to infinity, atic in quantum spacetime models (see, however, showing the independence of the spectral dimension [50]). We shall not explore this issue here since the on the IR regularization. On the other hand, there is considered examples are either Euclidean from the no reason to assume that a similar factorization start (quantum ) or have the well-defined κ would take place outside of the realm of manifolds. Euclidean counterparts ( -Minkowski momentum Although it can be demonstrated for specific exam- spaces [19]). ples, such as the κ-Minkowski spacetime which B. Dimension spectrum from asymptotic expansion we consider in Sec. IV, in general one should expect to encounter the notorious IR/UV mixing Let us now restart with the spacetime modeled by a problem [47]. Riemannian manifold M and turn toward the notion of the (3) The multiplicative factor 2 in Eq. (5) originates from dimension spectrum. the fact that the Laplacian is a second-order differ- Abundant information about the geometry of M can ential operator. This can be easily adapted if T is the be learned from the celebrated heat kernel expansion (pseudo)differential operator of any order η > 0 by [31,32,38]: redefining X∞ X∞ ðk−dÞ=η l PðσÞ ∼ a ðTÞσ þ blðTÞσ log σ; ð11Þ ∂ log PðσÞ σ↓0 k d ðσÞ ≔−η : ð10Þ k¼0 l¼0 S ∂ log σ where η is the order of the pseudodifferential operator T.A However, beyond the safe realm of smooth mani- few comments about formula (11) are in order. folds, the order of T is not a priori defined and First, the infinite series in formula (11) are asymptotic Eq. (10) becomes ambiguous. We shall illustrate this series, which are in general divergent for any σ > 0. problem in Sec. IV. Nevertheless, the formula has a precise meaning as an

086003-4 SPECTRAL DIMENSIONS AND DIMENSION SPECTRA OF … PHYS. REV. D 102, 086003 (2020) asymptotic expansion (cf. [51] and Sec. 2.5 of [33]). underlying manifold. In particular, if T is a differential It provides accurate information about the small-σ asymp- operator, then ord SdðTÞ¼1. In either case we have P σ η totic behavior of ð Þ, but in general it fails for larger dSd ¼ d= , as expected. values of σ [48]. Dimension spectra of higher order can be found Secondly, if T is a differential operator, the coefficients beyond the realm of classical pseudodifferential operators blðTÞ¼0 and akðTÞ are locally computable quantities of [33,42,53,54]. In particular, dimension spectra of order 3 geometricalR origin—that is, they can be expressed as were found for Fuchs-type operators on manifolds with αT akðTÞ¼ M k ðxÞ. On the other hand, if T is only pseu- conical singularities [42]. Surprisingly enough, the same dodifferential, then akðTÞ for ðk − dÞ ∈ ηN are not locally feature was discovered in the very different context of the computable [52]. quantum sphere [41,55] (see Sec. III). Meanwhile, the Thirdly, one sees that d ¼ dim M is encoded in Eq. (11) as presence of complex numbers in Sd, typical for theleadingsmall-σ behavior,whilea0ðTÞ ∝ volðMÞ.IfT isa spaces [56], is interpreted as a signature of the self-similar scalarLaplace-typeoperatorandM hasnoboundary,thenthe structure [57]. In view of formula (12) this implies in turn odd coefficients a2nþ1ðTÞRvanish.pffiffiffiffiffiffiffiffiffi The second coefficient that the heat trace exhibits oscillations [25,27]. 1 4π −d=2 d R R The above definition of dimension spectrum is borrowed reads a2ðTÞ¼6 ð Þ M d x gðxÞ ðxÞ, where is the scalar curvature, whereas a2 ðTÞ for n ≥ 2 involve from noncommutative geometry ala` Connes [58].The n A H D higher-orderinvariantsconstructedfromtheRiemanntensor. central notion of the latter is a spectral triple ð ; ; Þ A If T acts on a vector bundle E over M, then a ðTÞ involve also consisting of a noncommutative algebra of space(time) n H D the curvature of E—see [38] for a complete catalog. observables represented on and an unbounded operator H Finally, Eq. (11) extends to the noncompact setting. acting on , all tied together with a set of axioms. In this As mentioned earlier, this requires an IR regularizing context, one talks about the dimension spectrum of a spectral — triple SdðA; H; DÞ [30] (cf. also Sec. 1.4 of [33]), which is operator typically, a compactly supported smooth func- 1 tion f on M. In such a case, Pðσ;fÞ still admits an the union of dimension spectra of a family of operators. D asymptotic expansion of the form (11) but its coefficients These originate from the fluctuations of the bare operator . D now depend on f. On the physical side, considering fluctuated amounts to Let us now leave the domain of smooth manifolds and dressing it with all gauge potentials available for a given D trade the pseudodifferential operator for a positive spectral triple. Thus, one could say that Sdð Þ refers to unbounded operator acting on a separable Hilbert space the “pure gravity” scenario. The internal fluctuations of H. In this context, one expects a more general form of the geometry caused by the gauge fields will in general change heat trace expansion (cf. [33]): the dimension spectrum, including its order. They will not, however, change the maximal dimension dSd (see −σT PðσÞ¼TrHe Proposition 4.11 of [33]). Observe that the spectral dimen- “ ” X∞ X Xp sion will also change in presence of other nongravitational ∼ a ðlog σÞnσ−zðk;mÞ; ð12Þ fields as the bare Laplacian will get dressed by a potential. σ↓0 zðk;mÞ;n k¼0 m∈Z n¼0 It is also worth noting (see [33,59] for the full story) that if a positive operator T admits an expansion of the form for a (discrete, but possibly infinite) set of complex (12), then the associated spectral zeta function ζT defined as numbers zðk; mÞ. We define the dimension spectrum of −s the operator T as the collection of exponents (i.e., a set of ζTðsÞ ≔ TrT ; for ReðsÞ ≫ 0; ð15Þ numbers): admits a meromorphic extension to the whole complex SdðTÞ ≔∪zðk; mÞ ⊂ C; ð13Þ plane. This enjoyable interplay is revealed with the help of k;m the Mellin transform: whence the number (p þ 1), capturing the maximal power Z ∞ of log σ terms in Eq. (12), is called the order of the −σT s−1 Tre σ dσ ¼ ΓðsÞζTðsÞ; for ReðsÞ ≫ 0: ð16Þ dimension spectrum ord SdðTÞ. It is also useful to define 0 the maximal (real) dimension in the spectrum Formula (16) allows us furthermore to retrieve the ≔ ζ dSd sup ReðzÞ: ð14Þ complete structure of poles of T. In particular, the set ∈ z Sd SdðTÞ coincides with the set of poles of the function Γ · ζT and, moreover, From Eq. (7) one immediately reads out that if T is ⊂ a classical pseudodifferential operator, then SdðTÞ 1 1 − N − η ∈ N ≤ 2 The precise statement is: If ðA; H; DÞ is a spectral triple and η ðd Þ¼fðd kÞ= jk g and ord SdðTÞ , where jDj admits an expansion of the form (12), then SdðjDjÞ ⊂ η ¼ ηðTÞ is the order of T and d is the dimension of the SdðA; H; DÞ and ordSdðjDjÞ ≤ ordSdðA; H; DÞ.

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n n ∀ z ∈ Sd Resðs − zÞ ΓðsÞζTðsÞ¼ð−1Þ n!az;n: ð17Þ In the next section, we illustrate the (dis)similarities of s¼z the two notions of dimensionality via a careful analysis of Recall that the Gamma function has simple poles at Γ −1 l l! two classes of examples. nonpositive with Ress¼−l ðsÞ¼ð Þ = .In summary, a term of order ðlog σÞn−1σ−z in the small-σ III. QUANTUM SPHERE expansion of PðσÞ corresponds to a pole of Γ · ζT of order n ∈ C at z . A quantum sphere was first introduced by Podleś [39] as Let us now discuss the properties and problems of a quantum homogeneous space of the deformed group the dimension spectrum, as compared with the spectral SU ð2Þ. As a topological space it is described via the dimension: q complex -algebra A generated by A ¼ A;B and B (1) On top of the problem of checking whether, for a q subject to the relations given operator T, Eq. (7) is well defined, one needs to prove the existence of an asymptotic expansion of the form (12). This is guaranteed if T is a classical AB ¼ q2BA; AB ¼ q−2BA; elliptic pseudodifferential operator [52] (cf. also −2 1 − 1 − 2 Appendix A of [33]). On the other hand, beyond BB ¼ q Að AÞ;BB ¼ Að q AÞ; ð18Þ this realm it is a formidable task and no general results are available (see, however, [33,59]). for a deformation parameter 0

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A. Simplified Laplacian where F and G are periodic bounded smooth functions on S R, defined as The simplified Dirac operator Dq on a quantum sphere was introduced in [41]. Together with the algebra Aq and H X   the Hilbert space q, it satisfies all of the axioms of a πi ≔ 4γ − 4 Γ πijx= log q spectral triple. Its square—the simplified Laplacian—acts GðxÞ j e ; j∈Z log q on vectors of Hq as 1 FðxÞ ≔ ðπ2 þ 6γ2 − 4 log2 qÞ Δsm −ð2lþ1Þ 3 q jl; mi ¼ uq jl; mi;     X πi πi 4 Γ − ψ πijx= log q with u ¼ uðqÞ ≔ ðq−1 − qÞ−2. The simple exponential þ j j e ; ∈Z log q log q form of eigenvalues implies a self-similarity relation j Δsm ≔ DS 2 DS Δsm q ð qÞ ¼ uj q2 j. Note that q has no zero modes and does not have a well-defined classical limit q → 1. with Z ¼ Znf0g and sm The heat trace associated with Δq reads X∞ k 2k −σΔsm −1 Psm σ q ð Þ q k q ð Þ¼TrHq e R ðxÞ ≔ 4 x : sm ! 1 − 2k 2 1 k ð q Þ X X Xl k¼ sm −σΔq ¼ hl; mje jl; mi ;− ∈N 1 2 − þ Xl þ = m¼ l The symbol γ denotes the Euler-Mascheroni constant ¼ 2 ð2l þ 1Þ exp ð−σuq−ð2lþ1ÞÞ and ψ ¼ Γ0=Γ—the digamma function. All of the series l∈Nþ1=2 invoked in the above formulas are absolutely convergent X∞ on R. ¼ 4 n exp ð−σuq−2nÞ: ð19Þ From Eq. (21) we can quickly read out the dimension n¼1 spectrum [see Fig. 1(b)]:

−2 When σ tends to infinity, PsmðσÞ decays as e−σuq . q πi However, the small-σ behavior of PsmðσÞ cannot be easily SdðΔsmÞ¼ Z∪ð−NÞ q q logq deduced from the doubly exponential series in Eq. (19).   On the other hand, the associated zeta function is a πi ¼ kjk∈Z ∪f−njn∈Ng; with ordSd¼3: simple geometric series: logq X∞ ð22Þ sm −s −s 2ns ζ sm Δ 4 Δq ðsÞ¼TrHq ð q Þ ¼ u nq n¼1 It coincides with the set of poles of the function Γ · ζΔsm ,as 4 −s 2s 1 − 2s −2 q ¼ u q ð q Þ ; ð20Þ expected from general theorems discussed around Eq. (16). The third-order pole at s ¼ 0 yields the leading log2 σ term, which is meromorphic on the entire complex plane. It has the second-order purely imaginary poles of ζΔsm result in double poles located solely on the imaginary axis, for q s πij= log q with j ∈ Z. the oscillating behavior captured by F and G, whereas the ¼ Γ In this specific case one can deduce, via (the inverse of) simple poles of at negative integers give rise to the Eq. (16), the explicit “nonperturbative” formula for the heat remainder Rsm. trace (Theorem 4.13 of [41]): Let us emphasize that Eq. (21) is indeed a genuine equality valid for any σ > 0. This is in sharp contrast with a 1 typical situation of heat trace expansion on a manifold PsmðσÞ¼ ½2 log2ðuσÞþGðlogðuσÞÞ logðuσÞ ’ q 4 log2 q where one has only an asymptotic formula at one s disposal. We can thus compute explicitly the corresponding σ σ þ Fðlogðu ÞÞ þ Rsmðu Þ; ð21Þ spectral dimension:

½G0ðlogðuσÞÞ þ 4 logðuσÞþF0ðlogðuσÞÞ þ GðlogðuσÞÞ þ uσR0 ðuσÞ dq;sm σ −2 sm : S ð Þ¼ 2 2 σ σ σ σ σ ð23Þ log ðu ÞþGðlogðu ÞÞ logðu ÞþFðlogðu ÞÞ þ Rsmðu Þ

This function is plotted and analyzed in Sec. III D.

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sm sp B. Spinor Laplacian sm sp −σΔq −σΔq Pq ðσÞ − Pq ðσÞ¼Trðe − e Þ sp Let us now turn to the spinor Laplacian Δ . It arises as −σΔsm −σ Δsp−Δsm q ¼ Tre q ð1 − e ð q q ÞÞ the square of the Dirac operator D , introduced in [40]. The q −σ Δsp−Δsm ≤ 1 − e ð q q Þ Psm σ latter is a unique Uqðsuð2ÞÞ-equivariant operator, which k k q ð Þ renders a real spectral triple. The spinor Laplacian acts on ¼ Oðσlog2σÞ: basis vectors of H as q The last equality follows from an operatorial inequality 1 1 − −tX ≤ limt→0 t k e k kXk (see Remark 4.12 of [41]) and Δsp −ðlþ1=2Þ − ðlþ1=2Þ 2 Eq. (21). This means that q jl; mi ¼ uðq q Þ jl; mi; sp 1 2 Pq ðσÞ¼ ½2 log ðuσÞþGðlogðuσÞÞ logðuσÞ 4 log2 q with u q−1 − q −2, as previously. The operator Δsp also ¼ð Þ q σ σ does not have a zero mode. In the limit q → 1 it tends þ Fðlogðu ÞÞ þ Rspðu Þ: ð25Þ 2 sp (strongly) to the spinor Laplacian on S [cf. Eq. (30)]. Hence indeed the leading small-σ behavior of Pq ðσÞ is sp The spectral zeta function associated with Δq can easily captured by Eq. (21) for the simplified Laplacian. This be computed (cf. Propositions 1 and 2 of [41] and also the harmonizes with the fact that the n ¼ 0 term in the last Appendix): ζ sm formula in Eq. (24) is nothing but Δq . σ The structure of the remainder Rspð Þ can be inferred sp − −2 from Eq. (24) for the zeta function, in close analogy with ζ sp Δ s D s ζ 2 Δ ðsÞ¼TrH ð q Þ ¼ TrH j qj ¼ jD jð sÞ q q q q Theorem 4.4 of [41]. Observe that ζΔsp has poles located on ∞ q X q2ks 4 −2s a regular lattice in the left complex half-plane [see ¼ u k 2k 2s 1 − q Fig. 1(c)]. Consequently, Γ · ζΔsp has third-order poles at k¼1 ð Þ q 2 X∞ 2 negative integers—yielding σn log σ contribution and Γðn þ 2sÞ q n ¼ 4u−2sq2s : ð24Þ double poles elsewhere—giving rise to σπij= log qσn log σ n!Γð2sÞ 1 − q2ðnþsÞ 2 n¼0 ð Þ oscillatory terms. In total, X∞ R ðxÞ ∼ ½h log2 x þ G ðlog xÞ log x þ F ðlog xÞxn; The last formula provides a valid meromorphic extension sp σ↓0 n n n 1 to the entire complex plane. n¼ sp The full asymptotic expansion of Pq ðσÞ could again ð26Þ be deduced from formula (24) via the inverse Mellin where hn ∈ R and Fn, Gn are periodic bounded functions transform, along the lines of [41]. The rather tedious of a form similar to that of F and G. Let us stress that the sp computations can be bypassed by noting that Δq and sum over n need not a priori be convergent, which is just a Δsm q commute and differ by a bounded perturbation restatement of the fact that asymptotic expansions of heat sp sm sm −1 Δq − Δq ¼ −2 þðΔq Þ . Consequently, we can write traces are generically divergent.

(a) (b) (c)

FIG. 1. (a) A comparison of dimension spectra for different Laplacians on two-sphere SdðΔspÞ¼SdðΔscÞ and on quantum sphere sm sp sc (b) SdðΔq Þ, (c) SdðΔq Þ¼SdðΔq Þ. The symbols ×, and • denote points in Sd, corresponding to poles of the function Γ · ζ of order 1, 2 and 3, respectively, while φ ¼ π=ðlog qÞ.Annth-order pole of the function Γ · ζ yields a term proportional to ðlog σÞn−1 in the asymptotic expansion of PðσÞ [recall Eq. (17)].

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This analysis leads to the conclusion about the dimen- Observe the difference with respect to Eq. (22), illustrated sion spectrum: in Fig. 1. Even though we only have an asymptotic formula for πi SdðΔspÞ¼ Z − N Psp, we can deduce the leading behavior of the spectral q logq q   dimension associated with the spinor Laplacian. This is πi because formula (26) grants us an explicit control on the ¼ k − njk ∈ Z;n∈ N ; with ordSd ¼ 3: logq remainder. We thus have

½G0ðlogðuσÞÞ þ 4 logðuσÞþF0ðlogðuσÞÞ þ GðlogðuσÞÞ dq;spðσÞ¼−2 þ OðσÞ¼dq;smðσÞþOðσÞ: S 2 log2ðuσÞþGðlogðuσÞÞ logðuσÞþFðlogðuσÞÞ S

Hence, also the spectral dimensions associated with the coshð1 ð2j þ 1Þ log qÞ − coshð1 log qÞ Δsc 2 2 spinor and simplified Laplacians on the quantum sphere q jj; mi¼ 2 1 jj; mi 2 sinh ð2 log qÞ share the same leading behavior for small σ—see Fig. 2. pffiffiffi ¼ uð qÞq−1=2ðq−j − 1 − q þ qjþ1Þjj; mi: ð27Þ

C. Scalar Laplacian sc In the limit q → 1 the operator Δq tends (strongly) to the sc 2 The scalar Laplacian Δq , introduced in [20], originates standard scalar Laplacian on S [cf. (31)]. In contradis- U su 2 sp from the Casimir operator on the Hopf algebra qð ð ÞÞ. tinction with Δq , it does have a zero mode, which means H0 A sc sc It acts on a Hilbert space q (on which the algebra q can that Pq ðσÞ tends to dim KerΔq ¼ 1 as σ goes to infinity. also be faithfully represented) spanned by orthonormal The small-σ behavior of the heat trace can be deduced by ∈ − − 1 … ∈ N sc vectors jj; mi, with m f j; j þ ; ;jg and j as singling out the unbounded part of Δq , as in the case of sp Δq . Namely, we have

X∞ ffiffiffi sc −σΔsc p −1=2 −j jþ1 P σ 0 q 2 1 −σ − 1 − q ð Þ¼TrHq e ¼ ð j þ Þ expf uð qÞq ðq q þ q Þg j¼0 X∞ pffiffiffi ¼ 1 þ ð2n þ 1Þ expf−σuð qÞq−1=2ðq−n − 1 − q þ qnþ1Þg n¼1 X∞ pffiffiffi ¼ 1 þ ð2n þ 1Þ expf−σuð qÞq−1=2q−ngþOðσ log2 σÞ n¼1 X∞ 1 −1 2 pffiffiffi −1 2 − 2 1 Psmpffiffi σq = exp −σu q q = q n O σ log σ : ¼ þ 2 qð Þþ f ð Þ gþ ð Þ n¼1

sm FIG. 2. Left panel: the dark blue is the spectral dimension for the Laplacian Δq , computed numerically from Eq. (8) up to the 100th eigenvalue. The leading behavior in the UV is determined from Eq. (33) as −4=ðlog σÞ (light green line). Right panel: the function q;sm −4 σ dS after subtraction of the leading behavior =ðlog Þ clearly shows the log-periodic oscillations.

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The last series can be evaluated explicitly (Proposition This yields 12 of [59]) using the inverse Mellin transform 1 1 ffiffiffi technique: −1 2 p PscðσÞ¼ Psmpffiffiðσq = Þþ ½logðσuð qÞÞ þ γ q 2 q log q   ffiffiffi X∞ σ p −1=2 O σ 2 σ − 1 1 þ Hðlogð uð qÞq ÞÞ þ ð log Þ: ð28Þ e−xq n ¼ logx þ γ − logq þ HðlogxÞ þ OðxÞ; logq 2 n¼1 Similarly as in the spinor case, one can unfold the structure of the remainder by an inspection of the zeta ζ sc — with function Δq see the Appendix. Its meromorphic struc- sc ture is very similar to that of ζ sp . The conclusion is that Δ Δq q   sp X 2π and Δq have identical dimension spectra, both of order 3. i 2πikx= log q HðxÞ ≔− Γ − k e : Formula (28) allows us also to compute the spectral ∈Z log q k ζ sc dimension associated with Δq up to the terms of order OðσÞ at σ ¼ 0,

1 −1 2 −1 2 1 pffiffiffi −1 2 σq = ðPsmpffiffiÞ0ðσq = Þþ ½1 þ H0ðlogðσuð qÞq = ÞÞ q;sc σ 2 q log q O σ dS ð Þ¼1 −1 2 1 pffiffiffi pffiffiffi −1 2 þ ð Þ Psmpffiffi σ = σ γ σ = 2 qð q Þþlog q ½logð uð qÞÞ þ þ Hðlogð uð qÞq ÞÞ pffiffi q;sm −1=2σ O σ −2 ¼ dS ðq Þþ ððlog Þ Þ; ð29Þ where the last equality follows from the exact formula (23). SdðΔspÞ¼SdðΔscÞ¼1 − N; with ord Sd ¼ 1: ð32Þ It shows that the leading small-σ behavior of the spectral dimension for the scalar Laplacian is captured by the 1 Δsm In the classical case we have dSd ¼ , in agreement with rescaled spectral dimension for q . η 2 η 2 the general formula dSd ¼ d= ,asd ¼ and ¼ .In contrast, for all three operators on the quantum sphere we D. Comparison 0 have dSd ¼ . In order to interpret this fact as the dimension We now compare the results obtained for the three drop, i.e., d ¼ 0, we need to argue that the “quantum Laplacians and contrast them with their classical Laplacians” are of order η > 0. D counterparts. In the framework of spectral triples, the operator q Recall that the spinor Laplacian3 acts on spinor harmon- verifies the so-called first-order condition [40], which ics over the unit two-sphere as follows [66,67]: mimics the demand for a classical Dirac operator to be a   first-order differential operator [58,68]. On the physical 1 2 side, this condition limits the admissible fluctuations of an Δsp jl; mi ¼ l þ 2 jl; mi: ð30Þ operator D by gauge fields [69] and thus it is pertinent in building particle physics models from noncommutative In turn, for the scalar Laplacian (also known as the Laplace- geometry [70,71]. Since Dq is a first-order operator, for sp 2 S Beltrami operator) we have [45] Δq ¼ Dq we should set η ¼ 2. The operator Dq does not sc meet the first-order condition [41] and Δq does not come Δscjj; mi¼jðj þ 1Þjj; mi: ð31Þ sm from a “Dirac” operator at all. Nevertheless, Δq differs from Δsp by a bounded perturbation and so does Δsc after a Let us first have a look at the dimension spectra at Fig. 1. q pffiffiffi q suitable rescaling and reparametrization q↝ q. We can Both Δsp and Δsc are classical differential operators of η Δsc η Δsm 2 second order acting over a two-dimensional manifold. thus safely assume that ð q Þ¼ ð q Þ¼ and conclude Consequently, we have [32] (see also [48] for a direct that indeed the dimension of the quantum sphere is 0. The computation) issue of the order of an operator over a quantum space is more subtle for the κ-Minkowski space, as we will see in the next section. 3 ΔS In the mathematical literature [65] the spinor Laplacian is The existence of nonpositive numbers in the dimension slightly different from Δsp, which is the square of the Dirac operator =D2. The two operators are related through the Schrö- spectra of the two classical Laplacians on the two-sphere 2 S 1 certify the impact of the nontrivial Riemann on the dinger-Lichnerowicz formula =D ¼ Δ þ 4 R, with R being the scalar curvature. On the unit two-sphere with a round metric the heat trace [recall Eq. (11)]. The dimension spectra of difference amounts to a trivial shift Δsp ¼ =D2 ¼ ΔS þ 2. quantum Laplacians also contain negative numbers, which

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Fuchs-type operators [54]. Although the mathematical context here is very different, one might take it as a (not so surprising) indication that the geometry of the quantum sphere is not smooth. We now turn to the analysis of the spectral dimensions. Let us first have a look at the small-σ behavior of dSðσÞ sm for the simplified Laplacian Δq . From Eq. (23) one deduces the leading behavior in the UV:     −4 πi X πi dq;sm σ 1 Γ σ−πij= log q S ð Þ¼ σ þ j j log log q j∈Z log q −2 FIG. 3. The q dependence of the amplitude of (leading þ Oððlog σÞ Þ: ð33Þ sp frequency) oscillations in the spectral dimension for Δq (dark blue) and Δsc (light red). The former is given by the function A q q;sm 1 2 q ffiffiffi ð Þ The function dS for q ¼ = is illustrated in Fig. 2. p sm defined in Eq. (34), whereas the latter is Að qÞ, because of the Formula (33) shows that the spectral dimension of Δq relation (29). drops to zero in the UV. Observe that the slope −4=ðlog σÞ does not depend on q. On the other hand, the amplitude of suggest that quantum spheres are “curved” in some sense oscillations exhibits strong dependence on the value of q. 1 (see, e.g., [46,72–76] for a discussion of curvature of Concretely, for the leading frequency [j ¼ in the sum in quantum spaces). On top of that, these dimension spectra Eq. (33)]wehave   contain points outside of the real axis, which hint at some π πi kind of self-similar structure of the quantum sphere [41,57] AðqÞ¼ Γ : ð34Þ (see also [25,27]). Note also that, excluding the simplified log q log q sm case of Δq , the nonreal points appear all over the left complex half-plane. This suggests that the curvature and The amplitude of oscillations tends to 1 as q → 0 and 2 self-similar structure of a quantum sphere are deeply decays very rapidly, as eπ =ð2ðq−1ÞÞ, when q goes to 1— interwoven. see Fig. 3. Finally, the dimension spectra of quantum Laplacians are In the previous sections we have shown that the leading of order 3, which means that they are already beyond the UV behavior of the spectral dimension for the spinor q;sm realm of classical pseudodifferential operators, as the latter Laplacian is captured by dS . The same is true, after can only have ord Sd ¼ 2. Third-order poles in the dimen- suitable rescaling, also for the scalar Laplacian, though sion spectra have been detected in the context of manifolds with worse precision. The situation is illustrated in Fig. 4 with conical singularities [42]. They occur when one through the numerical summation in Eq. (8) up to the 100th studies the conical singularities (and, more generally, eigenvalue. stratified spaces) from the perspective of manifolds with A comparison of the spectral dimension associated with sp sc sp boundary [77]. Concretely, specific nonlocal boundary the operators Δq ; Δq and their classical counterparts Δ , conditions related to the singularity coerce the use of Δsc, respectively, is presented in Figs. 5 and 6. The main

FIG. 4. A comparison of the UV behavior of the spectral dimensions for three Laplacians on the quantum sphere for q ¼ 1=3. In the q;sm σ q;sp σ q;sm σ left panel the dark blue line corresponds to dS ð Þ and the light green one to dS ð Þ. In the right panel the dark blue line is dS ð Þ, q2;sc σ whereas the light red one shows dS ðq Þ.

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the κ-Poincar´e Hopf algebra. Both these mathematical objects can be defined in any number of dimensions [78] but κ-Poincar´e algebra was first derived [17,18] in the ð3 þ 1Þd case, as a contraction of the quantum- deformed anti–de Sitter algebra Uqðsoð3; 2ÞÞ, obtained by taking the limit of the (real) deformation parameter q → 1 and anti–de Sitter R → ∞, while their ratio R log q ¼ κ−1, κ > 0 is kept fixed. Hence the new defor- mation parameter κ has the dimension of inverse length (in contrast to the dimensionless q), which allows for −1 the geometrization of the Planck mass mP ¼ ℏc λP, expressed in terms of Planck length λP. This peculiar feature enabled application of κ-Poincar´e algebra in the construction of models of so-called doubly special rela- sc FIG. 5. The spectral dimension dSðσÞ for Δq (continuous sp tivity, which was subsequently recast as the relative locality decaying ) and Δq (continuous diverging) with q ¼ 0.15, 2 framework and serves as an important source for the and for S with the scalar (dashed decaying curve) and spinorial quantum gravity phenomenology [79]. The interest in κ- (dashed diverging) Laplacians. Poincar´e and κ-(anti–)de Sitter algebras is also motivated by results for the (2 þ 1)-dimensional gravity, where it is known that they may arise from the structure of classical theory, at least in certain special cases (see [80] and references therein). The (n þ 1)-dimensional κ-Minkowski noncommutative space is the dual of the subalgebra of translations of the [(n þ 1)-dimensional] κ-Poincar´e algebra, since the latter is naturally interpreted as the algebra of spacetime coordi- nates. The time X0 and spatial coordinates Xa, a ¼ 1; …;n satisfy the following commutation relations:

i 0 ½X0;Xa¼κ Xa; ½Xa;Xb¼ : ð35Þ

As a , the κ-Minkowski space is isomorphic to the ordinary Minkowski space in n þ 1 dimensions, which σ Δsc 0 1 FIG. 6. The spectral dimension dSð Þ for q , with q ¼ . can be recovered in the classical limit κ → ∞. The Lie 0 5 (bottom continuous curve), q ¼ . (middle continuous) and algebra generated by X0;X is usually denoted anðnÞ, 0 9 2 a q ¼ . (top continuous), and for S with the scalar Laplacian where the notation refers to n Abelian and nilpotent (dashed). generators Xa. The corresponding Lie group ANðnÞ has the geometry of conclusion is that the spectral dimensions of both quantum (n þ 1)-dimensional elliptic [81,82]. Group Laplacians on the quantum sphere exhibit log-periodic elements can be written as ordered exponentials, whose oscillations in UV. The amplitude of these oscillations ordering is equivalent to the choice of coordinates on the drops very rapidly when q tends to the classical value 1 (see group manifold. For example, in the time-to-the-right Δsc −0.01 ≈ 0 99 ordering a group element has the form Fig. 3). In particular, for q the valueffiffiffiq ¼ e . p ∼ 10−430 adopted in Fig. 1 in [20], we have Að qÞ , which a −iP X iP0X0 explains why the oscillations have been overlooked in g ¼ e a e ð36Þ that paper. and P0;Pa ∈ R are coordinates in the so-called bicross- IV. κ-MINKOWSKI SPACETIME basis. It is easy to notice that we can interpret ANðnÞ as momentum space if such exponentials are treated The κ-Minkowski space was introduced [16] in the as plane waves on (n þ 1)-dimensional κ-Minkowski context of quantum groups that describe deformed relativ- space. ANðnÞ is equipped with the structure of the algebra istic —it is the spacetime that is bicovariant of translations and is related with spacetime co- under the action and coaction (the former is determined by ordinate algebra via the group Fourier transform [83,84]. the algebraic and the latter by the coalgebraic structure) of Furthermore, the geometry of momentum space becomes

086003-12 SPECTRAL DIMENSIONS AND DIMENSION SPECTRA OF … PHYS. REV. D 102, 086003 (2020) evident in the classical basis (the name refers to the each of them. In this way we improve and extend the results classical form of expression for the dispersion relation in obtained by one of us in [19]. Furthermore, we compute the this case), which can be introduced via the transformation complete dimension spectra of the relevant Laplacians.   P0 1 κ P0=κ a A. Bicovariant Laplacian p0 ¼ sinh κ þ 2κ e PaP ; Let us first consider the Laplacian determined by the p ¼ eP0=κP ; κ a a  bicovariant on the -Minkowski space P0 1 [91,92], which we call the bicovariant Laplacian. In terms κ − P0=κ a p−1 ¼ cosh κ 2κ e PaP : ð37Þ of bicrossproduct coordinates, the Euclidean momentum space representation of this Laplacian is given by

The above relations lead to the constraints on fp0;pa;p−1g,   2 a 2 2 1 − κ 0 2 2 P0=κ a p0 þpap þp−1 ¼ and p0 þ p−1 > , which describe L ðP0;fP gÞ ¼ 4κ sinh P0 þ e P P 1 cv a 2κ a the embedding of ANðnÞ as half of an ðn; Þ-hyperboloid in     [ðn þ 1Þþ1]-dimensional Minkowski space. p−1 is just an 1 1 2 4κ2 2 P0=κ a auxiliary coordinate, diverging in the κ → þ∞ limit. þ 4κ2 sinh 2κP0 þ e PaP ; In order to consider the diffusion process (i.e., study the heat trace) determined by a Laplacian on κ-Minkowski space, ð39Þ one first has to perform the Wick rotation ðp0 ↦ip0; while in classical coordinates it acquires the familiar p−1 ↦ip−1;κ ↦iκ;P0 ↦iP0Þ. This leads to the Euclidean momentum space, which has the [19]. standard form The Euclidean version of κ-Minkowski space has also L 2 a been studied from the perspective of spectral triples cvðp0; fpagÞ ¼ p0 þ pap : ð40Þ [85–87] via the star-product realizations [88–90]. In this 3 1 framework the algebra (35) is faithfully represented on The return probability in þ dimensions reads [19] the Hilbert space L2ðRnþ1Þ through a left regular group P σ representation. ð3þ1Þð Þ 2 As mentioned in the Introduction, there are several π pffiffiffi pffiffiffi 2 pffiffiffi ¼ ð2κ2 σ − πeκ σð2κ2σ − 1Þð1 − erfðκ σÞÞÞ; possible choices for a Laplacian on (Euclidean) κ- 2σ3=2 Minkowski space. All of them have continuous spectra, ð41Þ which reflects the noncompactness of the underlying space. Therefore, in order to define the corresponding heat traces, where erfð·Þ is the error function, while in 2 þ 1 dimen- one first needs to choose an IR cutoff. A natural choice is a sions we have function f compactly supported on Rnþ1 and promoted to 2 nþ1   an operator on L ðR Þ through the Weyl-like quantiza- π3=2κ 1 P σ 0; κ2σ tion (cf. [86,89]). Fortunately, it turns out that the regu- ð2þ1Þð Þ¼ σ U 2 ; ; ð42Þ larizing function factors out in trace formulas and contributes just a multiplicative factor kfk 2 [86,87]. L where Uða; b; ·Þ is a Tricomi confluent hypergeometric This means that the return probability does not depend function. Furthermore, we may also consider the case of on the choice of the IR regularization and can be computed 1 þ 1 dimensions (not discussed in [19]), via the heat kernel formula (6). In the classical basis it 4 reads [19] 3=2 π κ 2 pffiffiffi P σ ffiffiffi κ σ 1 − κ σ Z ð1þ1Þð Þ¼ pσ e ð erfð ÞÞ: ð43Þ 1 κ −σL PðσÞ¼ dnþ p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ðp0;fpagÞ; ð38Þ 2 a κ2 p0 þ pap þ The exact formulas (41)–(43) for heat traces can be directly developed into series around σ ¼ 0, where L is the (Euclidean) momentum space representation of a Laplacian. κπ5=2 κ3π5=2 4κ4π2 3κ5π5=2 pffiffiffi In Secs. IVA–IV C we will calculate the spectral P σ ffiffiffi − ffiffiffi − σ O σ ð3þ1Þð Þ¼2pσ3 2pσ þ 3 4 þ ð Þ; dimensions of κ-Minkowski space equipped with three   2κπ κ distinct Laplacians, in 2, 3 and 4 topological dimensions for P σ κ3π σ κ3π 1 γ 2 O σ ð2þ1Þð Þ¼ σ þ log þ þ þ log2 þ ð Þ; 4 The factor κ in the numerator was missing in [19] but it did κπ3=2 pffiffiffi P σ ffiffiffi −2κ2π κ3π3=2 σ O σ not affect results for the spectral dimension; here we have also ð1þ1Þð Þ¼ p þ þ ð Þ; ð44Þ rescaled the integrand by 2. σ

086003-13 MICHAŁ ECKSTEIN and TOMASZ TRZEŚNIEWSKI PHYS. REV. D 102, 086003 (2020) where γ is the Euler-Mascheroni constant. From these expansions one can immediately read out the corresponding dimension spectra n o n o n o 3 ∪ 1−n ∈ N 3 1 0 − 1 −1 − 3 … 1 Sdð3þ1Þ ¼ 2 2 jn ¼ 2 ; 2 ; ; 2 ; ; 2 ; ; ord Sd ¼ ;

Sd 2 1 ¼ 1 − N ¼f1; 0; −1; −2; …g; ord Sd ¼ 2; ð45Þ ð þ Þ n o 1 1 − N 1 0 − 1 −1 … 1 Sdð1þ1Þ ¼ 2 ð Þ¼ 2 ; ; 2 ; ; ; ord Sd ¼ :

Let us recall that in the 2 þ 1 dimensional case the dimension In classical coordinates it becomes spectrum is of the second order because of the presence of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  logarithmic terms in the expansion of P 2 1 ðσÞ. ð þ Þ L 2κ 2 a κ2 − κ We will now turn to results for the spectral dimensions. cpðp0; fpagÞ ¼ p0 þ pap þ : ð50Þ From Eq. (41) one finds that in 3 þ 1 dimensions the spectral dimension is given by the expression It turns out that the corresponding return probability in 3 þ 1 dimensions is given by a simple rational function ð3þ1Þ σ dS ð Þ 2 pffiffiffi pffiffiffi 2 pffiffiffi 2κ σ 1 2κ σ − πeκ σð2κ2σ þ 1Þð1 − erfðκ σÞÞ P σ π2 þ 3 2κ2σ ð3þ1Þð Þ¼ 2 3 ; ð51Þ ¼ þ pffiffiffi pffiffiffi 2 pffiffiffi ; 2κ σ −2κ σ þ πeκ σð2κ2σ − 1Þð1 − erfðκ σÞÞ ð46Þ while in 2 þ 1 dimensions we obtain ([19] in this case had only numerical results) ffiffi ð3þ1Þ which has the IR and UV limits limκpσ→∞ d ðσÞ¼4 S 4πκ ffiffi ð3þ1Þ 2κ2σ 2 p σ 3 P 2 1 σ e K1 2κ σ ; and limκ σ→0 dS ð Þ¼ , respectively. Analogously, in ð þ Þð Þ¼ σ ð Þ ð52Þ 2 þ 1 dimensions one uses Eq. (42) to get where Kαð·Þ is a modified Bessel function of the second κ2σ 3 1 κ2σ 2 1 Uð2 ; ; Þ 1 1 dð þ ÞðσÞ¼2 þ ; ð47Þ kind. Finally, in þ dimensions we have just S Uð1 ; 0; κ2σÞ 2 π P σ 2 1 ð1þ1Þð Þ¼ : ð53Þ pffiffi ð þ Þ σ 3 σ whose IR and UV limits are limκ σ→∞ dS ð Þ¼ and 2 1 pffiffi ð þ Þ σ 2 1 1 limκ σ→0 dS ð Þ¼ , respectively. In þ dimensions Observe that in the latter case there is no dependence on the (43) leads to parameter κ.   The dimension spectra can again be obtained directly −κ2σ 1 1 1 e from the expansions around σ ¼ 0 of the exact formulas dð þ ÞðσÞ¼1 þ 2κ2σ pffiffiffi pffiffiffi pffiffiffi − 1 ; ð48Þ S πκ σ 1 − erfðκ σÞ (51) and (52) [and the trivial equation (53)]:

1 1 2 2 pffiffi ð þ Þ σ 2 π π with the IR and UV limits limκ σ→∞ dS ð Þ¼ and P σ ð3þ1Þð Þ¼ 2 3 þ 2 ; ffiffi ð1þ1Þ 2κ σ σ limκpσ→0 d ðσÞ¼1, respectively. S 2π 4κπ P σ 4κ3π σ ð2þ1Þð Þ¼κσ2 þ σ þ log B. Bicrossproduct Laplacian þ 2κ3πð1 þ 2γ þ 4 log κÞþOðσÞ: ð54Þ Another possible Laplacian on κ-Minkowski space, called the bicrossproduct Laplacian, corresponds to the Consequently, we have simplest mass Casimir of κ-Poincar´e algebra [18] (let us stress that any function of this Casimir that has the correct Sd 3 1 ¼f3;2g; ordSd ¼ 1; classical limit is also a Casimir). More precisely, it is the ð þ Þ 2 −N 2 1 0 −1 −2 … 2 Euclidean version of the momentum space representation Sdð2þ1Þ ¼ ¼f ; ; ; ; ; g; ordSd ¼ ; ð55Þ of the Casimir and has the form 1 1 Sdð1þ1Þ ¼f g; ordSd ¼ :   1 L 4κ2 2 P0=κ a If we use the standard formula (5), Eq. (51) leads to the cpðP0; fPagÞ ¼ sinh P0 þ e PaP : ð49Þ 2κ spectral dimension

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2 3 1 4κ σ In this case, none of the heat traces (60)–(62) can be dð þ ÞðσÞ¼6 − ; ð56Þ S 2κ2σ þ 1 expanded in a series of the form (12). This is because of the factor e1=σ, which yields an essential singularity at σ ¼ 0. 3 1 3 1 pffiffi ð þ Þ σ 4 pffiffi ð þ Þ σ 6 with limκ σ→∞ dS ð Þ¼ and limκ σ→0 dS ð Þ¼ ; Consequently, the dimension spectrum of the relative- and Eq. (52) leads to locality Laplacian does not exist in any of the three   considered topological dimensions. 2κ2σ – ð2þ1Þ σ 4 − 4κ2σ 1 − K0ð Þ On the other hand, given the exact formulas (60) (62), dS ð Þ¼ 2 ; ð57Þ K1ð2κ σÞ we can compute the corresponding spectral dimensions explicitly: 2 1 2 1 pffiffi ð þ Þ σ 3 pffiffi ð þ Þ σ 4 with limκ σ→∞ dS ð Þ¼ and limκ σ→0 dS ð Þ¼ . 1 2 κ2σ 3 3 erfð pffiffiÞ − 3e =ð Þerfð pffiffiÞ In both cases we observe the dimension growing at small ð3þ1Þ 2κ σ 2κ σ d ðσÞ¼1 þ 2 ; ð63Þ S 2κ2σ 3 1 ffiffi − 2=ðκ σÞ 3 ffiffi scales above the classical value (see, however, Sec. IV D), erfð2κpσÞ e erfð2κpσÞ which—from the perspective of a random walk process (1)—is the pattern of superdiffusion. Finally, the case of ffiffi ffiffi with limκpσ→∞ ¼ 4, limκpσ→0 ¼ ∞; similarly, in 2 þ 1 1 1 — þ dimensions is trivial with no dimensional flow. dimensions C. Relative-locality Laplacian 2 1 ð2þ1Þ σ 1 dS ð Þ¼ þ 2 2 ; ð64Þ The last Laplacian that is of our interest has been κ σ 1 − e−1=ðκ σÞ proposed in the framework of relative locality. This pffiffi 3 pffiffi ∞ 1 1 relative-locality Laplacian is determined by the square of with limκ σ→∞ ¼ , limκ σ→0 ¼ ; and in þ dimen- the distance from the origin in momentum space sions [93], which has the same form for both the Lorentzian and  ffiffiffi  p −1= 4κ2σ 2 1 1 1 2κ σe ð Þ Euclidean metric signatures [19], namely, d ðp0;paÞ¼ dð þ Þ σ 1 1 ffiffiffi ; 65 2 2 S ð Þ¼ þ 2κ2σ þ pπ 1 ffiffi ð Þ κ arccosh ðp−1=κÞ. In bicrossproduct coordinates it erfð2κpσÞ becomes ffiffi ffiffi with limκpσ→∞ ¼ 2, limκpσ→0 ¼ ∞. The UV divergence of L ðP0;fP gÞ σ rl a     dSð Þ might be considered problematic, but such behavior 1 1 of the dimensionality (sometimes also for the Hausforff 2 2 P0=κ a −κ P0 e P P ; ¼ arccosh cosh κ þ 2κ2 a ð58Þ dimension) has been encountered in other models of quantum spacetime and at least in commutative geometry while in classical coordinates, seems to be a consequence of the extremely high con- nectivity between points of space [94].  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 L −κ2 2 2 a κ2 rlðp0;fpagÞ ¼ arccosh κ p0 þ pap þ : ð59Þ D. Comparison For the sake of comparison, let us first recall [Eq. (4)] The return probability in 3 þ 1 dimensions is now given by that the standard return probability on Rn reads

P σ −n=2 ð3þ1Þð Þ P ðσÞ¼ð4πσÞ : ð66Þ      class π5=2κ3 3 1 ffiffiffi 1=ð4κ2σÞ 2=ðκ2σÞ ffiffiffi −3 ffiffiffi ¼ 4pσ e e erf 2κpσ erf 2κpσ ; Hence, the dimension spectrum consists of a single element fn=2g and is of order 1. The spectral dimension is a ð60Þ constant function equal to n. The first observation is that for the bicovariant Laplacian in 2 þ 1 dimensions by the dimension spectra contain multiple elements. By analogy with the , one could interpret π3=2κ2 1=ðκ2σÞ this as a signature of some sort of curved geometry of κ- P 2 1 ðσÞ¼ pffiffiffi ðe − 1Þð61Þ ð þ Þ σ Minkowski space. Furthermore, in the (2 þ 1)-dimensional case the dimension spectrum is of order 2. From the and in 1 þ 1 dimensions by perspective of (pseudo), this means   that the heat trace expansion involves nonlocal coefficients. π3=2κ 1 1=ð4κ2σÞ The situation is analogous for the bicrossproduct Laplacian, P 1 1 ðσÞ¼ pffiffiffi e erf pffiffiffi ð62Þ ð þ Þ σ 2κ σ apart from the (1 þ 1)-dimensional case, for which the dimension spectrum coincides with the classical one. The L ([19] had only numerical results for the Laplacian rl). relative locality Laplacian does not have a dimension

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FIG. 7. Spectral dimension dSðσÞ in 3 þ 1 dimensions (left panel) and in 2 þ 1 dimensions (right panel) for the Laplacians Lcv (bottom curve), Lcp (middle curve) and Lrl (top curve) (we set κ ¼ 1). spectrum at all, which suggests that the corresponding because the considered Laplacians do not originate from a geometry is infinite dimensional. Dirac operator. In stark contrast with the quantum spheres, none of the Observe that in classical coordinates the bicovariant studied Laplacians on κ-Minkowski space exhibit complex Laplacian acquires the same form (40) as the standard numbers outside of the real axis in their dimension spectra. Laplacian on Rnþ1. This justifies the assumption that its Consequently, there are no oscillations in the corresponding order equals 2. On the other hand, the bicrossproduct spectral dimensions. The latter is also true for the relative- Laplacian in classical coordinates (50) looks as if was locality Laplacian. of order 1. Even more curiously, the relative-locality Let us now inspect the spectral dimensions closer. Laplacian in classical coordinates (59) seems to be of σ 3 1 The profiles of dSð Þ for different Laplacians in þ order 0.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Indeed, its leading behavior for large values of 2 1 2 a and þ topological dimensions are compared in Fig. 7 jpj ≔ p0 þ p p is log jpj, which grows slower than 1 1 a [the situation in þ topological dimensions is qualita- any power of jpj. L tively the same apart from the case of cp, for which The problem is that in the quantum gravity models one σ 2— dSð Þ¼ see Subsec. IV B]. All in both plots usually considers the quantum spacetime to be a certain exhibit the strong discrepancy in the UV, while they deformation of the classical one (more accurately, it is the converge to the same IR limit, equal to the topological classical spacetime that is an approximation of the quantum dimension. Unless we have some extra reason to claim that one), with the deformation controlled by some parameter(s) only one Laplacian is physically correct, the spectral related to the Planck length or mass. In particular, it is dimension seems to be an ambiguous characteristic of expected that a generalized Laplacian becomes the standard κ-Minkowski space. Some further comments about the one in the classical limit, when the deformation vanishes, i.e., relations between specific UV limits visible in Fig. 7 and we actually have a parametrized family of operators, valid at various results obtained within the quantum gravity different scales. In this context, Eq. (5) allows one to track a research, which could single out one of the Laplacians, deviation from the IR value of spacetime dimension. Using can be found in [19]. We also note that a recent analysis Eq. (10) with η ≠ 2 would prevent us from recovering the [95] of a static potential between two sources on (3 þ 1)- correct IR limit, unless we allow the order of the operator to dimensional κ-Minkowski space brings evidence that the depend on the deformation parameter. physical dimension in the UV is equal to 3, in agreement If we assumed the order of the bicrossproduct Laplacian with the result for the bicovariant Laplacian. to be a continuous function of κ, such that limκ→0 ηðκÞ¼1 σ Let us note, however, that the spectral dimensions dSð Þ and limκ→∞ ηðκÞ¼2, then we would obtain the dimension in the previous subsections were calculated under an n in the UV and n þ 1 in the IR, as in the bicovariant case. implicit assumption that all three operators L are of order In the same vein, we could set the order of the relative- 2—recall the discussion around Eq. (10). This is based on locality Laplacian to be a function of κ, satisfying the fact that these operators are deformations of the limκ→∞ ηðκÞ¼2. The situation in the UV limit in this classical Laplacian, which is a second-order differential case is more ambiguous since the dimension diverges, operator. In the case of the quantum spheres, we could while the order should tend to zero and their ratio could provide an external argument for the order of quantum yield any number. In particular, the behavior of ηðκÞ close Laplacians, which is based on a rigorous first-order con- to κ ¼ 0 might be such that limκ→0 2=ηðκÞdSðσÞ¼n,as 2 dition in the theory of spectral triples. In the case of for other Laplacians. Taking dSðσÞ¼dSðκ σÞ given by κ 2 -Minkowski space such an argument is not available, Eqs. (63)–(65) and considering the ansatz ηðκÞ ∝ κ ,we

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η κ ∼ 4 κ2 κ → 0 find that the desired limit requires ð Þ n for . quantum sphere. These were overlooked in [20], because of Such a trick allows us to remove the discrepancy between their very small amplitude for the deformation parameter the UV behavior of the spectral dimensions of all three close to the classical value. It is remarkable that the complex Laplacians, while maintaining the correct classical limit. It dimensions occur here in a somewhat unexpected setting, also agrees with the viewpoint on the operator orders where no fractal properties or discrete scale invariance were suggested by the formulas in classical coordinates. a priori imposed. Nevertheless, let us stress that the deformation-param- In contrast, such oscillations are present on κ-Minkowski eter-dependent order is only a tentative hypothesis, intro- spacetime for none of the three considered Laplacians, at duced by us here. Furthermore, even if this hypothesis was least in the cases of two, three and four topological rigorously implemented and erased differences between dimensions that we studied. One might relate this to the spectral dimensions for the three Laplacians, the corre- fact the quantum sphere is compact, whereas κ-Minkowski sponding dimension spectra would still remain irreconcil- spacetime is not and interpret the oscillations as a peculiar ably different. For example, the dimension spectrum in IR/UV mixing effect. On the other hand, from a more the case of 3 þ 1 topological dimensions, depending on conservative standpoint, one could simply say that the the Laplacian, is infinite, has only two elements or does quantum sphere is “more quantum.” In either case, it seems not exist. worth looking for the complex dimensions in other models of quantum spacetime. V. SUMMARY Finally, we would like to stress that the spectral dimen- sion and dimension spectrum of a given quantum spacetime Our analysis shows that the UV behavior of the spectral strongly depend on the chosen Laplacian. On the math- dimensions of quantum spacetimes is fairly complex and ematical side, this is simply a consequence of their defi- not limited to a monotonic flow. To understand it we have nitions, arising from the trace of the heat operator, which is adopted from noncommutative geometry ala` Connes a constructed from a given Laplacian. As we discussed in the rigorous notion of a dimension spectrum. The latter last subsection, the discrepancy between the UV behavior of characterizes the UV behavior of a heat trace, from which κ the spectral dimension is deduced. The relationship the spectral dimensions of -Minkowski space with differ- between the spectral dimension and the dimension spec- ent Laplacians can perhaps be removed using the concept of trum of a given Laplacian (more generally, an operator of the deformation-dependent order of the Laplacian but even order η) is captured by the following dictionary: then, differences will survive in the dimension spectra. On (a) The dimension spectrum consisting of more than one the other hand, while the dimension spectra of the quantum element implies a nonconstant behavior of the spectral sphere with the spinorial and scalar Laplacians are identical, dimension and suggests a nontrivial geometry. their spectral dimensions do not overlap and actually behave in the opposite ways in the IR. Thus, the spectral dimension (b) If the UV limit σ → 0 is finite, then dSð0Þ¼ηd , Sd and dimension spectrum provide complementary informa- where dSd is the largest in the dimension tion about a quantum spacetime structure. spectrum (14). Note that, in general, dSd need not be a natural number. This happens routinely in fractal In terms of physics, we believe that it is of key importance to understand which aspects of dimensionality are genuine geometry, where dSd recovers the Hausdorff dimen- sion [56,57,96–98]. to the given quantum geometry and which are just math- (c) The order of the dimension spectrum determines the ematical artifacts of the chosen Laplacian. Our analysis leading behavior of the spectral dimension for small σ. suggests that such universal properties are the log-periodic If ord Sd > 1, then this behavior is logarithmic. oscillations for the quantum sphere, as well as the lack of − κ More concretely, if PðσÞ ∼ ðlog σÞpσ r as σ tends to these for the -Minkowski spacetime, because these features persist for all of the studied Laplacians. The same could be 0, with p ¼ ord Sd − 1 ∈ N and r ≥ 0, then dSðσÞ ∼ 2r − 2p=ðlog σÞ. This happens on the quantum sphere said about the appearance of third-order poles in the with p ¼ 2 and r ¼ 0. On the other hand, a subleading dimension spectra on quantum sphere. As pointed out, this − − −1 “ ” logarithmic behavior PðσÞ ∼ ασ r þ βσ r ðlog σÞp could be taken as an indication of some sort of singular or p “nonsmooth” character of this quantum spacetime. On the translates to dSðσÞ∼2ðr−1Þþ2α=ðαþβσðlogσÞ Þ. This is exemplified on the (2 þ 1)-dimensional other hand, the dimension spectra of κ-Minkowski space- κ-Minkowski spacetime, with p ¼ 1 and r ¼ 1 or time fit within the ones obtained in classical (pseudo) r ¼ 2 for the bicovariant and the bicrossproduct differential geometry. Interestingly enough, the second- Laplacians, respectively. order poles, and hence the log σ terms in the heat trace, (d) The presence of nonreal numbers in Sd signifies occur only in 2 þ 1 topological dimensions, but for both the presence of log-periodic oscillations in the UV bicovariant and bicrossproduct Laplacians. This suggests behavior of the spectral dimension. that the geometry of (2 þ 1)-dimensional κ-Minkowski With the help of the dimension spectrum we have detected spacetime departs more strongly from the classical case. the log-periodic oscillations of the spectral dimension for the However, further studies—possibly connected with the

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sc expected nonlocal nature of some coefficients of the operator Δq defined by Eq. (27). Before we start we heat trace expansion—would be needed to confirm this need to take care of the zero mode, as for any s ∈ C the sc −s conclusion. operator ðΔq Þ is not trace class on the full Hilbert space H0 . We thus set h ≔ ðker ΔscÞ⊥ and compute the zeta ACKNOWLEDGMENTS q q function without the zero mode. Note that after such a The work of M. E. was supported by the National Science truncation, it is still possible to use the inverse Mellin Centre in Poland under the Research Grant Sonatina transform technique for the computation of the correspond- (No. 2017/24/C/ST2/00322). The publication was supported ing heat trace (see [59] for the general method). Indeed, by the John Templeton Foundation Grant “Conceptual we have Problems in Unification Theories” (No. 60671). sc sc sc −σΔq −σΔq Pq ðσÞ¼TrH0 e ¼ 1 þ Trhe APPENDIX: THE SPECTRAL ZETA q FUNCTION OF THE OPERATOR Δsc q and simply use formulas (16) and (17) with h at the place 0 In this Appendix we provide an explicit meromorphic of Hq. extension of the spectral zeta function associated with the Now, for ReðsÞ > 0 we have

X∞ Xj sc −s sc −s ζ sc Δ Δ Δq ðsÞ¼Trhð q Þ ¼ hj; mjð q Þ jj; mi j¼1 m¼−j X∞ pffiffiffi ¼ uð qÞ−s ð2j þ 1Þqs=2ðq−j − 1 − q þ qjþ1Þ−s j¼1 X∞ pffiffiffi ¼ðuð qÞq−3=2Þ−s ð2k þ 3Þðq−k − q − q2 þ qkþ3Þ−s k¼0 X∞ pffiffiffi ¼ðuð qÞq−3=2Þ−s ð2k þ 3Þqksð1 − qkþ1Þ−sð1 − qkþ2Þ−s: k¼0

ζ sc In order to construct a meromorphic extension of Δq to With the help of formula (A1) we rewrite the zeta the entire complex plane we use the standard binomial function as follows: expansion formula ffiffiffi X∞ p −3=2 −s ζ sc 2 3 Δq ðsÞ¼ðuð qÞq Þ ð k þ Þ   k¼0 X∞    s þ n − 1 X∞ X∞ ð1 − xÞ−s ¼ xn ðA1Þ s þ l − 1 s þ m − 1 n × n¼0 l m l¼0 m¼0 × qðkþ1Þlqðkþ2Þmqks: valid for any s and any x ∈ C with For ReðsÞ > 0 all of the series are absolutely convergent 1 sþn−1 ΓðsþnÞ jxj < . The coefficients ð n Þ¼ n!ΓðsÞ are polynomials and we are free to change the order of summation and in s of order n. compute the sum over k. We thus have

   ffiffiffi X∞ X∞ l − 1 − 1 lþ2m 3 − lþmþs p −3=2 −s s þ s þ m q ð q Þ ζΔsc ðsÞ¼ðuð qÞq Þ q l 1 − lþmþs 2 l 0 0 m ð q Þ ¼ m¼    X∞ Xn pffiffiffi s þ n − m − 1 s þ m − 1 qnþmð3 − qnþsÞ ¼ðuð qÞq−3=2Þ−s − 1 − nþs 2 0 0 n m m ð q Þ n¼ m¼   ffiffiffi X∞ n 3 − nþs − 1 p −3=2 −s q ð q Þ s þ n ¼ðuð qÞq Þ F1ð−n; s; −n þ s þ 1; qÞ; 1 − nþs 2 n 2 n¼0 ð q Þ with a hypergeometric function 2F1.

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The last series over n can easily be shown Gamma function contribute additional poles at −n, n ∈ N, (cf. Proposition 3.2 in [41]) to be absolutely convergent hence the order of the dimension spectrum is 3. πi Z − N Formula (24) for the meromorphic extension of the for any complex s outside of the discrete set log q .We function ζ sp is proved along the same lines with the help Δq have thus obtained a meromorphic extension of ζΔsc to the q of the identity (A1). The two zeta functions have the same ζ sc entire complex plane. It has isolated double poles precisely meromorphic structure, though Δq has a more involved Δsc Δsp πi Z − N in the set Sd q ¼ Sd q ¼ log q . The poles of the form of the coefficients.

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