Table Of ContentTowards Spectral Geometric Methods for Euclidean Quantum Gravity
Mikhail Panine1 and Achim Kempf1,2
1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
2Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
The unification of general relativity with quantum theory will also require a coming together of
the two quite different mathematical languages of general relativity and quantum theory, i.e., of
differential geometry and functional analysis respectively. Of particular interest in this regard is
the field of spectral geometry, which studies to which extent the shape of a Riemannian manifold
is describable in terms of the spectra of differential operators defined on the manifold. Spectral
geometry is hard because it is highly nonlinear, but linearized spectral geometry, i.e., the task to
6
determine small shape changes from small spectral changes, is much more tractable, and may be
1
iteratedtoapproximatethefullproblem. Here,wegeneralizethisapproach,allowing,inparticular,
0
non-equalfinitenumbersofshapeandspectraldegreesoffreedom. Thisallowsustostudyhowwell
2
the shape degrees of freedom are encoded in the eigenvalues. We apply this strategy numerically
r to a class of planar domains and find that the reconstruction of small shape changes from small
a
spectral changes is possible if enough eigenvalues are used. While isospectral non-isometric shapes
M
are known to exist, we find evidence that generically shaped isospectral non-isometric shapes, if
existing, are exceedingly rare.
5
1
PACSnumbers: 04.60.-m,02.30.Zz,02.40.-k
]
h
p I. INTRODUCTION times assumed flat) manifold from spectra. A survey of
- such results can be found in [10] and see also [11–13].
h Examples are reflection-symmetric domains in Rn and
t A fundamentaldifficulty with the quantizationofgen-
a surfaces of revolution.
eralrelativityistoseparateinthemetricthetruedegrees
m
offreedomfromspurious degreesoffreedomthat merely Ifspectralgeometryistohelpwiththequantizationof
[ express choices of coordinates [1–3]. It is of interest, gravity, i.e., if eigenvalues are to serve as the dynamical
2 therefore, to obtain a description of curved manifolds in degrees of freedom of curvature, then the key question
v termsofcoordinatesystemindependentquantities. Such is under which circumstances the spectra of suitable dif-
0 a set of invariants could be provided by the spectra of ferential operators can fully describe the curvature of a
1 canonical differential operators such as the Dirac opera- manifold.
5
tororLaplacians,atleastinthecaseofEuclideangravity Indeed, there exist classes of manifolds that contain
7
[4–6]. This approachis interesting also because by relat- non-isometric but isospectral manifolds, i.e., there are
0
ing curvature to spectra it naturally translates between circumstancesinwhichtheshapecannotbeuniquelyde-
.
1 the differential geometric language of general relativity termined from spectra. There are techniques for con-
0
and the functional analytic language of quantum theory structing such isospectral non-isometric manifolds, see
6
[4]. [14]. These techniques apply only in special instances,
1
: The mathematical discipline concerned with the rela- however,anditremainsunknownhowprevalentisospec-
v
tionship between the curvature or ‘shape’ of a manifold tral nonisometric manifolds are among the generic man-
i
X and the spectra of operators defined on it is known as ifolds that are of interest in physics. A key question,
r spectral geometry. Its origins trace back to Weyl [7] and therefore, has remained open, namely whether the rela-
a predate quantum mechanics. Concretely, spectral geom- tionshipbetweenshapeandspectraisinthegenericcase
etryasks,towhatextentpropertiesofmanifolds,suchas uniqueorambiguous. Areisospectralnonisometricman-
theircurvatureandtheirboundaries(oralsotheirbound- ifolds the norm or the exception?
aryconditions)canbedeterminedbyspectraofoperators The reason why this question has been difficult to an-
on them. The case of the detection of the boundaries of swer is that the map from the curvature or shape of a
a manifold from spectra has been popularized by Kac’s manifoldtoitsspectrumishighlynonlinear,whichmakes
paper entitled “Can one hear the shape of a drum?” [8]. ithardto studyits invertibilityproperties. Tomakethis
Kac’squestionhasinspiredinvestigationsintoacoustical problem tractable, our approach here is to linearize the
engineering applications of spectral geometry, see, e.g., problembyapplyingperturbationtheory,sothatwecan
the recent [9]. For simplicity, we will refer to both, the then at least address the question of local invertibility:
“hearing” of curvature and the “hearing” of boundaries we ask if knowledge of a small change of spectrum suf-
as the quest to detect the “shape” of a manifold from ficestoreconstructthesmallchangeofshapethatcaused
spectra. it. Locally inverting the map between shape and spec-
Indeed, ithas beenshownthat, within suitable classes trum then becomes a question of (pseudo-) inverting a
of Riemannian manifolds, it is possible to determine the linear operator. If this is possible, i.e., if it is possible
curvatureand/ortheshapeoftheboundaryofa(some- to uniquely infer small shape changes from small spec-
2
tralchanges,thentheaimistoiteratetheseinfinitesimal metricd (, ). Thismetricisusedtoverifyifthespectra
σ
· ·
stepstoobtainfiniteshapechangesfromfinitedifferences areequaluptosomefixedfinite thresholdε . Moreover,
σ
inthespectrum. Thatshouldenableonetothenaddress the spectral map must be continuous and differentiable
theoriginalquestionabouttheoveralluniquenessoram- with respect to this metric. The usual Euclidean metric
biguity of shapes for given spectra. To summarize, the suffices for this task, so it is the one we shall employ
point of using the linearization of the map from shapes from now on. One can construct a spectral map σ :
to spectra is that it is local and easier to invert than the RM RN between the shape and spectral degrees of
→
full map. We thus trade a global inverse that one can- freedom. Since all the studied spectra come from some
not construct, and that may not even exist, for a local shape in , the notation for the spectraldistance can be
G
(pseudo-)inverse that one can construct. simplifiedbywritingd (A,B)insteadofd (σ(A),σ(B)).
σ σ
As aconcreteexampleofthisapproach,weherestudy Similarly, one can consider variants of the spectral map
the case of domains in the plane whose shape can be de- that use the spectrum of the Green’s operator (which is
scribed by a finite number of degrees of freedom. We thespectrumusedinwhatfollows)oranyotherfunction
then use finite element methods to compute the spectra of the Laplacian. The study of IISG is then reduced to
ofthosedomains. Ourobservationisthatitis almostal- the study of the map σ.
ways possible to locally determine shape from spectrum, Given A,B RM, IISG strives to construct a
i.e.,todeterminesmallshapechangesfromsmallspectral parametrized pa∈th P(t) in RM starting at A and end-
changes. Indeed, pairs of isospectral yet non-isometric ing at B. This path must be constructed only with the
manifolds appear to be of measure zero. knowledge of the desired spectrum σ(B) and the be-
haviour of σ in the neighborhood of the current point
P(t). Since we can not a priori guarantee that the tar-
II. INFINITESIMAL SPECTRAL GEOMETRY get spectrum will be reached every time, we impose the
milder condition that d (σ(P(t)),σ(B)) must be non-
σ
increasing along the path. This ensures that even if the
Most difficulties of spectral geometry can be ascribed
method used to constructthe path is imperfect and fails
to the fact that the map between shape and spectrum is
toreachashapewiththedesiredspectrum,theresulting
highly nonlinear. A tried and true strategy to simplify
shape is at worst as far from the target as the starting
suchproblemsistolocallylinearizethem. Inthecontext
shape was. The simplest way to achieve this is to use
of inverse spectral geometry, this amounts to trying to
a gradient descent optimization method on the spectral
determine changes in shape from changes in spectra in
distance. This is also the approach used in [15] to study
a small (infinitesimal) neighborhood of some reference
the infinitesimal inverse spectral geometry of Laplacians
shape. By iterating (similar to integrating) such steps,
on graphs.
one may even obtain finite changes in both shape and
spectrum. That is, given an initial shape A and a target
d
shape B one can deform A in small (infinitesimal) steps P(t)= grad(d (P(t),B)) (1)
σ
dt −
thattakeitsspectrumcloserandclosertothatofB. We
dub any such approaches infinitesimal inverse spectral
While simple, this approachpresents a number of disad-
geometry (IISG). Such an approach was used previously
vantages. First, straightforward numerical implementa-
in [15] for the spectra of a graph Laplacians on a special
tionsofthegradientdescentarepronetomeandering(see
family of graphs.
[16], among many others). More importantly though,
Here, our aim is to develop IISG in a setting that is
the conceptualmeaningofgradientdescentoptimization
suitablefornumericalinvestigations. Wewillusetheter-
straysfromtheintuitiveideaoflocallyinvertingthespec-
minologythatthewordshape shalldenoteaRiemannian
tralmap. Animprovedversionofthismethodcanbeob-
manifold picked from a suitable class, . Here, can
G G tainedasfollows. Letvσ =σ(B) σ(P(t))bethedesired
contain shapes that are equivalent by isometry. The set −
spectraldirectionandlet (t)bethe Jacobianmatrixof
of isometry equivalence classes of shall be denoted [ ]. J
G G σ atP(t). It is easyto show thatif the spectraldistance
We will assume that can be parametrized in a well-
behaved way by RM. GWe will call the M coordinates in Rdσn(,·,t·h)eipsacthhosPe(nt)todebfienethdebsyttahnedgarraddEieunctliddeesacnenmtemtreitchoond
RM the shape degrees of freedom. For brevity we will
ofEquation(1)isequivalenttothepathP(t)solvingthe
also refer to the points in RM as shapes. The space of
following equation, up to a reparametrizationof t:
shapes will be equipped with a metric dG(, ). This
G · ·
then allows us to verify if shapes match up to some pre-
d 1
determined finite threshold εG, as required by the inher- dtP(t)=−2grad(kvσk2)=JT(t)vσ (2)
ent limitations of numerical methods. Under additional
mild conditions one can obtain a metric d (, ) on [ ]
[G] This form suggests the following improvement. Let
· · G
by taking d[G]([A],[B])=infA∈[A],B∈[B]dG(A,B). (t)+ denote the pseudoinverseof the Jacobianand set:
Given a shape in one can compute a finite number J
N of lowest eigenvaluGes of its Laplacian. Let RN be the d
space of spectra and suppose that it is equipped with a P(t)= +(t)vσ (3)
dt J
3
In a sense, the pseudoinverse and gradient descent ap-
proaches are unified by equations (2) and (3). Most im-
portantly, these equations show that the pseudoinverse
method provides an optimal approximation to the local
inverse of the spectral map, unlike the gradient method.
Indeed, sincethe pseudoinverseencodes the solutionofa
least-squaresproblem,σ(P(t))willlocallyevolvetowards
σ(B) in a way as close as possible to a straight line in
the sense of the Euclidean metric in the space of spectra
RN.
Moreover, the pseudoinverse provides a useful canoni-
cal generalizationof the inverse of a linear map which is
applicable to maps also between vector spaces of differ-
ent dimensions. It allows one to study situations where
thenumbersofshapeandspectraldegreesoffreedom(M FIG. 1. An example shape for M =11.
andN)arenotequal. Insuchsituationstheusualinverse
function theorem is not applicable. However, the pseu-
testhowthesuccessratedependsonvariousfactorssuch
doinverse approach can still give information about how
as the number of considered eigenvalues and the initial
well the spectral degrees of freedom locally encode the
distance d (A,B) between shapes. Of particular inter-
shape degrees of freedom. This is a significant technical [G]
est will be the question whether the rate of success ap-
advancecomparedtotheresultsoftherelatedstudycon-
proaches1 as d (A,B) is decreased,providedthat N is
cerning the spectralgeometry of graphs reportedin [15], [G]
large enough, as compared to the number of shape de-
where, by construction, the number of shape degrees of
grees of freedom M.
freedom always matched the number of spectral degrees
We apply this method to a class of domains in R2
of freedom.
equipped with the standard Laplacian with Dirichlet
We should note that the partial derivatives of the
boundary conditions. The spectra were computed using
eigenvalues with respect to the degrees of freedom of
theFreeFem++[18]finite elementsolver. Theboundary
shapemaybeundefinedatshapeswhoseLaplacianshave
of the studied domains is given by a radialfunction r(φ)
degenerate spectrum. While at such points in shape
of the standard polar angle φ:
space,equation(3)doesnothold,fortunately,suchcases
are not generic. Indeed, it is known that, on a fixed dif-
ferentiablemanifold,metricsthatinduceLaplacianswith M−1
2
nondegenerate spectrum form a residual set in the space
r(φ)=a+bexp C + [C cos(kφ)+S sin(kφ)]
of all smooth metrics [17]. For the present paper, we 0 X k k
k=1
restrict our analysis to such generic cases.
(4)
The arbitrary positive parameters a and b were set to
a = 0.1, b = 0.9. The purpose of a is to set a small but
III. NUMERICAL SETUP
finite minimal radius to the studied shapes, a technical
condition that ensures proper functioning of the finite
Equation (3) is numerically integrated by the Euler element solver. The value of b is set so that r(φ) = 1 if
method with a variable step size. If the step results in alloftheFouriercoefficientsC andS vanish. Thespace
i i
a shape whose spectrum is closer to the target one, the of shape degrees of freedom is taken to be the space of
step size is increased. Otherwise, the step is cancelled the firstM Fouriercoefficients. The studiedrangeofthe
and the step size is decreased. The increase and de- coefficients yields shapes with diameter roughly between
crease of step sizes are accomplished by multiplying the 1 and 10. The shape tolerance threshold is set to be
step size by constant factors 1.1 and 0.7, respectively. εG = 0.005, which is well smaller than the typical size
The precise choice of those values is of course arbitrary. of the studied shapes. The spectral threshold is set to
Increasing the step size speeds up the execution of the ε =√10−9 3.16 10−5, a number that was chosen to
σ
≈ ·
algorithm, which can be quite lengthy if the distance becompatiblewiththethresholdεG,aswillbediscussed
d (A,B) is large. The decrease in step size helps the inmoredetailbelow. AexampleshapeisshowninFigure
[G]
algorithm converge. The algorithm is stopped when ei- 1.
ther σ(P(t)) becomes close enough to σ(B), as dictated A metric dG(, ) on this class of shapes is obtained
· ·
by the tolerance ε , or the step size becomes negligibly by using the Hausdorff distance between the boundaries
σ
small, indicating that the algorithm is stuck. The dis- viewed as subsets of R2 [19]. A corresponding notion of
tance d[G](P(t),B) is then computed and if it is smaller distanced[G](, )on[ ]isobtainedbyminimizingdG(, )
· · G · ·
than the tolerance threshold εG it is deemed that the al- overthe isometries of the plane. In practice,this is done
gorithm has succeeded. Otherwise, the run is deemed by translating both shapes so that their centers of mass
a failure. By generating random pairs (A,B) one can coincide with the origin and then minimize the distance
4
1
0.9
0.8 1
0.7 0.8
)
d,N200..56 d,N)0.6
d,(10.4 A(0.4
L
0.3 0.2
0.2 0
0 40
0.1 0.05 30
0.1 20
00 0.05 0.1d 0.15 0.2 0.1d5 0.2 0 10 N
FIG. 2. Success rate as a function of initial shape distance FIG. 3. Success rate for runs with initial shape distance less
for M =11 and N =40. than d for M =11 and varyingN.
over all rotations and reflections of one of the shapes.
the startingandtargetshapegoestozero,providedthat
N is large enough. This indicates a success of the pro-
gramofinfinitesimalinversespectralgeometry. The sec-
IV. NUMERICAL RESULTS ond feature is the threshold at which N becomes large
enough. Notice that the success rate is rapidly increas-
Rates of success of our approach were obtained by ing as N goes from 1 to roughly M, and then plateaus
fixing M−1 = 1...5 (i.e. M = 3,5,...,11), generat- near 1. This indicates that, at least for the considered
ing rando2m pairs (A,B) and running the algorithm for space of shapes, it is sufficient to attain an approximate
N = 1...40. The number of random pairs is between match between N and M to be able to reconstruct in-
1250 and 1750, depending on M. Random pairs whose finitesimal changes in shape from infinitesimal changes
initial shape distance was too small, i.e., less than the in spectrum. The reason this match is approximate is
shape tolerance threshold εG (i.e. automatic successes) that the description of the space of shapes possesses re-
were excluded. For a fixed M, the success rate of the dundancy. Indeed, since rotations about the origin are
algorithm was analyzed as a function of the isometry- isometries, one shape degree of freedom is pure gauge.
invariantshapedistanced (A,B) andofthe numberN ThissuggeststhepossibilitythatthelimitM,N of
[G] →∞
of considered eigenvalues. infinitely many shape degrees of freedom and full spec-
trumcouldbe successfullytreatedbythesameapproach
In our analysis we employ two ways to represent the
as the finite dimensional case, although we will not pur-
dependence of the success rate on the initial shape dis-
sue this limit here.
tance. The first is to present L(d ,d ,N) which is the
1 2
proportion of all pairs (A,B) with d < d (A,B) d Let us now investigate the cases where the algorithm
1 [G] 2
for which the algorithm succeeds for a given N. T≤hus, does not succeed. We start by considering the possible
it is simply the intuitive notion of success rate in a bin outcomes of the algorithm. The four possibilities are: 1)
(d ,d ]. This success rate is illustrated on Figure 2 for bothshapesandspectramatch,2)shapesmatchbutnot
1 2
M = 11 and N = 40. Notice that the success rate de- spectra, 3) spectra match but not shapes and 4) neither
caysrapidlywithd (A,B). Still,atshortdistances,the match. The first case is of course that of the algorithm
[G]
success rate is quite encouraging. succeeding and warrants no further explanation.
In order to probe the short distance behaviour more Thesecondcaseisanunavoidableartifactoffinite nu-
closely,weuse a second,specializedwayto representthe merical precision and of the fact that identical balls in
dependence of the success rate on the initial shape dis- shapespacewillcorrespondto domainsinspectralspace
tance. Let R(d) denote the number of pairs (A,B) such of significantly varying volume and shape. It is thus not
that d (A,B) d and let S(d,N) denote the number possible to relatethe size ofthose domains by the choice
[G]
of pairs (A,B)≤such that the algorithm has succeeded oftwoconstantthresholds. Inotherwords,havingpicked
in finding a shape isometric to B when using N eigen- a constant εG, one can not pick a constant εσ such that
values. We then define the accumulated success rate isometry up to the threshold εG always implies isospec-
A(d,N)=S(d,N)/R(d). This allows us to better repre- trality up to the the threshold εσ. We sidestep this is-
sentthesuccessrateasd (A,B)goestozero,asA(d,N) sueby,conservatively,countingthoseambiguouscasesas
[G]
is the success rate in a bin [0,d]. For M = 11 and failures. Thus,the successratesthatwereportarelower
N = 1...40, A(d,N) is illustrated on Figure 3. Two key bounds for the success rates for the chosen spectral and
features become apparent. First, the success rate of the shape tolerances.
algorithmindeedtendstowards1asthedistancebetween The third possible outcome is interesting, as it corre-
5
curvature of a boundary from knowledge of the eigen-
1 values by turning the usual question “Can one hear the
0.9 shape of a drum?” into the much more manageable lo-
0.8 cal question “Does a small change of sound tell the cor-
0.7 responding small change of shape?”. This method of
0.6 “infinitesimal” inverse spectral geometry allowed us to
0.5 study the invertibility properties of the nonlinear map
0.4 from shapes to spectra both a) in the neighborhood of a
0.3 shapeand,b)tosomeextentalsointhenonlinearregime
0.2 via iteration:
0.1
(a) We considered the “limit” behavior for decreasing
0
0 5 10 15 n2e0v 25 30 35 40 distancesbetweentheinitialandtargetshapeforvarying
numbers M,N of shape and spectral degrees of freedom
FIG. 4. Proportion of isometric runs among isospectral ones respectively. We found that as soon as the number N
for M =11. of spectral degrees of freedom matches or exceeds the
thresholdofM shapedegreesoffreedomthesuccessrate
tendstowards100%. Thisconfirmsanalyticexpectations
spondstodomainsthatarenon-isometricbutareisospec- on the basis of linearizing the nonlinear map from RM
tralontheirfirstN eigenvalues. Thatis,theyarepoten- to RN. Note that this reasoning applies equally to the
tial counterexamples to the general programof infinites- case of a curved manifold whose metric is parametrized
imal inverse spectral geometry. Figure 4 illustrates the by RM. The fact that the threshold is indeed M, rather
proportion of isometric runs among isospectral ones for than some function of M is encouraging regarding the
varying N. Notice that this proportion goes to 1 as N limit M,N .
→∞
increases. This indicates that, at least for the studied
space of shapes, non-isometric isospectral shapes form a (b) We investigated cases of considerable shape dis-
set of measure zero. tance by iterating the small steps. In particular, we fo-
Finally, the fourth outcome is the algorithm getting cused on those runs of the algorithm that found a final
stuck when the righthandside ofEquation(3) vanishes. shape with the desired target spectrum. We found that
Since the transpose and the pseudoinverse of a matrix as the number, N, of considered eigenvalues is increased
sharetheirkernels, +(t)v =0ifandonlyif T(t)v = beyondthenumberofshapedegreesoffreedomM,those
σ σ
0. It is straightforwJardto show that T(t)v J=0 if and runsthatalsofoundthedesiredshapebecamedominant.
σ
onlyifP(t)isacriticalpointofd (,σJ(B)). Thatis,our In fact, the proportion of such cases seems to rapidly go
σ
minimization algorithm gets stuck·in a local minimum, to 1 as N increases. This strongly suggests that coun-
as minimization algorithms are prone to do. It could be terexamples to inverse spectral geometry are of measure
useful, therefore, to employ more sophisticated methods zeroorevenabsentwithintheconsideredclassofshapes.
for overcoming trapping in local minima. This is consistent with the fact that, to the best of our
knowledge, all counterexamples to the spectral geome-
try program in the plane [20–22] are not in our class of
shapes because they are non-star-shaped domains with
V. CONCLUSIONS AND OUTLOOK
non-smooth boundaries. The counterexamples are thus
not only not part of the set of shapes we studied, but
It is of great interest for the quantization of grav- cannot even be approximated by shapes from the con-
ity to be able to separate the true degrees of freedom sidered set, no matter how many Fourier coefficients are
of a (pseudo-) Riemannian manifold (i.e., the curvature used. It will therefore be interesting to further study
of both the bulk and/or the boundary) from the spuri- the propertiesof this class ofshapes using, inparticular,
ous degreesof freedomin the metric that merely express methods inspired by those used in, for example, [11–13].
choices of coordinates. There,theproofsrelyonthefactthattheboundariesare
To this end, we started with the observation that the analytic functions, which is also true in our case.
eigenvalues of Laplacians and other natural differential
operators on the manifold are true degrees of freedom Returningtoouroriginalmotivation,inordertoapply
of the metric because they are geometric invariants, i.e., our infinitesimal inverse spectral geometry technique to
they are independent of diffeomorphisms. The key ques- the quantization of (Euclidean) gravity, it will be neces-
tion, however, is whether these eigenvalues encode all of sary to generalize our new pseudoinverse-based method,
the true degrees of freedom. Answering this question is see equation (3), to the degrees of freedom of the metric
complicated by the fact that the relationship between on the bulk of a compact manifold. Ultimately, this will
the metric and the associated eigenvalues is nonlinear. involvedevelopingafunctionalanalyticgeneralizationof
For this reason, we introduced a perturbative approach, the pseudoinverse-based method in order to handle in-
namely we re-examined the task of reconstructing the finitely many shape and spectral degrees of freedom.
6
ACKNOWLEDGMENTS also acknowledges support through the Ontario Gradu-
ate Scholarhip (OGS) program. This work was made
AKandMPacknowledgesupportthroughthe Discov- possible by the facilities of the Shared Hierarchical Aca-
ery,NSERC-CGS-MandNSERC-PGS-D(CanadaGrad- demic ResearchComputing Network(SHARCNET) and
uate Scholarship) programs of the Natural Sciences and Compute/Calcul Canada.
EngineeringResearchCouncil (NSERC) of Canada. MP
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