PHYSICAL REVIEW D 102, 086003 (2020)
Spectral dimensions and dimension spectra of quantum spacetimes
† Michał Eckstein 1,2,* and Tomasz Trześniewski 3,2, 1Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, 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 gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterized by a discrete complex set—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 sphere 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.
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 length manifold, has proven to be extremely fruitful in modeling scale or a regularization cutoff (see  for a conceptual gravitational phenomena. However, it is generally sup- discussion). In analogy to systems in condensed matter posed that the smooth geometry breaks down at small physics, configurations of the underlying “atoms of space- scales or high energies, due to the quantum effects. time” 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 general relativity. which we completely recover familiar manifolds 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 number the classical differentiable manifold, 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 quantum gravity models , 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 . 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) diffusion 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 spaces determined by noncommutative algebras of diffusion time) that can be identified with the length scale at observables—ought to be characterized by a discrete subset which the geometry is probed. Therefore, it is naturally of the complex plane—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  (updated in ), 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 , asymptotic safety transform (see [31,32] and also ). Both spectral zeta scenario , nonlocal quantum gravity , spin foam models functions and heat trace expansions are indispensable tools  (and kinematical states of loop quantum gravity in general in quantum field theory [34–36], also in its noncommutative ), causal sets [11,12] and multifractional spacetimes . version . The heat trace can be utilized to compute The almost universal prediction is the dimensional reduction 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 . 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 ). sion spectra an interesting example is provided by the Quantum spacetime often turns out to be described in Podleś quantum sphere . For a particular choice of terms of broadly understood noncommutative geometry, geometry , the dimension spectrum turns out to have which is also sometimes treated as a stand-alone approach to surprising features : 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 , associated with κ-Poincar´e algebra [17,18] Secondly, the associated spectral zeta function contains and widely considered in doubly/deformed special relativity poles outside the real axis, suggesting self-similarity. and relative locality. As it was shown in , the small-scale Thirdly, these poles are of second order, which is character- behavior of the spectral dimension of κ-Minkowski space- istic for spaces with conical singularities . Finally, time (first calculated in ) 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 geometries (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  presents strengthens the observation made in  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 theories , 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 entropy . The reason is that various physical quantities  and can possibly affect the the latter can be derived from the heat trace over the cosmic microwave background spectrum , introduce a boundary of some region of space . stochastic noise to gravitational waves  or modify the The purpose of this work is to revisit the concept of the thermodynamics of photons . 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 , as well as condensed matter physics, earth- we compute the dimension spectra and spectral dimensions quakes and financial markets, see, e.g., . of three Laplacians on the Podleś quantum sphere .We Meanwhile, it has already been recognized by Connes and show that the dimension drop observed by Benedetti  Moscovici in 1995  that quantum spaces—understood has a finer structure with the square-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 volume V. (It factorizes parameter q close to the classical value 1 the amplitude in the leading term and therefore can be taken to infinity 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 point x ∈ V ⊂ M. The presence is clearly attested for any q by nonreal numbers 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 . This example uncovers an ambiguity in In the case of M being a flat Euclidean space 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 size 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 equation 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 group, 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  or even a pseudodifferential ð2πÞ4 one (see ). 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 Hilbert space H (cf.  and Appendix A of ). 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 . (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  to remove the zero mode. infinite-dimensional spaces . 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 . 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 rotation 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 ). 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 spheres) 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 ). 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 . 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.  and Sec. 2.5 of ). 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 . 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 . 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 . 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 fractal theleadingsmall-σ behavior,whilea0ðTÞ ∝ volðMÞ.IfT isa spaces , is interpreted as a signature of the self-similar scalarLaplace-typeoperatorandM hasnoboundary,thenthe structure . 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 .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  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  (cf. also Sec. 1.4 of ), 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. ): the dimension spectrum, including its order. They will not, however, change the maximal dimension dSd (see −σT PðσÞ¼TrHe Proposition 4.11 of ). 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 integers 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  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−2B A; elliptic pseudodifferential operator  (cf. also −2 1 − 1 − 2 Appendix A of ). 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