From dispersion relations to spectral dimension - and back again
aa r X i v : . [ h e p - t h ] D ec From dispersion relations to spectraldimension — and back again
Thomas P. Sotiriou , , Matt Visser , and Silke Weinfurtner SISSA - International School for Advanced Studiesvia Bonomea 265, 34136 Trieste, Italy, and
INFN, Sezione di Trieste. Department of Applied Mathematics and Theoretical Physics, CMSUniversity of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK School of Mathematics, Statistics, and Operations ResearchVictoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
E-mail: [email protected] , [email protected] , [email protected] Abstract:
The so-called spectral dimension is a scale-dependent number associatedwith both geometries and field theories that has recently attracted much attention,driven largely, though not exclusively, by investigations of causal dynamical triangu-lations (CDT) and Hoˇrava gravity as possible candidates for quantum gravity. Weadvocate the use of the spectral dimension as a probe for the kinematics of these (andother) systems in the region where spacetime curvature is small, and the manifold is flatto a good approximation. In particular, we show how to assign a spectral dimension (asa function of so-called diffusion time) to any arbitrarily specified dispersion relation.We also analyze the fundamental properties of spectral dimension using extensions ofthe usual Seeley–DeWitt and Feynman expansions, and by saddle point techniques.The spectral dimension turns out to be a useful, robust and powerful probe, not onlyof geometry, but also of kinematics.Published as: Physical Review D (2011) 104018doi: 10.1103/PhysRevD.84.10401811 May 2011; 15 August 2011; Published 8 November 2011;L A TEX-ed October 29, 2018.
Keywords: dispersion relations; spectral dimension; Lorentz symmetry breaking ontents – 1 – eferences 25
The spectral dimension has been proposed as possible observable characterizing thegeometry in discrete quantum gravity approaches [1]. It can be viewed as an effectivenotion of dimension defined though a fictitious diffusion process on the discrete geom-etry. In simple terms (see reference [1] for more details), the diffusion process can bethought as a stochastic random walk and the spectral dimension is defined in terms ofthe average return propability P ( s ) of the fictitious diffusion as d S ( s ) = − P ( s )d ln s , (1.1)with s being the (fictitious) diffusion time. As opposed to the topological dimension d ,the spectral dimension d S ( s ) need not be an integer and furthermore is scale-dependent.The concept of spectral dimension has attracted considerable interest in causaldynamical triangulations (CDT) [1–6], since finding meaningful observables in discretetheories is not a trivial task. (See also references [7–9] where the spectral dimension isused in noncommutative spacetimes, Euclidean dynamical triangulations and randomcombs respectively.) Such observables can provide the much needed connection withcontinuum theories (at suitable limits). Indeed the spectral dimension can be definedin the continuum quantum gravity models as well and be used to characterize andunderstand their short-distance behaviour (see references [10–14]). Furthermore, it wasshown in reference [10] that both CDT and Hoˇrava gravity seem to lead to a spectraldimension of 2 in the ultraviolet limit, while the value of the spectral dimension matchesthat of the topological dimension at larger length scales. These first results suggest thatthe spectral dimension can be a useful potential link between discrete and continuumtheories.Clearly if one is to utilize the spectral dimension in this way, it is crucial to firstget a deep understanding one what kind of information it actually carries in continuumtheories and also develop a set of tools for extracting this information. Our goal hereis to make a first step in this direction.Given the definition of a spectral dimension, it is actually quite intuitive (and willbecome clearer shortly) that, in continuum theories, its behaviour is scale-dependentand one can expect that: – 2 –. At long distances, scales which are comparable to, or larger than, the radius ofcurvature, the spectral dimension will depend on the details of the geometry andthe presence gravitational sources. At these scales the deviation of the spectraldimension from the topological dimension can be essentially attributed to curva-ture effects. Alternatively, at these scales the spectral dimension probes aspectsof the geometry.2. At intermediate scales, large compared to the Planck scale but small comparedto the radius of curvature, spacetime is effectively flat, and one should obtain aspectral dimension whose value matches that of the usual topological dimensionof spacetime.3. At short distances, (typically Planckian distances, but sometimes one is insteadinterested in the near-horizon limit of a black hole), curvature effects can certainlybe neglected, and the deviation of the spectral dimension from the topologicaldimension should be attributed to different effects.We will not be directly interested in regime 1 for the purposes of the current analysis.That is, we will not attempt any thorough analysis of how the spectral dimension canbe used as a probe of geometry or how curvature affects the spectral dimension (thoughwe will comment upon it briefly). Instead, in this article we shall focus on the flowof the spectral dimension from flat spacetime, which is for our purposes the infraredlimit, into the ultraviolet limit (regime 2 above to regime 3 above). The correspondinglength scales are of particular interest in quantum gravity candidates. We shall developa number of theoretical results, and complement them with analytic and computationaltools that allow for both exact and approximate evaluation of the spectral dimensionin a number of interesting situations.In particular, we shall demonstrate that the spectral dimension is not necessarily intrinsically geometric. At scales small enough for curvature effects to be negligibleits deviations from the topological dimension it actually becomes an analytic propertyof the differential operator (ultimately equivalent to a dispersion relation) that oneis using as input to define the generalized diffusion process (as already suggested bythe analysis of reference [10] for the particular case of Hoˇrava gravity). In turn, thisoperator acts as the propagator of some dynamical degree of freedom in flat space. Inthis sense, the spectral dimension acts, at suitable scales, as a probe of the kinematicsof the particular degree of freedom. Within this framework we will show that, given aspecified topological dimension d , it is possible to define a scale-dependent notion of aspectral dimension d S ( s ) for any arbitrary dispersion relation ω = Ω( k ). We shall alsoderive a formal inversion process, so that from the spectral dimension, (as a function– 3 –f s ), and the topological dimension d , one can in principle reconstruct the dispersionrelation.For some specific dispersion relations the situation is sufficiently simple that it ispossible to explicitly evaluate the relevant integrals, and so explicitly determine thespectral dimension in terms of standard special functions (such as Bessel functions and[Gaussian] error functions). More generally, we shall develop a number of analytic tools,(using a generalization of the Seeley–DeWitt expansion in the ultraviolet, and a gener-alization of the Feynman expansion in the infrared), that give us good analytic controlof these two limits. We also develop an approximation based on saddle point tech-niques to express the spectral dimension in terms of the phase and group velocities ofthe underlying dispersion relation — this approximation agrees with asymptotic resultsin the ultraviolet and infrared, and can also be used in the intermediate regime. (Whensaddle point and exact integral techniques overlap, the saddle point approximation willbe seen to be a tolerably good global fit to the exact result.)Apart from those issues of relevance to the quantum gravity and field theory com-munities, we additionally point out that the formalism derived in this article may alsobe of relevance in completely non-relativistic situations (such as the propagation ofsurface waves on liquid-gas interfaces.) Overall, we wish to advocate the use of spectraldimension as a general, powerful, and robust probe of the kinematics (specifically thedispersion relation) of whatever system one has under consideration. Suppose we are working in flat ( d + 1) dimensional spacetime and have a (in generalLorentz-violating) dispersion relation ω = Ω( k ) . (2.1)Now define f ( k ) = Ω( k ) . (2.2)This dispersion relation can always be viewed as being completely specified by thesolutions of the (in general Lorentz-violating) differential equation D L Φ ≡ (cid:8) − ∂ t − f ( −∇ ) (cid:9) Φ = 0 . (2.3)The reason that we choose the differential operator to be second order in physical time t is a purely pragmatic one — with such a choice the differential equation encoding the– 4 –ispersion relation can typically be derived from a ghost free non-interacting Lagrangian L = 12 Φ D L Φ = 12 Φ (cid:8) − ∂ t − f ( −∇ ) (cid:9) Φ . (2.4) First we Wick-rotate in physical time t to consider the Euclideanized differential oper-ator D E in ( d + 1) topological dimensions D E Φ ≡ (cid:8) − ∂ t + f ( −∇ ) (cid:9) Φ . (2.5)Now consider the generalized diffusion process (cid:26) ∂∂s + D E (cid:27) ρ ( x, x ′ , s ) = 0; ρ ( x, x ′ ,
0) = δ d +1 ( x − x ′ ) , (2.6)where D E is the differential operator defined above. We set x = ( t, ~x ), and s is bestthought of as an auxiliary diffusion time. (That is, s is a “fake time”, a Lorentz-violatinggeneralization of Schwinger–De Witt proper time — as in the usual Schwinger–De Wittproper-time regularization of spacetime propagators.) We shall soon see that s can alsobe thought of as the scale at which one is probing the diffusion process — we shall seethat s → s → ∞ probes the infrared. The generalsolution of equation (2.6) is given by ρ ( x, x ′ , s ) = Z d d k d ω (2 π ) d +1 exp { i [ ~k · ( ~x − ~x ′ ) + ω ( t − t ′ )] } exp {− s [ ω + f ( k )] } , (2.7)as can easily be verified by differentiation and by noting that the boundary conditionis satisfied at s = 0. The quantity ρ ( x, x ′ , s ) is often called the heat kernel and canbe thought of as the probability density that a particle originally at x ′ at s = 0 willdiffuse to x in diffusion time s . Now consider the average return probability density P ( s ) (also called the diagonal part of the heat kernel): P ( s ) = ρ ( x, x, s ) = Z d d k d ω (2 π ) d +1 exp {− s [ ω + f ( k )] } , (2.8)(We are working in flat space where, due to translational invariance the average returnprobability equals the return probability.) The spectral dimension is defined as d S ( s ) = − P ( s )d ln s = − ρ ( x, x ; s )d ln s . (2.9)The reason for this definition is that if we consider the particularly simple differentialoperator D = (cid:3) = −∇ , then in ( d + 1) topological dimensions then one can show that P −∇ ( s ) = ρ −∇ ( x, x ; s ) ∝ s − ( d +1) / (2.10)– 5 –o that in this specific case d S ; −∇ ( s ) = − P −∇ ( s )d ln s = − ρ −∇ ( x, x ; s )d ln s = d + 1 . (2.11)That is: The spectral dimension d S ; D E ( s ) of the complicated diffusion process governedby the differential operator D E is the (fictitious) topological dimension of that sim-ple diffusion process governed by −∇ that most closely approximates P D E ( s ) at theindicated value of the diffusion time s . We emphasise that this is a definition, not atheorem. This definition is useful only insofar as it leads to interesting results. Factorize P ( s ) into “time” and “space” contributions P ( s ) = (cid:20)Z d ω π exp {− s ω } (cid:21) × (cid:20)Z d d k (2 π ) d exp {− s f ( k ) } (cid:21) . (2.12)The first integral is elementary, so one has P ( s ) = (cid:20) √ πs (cid:21) × (cid:20)Z d d k (2 π ) d +1 exp {− s f ( k ) } (cid:21) . (2.13)Dropping irrelevant constants of proportionality (they would in any case drop out inthe calculation of the spectral dimension) we see P ( s ) ∝ √ s Z k d − exp {− s f ( k ) } d k. (2.14)That is ln P ( s ) = −
12 ln s + ln Z k d − exp {− s f ( k ) } d k + C. (2.15)(Note that C will be used to denote a generic constant, when we re-use the symbolbelow it will not necessarily represent the same constant.) This expression lets youcalculate the exact spectral dimension as d S ( s ) = 1 + 2 s R f ( k ) k d − exp {− s f ( k ) } d k R k d − exp {− s f ( k ) } d k . (2.16)Note that the leading “1” comes from the fact that the physical time part of thedifferential operator D E is trivial — it is just ∂ t . All the complications come from the d spatial dimensions. If we now write the dispersion relation as ω = f ( k ) = Ω( k ) , (2.17)– 6 –hen, still as an exact result: d S ( s ) = 1 + 2 s R Ω( k ) k d − exp {− s Ω( k ) } d k R k d − exp {− s Ω( k ) } d k . (2.18)It is now useful to define a partition function Z ( s ) = Z k d − exp {− s f ( k ) } d k = Z k d − exp {− s Ω( k ) } d k . (2.19)Now by construction Z ( s ) ∝ P ( s ), so on the one hand it carries no extra information.On the other hand it lets us formally rewrite the spectral dimension as an ensembleaverage: d S ( s ) = 1 − s d ln Z ( s )d s = 1 + 2 s h Ω( k ) i s . (2.20)Note that this partition function encodes relatively simple information concerning thedispersion relation of the specific degree of freedom under consideration. It is by nomeans the partition function of the entire system.This completes the first (and simplest) part of the programme: From any arbitrarydispersion relation ω = Ω( k ), and specified topological dimension d , we have seen howto construct suitable differential operators D L amd D E that encode this dispersionrelation, and how to use the generalized diffusion process associated with the differentialoperator D E in order to define the corresponding spectral dimension d S ( s ). The definitions given above allow one to determine the spectral dimension for a givendispersion relation. This is the usual case when one deals with a continuum theory andcan linearize the field equations in order to determine the dispersion relation. In suchcases the spectral dimension acts as an additional probe of the kinematics describedby this dispersion relation (and potentially the geometry, see also section 2.5.2 below).There are, however, cases where one can calculate the spectral dimension by othertechniques and it then becomes useful to determine a corresponding (effective) disper-sion relation. A typical example arises in discrete theories, such as CDT, where onecan define and calculate the spectral dimension directly from a discretized diffusionprocess [1, 6]. Clearly, in such theories the spectral dimension is not just an alternativeto studying directly the propagator. Since the latter is simply not available, the formerbecomes a far more essential tool for obtaining information about the theory. Deter-mining a corresponding (effective) dispersion relation could facilitate the comparisonto some candidate continuum effective theory, see reference [15] for an example.– 7 – .4.1 Dispersion relation as inverse Laplace transform
Formally one can invert the process above and from the spectral dimension (as a func-tion of s ), plus the specified topological dimension, reconstruct the dispersion relation.This is best done in two steps. Fromd ln Z ( s )d s = − d S ( s ) − s , (2.21)we have Z ( s ) = Z ( s ) exp (cid:26) − Z ss d S (¯ s ) − s d¯ s (cid:27) . (2.22)That is, up so some constant of proportionality Z ( s ), we can certainly reconstruct thepartition function Z ( s ) from the spectral dimension. Can we take this any further?Can we reconstruct the dispersion relation Ω( k ) from the spectral dimension d S ( s )?After some mathematical manipulations one can bring equation (2.19) in the form Z ( s ) = 1 d Z ∞ exp {− s Ω } d { k (Ω) d } d[Ω ] d[Ω ] . (2.23)Performing an integration by parts yields Z ∞ exp {− s Ω } { k (Ω) d } d[Ω ] = ds Z ( s ) . (2.24)This has the form of a Laplace transform, in the variable Ω , of the function k (Ω) d .Implementing the inverse Laplace transform via a complex integration we have k (Ω) d = 12 πi Z C ds Z ( s ) e Ω s d s, (2.25)where C is an appropriate contour in the complex plane and Z ( s ) is given by equa-tion (2.22). Therefore, in principle one can calculate the dispersion relation when thespectral dimension is analytically known on the complex plane as a function of s . Though it settles an important issue of principle, the method of calculating the disper-sion relation from the spectral dimension presented above is usually of little practicalvalue. This is because, in either discrete or continuum theories, when one might be in-terested in determining a corresponding dispersion relation from a spectral dimension,the latter is typically not known analytically as a function of s . Hence, an algorithmthat requires analytic continuation of the spectral dimension to complex values of s isimpractible. This problem can be circumvented in various ways:– 8 – Direct inversion via semi-analytic model building:
If one has somehow, (for in-stance, from some numerical simulation followed by some data-fitting), built asemi-analytic model for the spectral dimension d S ( s ), then integrating to findthe partition function Z ( s ) is typically easy. This semi-analytic model partitionfunction Z ( s ) can be analytically continued to the complex plane, and sometimesone is lucky — one may encounter a model for the partition function for whichthe inverse Laplace transform is explicitly known. Even when this works, weemphasise that one is analytically continuing the model, not the data. • Direct inversion via infinite differentiation:
There is a little-known method dueto Post [16], see also Bryan [17], that allows for inversion of Laplace transforms bytaking arbitrarily high derivatives. Specifically, if G ( s ) is the Laplace transformof g ( t ) then g ( t ) = lim n →∞ ( − n n ! (cid:16) nt (cid:17) n +1 G ( n ) (cid:16) nt (cid:17) . (2.26)This algorithm may not always be practical, since one needs arbitrarily highderivatives. Even if not always practical, it again settles an important issue ofprinciple — — knowledge of the spectral dimension d S ( s ) in principle allows oneto reconstruct an equivalent dispersion relation Ω( k ). • Indirect inversion via nonlinear regression:
In contrast, a practical algorithm forextracting (or rather, approximating) the dispersion relation is to build sometheoretically appealing model for Ω( k ) with several adjustable parameters, andto then use the technique of nonlinear regression to find a best fit within thatclass of dispersion relations. In a companion paper [15] we have applied this ideato the measured spectral dimension function arising in CDT models of quantumgravity. Let us now indicate some possible generalizations and extensions of this formalism.
If one is (perhaps foolishly) willing to play with ghost fields then there is no need torestrict to second order derivatives in physical time. Consider the operator D Φ ≡ f ( − ∂ t , −∇ ) Φ . (2.27)The relevant dispersion relation is now only implicitly defined by f ( ω , k ) = 0 . (2.28)– 9 –n the other hand the heat kernel becomes ρ ( x, x ′ , s ) = Z d d k d ω (2 π ) d +1 exp { i [ ~k · ( ~x − ~x ′ ) + ω ( t − t ′ )] } exp {− s [ f ( ω , k )] } , (2.29)and the diagonal part of the heat kernel specializes to P ( s ) = ρ ( x, x, s ) = Z d d k d ω (2 π ) d +1 exp {− s f ( ω , k ) } . (2.30)So the spectral dimension is d S ( s ) = 2 s R f ( ω , k ) exp {− s f ( ω , k ) } d ω k d − d k R exp {− s f ( ω , k ) } d ω k d − d k = 2 s h f ( ω , k ) i s (2.31)and depends on the whole of the function f ( ω , k ), while the dispersion relation isgiven by the solution of equation (2.28) and uses only limited information about thefunction f ( ω , k ).If the differential operator can be put in the form D Φ ≡ (cid:8) f ( − ∂ t ) + f ( −∇ ) (cid:9) Φ , (2.32)then the space and time contributions to the spectral dimension become separable andone has d S ( s ) = 2 s R f ( ω ) exp {− s f ( ω ) } d ω R exp {− s f ( ω ) } d ω + 2 s R f ( k ) k d − exp {− s f ( k ) } d k R k d − exp {− s f ( k ) } d k . (2.33)Clearly, when f ( − ∂ t ) = − ∂ t , one recovers the result given in equation (2.16). A perhaps more interesting question is to ask what happens if spacetime is allowedto be curved, in addition to having a nontrivial dispersion relation. Extending thedefinition of the spectral dimension to dispersion relations describing propagators in a given curved background is actually rather straightforward: D E in equation (2.6) willbe a differential operator in curved space, but the solution of this equation can still beused to calculate the return probability. The only subtlety is that when one has to dealwith spaces which do not respect translational invariance, the return probability wouldbe space dependent, P ( x, s ). Thus, so long as one does not want the spectral dimensionto depend on the spacetime point, one has to actually use the average return probabilityin the definition of equation (2.9) (as was mentioned already in the the introduction).An Arnowitt–Desser–Misner decomposition provides a natural setting for constructingdifferential operators which include curvature and at the same time lead to nontrivialdispersion relations even around maximally symmetric spaces.– 10 – .6 Space versus spacetime The d S ( s ) we have so far been considering is the spectral dimension of spacetime . Ifone is only interested in the spectral dimension of space , (as is usual for most classical[non-relativistic] physicists), then the appropriate thing to do is to simply drop the t coordinate and define D space = − f ( −∇ ) . (2.34)One would then consider the diffusion process (cid:26) ∂∂s − f ( −∇ ) (cid:27) ρ ( x, x ′ , s ) = 0; ρ ( x, x ′ ,
0) = δ d ( x − x ′ ) . (2.35)In this case (now without the leading “1”) we have d S, space ( s ) = 2 s R f ( k ) k d − exp {− s f ( k ) } d k R k d − exp {− s f ( k ) } d k , (2.36)or equivalently d S, space ( s ) = 2 s R Ω( k ) k d − exp {− s Ω( k ) } d k R k d − exp {− s Ω( k ) } d k = 2 s h Ω( k ) i s . (2.37)Note that (assuming physical time only shows up in the simple form ∂ t ) the onlydifference between the spectral dimension of spacetime, and that of space itself, is theleading “1”. Furthermore, since t has been eliminated from the formalism, the diffusiontime s can in this situation be identified with physical Newtonian time. We shall now explore some calculational techniques for determining the spectral di-mension d S ( s ). In simple situations we can obtain exact analytic results, but moretypically one has to resort to some form of approximation. We shall show that makinguse of asymptotic expansions can be fruitful in both the infrared and ultraviolet, andthat saddle point techniques can also be employed — typically over the whole range ofinterest.In the following discussion we will also present several examples. For simplicity wewill largely focus on polynomial dispersion relations, where ω = c w k w + c w +1 k w +2 + · · · + c z − k z − + c z k z . (3.1)– 11 –ere w ≤ z are integers potentially unrelated to d . As additional examples (better dealtwith using asymptotic or saddle point techniques) we will also use rational polynomialdispersion relations of the form ω = p ( k ) q ( k ) , (3.2)where p ( k ) and q ( k ) are polynomials in k , and surface-wave dispersion relations thatsometimes include trigonometric functions.The choice of these specific types of dispersion relations as examples is motivatedmore by mathematical simplicity and their demonstrative power, than by their originor genericity. Nevertheless, all of them are common (to different extents). Polynomialdispersion relations are typical in (truncated) effective field theories that exhibit Lorentzviolations. They also appear in projectable Hoˇrava gravity [18–21]. (Projectabilityrequires that the lapse function is a function of time only, N = N ( t ).) In the lattercase, the dispersion relation is actually polynomial in k with the highest power being( k ) d , that is, ω = c k + c k + · · · + c d k d . (3.3)and with particularly pleasant renormalizability properties being associated with thechoice z = d [18, 22, 23]. Rational polynomial dispersion relations are generic in moregeneral, non-projectable Hoˇrava gravity [24, 25]. (See reference [26] for a brief reviewon Hoˇrava gravity.) Finally, gravity-driven surface waves have dispersion relationsthat include trigonometric functions. In any case, the techniques we present here areapplicable to more general dispersion relations. Useful exact results seem to be limited to special cases of the two-term dispersionrelation ω = c w k w + c z k z . (3.4)More complicated dispersion relations, or even the general case above, typically requireone to invoke some sort of series expansion or approximation technique. The so-called Bogoliubov dispersion relation can conveniently be put in the form f ( k ) = k + k d K ; Ω( k ) = k r k d K . (3.5)(The factor of 2 d is there just to minimize irrelevant numerical factors in the resultsbelow.) – 12 – wo space dimensions: The partition function for the Bogoliubov dispersion rela-tion can be explicitly evaluated in terms of the error function Z ( s ) = Z k exp (cid:26) − sk (cid:18) k K (cid:19)(cid:27) d k (3.6)= K r πs exp (cid:0) sK (cid:1) h − erf (cid:16) √ sK (cid:17)i . (3.7)The spectral dimension is then explicitly evaluable as d S ( s ) = 2 − sK + 2 r sK π e − sK − erf( √ sK ) . (3.8)Note the limits d S ( s →
0) = 2; d S ( s → ∞ ) = 3 . (3.9) Three space dimensions:
In this situation the partition function for the Bogoliubovdispersion relation can be explicitly evaluated in terms of Bessel functions: Z ( s ) = Z k exp (cid:26) − sk (cid:18) k K (cid:19)(cid:27) d k (3.10)= √ K exp (cid:0) sK (cid:1) (cid:2) K / (cid:0) sK (cid:1) − K / (cid:0) sK (cid:1)(cid:3) . (3.11)The spectral dimension is then explicitly evaluable as d S ( s ) = 2 − sK − (cid:20) K / ( sK ) + K / ( sK ) K / ( sK ) − K / ( sK ) (cid:21) . (3.12)Note the limits d S ( s →
0) = 52 ; d S ( s → ∞ ) = 4 . (3.13) Consider the specific dispersion relation f ( k ) = k (cid:18) k K (cid:19) = k + k K ; Ω( k ) = k r k K . (3.14)(The “27” is there just to minimize irrelevant numerical factors in the results below.)This dispersion relation is potentially interesting as can serve as a good approximationfor both the high-momentum and low-momentum limit of Hoˇrava gravity [10] (it inter-polates between low-momentum Lorentz invariance and a z = 3 Lifshitz point at large– 13 –omentum). It is not, however, the exact dispersion relation of any of the versions ofthe model in 3+1 dimensions, (there is no k term, and it is not a rational polynomial).The partition function can be explicitly evaluated in terms of Bessel functions Z ( s ) = Z k exp (cid:26) − sk (cid:18) k K (cid:19)(cid:27) d k (3.15)= − π / K √ s (cid:2) J / (cid:0) sK (cid:1) J − / (cid:0) sK (cid:1) + Y / (cid:0) sK (cid:1) Y − / (cid:0) sK (cid:1)(cid:3) . The spectral dimension is then explicitly evaluable as d S ( s ) = 2 + 2 sK " J / ( sK ) − J − / ( sK ) + Y / ( sK ) − Y − / ( sK ) J / ( sK ) J − / ( sK ) + Y / ( sK ) Y − / ( sK ) . (3.16)Note the limits d S ( s →
0) = 2; d S ( s → ∞ ) = 4 . (3.17) As a final exact example, albeit one where the results are too messy to actually explicitlywrite down, in (3+1) dimensions consider the dispersion relation f ( k ) = 3 k + 2 k K . (3.18)The numerical factors are again chosen to keep expressions as simple as possible. Thepartition function can then be evaluated as a sum of generalized hyper-geometric func-tions, specifically as a sum of three F ( − sK ) hyper-geometric functions. We have Z ( s ) = Z k exp (cid:26) − s (cid:18) k + 2 k K (cid:19)(cid:27) d k = 112 r πK s " F / , / , [1 / , / ( − sK ) − Γ( )4 √ π (2 sK ) / F / , / , [2 / , / ( − sK )+ 15Γ( )16 √ π (2 sK ) / F / , / , [4 / , / ( − sK ) . (3.19)The resulting spectral dimension is too messy to be worth writing down explicitly, butthe limits are straightforward to evaluate: d S ( s →
0) = 2; d S ( s → ∞ ) = 52 . (3.20)– 14 – .1.4 Summary of exact results It should be obvious by now that even relatively simple dispersion relations, such as ω = c w k w + c z k z , (3.21)lead to rather complicated expression for the spectral dimension. Therefore, it is inter-esting to develop techniques that could allow us to determine the spectral dimensionto some required accuracy based on a suitable approximation. This will be the aim ofthe in the following sections.Apart from demonstrating the difficulty of determining the spectral dimension fora given dispersion relation, the exact results presented above provide also some insightinto how the spectral dimension changes with the length scale. Given that s is a(fictitious) diffusion time, small values of s probe small length scales and large valuesof s probe large scales. This is the reason why we paid particular attention to the limits s → s → ∞ . For dispersion relations of the form given above and whenever wehave been able to explicitly carry out the exact calculation we have found that d S ( s →
0) = 1 + dz ; d S ( s → ∞ ) = 1 + dw , (3.22)in ( d + 1) topological dimensions.Note that, from a field theory point of view it is customary to identify the smalllength scale limit with k → ∞ and the large length scale limit with k →
0. Indeed,it is intuitive that there is generically correspondence between s → k → ∞ orbetween s → ∞ and k →
0. In fact, if one is interested in calculating only theselimits, it is much easier to suitably truncate the dispersion relation for small or large k and then directly calculate the spectral dimension in these limits (as was done inreference [10]).It is worth mentioning that the correspondence between the two limits (in s and k ) can be readily seen from the end result of the three examples we gave above. Noticethat in all three cases the spectral dimension is actually a function of sK n , where K is the energy scale suppressing the higher order momentum terms in the dispersionrelation (and n is the exponent of the lower order term based on dimensional analysis).Hence, clearly the limit s → s → ∞ ) will give no different results than the limit K → K → ∞ ). However, given the form of dispersion relations we are considering,driving K to infinity is no different than taking k →
0, and driving K to zero is notdifferent than taking k → ∞ .The general pattern relating the small/large length scales (ultraviolet/infrared lim-its) of the spectral dimension suggested by our analytic examples will be be investigatedin the next two sections using both asymptotic techniques and saddle point methods.– 15 – .2 Asymptotic expansions Since obtaining explicit analytic results for the spectral dimension seems to be limitedto very simple dispersion relations, it is worth developing perturbative expansions thatcould help us determine the former from the latter to some desired accuracy in aregime of interest. Specifically, if one is interested in studying the very small lengthscale (ultraviolet) or the very large length scale (infrared) behaviour of the spectraldimension, then asymptotic techniques are particularly useful. These are based on theidea of dividing the exponential appearing in the partition function, e − sf ( k ) , into adominant piece plus perturbative corrections. Exactly how one does the division willdepend on which of the two asymptotic regimes one is interested in probing.Consider a dispersion relation of the form ω = f ( k ) = c w k w + c w +1 k w +2 + · · · + c z − k z − + c z k z , (3.23)with 1 ≤ w < z . Then one can certainly write bothexp[ − sf ( k )] = exp( − sc w k w ) (cid:8) − sc w +1 k w +2 + . . . (cid:9) , (3.24)and exp[ − sf ( k )] = exp( − sc z k z ) (cid:8) − sc z − k z − + . . . (cid:9) , (3.25)by performing expansions around k = 0 and s = 0 respectively. The series in bracesconsist of various positive powers of s and k and are guaranteed convergent for allvalues of s and k . The sub-leading terms appearing in equation (3.24) are all of theform s p k q with q ≥ p ( w + 1). Similarly sub-leading terms appearing in equation (3.25)are all of the form s p k q with q ≤ p ( z − ω = f ( k ).If f ( · ) does not have a pole at k = 0, then one can always write ω = f ( k ) = c w k w + c w +1 k w +2 + . . . , (3.26)which is sufficient to obtain equation (3.24). On the other hand, as long as f ( · ) hasa pole of finite order at k → ∞ , one can always write the dispersion relation for verylarge k as a Taylor series of descending powers of k , i.e. ω = f ( k ) = c z k z + c z − k z − + . . . . (3.27)Again, this is sufficient for obtaining equation (3.25). Both requirements are fulfilledby reasonable and phenomenologically interesting dispersion relations.– 16 –e are now ready to proceed to suitable ultraviolet and infrared perturbativeexpansions. Before doing so, it is worth mentioning the following. We have restrictedour discussion so far to dispersion relations that respect parity invariance. This is avery well motivated simplifying assumption: The lowest order term that violates parity( ω ∝ k ) is important at low energies and would, therefore, lead to severe disagreementwith observations. However, one could relax parity invariance and introduce some othersymmetry that would prohibit the presence of lower-order but not higher-order parityviolating terms (one example seems to be Hoˇrava’s detailed balance condition [18]). Inany case, even though we will retain parity invariance for the sake of brevity, it is notcrucial for our approach. After deriving the asymptotic expansions, we will commenton how they would be modified if parity invariance were to be abandoned. We now argue that there is an appropriate generalization of the usual Seeley–De Wittexpansion beyond ordinary second-order differential operators. If f ( −∇ ) is polynomialand contains terms up to order ∇ z , then there will exist an asymptotic expansion (interms of positive fractional powers of s ) of the form Z ( s ) = Cs d/ (2 z ) ( N X n =0 a n s n/z + O (cid:0) s ( N +1) /z (cid:1)) , (3.28)where we normalize to a = 1, (and C is for our current purposes uninteresting). Theusual Seeley–De Witt expansion corresponds to z = 1. Note that we have alreadypeeled off physical time and are only dealing with the spatial part of the heat kernel.To convince yourself that this expansion holds, proceed as follows: Focus on the highest-derivative piece. A straightforward computation (using the definition if the Γ function)yields Z ( −∇ ) z ( s ) = Z k d − e − sk z d k = Γ( d z )2 z s − d/ (2 z ) . (3.29)Note that the sub-leading terms are all of the form s p k q and so will contribute Z k d − { s p k q } exp( − sk z ) d k ∝ s − d/ (2 z ) · s p · s − q/z . (3.30)So in particular, for the first sub-leading term in equation (3.25) which is proportionalto sk z − , we have Z k d − { sk z − } exp( − sk z ) d k ∝ s − d/ (2 z ) · s · s − · s +1 /z = s − d/ (2 z ) · s +1 /z . (3.31)– 17 –hen, at the very least Z ( s ) will contain the terms Z ( s ) = C { a s − d/ (2 z ) + a s − d/ (2 z ) · s /z + . . . } . (3.32)Higher order terms s p k q contribute to the partition function with relative strength s p · s − q/z , that is s ( pz − q ) /z . But pz − q ≥ p is a positive integer, so these are all positivepowers s n/z of s /z . That is, this series for Z ( s ) (which can at best be an asymptoticseries) continues with increasing powers of ( s /z ) n . This implies that Z ( s ) is indeedgiven by equation (3.28) as claimed.Now, turning to the spectral dimension, one can straightforwardly derive d S ( s ) = 1 + dz − a z s /z + O ( s /z ) . (3.33)So, in the deep ultraviolet ( s →
0) the spectral dimension flows generically (moduloour assumptions about parity invariance) to d S ( s ) → dz , (3.34)with a prescribed rate of O ( s /z ), where 2 z is the order of the pole of the dispersionrelations as k → ∞ (and d the topological spatial dimension). We are (for currentpurposes) not particularly interested in the specific value of the coefficient a , thoughit can be calculated without any particular difficulty. When we normalize to a = 1 abrief calculation yields a = − c z − ( c z ) ( z − /z Γ( d − z + 1)Γ( d z ) . (3.35)This quantity a is not, in any sense, universal like the usual Seeley–De Witt coefficients— a will depend explicitly on the coefficients in the dispersion relation f ( k ). Had weallowed for parity violations the only qualitative modification in the spectral dimensionas given by equation (3.33) would be that z would be allowed to take half-integer values. Turning to the infrared, we now argue that there is an appropriate “inverted” gen-eralization of the Seeley–De Witt expansion beyond ordinary second-order differentialoperators. (This is very close in spirit to the usual Feynman diagram expansion arounda Gaussian integral.) Specifically, if w = 1 so that the low-momentum dispersion rela-tion starts off as f ( k ) = k + ... , we assert the existence of an asymptotic expansion(now in inverse integer powers of s ) of the form Z ( s ) = Cs d/ ( N X n =0 ˜ a n s − n + O (cid:0) s − ( N +1) (cid:1)) , (3.36)– 18 –here we can normalize to ˜ a = 1. Even if we do not have Lorentz invariance in thelow energy limit ( w > Z ( s ) = Cs d/ w ( N X n =0 ˜ a n s − n/w + O (cid:0) s − ( N +1) /w (cid:1)) . (3.37)To convince yourself that this is true, proceed as follows (many of the technical stepsare very similar to those for the ultraviolet limit and some details will be suppressed).Focus on the lowest-derivative piece, for which we have Z ( −∇ ) w ( s ) = Z k d − e − sk w d k ∝ s − d/ w . (3.38)Furthermore, note that all the sub-leading terms are of the form s p k q , and so by aminor variant of the previous argument they contribute quantities of relative magnitude s p · s − q/w to the partition function Z ( s ). But the first sub-leading term in equation(3.24) is proportional to sk w +2 , so at the very least Z ( s ) will contain the terms Z ( s ) = C { a s − d/ (2 w ) + s · a s − d/ (2 w ) · s − ( w +1) /w + . . . } . (3.39)Higher order terms contribute with relative strength s p · s − q/w = s ( pw − q ) /w , butfrom equation (3.24) we know q > p ( w + 1) so pq − q < − p is a negative integer.So higher order terms contribute with relative strength s − n/w , that is, with positiveinteger powers of s − /w . Thus, Z ( s ) is indeed given by equation (3.37) as claimed. Thecorresponding spectral dimension is then d S ( s ) = 1 + dw + 2˜ a s − /w + O ( s − /w ) . (3.40)Thus deep in the infrared ( s → ∞ ), as long as the low-momentum dispersion relationis of the form f ( k ) = k + k + ... , (that is w = 1), the spectral dimension will flow to d + 1, the topological dimension of spacetime, at a rate O (1 /s ). More generally, it willflow to 1 + d/w at a rate of O ( s − /w ). As before, had we given up parity invariance, w would be allowed to take half-integer values. Finally, ˜ a is not in any sense universaland can be calculated without difficulty. Normalizing to ˜ a = 1, a brief calculationyields ˜ a = − c w +1 ( c w ) ( w +1) /w Γ( d +22 w + 1)Γ( d w ) . (3.41)– 19 – .2.3 Example: Rational polynomial dispersion relation As previously mentioned, non-projectable Hoˇrava gravity models lead to dispersionrelations that are rational ratios of polynomials in k [24, 25] so that ω = p ( k ) q ( k ) . (3.42)Since p ( k ) and q ( k ) are both polynomials in k we can argue that p ( k ) q ( k ) → c z k z + O ( k z − ) as k → ∞ . (3.43)Then we have f ( k ) = c z k z + O ( k z − ) as k → ∞ . (3.44)Therefore for the spectral dimension we have d S = 1 + dz + O ( s /z ) . (3.45)A similar procedure works in the infrared. At low k any rational ratio p ( k ) /q ( k ) canbe expanded as a formally infinite power series in k . Ignoring issues of convergence,we can then still apply the generalized Feynman expansion to argue for behaviour ofthe form d S = 1 + dw + O ( s − ) , (3.46)where we define w by p ( k ) /q ( k ) → k w + O ( k w +2 ) as k → We now have good control of both asymptotic limits, which (where they overlap) arein complete agreement with the exact results obtained previously. However, in theintermediate regime neither asymptotic expansion need be reliable — neither general-ized Seeley–De Witt nor generalized Feynman expansions are guaranteed to be useful.Fortunately, in this situation saddle point techniques can provide significant and usefulapproximate information.Consider any integral of the form J = Z + ∞−∞ exp[ h ( x )] d x. (3.47)The saddle point approximation consists of first locating the maximum of the argument h ( x ) by solving h ′ ( x ∗ ) = 0 , (3.48)– 20 –nd then approximating J ≈ Z + ∞−∞ exp (cid:20) h ( x ∗ ) + 12 h ′′ ( x ∗ )( x − x ∗ ) (cid:21) d x (3.49)= exp [ h ( x ∗ )] Z + ∞−∞ exp (cid:20) h ′′ ( x ∗ )( x − x ∗ ) (cid:21) d x (3.50)= exp [ h ( x ∗ )] × s π − h ′′ ( x ∗ ) . (3.51)For most purposes (especially when looking at ratios of integrals) it is sufficient to usethe even more brutal approximation Z + ∞−∞ exp[ h ( x )] d x ≈ exp [ h ( x ∗ )] . (3.52)See for example reference [27], where similar considerations are applied to the stationaryphase approximation. Taking the change of variable u = ln k we can express the partition function as Z ( s ) = Z exp { ud − s f ( e u ) } d u. (3.53)Note that { ud − s f ( e u ) } ′ = d − s f ′ ( e u ) e u , (3.54)and so the location of the saddle point is at d = 2 sf ′ ( e u ) e u = 2 sf ′ ( k ∗ ) k ∗ , (3.55)which implicitly defines k ∗ ( s ). We can rewrite this equation in terms of the phase andgroup velocities as d = 2 s Ω ∗ k ∗ dΩd k ∗ k ∗ = 2 s v phase ( k ∗ ) v group ( k ∗ ) k ∗ , (3.56)implying that the saddle point estimate of the diffusion time can be specified as afunction of the wavenumber s ( k ∗ ) = d v phase ( k ∗ ) v group ( k ∗ ) k ∗ . (3.57)– 21 –e now see Z ( s ) ≈ exp[ { ud − s f ( e u ) } ∗ ] = k d ∗ exp {− s f ( k ∗ ) } . (3.58)Therefore, in view of the saddle point condition in equation (3.55), the leading term inthe saddle point approximation yields for the spectral dimension d S ( s ) ≈ sf ( k ∗ ) = 1 + d f ( k ∗ ) f ′ ( k ∗ ) k ∗ . (3.59)That is, expressing s in terms of k ∗ , d S ( k ∗ ) = 1 + d v phase ( k ∗ ) v group ( k ∗ ) + ... (3.60) s ( k ∗ ) = d v phase ( k ∗ ) v group ( k ∗ ) k ∗ . (3.61)If desired, the curve d S ( s ) can be explicitly constructed by parametrically plotting thepair { d S ( k ∗ ) , s ( k ∗ ) } .To once more verify our asymptotic results, let us now consider the situationΩ( k ) ∼ k w as k →
0; and Ω( k ) ∼ k z as k → ∞ . (3.62)We then have s ( k ∗ ) ∼ d wk w ∗ → ∞ as k ∗ →
0; and s ( k ∗ ) ∼ d zk z ∗ → k ∗ → ∞ , (3.63)as expected. Additionally d S ( k ∗ ) ≈ dw as k ∗ →
0; and d S ( k ∗ ) ≈ dz as k ∗ → ∞ , (3.64)but now there is no requirement that w and z be integers. So, the infrared and ultravio-let limits for d S ( s ) that we first encountered in several exact examples, and then verifiedfor general polynomial dispersion relations using asymptotic expansions, are now seento have even more generality within the framework of the saddle point approximation. Suppose we have a polynomial dispersion relationΩ( k ) = z X a = w c a k a . (3.65)– 22 –hen v group ( k ) = P a ac a k a − pP a c a k a ; v phase ( k ) = pP a c a k a k ; (3.66)so the saddle point approximation implies d S ( k ∗ ) ≈ d P a c a k a ∗ P a ac a k a ∗ ; s ( k ∗ ) = d P a ac a k a ∗ . (3.67)Then: d S ( s → ∞ ) = d S ( k ∗ → ≈ dw ; d S ( s →
0) = d S ( k ∗ → ∞ ) ≈ dz ; (3.68)which agrees with our explicit limits via the asymptotic analyses — but now also givesus approximate information at intermediate regimes. It is worthwhile to explicitly consider the simple two-term dispersion relationΩ( k ) = c w k w + c z k z , (3.69)since then we have particularly simple formulae d ( k ∗ ) ≈ d c w k w ∗ + c z k z ∗ wc w k w ∗ + zc z k z ∗ ; s ( k ∗ ) = d wc w k w ∗ + zc z k z ∗ ) . (3.70) Consider the specific case of the Bogoliubov dispersion relation written in the formΩ( k ) = k (cid:18) k K (cid:19) . (3.71)Then the saddle point approximation yields d ( k ∗ ) ≈ d k ∗ K k ∗ K ; s ( k ∗ ) = dk ∗
12 + k ∗ /K . (3.72)In this particular case we can explicitly invert s ( k ∗ ) to find k ∗ ( s ): k ∗ ( s ) = K r dsK + 1 − ! . (3.73)Thence d S ( s ) ≈ d sK + p s ( sK + d ) sK + p s ( sK + d ) . (3.74)– 23 –hat is d S ( s ) ≈ d p d/ ( sK )1 + p d/ ( sK ) . (3.75)This compact and explicit (though approximate) formula has the correct limits. ( d S ( s →∞ ) = d +1; while d s ( s →
0) = 1+ d/ Take a two-dimensional liquid-gas interface, (topological dimension d = 2), and considerthe surface waves. For example, finite-depth ocean waves ignoring surface tension.Then the dispersion relation is well known to beΩ( k ) = gk tanh( kh ) , (3.76)where h is the depth of the ocean. Since this an intrinsically non-relativistic system, aswe have previously argued it is best to simply consider the spectral dimension of space(rather than spacetime). The saddle point approximation leads to d S, space ( k ∗ ) ≈ k ∗ h )sinh(2 k ∗ h ) + 2 k ∗ h = 41 + k ∗ h sinh(2 k ∗ h ) , (3.77)while s ( k ∗ ) = 2 gk ∗ (tanh( kh ) + k ∗ h sech ( kh )) . (3.78)Then in the asymptotic regimes we have d S, space ( s → ∞ ) = d S ( k ∗ → ≈ d S, space ( s →
0) = d S ( k ∗ → ∞ ) ≈ . (3.79)So in the infrared one recovers the topological dimension, as expected — but in theultraviolet something slightly unusual happens in that the spectral dimension increases .This is ultimately due to the fact that for large k we have Ω( k ) ∝ k / , with an exponentthat is lower than the Ω( k ) ∝ k behaviour characteristic of the infrared.Equivalently we could work with fixed k ∗ and consider the effect of varying h . Wehave d S, space ( h → ≈ d S, space ( h → ∞ ) ≈ . (3.80)This implies that for shallow water surface waves the spectral dimension is 2, whereasfor deep water surface waves the spectral dimension is 4.If you include surface tension the dispersion relation is modified and eventuallyat very short distances a different asymptotic behaviour dominates: Ω ∝ k / at very– 24 –arge k . So in this region the surface tension dominated deep water waves exhibit aspectral dimension: d S, space ( s → ≈ dz = 23 / . (3.81) We have analyzed the concept of the spectral dimension in the continuum and focusedmainly on scales where curvature effects can be neglected. We have argued that in thisregime the spectral dimension characterizes a differential operator that is used to definethe (fictitious) diffusion process, or better yet the dispersion relation associated withthis operator. We exhibited a natural way of assigning a spectral dimension to anydispersion relation, and shown how one can in principle invert the relationship. Theseresults establish that the spectral dimension acts as probe of the kinematics of a givendegree of freedom in this regime.We have considered some simple examples of dispersion relations for which one cananalytically determine the spectral dimension. However, for more general (and realis-tic) dispersion relations this analytic determination is rarely feasible. To address thisissue we have developed approximate techniques that allow one to determine the be-haviour of the spectral dimension to a desired accuracy in various regimes of interests.In particular, we have presented asymptotics expansions (that resemble Seeley–DeWittand Feynman expansions) which allow one to obtain the infrared and ultraviolet be-havior of the spectral dimension (in flat space), and a technique to calculate the flowof the spectral dimension at intermediate scales using a saddle point approximation.As a major application of the theoretical and technical developments presentedhere, we consider the use of the concept of the spectral dimension as a link betweendiscrete and continuum quantum gravity theories. Given that in the former the spectraldimension is one of the few known observables, defining it and providing the techniquesfor calculating in the latter is an important step in understanding the continuum limitof discrete models. In a companion article [15] we shall focus on this issue in moredetail.
Acknowledgements
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