Discrete Gravity in One Dimension
hep-th/9412089 10 Dec 94 c Street Cam cosmological constan Regge s on dimension results obtained Finally scalar August CERN TH amined feomorphisms hange 2 1 A P the P ermanen ermanen mo del functional in partition the discrete form After bridge CB EW England the Discrete A simple decimation renormalization group transformation is p erformed t address t prop erties for address and measure in for a quan tro ducing in function Herb ert W Ham the tegral t due to the scalar eld uctuations v Ph ulation of tum P Departmen of implications ysics Departmen oissonian is CH the is determined Gra gra and a p erformed Theory Division CERN massless sp ectrum vit t the gra y vit of distribution of edge lengths ABSTRA Gen in vit b er results Applied for t Univ y y one ev scalar exactly It a of The and e in regularized is time the ersit are Mathematics found nature CT Ruth M Williams eld One Switzerland scalar y of California at Irvine Irvine Ca compared on dimension coupled a that of nite Laplacian gra Dimension exact and the vitational anishes iden with to lattice Theoretical renormalization is con the discussed CERN TH the are tin edge and measure uous compared exact tically in one Ph lengths the lattice dif ysics based solution ensuing are of Silv with the the ex on er
Intro duction
In the simplicial formulation of Quantum Gravity the metric prop erties of space
time are describ ed by a simplicial complex with varying edge lengths While
the classical equivalence b etween the simplicial and the continuum formulations has
b een established much less is known ab out the quantum equivalence b etween
the two formulations Also a numb er of ambiguities present in the continuum for
mulation such as the prop er de nition of the gravitational measure lead to
a p ossible lack of uniqueness of the lattice formulation The b est one can do is con
struct a lattice theory that mimics as much as p ossible the structure and symmetries
of the continuum formulation and hop e on the basis of universality that the correct
theory with the desired phenomenological prop erties will emerge at distance much
larger than the lattice cuto If a nontrivial lattice continuum limit can b e found
it should b e unique in the sense that only one theory of a massless spin two particle
can b e written down for su ciently large distances namely General Relativity One
can in fact go as far as arguing that a consistent lattice formulation provides a basis
for a constructive de nition of the continuum theory
In physical dimensions the only metho ds that have b een used so far to study
the lattice mo dels are the weak eld expansion and large scale numerical
simulations The former approach based on p erturbation theory su ers from
the well known problems asso ciated with the lack of p erturbative renormalizability
of the Einstein Hilb ert action The second approach has the advantage that it allows
one in principle to investigate the regime in which the bare couplings are not small
and where uctuations in the metric at short distances can b e rather large The
high price one pays is that numerical results are approximate and require a correct
interpretation for example the large cuto in nite volume limit has to b e reached
by a judicious extrap olation
In this pap er we will address the problem of Quantum Gravity in one di
mension one time and zero space dimensions The usual p erturbative counting
of gravitational degrees of freedom in d dimensional space gives after imp osing the
gauge conditions d d d d d which reduces to in one dimension
The theory is therefore quite trivial in such a low unphysical dimension and one
do es not exp ect deep implications for the higher dimensional case the same is true
in a sense in two dimensions due to the absence of gravitational waves Still there
is the advantage that all calculations can b e done analytically in complete fashion
in a nite b ox for example and all nite corrections can b e determined While
the only invariant asso ciated with a one dimensional universe is its length in time
di eomorphisms still play a role and should ensure that no unwanted terms app ear
in the e ective action irresp ectively of what the matter content might b e
Other features of interest which can emerge are the e ects on the measure of
integrating out the scalar degree of freedoms and the nature of coupling constant
renormalizations which can b e obtained for example from a decimation pro cess to
b e discussed b elow Although the mo del is very simple in many ways the calcula
tion of the eigenvalues of the Laplacian and the determination of the sp ectral density
function is quite complex as rst p ointed out in Ref We should add
that one dimensional gravity has already b een considered b efore but from rather
di erent p oints of view The authors of Ref have considered asp ects of a pre
geometric mo del for gravity while Ref discusses the canonical quantization of
such a mo del
In Section we discuss the action for a discretized line its invariance prop erties
and the corresp onding functional measure taking into account the restrictions im
p osed by the lattice analog of continuous di eomorphism invariance A scalar eld
is intro duced in Section and the invariance prop erties under continuous lattice
di eomorphisms are extended to this case Sum rules are written down for various
op erators including p owers of the edge lengths In Section the scalar functional
integral is p erformed exactly and the e ect on the measure is investigated A simple
decimation pro cess on the gravitational and scalar degrees of freedom is p erformed
in Section and the resulting renormalization of the couplings is discussed Section
consists of a comparison of the sp ectrum of the Laplacian calculated here with the
results obtained on a quenched Poissonian random lattice Section contains a few
concluding remarks
Discrete Gravity in One Dimension
In four dimensions the action for pure Euclidean gravity on the lattice can b e
written as
h i
X
I l k A V a A V V
g h h h h
h h h
hinges h
where V is the volume p er hinge a triangle in four dimensions A is the area of
h h
the hinge and the corresp onding de cit angle prop ortional to the curvature at
h
h All geometric quantities can b e evaluated in terms of the lattice edge lengths
l which uniquely sp ecify the lattice geometry for a xed incidence matrix In the
ij
continuum limit such an action is equivalent to
Z
i h
p
k R a R R g I g d x
g
As usual the Minkowski version of the theory is obtained by analytic continuation
In one dimension we discretize the line by intro ducing N p oints with lengths l
n
asso ciated with the edges and p erio dic b oundary conditions l l We de ne
N
here l to b e the distance b etween p oints n and n The only surviving invariant
n
term in one dimension is then the length of the curve
N
X
l L l
n
n
which corresp onds to
Z Z
q
dx g x dx e x
with g x g x in the continuum Here e x is the einb ein and satis es
q
g x e x In this context the discrete action is the obvious constraint
unique preserving the geometric prop erties of the continuum de nition From the
expression for the invariant line element ds g dx one asso ciates g x with l
n
and therefore e x with l One can further take the view that distances can only
n
b e assigned b etween vertices which app ear on some lattice in the ensemble although
this is not strictly necessary as distances can also b e de ned for lo cations that do
not coincide with any sp eci c vertex
The gravitational measure then contains an integration over the elementary lat
tice degrees of freedom the lattice edge lengths For the edges we write the lattice
integration measure as
Z Z
N
1
Y
dl l d l
n n
n
where is a parameter interp olating b etween di erent lo cal measures The
p ositivity of the edge lengths is all that remains of the triangle inequality constraints
plays a role analogous to the g which app ears in one dimension The factor l
n
for continuum measures in the Euclidean functional integral In d dimensions the
purely gravitational measure reads
Z Z
Y Y
g x dg x d g
x
The parameter app ears to dep end on the regularization pro cedure and the values
d d and d have b een prop osed Thus
determines a one parameter family of gravitational measures the hop e is that
physical predictions of lattice gravity close to the continuum limit do not dep end on
appreciably on In one dimension the gravitational measure is then simply
Z Z
Y
d g g x dg x
x
We see no comp elling reason at this p oint for considering lattice measures which are
non lo cal just as it would seem equally unattractive to consider action contributions
which are non lo cal Up to the stated ambiguity in the gravitational measures in
Eqs and are unique
The functional measure do es not have compact supp ort and the cosmological
term with a co e cient is therefore necessary to obtain convergence of the
functional integral as can b e seen for example from the expression for the average
edge length
Z
N N N N
1
X X Y X
l l dl l exp l i Z N hL l i h
n n n
n n N
n n n n
with
N
Z
N N
1
X Y
l exp l dl Z
n N
n n
n n
q
Similarly one nds for the uctuation in the total length L L N
which requires Di erent choices for then corresp ond to trivial rescalings
of the average lattice spacing l hl i One could go as far as considering
the cosmological term as an integral part of the measure as an integration over the
measure is not well de ned in its absence and b ecause such a term is known not to
contain derivatives of the metric in any dimensions As in higher dimensions it is
useful to consider the density of states N L de ned by
Z
i1
N L
L d e Z N L
N
L!1
i
i1
One notices that the measure has some in uence on the large L b ehavior of N L
In the continuum the action of Eq is invariant under continuous reparametriza
tions
0
x x x x x
d dx
0 0
O g x g x g x g x g x
0
dx dx
or equivalently
0 0
g x g x g x g
0 0
and we have set d dx A gauge can then b e chosen by imp osing g x
q
R
0
g x which can b e achieved by the choice of co ordinates x dx
We shall now write the discrete analog of the transformation rule as
0
n n
l l
n n
l
n
or simply
l
n n n
where the s represent gauge transformations de ned on the lattice vertices In or
n
der for the edge lengths to remain p ositive one should further require l
n n n
which is certainly satis ed for su ciently small s The ab ove continuous symmetry
is an exact invariance of the lattice action of Eq since
N N N
X X X
l L
n n n
n n n
and we have used Moreover it is the only lo cal symmetry of the action of
N
Eq Had we discretized the derivative of x di erently by setting for example
0
then the discrete action would no longer b e invariant l l
n n
n n
As in the continuum one has the freedom to cho ose a gauge by imp osing for
0
example l L N which can b e achieved by the following gauge transformation
n
n
X
l L N
m n
m
Q
l L N The corresp onding lattice gauge xing term would then simply b e
n
n
The in nitesimal invariance prop erty de ned in Eq formally selects a
Q
unique measure over the edge lengths corresp onding to dl in Eq
n
n
as long as we ignore the e ects of the lower limit of integration in the measure
On the other hand for su ciently large lattice di eomorphisms the lower limit of
integration comes into play since we require l always and the measure is
n
R
Q
1
no longer invariant We also note that a measure dl is not acceptable on
n
1
p
physical grounds It corresp onds to violating the constraint g or e which
is known for example to give rise to acausal propagation
The same functional measure can b e obtained from the following physical con
sideration De ne the gauge invariant distance b etween two con gurations of edge
0
lengths fl g and fl g by
n
n
N N N N
X X X X
0 0 0 0
0 0
l l M l l d l l L l L l
0
n n n n n
n
0 0
n n
n n
0
0
the ensuing measure is again Since M is indep endent of l and l with M
0
n n n
n
Q
simply dl Any constant metric indep endent factor multiplying the functional
n
integral drops out when computing exp ectation values We notice that the ab ove
metric over edge length deformations l is non lo cal
In the continuum the functional measure is usually determined by considering
the following lo cal norm in function space
Z
q
jj g jj dx g x g x G x g x
and di eomorphism invariance requires G x g x The volume element in
q
Q
G x dg x function space is then an ultraviolet regulated version of
x
Q Q
dg x g x Its naive discrete counterpart would b e dl l which is
n n
x
clearly not invariant under the transformation of Eq it is invariant under
l l which is not an invariance of the action
n n n n
One might think that it should b e p ossible to write an interaction term involving
di erent edges but this is not so easy Consider for example the following natural
expression describing a lo cal coupling b etween neighb oring edge lengths
N
X
l l
n n
l l
n n
l l
n n
n
It can b e regarded as natural since it involves a nearest neighb our coupling b etween
edge lengths weighted by the appropriate lattice volume element This action is
clearly not invariant under the gauge transformation of Eq In the continuum
it is not p ossible either to write down a lo cal invariant coupling term for the metric
due to the fact that the only intrinsic invariant asso ciated with a curve is its length
Consider for example a coupling term of the typ e
Z
q
dg x
dx g x g x
dx
Under a general co ordinate transformation
0 0
dg x dg x d dx dx dg x
g x log
0 0 0
dx dx dx dx dx dx
or in in nitesimal form
g g g O
Similarly
g g g g O
and the app earance of the higher derivatives of make it imp ossible to obtain a
lo cal invariant term
Scalar Field
In order to make the mo del less trivial intro duce a scalar eld de ned on the
n
sites with action
N N
X X
n n
I V l V l
n n
n
l
n
n n
with N It is natural in one dimension to take for the volume p er
edge V l l and for the volume p er site V l l l Here
n n n n n
plays the role of a mass for the scalar eld m The ab ove is of course the
one dimensional analog of the prescription of Ref for constructing the scalar
eld action on a quenched random lattice the only di erence b eing that the lattice
here is dynamical as the edge lengths represent gravitational degrees of freedom
Similar scalar eld actions have also b een discussed previously in Refs
The continuum analog of the scalar eld action would b e
Z
q
h i
dx g x g x x x I
c
As usual the discrete expression in Eq should b e regarded as the precise de ni
tion for what is intended when the continuum expression is inserted into a functional
integral since the quantum elds are known to b e nowhere di erentiable Let
us add that since the average of pro ducts of scalar propagators is not the same as
the pro duct of averages one exp ects to nd residual gravitational interactions even
in one dimension
In addition we keep a term
N
X
l L l
n
n
in the action corresp onding to a cosmological constant term and which is nec
essary in order to make the dl integration convergent at large l
n
Varying the action with resp ect to gives
n
n n n n
n
l l l l
n n n n
This is the discrete analog of the equation g g The sp ectrum of
the Laplacian of Eq corresp onds to Variation with resp ect to l
n
gives instead
n n
n n
l
n
For it suggests the well known interpretation of the elds as co ordinates
n
in emb edding space In the sp ectral prop erties of the scalar Laplacian of
Eq were investigated on a quenched random lattice constructed according
to the prescription of the authors of Ref Here we shall consider instead the
edge lengths as dynamical variables In the following we shall only consider the case
corresp onding to a massless scalar eld
It is instructive to lo ok at the invariance prop erties of the scalar action under
the continuous lattice di eomorphisms de ned in Eq The scalar nature of
0
the eld requires that under a change of co ordinates x x
0 0
x x
0
where x and x refer to the same physical p oint in the two co ordinate system On
the lattice di eomorphisms move the p oints around and at the same vertex lab elled
by n we exp ect
n n
0
n n n
n
l
n
One can determine the exact form of the change needed in by requiring that the
n
lo cal variation
n n n n n n
l l
n n n n
n n n n
l l
n n
b e zero Solving the resulting quadratic equation for one obtains a rather
n
unwieldy expression which is given to lowest order by
n n n n n
n
l l
n n
n n n n n n n
n
O
n
l l l l
n n
n
n
and which is indeed of the exp ected form as well as symmetric in the vertices n
and n For elds which are reasonably smo oth this correction is suppressed if
j j l On the other hand it should b e clear that the measure d is
n n n n
no longer manifestly invariant due to the rather involved transformation prop erty
of the scalar eld
The partition function for N sites then reads
Z Z
N N N
1 1
X Y X
l dl l d exp Z
n n N n n
n n
l
1
n
n n n
P
N
The trivial translational mo de in can b e eliminated for example by setting
n
n
Under a rescaling of the edge lengths l l one can derive the following iden
n n
tity for Z
N
N N
Z z z Z
N N
where we have replaced the co e cient of the scalar kinetic term by z It
follows then that
N
X
l i h hl i
n
N
n
and
N
X
h i
n n
N l
n
n
Without loss of generality we can x the average edge length to b e equal to one
It should b e clear that now we have to hl i which then requires
require in order for the mo del to b e meaningful
In the absence of the scalar eld the distribution of edge lengths is determined
by the generalized Poisson distribution
l
l e P l
with averages
n
n n n
hl i n
In order for the averages to b e de ned one imp oses therefore the restriction
in the absence of a scalar eld hl i requires The parameter can
then b e regarded as controlling the uctuations in the edge lengths since
hl i hl i hl i
The Poisson distribution is then obtained as a sp ecial case when the edge
lengths are determined from the distances of random co ordinate vectors joined to
d
form a line in R l j x x j It should b e clear that in general one would like
ij i j
to avoid frequent app earances of degenerate con gurations of edge lengths such as
the one depicted in Fig where one edge length has b ecome of order L
Fig Non degenerate and degenerate con gurations of edge lengths
Fig shows a graph of the normalized single site edge distribution P l as a
function of One notices that as gets large the distribution gets increasingly
p eaked around l For the distribution is singular at the origin
Decimation Transformation
We illustrate here a simple decimation transformation on the original N site par
tition function Z in which the degrees of freedom asso ciated with the o dd sites
N
l n o dd are integrated out It corresp onds to a real space renormalization
n n
group transformation in which the average edge length hl i and therefore the lattice
spacing is increased by a factor of two It represents a simple application of the
real space renormalization group ideas for gravity intro duced in Ref
In order to p erform the integration over the decimated l s we set l g
n n n
where the expansion parameter g is small and expand the resulting e ective action
in p owers of g At the end we will set g
Integration over one leads to a factor
n
Z
1
p
exp l l d
n n n n n
l l l l
1
n n n n
while the integration over leads to a factor
n
Z
q
n o
1
d a exp b a
n
1
with a and b given by
a g
n n
P l
l
Fig Normalized single edge distributions as a function of the measure parameter
Bottom to top at l For the distribution b ecomes
singular
g g b g
n n n n n n
Expanding the resulting expressions in and and rep eating the pro
n n n
cedure for every other o dd site leads to the following results for the renormalization
of the couplings
z z
Recall that z is the co e cient of the kinetic term for the scalar eld and was
originally Of course the factor of z can b e re absorb ed by rescaling the scalar eld
Also to this order we cannot distinguish a renormalization of from a renormaliza
tion of On the other hand the two are related since by expanding ab out l
n
we are tacitly assuming that hl i which xes with resp ect to as discussed
previously It is therefore su cient to lo ok at the renormalization of the parameters
and z
A graph of and z is shown in Fig An instability develops for a
su ciently singular measure here for In the full theory this value is
presumably shifted to as can b e seen for example from the sum rule
of Eq The signi cance of this particular value has already b een discussed
previously For the decimation transformation to b e meaningful has to b e such
that the correction is not to o large or in which case we conclude
that as the scale of the system is increased infrared or long distance limit the
measure parameter increases in value making the measure less singular while
the co e cient of the scalar action remains of the same sign for large the deviation
of z from unity approaches zero after we set
We should add that in this approximation if the eld would have had n
n f
comp onents the instability would have moved to n In the full
f
theory this should o ccur for n Since we have only p erformed the
f
decimation transformation using an expansion in weak disorder g we do not
exp ect this result to b e exact In the next section we will show that in fact one can
integrate the scalar eld out exactly
z
Fig Renormalized couplings and z after one decimation
Scalar Field Determinant
The scalar eld action of Eq with can b e written as
N N N
X X X
0 0
l M
n n n n n n
l
0 n
n n
n
0
l given by the matrix with M
n n
l l l l
1 1
N N
C B
C B
l l l l
1 1 2 2
C B
C B
C B
l l l l
2 2 3 3
C B
C B
C B
M l
N
C B
C B
C B
C B
C B
C B
A
l
N 1
l l l l
N N 1 N 1 N
The determinant of M l can b e computed exactly It is given by
N
P
N
N l
n
n
0
det M l
Q
N
N
l
n
n
where the trivial zero mo de corresp onding to an overall translations of the s has
n
P
N
The scalar eld b een factored out by requiring for example that
n
n
contribution to the e ective action is therefore
p
0
I l N log Tr log M l
ef f N
q N N
p
Y X
l log log N log l N log
n n
n n
The third term can b e expanded by setting l l
n n
N N
X X
l log log N log l log
n n
N
n n
and
N N N
X X X
O N log
n n n
N N N
n n n
Therefore in the in nite volume limit N one gets simply
q N
Y
N
exp f I l g l l exp f log N O N g
ef f n
n
q N
Y
l const
n
n
where const denotes an l indep endent expression We note that the last result
n
Q
p
corresp onds to a factor g x in the continuum with the identi cation g l
n
x
In other words in one dimension we nd the exact result
Y
p
const g x g g det
x
It should b e emphasized that the integration over the scalar eld generates in
teractions b etween the edges but that these interactions are suppressed by factors
of N and do not therefore contribute in the continuum limit As in the absence
of the scalar eld one nds that no unwanted correlations b etween di erent edges
are induced
i h
O N hl i hl i hl l i hl i
n m n m
This result is of course consistent with the exp ectation that no di eomorphism
invariant interaction term for the metric can b e written down in one dimension
P
N
l app ears in the exp onent of the Furthermore since no term of the typ e
n
n
nal answer one concludes that there is not even a nite renormalization of the
cosmological constant term in the in nite volume limit the renormalization is
N and vanishes in this limit
The only e ect of the scalar eld in this mo del is to change the measure by a
p
Q
N
factor l which can b e in fact b e cancelled by shifting the measure parameter
n
n
by in the original partition function If we insist as we argued in Section
Q
that the correct measure b e dl then this requires that we take
n
Poisson distribution
It is amusing to note that if one retains only a mass term for the scalar eld
which is the term prop ortional to in Eq and integrates over the scalar
eld one obtains quite a di erent weighting factor for the edges namely
N
q
Y
N
l exp f I l g l
ef f n n
n
Q
and which corresp onds to a factor g x in the continuum It can b e cancelled
x
by an appropriate re de nition of the functional measure over the edges or the
metric This discussion shows again clearly that a priori should b e treated as a free
parameter of the theory and is therefore either irrelevant or has to b e determined
empirically
It is instructive to compare the previous exact results with an analogous calcula
tion in the continuum We shall see that while it app ears attractive to develop the
continuum theory ab initio one inevitably encounters a numb er of ambiguities as
so ciated with the regularization pro cedure In Ref the determinant asso ciated
with the integration over the scalar eld is shown to lead to a factor
d
0
exp f I g g L det
ef f
dt
The determinant is then computed using
Z
1
X
ds d
0
exp n s L log det
2
dt s
n6
q
R
g d is again the where here represents a short distance cuto and L
length of the lo op For small cuto one obtains the result
L L
p
log
and therefore
d
0
I g log L log det
ef f
dt
L L
p
log
The signi cant di erence with the exact lattice result of Eq is in the cosmo
logical constant renormalization The formal continuum calculation gives a spurious
p
while the lattice renormalization vanishes in the renormalization
large N limit as we p ointed out previously The correctness of the lattice b ehavior
can b e traced back to the correct invariance prop erties see Eq of the scalar
eld action under continuous lattice di eomorphisms as describ ed in Eq
Sp ectral Prop erties
It is of interest to investigate the sp ectral prop erties of the one dimensional scalar
eld Laplacian of Eq with an edge length probability distribution given by
the generalized Poisson distribution of Eq
The sp ectral prop erties of the scalar Laplacian on a quenched random lattice
for which the separations l are indep endently distributed Poissonian variables
i
were rst discussed in Ref For the mo del we are considering in this pap er
their results can b e extended by considering the mo di cations due to the di erent
form of the probability distribution
Denote the average density of states p er unit length and unit frequency range by
In the presence of the ultraviolet lattice cuto it can b e normalized to unity
R
1
d The sp ectral density can b e obtained from the function
Z
1
d log
via the relationship
d
Im i
d
with Extending the weak disorder low frequency small expansion to the
distribution of Eq one obtains
O
By analytic continuation one then nds
p
O
It is clear that at least here the leading sp ectral prop erties of the Laplacian for
small frequencies do not dep end on the gravitational measure parameter which
app ears only in the rst lattice correction As already p ointed out in Ref on a
regular lattice
p
O
which corresp onds here to the choice while in the continuum the correction
terms vanish The same result can b e achieved for the rst lattice correction term
p
which is quite close to the Poisson distri if one cho oses
bution value As one approaches the singular p oint the corrections
diverge and the weak disorder expansion breaks down which lends further supp ort
to the conclusion that this p oint is clearly pathological
The lo calization length L determines the spatial spread of the eigenfunctions
of the random Laplacian and can b e computed from
L Re i
leading to
L O
On a regular lattice i is purely imaginary on the sp ectrum and the lo cal
ization length is in nite Here one nds instead that b ounded solutions to Eq
decrease exp onentially with distance with a lo calization length that vanishes in the
continuum limit unless approaches in which case the lo calization
length vanishes for all s
Conclusions
There are a numb er of conclusion which might b e drawn from the ab ove discussion of
one dimensional gravity and which might or might not b e relevant for the physical
theory four dimensional gravity We have rep eatedly emphasized the fact that there
are no gravitons in low dimensions and one has to limit the analogy to facets which
are not tied to sp eci c asp ects of the gravitational action We have argued that
items such as the simplicial lattice discretization the continuous di eomorphism
invariance and the issue of the invariant or nearly invariant functional measure have
their legitimate one dimensional counterparts and share some of the problems of the
higher dimensional formulation if one is willing to lo ok b eyond the sp eci c details
such as the choice of underlying lattice structure As usual the discrete expressions
for the action and the measure and their symmetries are highly constrained by
the fact that they have to repro duce their formal continuum counterparts and
should b e regarded as the precise de nition for what is intended when the continuum
expression is inserted into a functional integral since the quantum elds are known
to b e nowhere di erentiable
Let us summarize here the salient p oints of our analysis The lattice mo del
for pure one dimensional lattice we consider is the most natural one and is essen
tially unique maintaining b oth lo cality and p ositivity We have shown that the
discrete lattice action prop ortional simply to the length of the curve has an exact
lo cal continuous invariance Strictly sp eaking this invariance is only valid for small
deformations of the lattice as large deformations cannot b e p erformed without vi
olating the p ositivity of the edge lengths which should b e imp osed as a natural
physical requirement If these considerations are set aside then there is a unique
Q
lattice measure namely dl but the measure again violates the invariance due
n
n
to the presence of a lower limit in the edge length integration This should not
come as a surprise since the very presence of a lattice violates translational rota
tional and scale invariance all of which are sp ecial cases of di eomorphisms The
result is therefore in accordance with the general exp ectation that no fully invariant
discretization of gravity can exist as an ultraviolet cuto necessarily breaks scale
invariance The one dimensional mo del shows explicitly how close one can get to
a continuous lo cal invariance Considerations on the functional measure based on
the intro duction of a metric over metric deformations do not seem to provide fur
ther insights in fact we have argued that it suggests an incorrect measure in
the sense that the resulting naively discretized measure is not even invariant un
der in nitesimal di eomorphisms Intro ducing a more physically motivated but
non lo cal notion of distance b etween one dimensional manifolds we recover the
correct lattice measure We have further argued that as in the continuum it is
not p ossible to write down an interaction term b etween the lattice analogs of the
metric eld the edge lengths as such a term would violate the continuous lattice
invariance describ ed previously
The intro duction of a scalar eld coupled to the gravitational degrees of freedom
necessarily re prop oses some of the same problems Now the lattice scalar eld
action will b e exactly invariant under continuous lattice di eomorphisms provided
the scalar eld itself ob eys a certain natural transformation rule which is a discrete
form of the continuum eld transformation law In other words the action remains
invariant once the appropriate transformation laws for the elds are identi ed The
lesson here seems to b e that in general the continuous transformation laws in the
discrete case can b e rather involved Furthermore since the average of pro ducts
of scalar propagators is not the same as the pro duct of averages one nds residual
gravitational interactions even in one dimension
In the one dimensional case we have implemented a simple real space renormal
ization group transformation based on the concept of decimation in which a partial
trace is p erformed in the partition function The renormalization group transforma
tion determines the ow of coupling constants as the scale of the system is halved
In the case of one dimensional gravity it exhibits an instability of the theory which
app ears if the gravitational measure is to o singular leading to a collapse of the lat
tice structure This phenomenon can b e considered a remnant of the p otential for
collapse found in higher dimensions in which the lattice degenerates into a lower
dimensional ob ject
In one dimension one can go as far as integrating out exactly the scalar eld
and compute in closed form the resulting determinant The results one obtains are
of interest for a numb er of reasons As a consequence of the residual co ordinate
invariance built into the lattice action one nds that no e ective interaction is
generated b etween the edge lengths in the in nite volume limit This results agrees
with the exp ectation that no co ordinate invariant metric interaction term can b e
written down in one dimension all the e ects of the scalar eld can b e reabsorb ed
into a re de nition of the purely gravitational measure Furthermore when the
exact lattice computation is compared with a formal continuum calculation one
nds that a characteristic feature of the lattice result is that the renormalization of
the cosmological constant vanishes identically in the in nite volume limit contrary
to the continuum result
Finally we have brie y discussed the sp ectral prop erties of the scalar Laplacian
with uctuating edge lengths The sp ectral density and lo calization length can b e
computed for small uctuations of the edge lengths For small frequencies the density
of states agrees with the uniform lattice and continuum result with corrections that
diverge for singular choices of measure Similarly the lo calization length asso ciated
with the eigenfunctions of the Laplacian diverges for regular lattices and vanishes for
singular measures We have found it encouraging that the leading sp ectral prop erties
of the Laplacian for small frequencies do not dep end on the gravitational measure
parameter
Acknowledgements
The authors thank Gabriele Veneziano and the Theory Division at CERN for
hospitality and supp ort during the completion of this pap er This work was sup
p orted in part by the UK Science and Engineering Research Counsel under grant
GR J
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