Spectral Dimension of kappa-deformed space-time

0.32 MB·English

by  Anjana. V

Spectral Dimension of kappa-deformed space-time
Anjana. V1andE.Harikumar2
SchoolofPhysics,UniversityofHyderabad,CentralUniversityPO,Hyderabad-500046,India
Abstract
We investigate the spectral dimension of κ-space-time using the κ-deformed diffusion equation. The
deformedequationisconstructedfortwodifferentchoicesofLaplaciansinn-dimensional,κ-deformedEu-
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1 clideanspace-time. WeuseanapproachwherethedeformedLaplaciansareexpressedinthecommutative
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2 space-time itself. Using the perturbative solutions to diffusion equations, we calculate the spectral dimen-

r
a sion of κ-deformed space-time and show that it decreases as the probe length decreases. By introducing a
M
boundonthedeformationparameter,spectraldimensionisguaranteedtobepositivedefinite. Wefindthat,

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2 foroneofthechoicesoftheLaplacian,thenon-commutativecorrectiontothespectraldimensiondepends

] on the topological dimension of the space-time whereas for the other, it is independent of the topological
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t dimension. We have also analysed the dimensional flow for the case where the probe particle has a finite
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p
e extension,unlikeapointparticle.
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1e-mail:[email protected]
2e-mail:[email protected]
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1. INTRODUCTION
Many different paradigms have been developed and employed to unravel the structure of space-time at
microscopic scale, in order to obtain the consistent quantum mechanical description of gravity. Dimen-
sional flow [1] is a characterising behaviour that is common to these approaches such as causal dynamical
triangulations[1],asymptoticallysavegravity[2],loopgravity[3],deformed(ordoubly)specialrelativity[4],
non-commutative space-times[5–7], Horava-Lifschits gravity[8], and relative-locality[9–12]. All these ap-
proaches show that the effective dimensions felt by a fictitious test particle reduces at high energies (i.e.,
when the length scale probed is very small). This variation of dimensions with the probe scale at high
energies suggest the possibility of a fractal structure of the space-time at extremely short distances. This
reductionoftheeffectivedimensionsisofimportantconsequencesasgravityisknowntoberenormalisable
intwodimensions.
Prediction of the existence of a fundamental length scale is another common trait shared by vari-
ous frameworks developed to investigate microscopic theory of space-time. Non-commutative geometry
provides an elegant mathematical framework to incorporate this fundamental length scale[13]. In non-
commutativesetting,usualnotionsofspace-timepointsgetblurredandspace-timebecomesfuzzy. Apar-
ticle undertaking a random walk in such a fuzzy space-time may not be able to access all the dimensions.
Thus, the effective dimensions felt by this test particle can be different from the usual topological dimen-
sionsofthespace-time. Spectraldimensionhasbeenusedtostudythischangeintheeffectivedimensions
athighenergies.
The usual notion of the dimension of space is that it is the exponent that characterising the change in
the volume of an object with change in its size. Thus one defines the dimension as d = LT logvolme.
size→0 logsize
Geometricalaspectsofspaceatshortdistancesarestudiedusingthespectraldimension,whichinthelarge
length scale produces the same value as the usual dimension. Spectral dimension is calculated from the
solution to the diffusion equation defined in a Euclidean space with a given metric. The motion of the
particleinsuchaspaceisgovernedbythediffusionequation
∂
U(x,y;σ) = LU(x,y;σ) (1)
∂σ
where σ is fictitious diffusion time with dimension of length squared, L is the Laplacian and U(x,y;σ) is
theprobabilitydensityofthetestparticletodiffusefromtheinitialpositionxtoanotherpointy,indiffusion
time σ. The spectral dimension of this space is related to the trace of the diffusion probability known as
return probability which measures the probability to find the particle returning back to the starting point
afteradiffusiontimeσ. Thus,usingthesolutionforthediffusionequation,thereturnprobabilityisdefined
2
as
(cid:82) (cid:112)
dnx detg U(x,x;σ)
P(σ) = (cid:82) (cid:112)µν (2)
dnx detg
µν
whereg isthemetricoftheunderlyingspace. SpectraldimensionD isextractedbytakingthelogarithmic
µν s
derivativeofreturnprobabilityP(σ)
∂lnP(σ)
D = −2 . (3)
s
∂lnσ
It is easy to see that for usual n-dimensional Euclidean space(where the metric g = δ ), the spectral
µν µν
dimension turns out to be n itself. It is well known that the spectral dimension of quantum gravity models
depends on the diffusion time σ and it decreases smoothly to a lower value from four, at small values of
σ and this is known as dimensional flow[1]. It was shown that the spectral dimension reduces to 2 in the
UV (i.e., when probed at extremely short length scales) from 4 at IR in many cases[1–3, 8] whereas, for
non-commutative space-times, for certain choices of the Laplacian, the spectral dimension reduces to a
lowervalue(butnot2)andforotherchoices, onegetshighervaluesforspectraldimensionshowingsuper
diffusion[4–7]. In recent times a novel approach to understand quantum gravity known as the principal of
relative locality [9–12] was introduced. In this approach locality of an event is also observer dependent as
simultaneity in special and general relativity. In this approach phase space is at a fundamental level than
thespace-timeandarguethatthephasespacerelevantfordiscussionofquantumgravityshouldhaveanon
trivialgeometryasfirstsuggestedin[14].
Dimensional flow for a model compatible with deformed special relativity principle (DSR) has been
analysed in [4]. Starting with the metric in momentum space and with a specific choice of Laplacian
(constructed using an invariant scalar), return probability in DSR compatible space-time was calculated.
Using this, the spectral dimension was obtained and its variation with respect to parameters characterising
the deviation of the Laplacian from the usual case was studied numerically. It was shown that the spectral
dimension increases to 6 in the high energy scale showing super diffusion and take the expected value of
4 in the low energy scale. It was also noted that the spectral dimension do become non-integer for certain
valuesoftheparameters.
In [5], using numerical methods, fractal nature of the κ-Minkowski space-time was studied. Here, the
spectral dimension is calculated from the return probability obtained using a specific form of κ-deformed
Laplacian, inthemomentumspace. Here, amodifiedintegrationmeasure, inthemomentumspace, which
is invariant under the κ-deformed Lorentz algebra was also used. After carrying out the integrations, nu-
merically,itwasshownthatthespectraldimensionreducedtothree(3)inthelimitofdiffusiontimegoing
to zero. Further, using a different basis for the κ-deformed Lorentz algebra, same behaviour of spectral
dimensionwasobtained,butwithadifferentchoicefortheLaplacian.
3
Variation of spectral dimension with the probe scale in a non-commutative space-time was analysed
usinganotherapproachin[6]. Inthispaper,theentireeffectofnon-commutativitywasintroducedthrough
a modification of the initial condition satisfied by the solution to the diffusion equation. Since non-
commutativity will lead to smearing of point objects, the initial condition was taken as a Gaussian(instead
ofaDiracdeltafunction)withwidthcontrolledbytheminimallengthintroducedbythenon-commutativity
ofthespace-time. Inthiscase,thespectraldimensionwasshowntobeafunctionofdiffusiontimeandthe
minimallengthparameter. Inthiscase,atveryhighenergies,space-timedimensionreducestozero. Itwas
shown that in the limit where the diffusion time (s) is of the same order as the (square of ) minimal length
(l ), the spectral dimension becomes 2 , For trans-Planck regime (where s < l ), it was argued that the
min min
space-timedissolvescompletely,leadingthespectraldimensiontobezero.
In[7],changeofspectraldimensionofκ-Minkowskispace-timewasstudied. StaringfromtheEuclidean
momentumspaceassociatedwiththeκ-Minkowskispace-time,possibleLaplaciansinthemomentumspace
were constructed. These Laplacians were constructed by demanding them to be the Casimirs of the κ-
Poincare algebra in the momentum space. These Laplacians were constructed as the Casimirs using bi-
covariant differential calculus and bi-crossproduct basis, respectively. A third form for the Lpalacian was
derived as the geodesic distance in the κ-momentum space. By calculating the return probability, using
these Laplacians in the momentum space, spectral dimensions for these three cases were calculated. For
the first case, i.e., for the Laplacian written in bi-covariant differential calculus, it was shown that the
spectraldimensionflowfrom4atlowenergiesto3inthehighenergies. Forthesecondsituationwherethe
Laplacianwaswritteninbi-crossproductbasis,spectraldimensionwasshowntoincreasefrom4to6with
energy. In thethirdcase, Laplacianwas constructedusingthe notionof relativelocality, anditwas shown
that the spectral dimension goes to infinity (∞) as energy increases. Thus in the first case, the spectral
dimension flows to a lower value (of 3), replicating the general feature of the dimensional flow in other
studies, whereas, in the second case, spectral dimension increase to a higher, but finite value indicating
super diffusion. In the third situation, notion of diffusion breaks down completely at high energies as the
spectraldimensiongoestoinfinity. Asimilarbehaviourwasalsonotedin[15]forcertaineffectivemodels
involvingnon-locallaplacians. Inthefirsttwocasesconsideredin[7],thebehaviourofspectralflowswere
argued to be due to the modified integration measure, necessitated by the κ-deformation. In the third case
studied in [7] where the spectral dimension is calculated for a model compatible with relative locality, the
effects of this novel notion of locality leads to higher powers of momenta in the dispersion relation. By
numericalmethods,spectraldimensionwascalculatedandshownthatitdivergesastheprobescalegoesto
zero.
Wenotethatthedimensionalflowinthenon-commutativespace-time,andinparticularforthecaseofκ-
4
deformedspace-timedependsonthechoiceofLaplacians. Also,allthesestudies[4–7]wherethevariation
of spectral dimension of κ-space-time was analysed, used the Laplacians in the momentum space. Also,
the invariant measure in the momentum space had an important role in deciding the behaviour of spectral
dimensionasafunctionoftheprobescale[5,7]. In[1–3],solutionstodiffusionequationwereobtainedin
thecoordinatespaceandthusitisofinteresttoanalysethechangeinthespectraldimensionofκ-space-time
also usingthe probabilitydensity of thetest particleundergoing diffusion incoordinate space. Wetake up
thisissuehere.
Inthispaper,wederivethespectraldimensionoftheκ-space-time,bysettinguptheκ-deformeddiffusion
equation in terms of commutative co-ordinates. This is achieved by first writing down the κ-deformed
Laplacian in the Euclidean version of the n-dimensional κ-space-time, in terms of the derivatives with
respecttocommutativecoordinates. Wethenobtainthesolutiontothisheatequation,perturbatively. Using
thissolution,wecalculatethereturnprobability,validuptosecondorderinthedeformationparameter. We
thencalculatethespectraldimension,asafunctionofdiffusionlengthandthedeformationparametera. We
thenanalysethevariationofthespectraldimensionasafunctionofdiffusionlength. Wehaveanalysedthis
foranotherpossiblerealisationoftheκ-deformedLaplacianoperatoralso. Sincetheframeworkweusehere
allow us to set up the diffusion equation valid for the κ-deformed space-time, entirely in the commutative
space-time,wecanusethewellestablishedmethodstosolvethisdeformedheatequation. Also,wecanuse
thesameinitialconditionofdemandingthetestparticletobelocalisedatafixedpointinspaceattheinitial
time,asinthecommutativespace-time. Wehavealsoinvestigatedtheeffectoffiniteextensionoftheprobe
onthespectraldimension,byusingamodifiedinitialcondition.
This paper is organised as follows. In the next section, a brief summary of the κ-deformed Laplacian
operators written in commutative space-time is given[16]. In section 3, we set up the heat equation for
a fiducious particle in this κ-deformed space-time and obtain its solution. This solution is obtained as a
perturbative series in the deformation parameter and we obtain the solution valid upto first non-vanishing
terms in the deformation parameter. Using this solution, we calculate the return probability and spectral
dimension which is valid upto second order in the deformation parameter. We also discuss various limits
of the spectral dimension. Using a different initial condition, we calculate the spectral dimension with a
probehavingfiniteextension. Wefindthatthegenricfeatureofthespectraldimensionisnotalteredbythe
extended nature of the probe. In section 4, we start with a different, possible Laplacian in the κ-deformed
space-timeandevaluatecorrespondingspectraldimensionanddiscussitslimits. Hereagain,weinvestigate
theeffectofaprobewithafiniteextensiononthespectraldimension. Ourconcludingremarksaregivenin
section5.
5
2. KAPPA-DEFORMEDLAPLACIANINEUCLIDEANSPACE-TIME
Theapproachwetakehereistoexpresstheκ-deformedLaplacianintermsofthederivativeswithrespect
to the commutative coordinates and deformation parameter, developed in[16, 17]. In this framework, one
first re-express the coordinates xˆ of the κ-deformed space-time in terms of the commutative coordinates
µ
x and their derivatives. By demanding that this mapping preserves the defining relations satisfied by the
µ
non-commutativecoordinatesgivenby
[xˆ ,xˆ ] = i(a xˆ −a xˆ ) (4)
µ ν µ ν ν µ
where a are constant real parameters, which describe the deformation of Euclidean space. We choose
µ
a = 0,i=1ton-1,anda = a. Thenthealgebraofthenon-commutativecoordinatesbecomes
i n
[xˆ,xˆ ] = 0, [xˆ ,xˆ] = iaxˆ, i, j = 1,2,...,n−1 (5)
i j n i i
Theexplicitformof xˆ and xˆ are
n i
xˆ = x ψ(A)+iax∂γ(A), (6)
n n i i
xˆ = xϕ(A), (7)
i i
where A = ia∂ . By demanding that the deformed Poincare transformations must be linear in the coordi-
n
nates and their derivatives, one obtain the generators of the underlying symmetry algebra. These modified
generators satisfy the same relations as the generators of the usual Poincare algebra and known as unde-
formed κ-Poincare algebra. These deformed generators are given in terms of the commutative coordinates
x and their derivatives ∂ and depend on the deformation parameter a. In the limit of vanishing a, one
µ µ
recoverthePoincarealgebra.
Thederivativeswhichtransformasvectorunderκ-Poincarealgebra,calledDiracderivatives,areexplic-
itly
e−A sinhA e−A
D = ∂ , D = ∂ +ia∇2 . (8)
i i n n
ϕ A 2ϕ2
ThequadraticCasimirofthisalgebraisD Dµ [16,17]isgivenby
µ
a2
D D = DD +D D = (cid:3)(1− (cid:3)), (9)
µ µ i i n n
4
where
e−A 2(1−coshA)
(cid:3) = ∇2 −∂2 . (10)
ϕ2 n A2
6
Here A = ia∂n andwithoutloseofgenerality,wechooseϕ = e−A2. Notethatthe(cid:3)-operatorisquadraticin
spacederivativesandthusD D hasquarticspacederivatives.
µ µ
GeneralisingthenotionofLaplacianbeingtheCasimirofthePoincarealgebratotheκ-deformedspace-
time,weuseD D astheκ-deformedLaplacian. Inthecommutativelimit(a → 0),werecoverthestandard
µ µ
Laplacianinthecommutativespace-time. Butifwerelaxthisconditionandrequireonlythattheκ-deformed
operator should reduce to the usual Laplacian in the commutative space-time, we can use the (cid:3)-operator
defined in eqn.(10) also as the κ-deformed Laplacian. Both D D and (cid:3) operators have been analysed as
µ µ
possiblegeneralisationsofKlein-Gordonoperatorinκ-space-time[18,19].
It is clear from eqn.(10) that the (cid:3)-operator has higher derivatives in time. Further, we note that this
operatoralsohastermsinvolvingproductsofderivativeswithrespecttotimeandspacecoordinates. Since
D D isexpressedintermsofthe(cid:3)-operator,thesepropertiesarecarriedovertoD D also. Thiswillhave
µ µ µ µ
importantconsequencesinthecalculationofspectraldimensionoftheκ-deformedspace-time.
NotethatboththeoperatorsD D and(cid:3)arewrittencompletelyinthecommutativespace-timeandthus
µ µ
theLaplaciansweuseareexpressedinthecommutativespace-time.
3. SPECTRALDIMENSIONFROMD D OPERATOR
µ µ
In this section, we calculate and analyse the spectral dimension of the κ-deformed space-time, using
the Casimir of the undeformed κ-Poincare algebra D D as the Laplacian operator, in the n-dimensional
µ µ
κ-deformed Euclidean space. For this, we start with the diffusion equation in the κ-deformed space and
derive its solution. We solve this deformed diffusion equation, perturbatively and obtain the solution valid
uptosecondorderinthedeformationparameter. Usingthisdeformedprobabilitydensity, wecalculatethe
returnprobabilityandthenspectraldimension.
Westartwiththediffusionequationinann-dimensional,κ-deformedEuclideanspace,givenby
∂
U(x,y;σ) = D D U(x,y;σ). (11)
µ µ
∂σ
Werestrictourattentiontofirstnon-vanishingcorrectionsduetonon-commutativity. Thuswere-writethe
aboveequation,validuptothesecondorderinthedeformationparameteraas
∂U a2 a2 a2
= ∇2U +∂2U − ∂4U − ∇2∂2U − ∇4U. (12)
∂σ n 3 n 2 n 4
Note that all the a dependent terms are of higher derivatives; two of them having quartic derivatives while
anotherinvolvesproductofquadraticderivativesinspaceandtime.
ForconveniencewedefinetheLaplacianinthen-dimensionalcommutativeEuclideanspace-timeas
∇2U +∂2U = ∇2U. (13)
n n
7
Usingthis,eqn.(12)isre-writtenas
∂U a2 a2 a2
= ∇2U − ∂4U − ∇2∂2U − ∇4U. (14)
∂σ n 3 n 2 n 4
WeuseperturbativeapproachtosolvethisequationtoobtaintheheatkernelU(x,y;σ). Thuswestartwith
theprobabilitydensity,validuptosecondorderinthedeformationparameteraas
U = U +aU +a2U . (15)
0 1 2
Notethatwehavethefollowingrelationsbetweendimensionsoftermsintheaboveperturbativeseries,
1
[U ] = [U ] (16)
1 0
L
and
1
[U ] = [U ]. (17)
2 0
L2
Using eqn.(15) in eqn.(14) and equating the zeroth order terms in a, we find that U satisfy the usual heat
0
equation,
∂
U (x,y;σ) = ∇2U (x,y;σ). (18)
∂σ 0 n 0
Thesolutiontoaboveequationis
U0(x,y;σ) = 1 ne−|x4−σy|2. (19)
(4πσ)2
Next,weequatethefirstordertermsin‘a’ineqn.(14)toobtain
∂
U (x,y;σ) = ∇2U (x,y;σ) (20)
∂σ 1 n 1
showingthatU alsosatisfythesameheatequationasU . ThuswefindU tobeofthesameformasU ,
1 0 1 0
i.e.,
U1(x,y;σ) = α ne−|x4−σy|2 (21)
(4πσ)2
where α has the dimensions of L−1. Thus, for solving both U and U , we have used the usual initial
0 1
condition, i.e., U (x,y;0) = δn(x−y) = U (x,y;0). This is possible because we could write the deformed
0 1
diffusionequationinthecommutativespace-time.
Nowwesolveforthenexttermineqn.(15). Forthisweequatethesecondordertermsin‘a’ineqn.(14)
andobtain
∂ 1 1 1
U (x,y;σ) = ∇2U (x,y;σ)− ∂4U (x,y;σ)− ∇2∂2U (x,y;σ)− ∇4U (x,y;σ) (22)
∂σ 2 n 2 3 n 0 2 n 0 4 0
8
NotethatthelastthreetermsontheRHSofaboveequationshowthechangeinthediffusionequationdue
totheκ-deformation. UsingthesolutionforU obtainedineqn.(19)intheabove,andafterstraightforward
0
simplification,wegettheequationsatisfiedbyU as
2
∂ (n+1)2 (x −y )2 n+2 (x −y )4
U (x,y;σ) = ∇2U (x,y;σ)+[− + n n + Σn (x −y)2− n n
∂σ 2 n 2 16σ2 16σ3 16σ3 i=1 i i 48σ4
(23)
−(xn32−σy4n)2Σni=−11(xi−yi)2− 641σ4(Σni=−11(xi−yi)2)2](4π1σ)n2e−|x4−σy|2.
UsingDuhamel’sprinciple[20],wesolvetheaboveequationwhichisofthegenericform
∂
U (X,σ) = ∇2U (X,σ)+ f(X,σ) (24)
∂σ 2 n 2
where X = x−y. Withtheinitialcondition
U (X,0) = g(X), (25)
2
thesolutiontoeqn. (24)isgivenby
(cid:90) (cid:90) σ(cid:90)
U (X,σ) = Φ(X−X(cid:48),σ)g(X(cid:48))dX(cid:48)+ Φ(X−X(cid:48),σ−s)f(X(cid:48),s)dX(cid:48)ds (26)
2
Rn 0 Rn
where
Φ(X,σ) = 1 e−|X4σ|2. (27)
n
(4πσ)2
Inourcasewehavetheinitialconditionsatisfiedbythesolutionas
U (X,0) = g(X) = δn(X). (28)
2
Note that we are using the same boundary condition as in the usual diffusion equation. This is to be
contrastedwiththeapproachtakenin[6]. Since,theκ-deformedLaplacianandhencethediffusionequation
are written fully in the commutative space-time and all the effects of the non-commutativity is included in
themodifiedLaplacianandthuswearejustifiedinusingthisinitialcondition.
Usingeqn(27)andeqn.(28),thefirsttermonRHSofeqn.(26),U iscalculatedas
21
(cid:90)
U21(x,y;σ) = Φ(X−X(cid:48),σ)g(X(cid:48))dX(cid:48) = β ne−|x4−σy|2, (29)
Rn (4πσ)2
9
whereβhasthedimensionsof L−2. ThesecondtermonRHSofeqn.(26),U isevaluatedas
22
(cid:90) σ(cid:90)
U (x,y,σ) = Φ(X−X(cid:48),σ−s)f(X(cid:48),s)dX(cid:48)ds
22
0 Rn
(cid:32) (cid:33)
= (4π1σ)n2e−|x4−σy|2[ −(xn48−σy4n)4 − (xn32−σy4n)2Σni=−11(xi−yi)2− 641σ4 (cid:16)Σni=−11(xi−yi)2(cid:17)2 (σ−(cid:15))
(cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33)
(x −y )2 n+2 (n+1)2 1 1
+ n n + Σn (x −y)2 ln(σ/(cid:15))+ −
16σ2 16σ2 i=1 i i 16 σ (cid:15)
(cid:32) (cid:33)
(x −y )2 1 (n−1) (n+1)
− n n + Σn−1(x −y)2+ (x −y )2+ Σn−1(x −y)2 A
4σ3 16σ3 i=1 i i 16σ3 n n 16σ3 i=1 i i
1 (cid:16) (cid:17)
− (x −y )Σn−1(x −y)+(x −y )Σn−1(x −y)+...+(x −y )(x −y ) A
2πσ3 n n i=1 i i 1 1 i=2 i i n−2 n−2 n−1 n−1
(cid:32) (cid:33)
(x −y )3 (x −y ) (x −y )2
− n √n + n √ n Σn−1(x −y)2+ n √n Σn−1(x −y) B
6σ3 σπ 8σ3 σπ i=1 i i 8σ3 σπ i=1 i i
(cid:32) (cid:33)
1
− √ Σn−1(x −y)Σn−1(x −y )2 B
8σ3 σπ i=1 i i j=1 j j
(cid:32) (cid:33)(cid:32) (cid:114) (cid:114) (cid:33)
2(x −y ) (n+1) (n−1) σ σ
+ n√ n − √ Σn−1(x −y)− √ (x −y ) (2σ+(cid:15)) −1−3σtan−1 −1
3σ2 σπ 4σ2 σπ i=1 i i 4σ2 σπ n n (cid:15) (cid:15)
(n+1)2 (cid:32) σ2 (cid:33) (n+1)2 (cid:20) σ (cid:21)
− −2σln(σ/(cid:15))+ −(cid:15) + −1+ −ln(σ/(cid:15))
16σ2 (cid:15) 8σ (cid:15)
(cid:32) (cid:33)
(x −y ) n+2 (cid:16) (cid:112) (cid:112) (cid:17)
+ 2 n √ n + √ Σn (x −y) tan−1( σ/(cid:15) −1)− σ/(cid:15) −1 ] (30)
4σ σπ 4σ σπ i=1 i i
(cid:113) (cid:113)
whereA = σln(σ/(cid:15))−σ+(cid:15) andB = (σtan−1 σ −1−(cid:15) σ −1).
(cid:15) (cid:15)
Note that U treats Euclidean time and space coordinate on a different footing. This is apparent from
22
the fact that we have terms depending on (x −y ) and (x −y), separately. The (cid:15) appearing in the above
n n i i
is a lower cut-off introduced in evaluating the integral in eqn.(26) and we will set the limit (cid:15) → 0 after
calculatingthespectraldimension.
Using eqn.(29) and eqn.(30), we get the solution to the second order correction as U (x,y;σ) =
2
U (x,y;σ) + U (x,y;σ). Using eqns. (19), (21), (29) and eqn.(30) in eqn.(15), we find heat kernel,
21 22
validuptosecondorderina.
Usingthisineqn.(2),weobtainthereturnprobabilityas
(cid:34) (cid:32) (cid:33)(cid:35)
1 (n+1)2 (n+1)2
P(σ) = 1+aα+a2β+a2 − + (cid:15) . (31)
(4πσ)n2 16σ 16σ2
Notethatthereturnprobability,calculatedabove,dohavefirstorderaswellassecondordercorrectionsdue
tothenon-commutativity.
Usingthisineqn.(3),weevaluatethespectraldimensionas
n+naα+na2β−(n+2)(n+1)2 a2 +(n+4)(n+1)2 a2 (cid:15)
D = 16σ 16σ2 . (32)
s 1+aα+a2β−a2(n+1)2(1− (cid:15))
16σ σ
10