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Non perturbative quantum gravity
Nuclear Physics B (Proc. Suppl .) 17 (1990) 29-38 North-Holland
29
NON PERTURBATIVE QUANTUM GRAVITY Pietro MENOTTI Dipartimento di Fisica della Università di Pisa and INFN sezione di Pisa
A review is given of the lattice approach to non perturbative quantum gravity I shall consider here quantum gravity in four dimensions only ; moreover I shall restrict myself to the approaches which gave rise to some actual non perturbative calculation . There are already two reviews on the subject, one by Hamber Ill and the other by Berg [2]. After recalling a few results of the continuum formulation I shall give a brief review of the discrete regularizations; then I shall discuss the problem of the integration measure. Next the available numerical results will be discussed and an assessment of the present situation attempted.
with
To fix the ideas I intervenes in the euclidean functional integral in the form e-1. When expanded around a flat background (À = 0) such an action describes a spin-two massless particle which self interacts through a non renormalizable [4[ interaction . Terms quadratic in the Riemann tensor can be introduced ; in four dimensions the following equality holds (we refer to the euclidean case) _
1. Continuum formulation
The Einstein - Hilbert action in presence of a cosmological term is given by
2 R~r.ArcATd d4 x(mpR_A)= 1 [ I= J 2 -4 -r" A rb A 7` A rd1 Eabcd
=
(1.1
where mp V1116irGN = 0.17 1019 GeV, being GN Newton's constant and ra are the vierbein one forms related to the metric g, , by 1.2
;`v = iabrf` rV " 9 wab is the spin connection and . 4a`6 Rab = dwab -}- wa A
For the euclidean action Î = -il after rotation in the complex plane [3] we have Î= -
J
,fg- d4 x(mpR-A) = -
4
f[ 2 2
ra A rb Arc A rd] Eabcd
RabAr`ATd (1 .4
0920-5632/90/$3.50 © Elsevier Science Publishers B.V . North-Ilolland
Tara
1
g d4x_
32~r2
PRE
[R,V~
`Ap _ 4R, ,Rw"
+ R2]
(1 .6)
where X is the Euler signature. Thus for fixed topology the independent terms which are quadratic in the Riemann tensor reduce to two and the general action takes the form I=_ %
d4x [mpR-A-a(R,VRu'-1R2)-3R2]
or equivalently _
fV
1g d4x [m",R-a-4R~`~PR, .VaP_3(b-4a)R2] .
(1.8)
In Einstein's theory one can consider the connections wab as indepedent of the vierbeins in the so called first order formalism; the zero torsion relation Ra -dra+w%Arc=0
(1 .9)
is an equation of motion which when solved gives for the wab the usual Levi-Civita connections in the vierbein basis. For higher derivative theories (1 .9) is not an equation of motion and it is usually imposed as a constraint defining wab . However formulations of higher derivative gravity with wab and
P. Menotti/Non perturbative quantum gravity
30
r° independent have also been proposed [51. There is a general reasoning due to Hawking [31 proving the lower unboundedness of Einstein's gravitational action also in presence of a cosmological term : perform the following conformal transformation 9. .
~. _ 9, . = fj 2g
(l.10)
under which R --> R =
fl-2
R - 6TE-'OSE .
(1 .11)
Thus the action becomes
--
f
i=-f,
d4a(pnpR - A) --+
d4ae('np[fJ 2 R + 6St;QTE;69ab1- ATE') . (1 .12)
For rapidly varying conformal factor Î becomes
arbitrarily large and negative. As remarked by Weinberg 161 the R2 theory can be viewed in two different ways 1. As a renormalizable theory [71 but in presence of ghosts. In fact [71 the quadratic part of the lagrangian when expanded around a flat background gives rise to a rapidly decreasing propagator (like k'') which however contains ghosts. In addition to the usual graviton we have a spin 2 massive MP negative metric particle of bare mass m2 = Vfai MP and a massive scalar of mass rn o = -2b 2. As a unitary theory in presence of a non renamalizable interaction ; here one considers as free lagrangian the quadratic part arising from R only. Independently of the viewpoints, in presence of RZ terms, for proper chooce of a, b and A the action Î can be made definite positive for X = 0. In fact using the identity
f
2
f
~d4x~i,,vapCpvap !
f
~d4x(R, ,R,"-3 R2)
(1 .13) with C, ,ap Weyl's tensor, it is immediate to prove that fora > 0, b > 0 and A>
3 4h4
(1.14)
the action Î is definite positive . The last inequality considered for renormalized values of A and mp gives rise, due to the smallnes
) of the cosmological constant ( =4- < 10 -122 to MP a completely unphysical situation; fact we would be in presence of a scalar tachion of essentially zero mass Ime l < 10 -e1MP . One has to keep in mind however that the values appearing in Î are bare and not renormalized values. The theory with terms quadratic in the Riemann tensor, treated as a renormalizable theory shows for proper choice of a and b asymptotic free dom [8,91 (see 191 for the last result). Summarizing the result of paper [91 we have asymptotic freedom for the dimensionless coupling constants a -1 and b -1 small and positive, with b-1 > a-1/5 .46.... Then at high energy we have a-' -+ +0 and b-1 -+ 43 .6 .. a -1 . Several attempts [101 have been produced to give to the perturbative R2 theory a unitary status either by proper summation of graphs or by trying to remove the ghosts by proper choice of gauge or by using non conventional evaluation of Feynman graphs . None of these attempts however gave rise to a well defined procedure proved to be unitary and Lorentz invariant to all loops. Weinberg 161 has proposed to study gravity as an "asymptotically safe" theory, i.e. to look for an U.V . fixed point different from zero and renormalize around it S la Wilson . There are by now a few examples of theories which are renormalizable in non conventional way, like the Gross - Neveu model in dimension 2 < D < 3 [111 or the O(n) sigma model in 3 dimensions and thus in is interesting to look whether gravity can be defined as a non conventionally renormalizable theory. Moreover there are analytical hints for such a fixed point in 2 -f- E dimensions (see ref.[61 last section) . In order to examine such a possibility we need a non perturbative treatment of gravity and to start, a non perturbative regulator. In the next section we shall describe two non perturbative regularization ; othefs may be discovered in the future .
2. Discrete formulations
1. Regge calculus . The idea [121 is to replace a smooth manifold by a piecewise flat simplicial manifold . The curvature in concentrated on D - 2 simplices thus for D = 4 on two dimensional triangles, usually called hinges . Given a hinge and a four dimensional simplex which possesses such hinge as a face one can define an
P. Menotti/Non perturbative quantum gravity opening angle relative to it ; such an angle E salifies the (intuitive in 2 and 3 dimensions) relation sin ®D
D VDVD-2 D - 1 VD _IVD-1
(2 .1)
where VD is the volume of the D dimensional simplex and VD-1, VD_ i are the volumes of the two D -1 dimensional simplices which share the given hinge (for a complete characterization see 11,12)). Given a hinge one can define the deficit angle relative to it by bh=21r-S6, (2 .2) .Ch where a are the four dimensional simplices which possess h as an hinge. Regge [12) postulated the following action 2 E Ah6h h
(2.3)
where the sum is extended to all hinges . The relation of action (2 .3) to the continuum theory has been examined in several papers 1131 . Regge's action turns out to be unique in the following sense: in we consider a sequence of smooth manifolds approaching the piecewise flat manifold we have
f V9- d"a R -+ 2 Eh Ahbh
(2 .4)
(see references 113) for the exact meaning of "approachin,,' ). On the other hand it is not unique in the following sense: if we consider a sequence of piecewise flat manifolds approaching a given smooth manifold there are actions which differ from Regge's (e.g . by adding higher powers of bh) which in the limit converge to the usual continuum Einstein's action . There is no problem in extending Regge's action to include the cosmological term a& V® . Hamber and Williams [141 gave the transcription of terms quadratic in R,,, .\, on the Regge skeleton . To do this it is necessary to associate to each (two dimensional) hinge a four dimensional volume Vh . This can be done in a variety of ways . The simplest higher derivative term is h AVhh
-~
4
f
r,- d"a R,x,R" xP .
(2 .5)
This association is easily understood as the vanishing of all bh imposes to the Regge manifold to be flat, in the same way as the identical vanishing of
31
the Riemann tensor implies the flatness of the four dimensional space. Hamber and Williams 114) gave also the translation on the Regge skeleton of the other invariants quadratic in the Riemann tens. They associate to each four dimensional simplex s the Riemann tensor [R,.%,] .
Ab
= E w.,h[ V U, .UAP)h hC .
(2 .6)
with U, , functions of the links 1(.) 1(b) which specify the hinge h U,.~
_ _1 lx e,. aP (a)l(b) . 2A
(2 .7)
A, b, V are the area, the deficit angle and the volume associated to the hinge h and Vh N.IhV.
(2 .3)
with N.1h the number of simplices per hinge. In all cases the transcriptions are not unique and they become equivalent in the continuum limit i.e . when the maximum link itngth goes to zero relative to some characteristic radius of curvature of the continuum manifold . 2. Gauge approach . Here one exploits the (incomplete) analogy which gravity shares with gauge theories . The attitude is akin to considering gravity as a field theory in Minkowski space (or euclidean space) whose lagrangian is invariant under a group of local transformations (reparametrization transformations) . Several peoples 1151 introduced along these lines discretized versions of gravity. I shall here breefly describe the formulation due to Smolin 115) which can be considered as the prototype of these gauge formulations. Essentially Smolin's formulation is the discretized version of the MacDowell - Mansouri [161 gauge formulation of De Sitter gravity. Let us consider the De Sitter group 0(4,1) which goes over to 0(5) in the euclidean, and introduce the usual hypercubic lattice familiar to gauge theories . Associate now to each link of the hypercubic lattice a finite element of 0(5)
U(n, n + p) = exp( 1aJbcwb + aPce')
(2.9)
with Jab and Pa obeying the following 0(5) commutation relations
P. Menotti/Non perturbative quantum gravity
32
[-7&bv Jed] = &Jad - 6ac .7bd - bbd-lac + bad.lbc
gauge transformations
V.6, PC] = Sbl- P. - b, [Pa , P6] = The action is
(2.10)
-J b -
ritten oyapU
here a =
hw--,hw+k,,f +Lf, .
4
3~â
(n)Ua,D(n)£ABCDS
(2 .11)
and
Ur (n) = UAC (n, n + j&)UCD (n UDE (n + I& + v, n + v)UEB(n
+,u9 n + jL + v, n)
+ v)
(2 .12)
is the plaquette in the vector representation . The action is obviously invariant under local O(4) transformations, but not under O(5) transformations . This is the result of the incomplete analogy between gravity and gauge theories. Going over to the formal continuum limit one gets the ac Dowell - ansouri action z [det -r (m pR - :+) é1~Ap
$-
XpE
(2 .13)
38a which is the usual Einstein's action in presence of the cosmological term to which a Gauss - Bonnet term has been added. The discrete action is invariant under reparametrization only in the formal continuum limit. Extension of this treatment to the Poincaré group can be easily given [17] . The written discrete action is obviously non symmetric with respect to reflections because it favors the sitive directions on the lattice. One has to perform the symmetrization with respect to directions otherwise one ends up with propagators containin complex terms of the form exp(iak,,) -1, when expanding e.g. the Poincaré version around a flat background . When such an expansion is performed using the symmetrized lagrangian one obtains at the quadratic level the quadratic continuum Einstein's lagrangian 2kAkahm~(-k)~~(k) - 2kak~h~~(-k)hv~(k) +kake"ha~"(®k)hw(-k) ° kAkVha~(°k)hev(k)
(2 .14)
here however the continuum momentum k,, is replaced by k,, -- sin(ak,,)/a. Such a quadratic la rangian is invariant under the following linear
(2 .15)
However the appearance of k, = sin(ak,,)/a and not of 2sin(ak,,/2)/a implies the phenomenon of graviton doubling (16 gravitons !). H. Nielsen [18] suggested us that if a point is reached where reparametrization invariance is recovered only one graviton should remain massless while the others should get a mass divergent with the cut - off, because reparametrization invariance protects the mass of one and not of 16 particles. I shall come back to this problem in the next section. We have given a euclidean formulation and thus we should care about reflection positivity which is the property which assures unitarity in the reconstruction of the theory with Minkowski signature [19] . Symmetrization as already noticed, is necessary but it is not sufficient for having reflection positivity. In fact the written action is pseudoscalar in the internal space ar-d thus not equivalent to the original action . We have that 2 ,J
Rab n r` A rdeabcd
=
f det r R d4x
(2 .16)
is not equal to
f ~,fj_ R d°x = f
1
det-rIR d4a
(2 .17)
unless we multiply the lagrangian in (2.16) by sign(det r) . In order to do so we must extract from the fundamental link variables the local vierbeins,
i.e . vierbeins which transforms locally under O(4) . Such vierbeins are also necessary to write down the Poincaré formulation of gravity. These local vierbeins in the le Sitter case can be extracted through the formula [17] T,;(n)
_ -j atr[HP°U(n, n + p)HU(n + Ftv)] (2 .18)
where lip = diag(1,1,1,1, -1). After introduction of sign(det T) in (2 .11) reflection positivity can be proven for all O(4) invariant abservables (which are the only physical ones). Moreover given the local vierbeins one can easily translate all RI terms of the continuum on the lattice and again reflection positivity proven [17] . An extension to N = 1 supergravity has also been given 1201, but here no reflection positivity result is yet available. The gauge approach is strictly bound to the
P. Menotti/Non perturbative quantum gravity program outlined by Weinberg of finding a fixed point around which to renormalize the theory. The Re e approach can also be used to investigate this problem . On the other hand' the Regge approach has been proposed [211 as a way to introduce a fundamental length in physics by putting an upper bound on the number of sites of the random lattice in a given four dimensional voulme . We shall come back to this issue in section 5.
3. The problem o the measure There have been several proposal for the non perturbative integration measure for quantum gravity on the continuum . They are all based of formal arguments, which explains the variety of the obtained results. I shall limit myself here in outlining the origin of a few of them without any pretence of completeness . De Witt [221 looks for a definition of distance between two nearby geometries whose difference is characterized by bg,,r, which is invariant under reparametrizations and ultralocal, i.e function only of 9,," and not of its derivatives . The expected result, whose uniqueness is proven by De Witt is given 1169""II' where
= 1&"z d"y
G"AP(x, y) is
G"A"(x, y)
b9""(x)G ""AP(x, y)69AP(y)
(3.1)
the local bitensor density
9`9A°WZ-y) = V9_(9"A9"P+9"P9"A+-/ (3 .2)
and y is an arbitrary constant. This is also the starting point in the treatment given by Polyakov in string theory [231. One defines as usual the volume element as the square root of the determinant of the metric ; the determinant of the infinite continuous matrix G""AP(x, y) can be easily computed at the formal level to get
G(x)
det GI"P(x, y) = with
G(x) = const . 9(x)
(D-®1(D+1)
(3.3) (3.4)
and thus for the measure in 1) dimensions we have const.
.1(D+o1 41 9(x)(D`
11 /sÇ"
d9""(x)
(3 .5)
33
which in four dimensions is simply a constant . Other reasonings [241 are based on the imposition of invariance under local confamal tranformation noticing that ;7, , under confamal tranformation behaves formally like a dimension 2 object 89w(z) := é(x)(x - 8 + 2)9,,,(x) .
(3.6)
Starting from this remark De Witt's result is again obtained; using on the other hand the vierbeins a similar reasoning [241 gives for the measure in four dimensions dr, (x) (3.7) which is not the simple jacobian transformation of (3.5), due to the fact that under a dilatation the density of points itself varies . The integration over vierbeins is preferable at the non perturbative level because as pointed out by Hawking [251 it assures the correct signature of 9,u, . Faddeev and Popov [261 using the hamiltonian approach obtain 9(x)6/=
JUL
d9w,(z).
(3 .6)
"<"
Other results re obtained in the hamiltonian approach by Leutwyler and by Fradkin and Vilkovisky [271. (3.5) is not a measure in the Lebesgue sense, and it is an interesting question whether a Lebesgue measure for Riemann geometries can e defined at all. Gromov 1251 was able to give a rigorous definition of measure on the space of geometries; his main result is that it is possible to define a distance between geometries and that according to such a distance the set of eometries of bounded diameter and with Ricci tensor bounded from below, R,, , > -cg. is (pre-)compact thus allowing, going over to the dual of continuous functions the introduction of an infinity of true Lebesgue measures. Such ideas are very profound but they do not solve the problem of which is the correct measure for quantum gravity, a problem again handled over to physicists. to discrete formulations the mea Coming n sure employed in actual Re e calculus computations is the so called scale invariant measure
inspired by the Fadd v - Popov result (as many
P Menotti/Non perturbative quantum gravity
34
rs of 9 in the numerator as many in the denominator) . Hamber [29] has tried the measure (3 .9) on the two dimensional model getting within 10% the exact critical index -fme derived by ICnizhnik, Polyakov and Zamolodchikov [30] . This lends to the measure (3 .9); however such a measup s'_-!re is not immune from critics. I report here a typical one by Jevicki and Ninomiya [31] . Their point is that the link lengths are invariants which characterize completly the geometry of space; due to reparametrization invariance there are many 9,", which describe the same geometry. Thus given the {k-) one can construct in some fixed gauge some g,(a ; {h)) to which a reparametrization transformation g,, can be applied, thus obtaining the whole family of equivalent gw(z ; {h), &,). In the variation bg. one should consider both the one arising from the variation of the geometry i.e . of the {1s) and that arising from the variation of the ~,,. Starting from De Witt result (3.X) one can rewrite it as
(b9, b9) _
bl; K;; bl; + ff, 0)" b&, (3.l0)
where G(t) is easily proven from (3 .2) on the continuum to be
G(E) _ -4vfg-(9"V a + Rl')6'(z - y) = -4,1g-(-L" + 2R")b4 (z - y).
(3.11)
Rew is the Ricci tensor and LIv the Lichnerowicz
operator
LI'
=
-(g"'®'
-
R,`~),
(3 .12)
with Va the vector covariant laplacian. Going over to the lattice the Lichnerowicz operator has a simple translation in the form of L = b d + d b [32], being d and b the boundary and coboundary operations. Also as we saw above, Rw can be translated on the discrete and thus dl; (det K)=(det G( 4)) 12'
(3 .13)
where det K turns out to be an essentially local object while det G(E) is a non local object of the Faddeev - Popov type . Turning over to the gauge formulation of gravity we saw how on the continuum the proposed measures in the 9, , or in the vierbeins ra, are of the type (det r)M . To throw some light on the measure
in presence of the already described lattice cut - off an explicit one loop computation of the SlavnovTaylor identities was performed and the local measure determined such as to satisfy for a --4, 0 the above mentioned identities [33] . The calculation can be performed in completeley analytical way and it gives for the local measure (det r)N
f(E r1,rl) a
3.14
where the O(4) invariant function f(E. T~asr) has to be allowed as it happens in Yang - Mills theory, and if one likes, can be absorbed in the definition of the vierbeins themselves. The result is N= 3+3t with t = 2
f
dz [e - ZIo (z)] 4 = 0.619 . ..
(3 .15) The result is rather disappointing as a flat integer value would have been more appealing. Higher loop can introduce corrections to the result and from the structure of the calculation one also gets the impression that the obtained value is strictly regularization dependent; using e.g . a different lattice would have given a different result . On the other hand the dependence of the measure on the regularization is not a new fact . Imposition of the Slavnov - Taylor identities is equivalent to imposing that the (0, 0, 0, 0) graviton does not acquire a mass. On the other hand the three graviton vertex which can be worked out completly [33] is not invariant under the transformation ak,, --+ 7r-ak,, as it contains terms with sin(ak,,/2) and cos(ak,/2) . Thus in agreement with Nielsen's suggestion the other 15 gravitons should take a mass of order 1/(mp as), a calculation on the way of being performed. The measure in lattice gravity is a major problem and some attention should be devoted to it either by putting on rigorous grounds existing formal arguments or by performing experiments on exactly known results.
4 . Numerical simulations for lattice gravIn four dimensions we have two numerical simulations for lattice gravity in the Regge formulation, one by Berg [34] and the other by Hamber and Williams 1351 and one simulation by Caracciolo and Pelissetto [36] for De Sitter lattice gravity in the gauge formulation . We shall give here a short re-
P Menotti/Non perturbative quantum gravity
view of their result : referring to the original paper for full details. Berg [341 considers the pure Einstein case where the partition function is given by dl
1
e
2rc2
r h
Ahbh
(4.1)
The simulation is performed at fixed total four dimensional volume which corresponds to introducinga kind of microcanonical cosmological constant. Such a constraint is necessary to get a meaningful functional integral. We recall that a bare cosmological constant is necessary [371 even in perturbative calculations [331 to kill the quartic divergence which arises already to one loop (except in the dimensional regularization scheme where it is put to zero by hand). The meaning of the pure number rc2 in the numerical simulation is that the value of the total volume is 0 Vro/mp if during the simulation the total volume is kept equal to the pure number Yo. A random Regge lattice i.e. with arbitrary incidence matrices appears at present too complex for a numerical computation . Thus Berg starts from a regular hypercubic lattice with positive diagonals or with a kind of centered hypercubic lattice and considers variations of the link lengths. It appears that such structures provide a good approximation to the random lattice even if here the topology is kept fixed. The simulation is performed on 2s and 34 lattices which already have a large number of link variables (15 for each hypercube in the hypercubic model with the positive diagonals) . Berg computes the average values of several quantities : the average action density S = (2bhAh), the average area of the hinges (Ah) and the average deficit angle (bh). First the simulation is performed putting rc= = 0, thus one measures here only the effect of the entropy of the configurations. The main result here is that the average action density is small and negative S = -0.056 f 0.006 . The simulation is then repeated increasing rc2 away from zero up to the value 0.3. The main result found by Berg is a sudden transition in all the above listed average values at a critical value rc~ estimated in the range 0.02 <; rc, G 0.04 . For values below rcc' i.e. in the so called entropy dominated region the situation is very similar to the pure entropy case, while across the transition S changes sign and becomes S " 102 i.e.orders of magnitude larger than below rc2 . More-
35
over one ends up into non reproducible situations thus pointing to the existence of many metastable states for r62 > r .,2 similar to a spin glass model. A similar discontinuous behavior is experienced by the other quantities like (bh) and (Ah). A notable fact in the region rc2 > rc;2 is a positive value of S coupled with a negative value of (bh) which shows a correlation between the deficit angles and the area of the hinges. Hamber and Williams [351 consider gravity in the Regge formulation in presence of terms quadratic in the Riemann tensor. The general structure of the action they use in the numerical simulation is the lattice equivalent of J
~ dz'' [mpR - A - 4R, ,i,R"-% p]
(4.2)
thus setting a = 4b in (1.8) . Due to the scale invariance of the adopted measure all resiAt r are functions of the adimensional constant mp%a and a = 4b. Thus in Hamber and Williams' simulation the cosmological constant is explicitly introduced ; the term R.a,R'--% P is expected to stabilize the functional integral. They consider the following adimensional quantities
and
R = ( 2 Eh Ahbh) (12)
(4 .3)
R2 = (4 Eh 461/Îrh) (12)2 .
(4.4)
(Eh Vh)
(Eh Vh)
Again putting all couplings to zero Hamber and Williams find a negative average curvature R -10 and 7i.2 se 58200. Thus we have a large adimensional curvature with large fluctuations . For non zero action two typical cases are explored the first with a small value of the R2 term i.e. a = 0.005 and the second with a = mp/A i.e equal to the other adimensional constant. Fixing a = 0.005 they find, varying mp/A, the following results for R and R2 R R2 mpla
0.125 0.167 0.250 0 .500
-0.612 8900 -0.420 8660 132 134000 123 143000
Thus in presence of a small R$ term again one finds for small mP/a an average curvature which is small
P. Menotti/Non perturbative quantum gravity
36
and negative and with large roughness Rs/R >> 1. Above the critical value m4/,X = 0.2 f 0.04 one has a sudden transition to large postive average curvature and the jump is so large that appears to indicative of a discontinuous transition . Values of R and Ra much smaller than 1 would signal a transition toward a continuum limit; from the a ve we see that for small a (a = 0.005) such a situation is not achieved. For a sizable R2 term i.e. for a = rnp/a the situation changes considerably
mP/A
0.167 0.250 0.500
R -0.044 -0-031 -0-030
Ra 496 432 208
e see n that R is consistently small though with large roughness. Thus with regard to a continuum limit the situation has definitely improved even though cannot say having reached the continuum limit. I come now to the simulation performed by Caracciolo and Pelissetto 1361 of the De Sitter model in the gauge formulation. They made ex tensive simulations on an 8s lattice . The reflection positive action is given by
4
sign(det T)E~A°tIjIIAPD EABCD5
(4 .5)
M4
3 where Q = 32 T and one sums over positive and negative directions . The factor sign(det T) is rather time consuming and thus the first simulation by Caracciolo and Pelissetto has been performed in absence of such a factor i.e. in the pure Smolin model . With regard to the measure they adopted the general local form (det T)N and the results were examined in a large range of N (0 < N < 150). Due to the compact nature of the group SO(5) and the presence of the lattice cut - off the action is bounded and thus the theory must exhibit a well defined ground state. For small Q the vacuum is similar to the QCD vacuum in the strong coupling region . The large Q region is characterized by the vanishing of the vierbein so that the action becomes dominated by the topological term . An interesting order parameter is given by P~
RaaV6 Ra6 pv
(4.6)
whose distribution is peaked around zero in the small Q region and around the extremes ±1 in the large ,6 region. A sudden transition is observed between the two phases at Q, = 0.08 f 0.01 . Around such a value all measured quantities like the mean value of the action per site, the trace of the metric, the O(4) curvature and P, , exhibit very large hysteresis cycles thus showing that the transition is first order. Moreover they also performed simulation with mixed phases showing definitively the first order nature of the transition. For Q > & the system ends up in non reproducible states showing the existence of many metastable states above & recalling a situation similar to a spin glass. The existence of a dense set of topologically metastable states is due to the dominance of the Gauss - Bonnet term in the action for Q > Q. and renders the thermalization of the system above Q. problematic even if the existence of the thermal average is assured by the boundedness of the action. Preliminary results [38] of simulations with the full action (4.5) i.e. with the term sign(det T) included show still greater problem in the thermalization above the critical coupling and appear not to alter the nature of the transition. 5. Conclusions
It is clear from the above review that the computational approach to quantum gravity is still in its infancy. We have only three numerical simulation on relatively small lattices . One conclusion that can be drawn is that despite the difference in the formulations (Regge calculus, gauge formulation) the different treatment of the cosmological constant and the uncertainty in the measure, the three simulations agree on the absence of a second order phase transition for the pure Einstein plus cosmological term model. If this result is going to be confimed as we expect, the above described treatements are unable to define a continuum limit for the pure Einstein gravity at the critical coupling constant . Terms quadratic in the Riemann tensor can be added and as we saw the investigation of their effect has already been started by Hamber and William in the Regge formulation . As we mentioned in sec.(2) the Regge approach has been suggested as a way to introduce in physics a fundamental length [21]. If one accepts this viewpoint the theory becomes more elastic and different
P. Menotti/Non perturibative quantum gravity questions must be addressed. For example Berg [2] suggests to couple gravity to a matter field e.g . to a Yang - Mills field (we know alread how to write down a Yang - Mills field on a random lattice [39]) and to look for a point in the parameter space where the mass gap goes to zero compared to the Planck mass . An other possibility as suggested by Hamber
and William [35] is to extract the renormalized Newton constant by computing correlation func tions at geodesic distances (here one needs somewhat larger lattices ) and look for a point in the parameter space where the renormalized cosmological constant, defined in terms of the average curvature of the universe vanishes compared to the square of the inverse of Newton's constant . The study of correlation functions at geodesic distances appear particularly significant in quantum gravity and correspond to the introduction of the "radial" gauge [40] . Another crucial question in the Regge formulation in presence of terms quadratic in the Riemann tensor is checking the unitarity of the theory a problem about which very little is known.
Acknowledgements I am grateful to S. Caracciolo, H. Hamber, M . Martellini, A. Pelissetto, C. Rovelli and M. Toller for useful discussions . References 1. H.W . Hamber, Simplicial Quantum Gravity, in Les Houches Summer School, Session XL111,1984, Critical Phenomena, random systems, gauge theories ; K. Osterwalder and R. Stora eds. Elsevier Science Publishers B.V., 1986.
2. B.A . Berg, Quantum gravity motivated computer simulations ; XXVII. Internationale Universitätswochen für Kernphysik, Schadming 1988 . 3. S .W. Hawking in General relativity - An Einstein centenary survey, S.W . Hawking and W. Israel eds. Cambridge University Press (1979) 746. 4. S. Deser and P. van Nieuwenhuizen, Phys .Rev D 10 401(1974); G. 't Hooft and M . Vcltman, Ann.lnst. H . Poincaré 20 69 (1974) ; M .H .
37
Goroff and A. Sagnotti, Nucl .Phys. 3 288 709 (l986) . 5 . F.W. Hehl, P. von der Heide, G.D . Kerlich and J.M . Nester, Rev.Mod .Phys. 48 393 (1976) ; E. Sezgin and P. van Nieuwenhuizen,
Phys.Rev . D 213269 (1980); R. Battiti and M. Toller, Lett .Nuovo Cimento 44 35 (1985) .
6. S. Weinberg in General relativity - An Einstein centenary survey, S.W . Hawking and W. Israel eds. Cambridge University Press (1979) 790 . 7. K.S. Stelle, Phys .Rev.
10 953 (1977) .
8. J . Julve and M. Tonin, Nuovo Cimento 48 137 (1978); E.S . Fradkln and A.A . Tseytlin, Phys .Lett. 104 377 (1981) ; Nucl .Phys. 201469 (1982) ; Phys .Lett.100 63 (1981) . 9. I .G . Avramidy and A.0 . Baravinsky, Phys.Lett. 159 269 (1985) . 10 . See e.g . I. Antoniadis and E.T . Tomboulis, Phys .Rev. 3 2756 (1986) ; I. Antoniadis, J. Iliopoulos and T.N . Tomaras, Nucl .Phys. 267 497 (1986) and references therein; see also references in ref.[6] . 11 . K. Gawedzki and A. Kupiainen, Nucl .Phys.
262 33 (1985) .
12. T. Regge, Nuovo Cimento 19 558 (1961) . 13 . H. Cheeger, W. Müller and R. Schrader, Comm .Math.Phys. 92 405 (1984) ; R. Friedberg and T.D . Lee, Nucl .Phys. R 242 145 (1984) ; G.Feinberg, R. Friedberg, T.D . Lee and H .C. Ren, Nucl .Phys. R 245 343 (1984) ; M . Rocek and R.M . Williams, Z .Physik C 21 371 (1984) .
14. H.W . Hamber and R. Williams, Nucl .Phys. 248 392 (1984) ; Nucl .Phys. R 289 712
(1986) ;
15 . L. Smolin, Nucl .Phys. R 148 333 (1979); A. Das, M . Kaku and P.K. Townsend, Phys.Lett. 81 11 (1979); C. Mannlon and J .G . Taylor, Phys .Lett. 100 R 261 (1981) ; E .T. Tomboulis, Phys .Rev .Lett. 52 1173 (1984); K. Kondo, Prog .Theor.Phys. 72 841 (1984); P. Menotti and A. Pelissetto, Ann.Phys .
P Menotti/Non perturbative quantum gravity
38
3 119 ( .Y.) 0 287 (1986) ; Phys .Rev . D'Adda and L. agnea, (1987); . Caselle, A. . 06, 411 (1987) Phys.Lett. 192 II and F. ansouri, Phys. Rev. 16 . S.W. acD Lett . 38 739 (1977) ; Phys . Rev. Lett . 38 1376 (1977) . 17 .
. enotti and A. Peli 1194 (1987).
tto, Phys .Rev.
35
18 . H.B. Nielsen, private communication. 19.
. Oste alder and R. Schrader, Comm . ath. Phys. 31 83 (1973) ; ibid . 42 281 (1975) .
29. H .W . Hamber, private communication. 30 . V.G. Knizhnik, A.M . Polyakov and A.B . Zamolodchikov, Mod. Phys. Lett . A 3 819 (1988) . 31 . A. Jevicki and 1634 (1986) .
.Ninomiya, Phys.Rev .
33
32 . R. Sorkin, J. ath.Phys. 18 2432 (1875) ; D. Weingarten, J.Math. Phys. 18 165 (1977) ; 383 359 . Warner, Proc .Roy.Soc .London (1982) . 33 . P. Menotti and A. Pelissetto, NucI .Phys. 288 813 (1987).
20. S. Caracciolo, P. enotti and A. Pelissetto, ucl.Phys. 293 868 (1988) .
34. R. Berg, Phys . Rev. Lett . 55 904 (1985) ; Phys . Lett . 173 39 (1986) and ref. 2.
21 . G. Feinberg, R. Friedberg, T.D . Lee and H.C. Ren, NucI .Phys. 245 343 (1984) ; T.D . Lee, Discrete mechanics, in Proceedings of the 1983 Erice Summer School, A. Zichichi ed . Plenum Press.
35 . H .W. Hamber and R. Williams, Phys.Lett. 157 368 (1985) ; Nucl .Phys. 248 392 (1984); NucI .Phys. 289 712 (1986) and ref. 1.
22 . 8. De itt, Phys.Rev . 160 1113 (1967) and in General Relativity - An Einstein centenary survey, S.W. Hawking and W. Israel . eds. Cambridge University Press (1979) 680. 23 . A.M. Polyakov, Phys.Lett. 103
207 (1981) .
24 . V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cim. 57 227 (1981) ; K. Fujikawa, Nucl .Phys. 226 437 (1983) .
36 . S. Caracciolo and A. Pelissetto, Phys .Lett . B 193 237 (1987) ; Nucl .Phys. 299 693 (1988); Nucl .Phys. (Proc.Suppl .) 4 78 (1988), Proceedings of the International Symposium on Field Theory on the Lattice, Seillac 1987; Phys.Lett 207 468 (1988) . 37. V. de Alfaro, S. Fubini and G. Furlan, ref. 24 . 38. S. Caracciolo and A. Pelissetto, private communication .
25 . S.W.Hawking in Recent developments in gravitation, Cargèse Summer School 1978, M. Lévy and S. Deser eds. Plenum Press, pag.145 .
39 . N .H .Christ, R. Friedberg and T.D . Lee, NucI .Phys. 202 89 (1982) ; ibid. 210 (FS 61337 (1982) .
26 . L. Faddeev and V. Popov, Sov.Phys.Usp . 18 777 (1974).
40 . G. Modanese and M. Toiler "Radial gauge in Poincar6 gauge field theories" preprint U.T.F . 184 - 1989 to appear in J . Math . Phys .
27 . H. Leutwyler, Phys.Rev . 134 1155 (1964) ; E . Fradkin and G. Vilkovisky, Phys . Rev. I) 8 4241 (1973). 28.
. Gromov, Structures métriqes pour les variétés riemanniennes, CEDIC, Paris 1981 ; for a readable introduction see: 61A . Carfora and A. arzuoli, Phys.Rev .Lett. 82 1339 (1989) ; Class.Quantum Grav. 5 659 (1988).
29
NON PERTURBATIVE QUANTUM GRAVITY Pietro MENOTTI Dipartimento di Fisica della Università di Pisa and INFN sezione di Pisa
A review is given of the lattice approach to non perturbative quantum gravity I shall consider here quantum gravity in four dimensions only ; moreover I shall restrict myself to the approaches which gave rise to some actual non perturbative calculation . There are already two reviews on the subject, one by Hamber Ill and the other by Berg [2]. After recalling a few results of the continuum formulation I shall give a brief review of the discrete regularizations; then I shall discuss the problem of the integration measure. Next the available numerical results will be discussed and an assessment of the present situation attempted.
with
To fix the ideas I intervenes in the euclidean functional integral in the form e-1. When expanded around a flat background (À = 0) such an action describes a spin-two massless particle which self interacts through a non renormalizable [4[ interaction . Terms quadratic in the Riemann tensor can be introduced ; in four dimensions the following equality holds (we refer to the euclidean case) _
1. Continuum formulation
The Einstein - Hilbert action in presence of a cosmological term is given by
2 R~r.ArcATd d4 x(mpR_A)= 1 [ I= J 2 -4 -r" A rb A 7` A rd1 Eabcd
=
(1.1
where mp V1116irGN = 0.17 1019 GeV, being GN Newton's constant and ra are the vierbein one forms related to the metric g, , by 1.2
;`v = iabrf` rV " 9 wab is the spin connection and . 4a`6 Rab = dwab -}- wa A
For the euclidean action Î = -il after rotation in the complex plane [3] we have Î= -
J
,fg- d4 x(mpR-A) = -
4
f[ 2 2
ra A rb Arc A rd] Eabcd
RabAr`ATd (1 .4
0920-5632/90/$3.50 © Elsevier Science Publishers B.V . North-Ilolland
Tara
1
g d4x_
32~r2
PRE
[R,V~
`Ap _ 4R, ,Rw"
+ R2]
(1 .6)
where X is the Euler signature. Thus for fixed topology the independent terms which are quadratic in the Riemann tensor reduce to two and the general action takes the form I=_ %
d4x [mpR-A-a(R,VRu'-1R2)-3R2]
or equivalently _
fV
1g d4x [m",R-a-4R~`~PR, .VaP_3(b-4a)R2] .
(1.8)
In Einstein's theory one can consider the connections wab as indepedent of the vierbeins in the so called first order formalism; the zero torsion relation Ra -dra+w%Arc=0
(1 .9)
is an equation of motion which when solved gives for the wab the usual Levi-Civita connections in the vierbein basis. For higher derivative theories (1 .9) is not an equation of motion and it is usually imposed as a constraint defining wab . However formulations of higher derivative gravity with wab and
P. Menotti/Non perturbative quantum gravity
30
r° independent have also been proposed [51. There is a general reasoning due to Hawking [31 proving the lower unboundedness of Einstein's gravitational action also in presence of a cosmological term : perform the following conformal transformation 9. .
~. _ 9, . = fj 2g
(l.10)
under which R --> R =
fl-2
R - 6TE-'OSE .
(1 .11)
Thus the action becomes
--
f
i=-f,
d4a(pnpR - A) --+
d4ae('np[fJ 2 R + 6St;QTE;69ab1- ATE') . (1 .12)
For rapidly varying conformal factor Î becomes
arbitrarily large and negative. As remarked by Weinberg 161 the R2 theory can be viewed in two different ways 1. As a renormalizable theory [71 but in presence of ghosts. In fact [71 the quadratic part of the lagrangian when expanded around a flat background gives rise to a rapidly decreasing propagator (like k'') which however contains ghosts. In addition to the usual graviton we have a spin 2 massive MP negative metric particle of bare mass m2 = Vfai MP and a massive scalar of mass rn o = -2b 2. As a unitary theory in presence of a non renamalizable interaction ; here one considers as free lagrangian the quadratic part arising from R only. Independently of the viewpoints, in presence of RZ terms, for proper chooce of a, b and A the action Î can be made definite positive for X = 0. In fact using the identity
f
2
f
~d4x~i,,vapCpvap !
f
~d4x(R, ,R,"-3 R2)
(1 .13) with C, ,ap Weyl's tensor, it is immediate to prove that fora > 0, b > 0 and A>
3 4h4
(1.14)
the action Î is definite positive . The last inequality considered for renormalized values of A and mp gives rise, due to the smallnes
) of the cosmological constant ( =4- < 10 -122 to MP a completely unphysical situation; fact we would be in presence of a scalar tachion of essentially zero mass Ime l < 10 -e1MP . One has to keep in mind however that the values appearing in Î are bare and not renormalized values. The theory with terms quadratic in the Riemann tensor, treated as a renormalizable theory shows for proper choice of a and b asymptotic free dom [8,91 (see 191 for the last result). Summarizing the result of paper [91 we have asymptotic freedom for the dimensionless coupling constants a -1 and b -1 small and positive, with b-1 > a-1/5 .46.... Then at high energy we have a-' -+ +0 and b-1 -+ 43 .6 .. a -1 . Several attempts [101 have been produced to give to the perturbative R2 theory a unitary status either by proper summation of graphs or by trying to remove the ghosts by proper choice of gauge or by using non conventional evaluation of Feynman graphs . None of these attempts however gave rise to a well defined procedure proved to be unitary and Lorentz invariant to all loops. Weinberg 161 has proposed to study gravity as an "asymptotically safe" theory, i.e. to look for an U.V . fixed point different from zero and renormalize around it S la Wilson . There are by now a few examples of theories which are renormalizable in non conventional way, like the Gross - Neveu model in dimension 2 < D < 3 [111 or the O(n) sigma model in 3 dimensions and thus in is interesting to look whether gravity can be defined as a non conventionally renormalizable theory. Moreover there are analytical hints for such a fixed point in 2 -f- E dimensions (see ref.[61 last section) . In order to examine such a possibility we need a non perturbative treatment of gravity and to start, a non perturbative regulator. In the next section we shall describe two non perturbative regularization ; othefs may be discovered in the future .
2. Discrete formulations
1. Regge calculus . The idea [121 is to replace a smooth manifold by a piecewise flat simplicial manifold . The curvature in concentrated on D - 2 simplices thus for D = 4 on two dimensional triangles, usually called hinges . Given a hinge and a four dimensional simplex which possesses such hinge as a face one can define an
P. Menotti/Non perturbative quantum gravity opening angle relative to it ; such an angle E salifies the (intuitive in 2 and 3 dimensions) relation sin ®D
D VDVD-2 D - 1 VD _IVD-1
(2 .1)
where VD is the volume of the D dimensional simplex and VD-1, VD_ i are the volumes of the two D -1 dimensional simplices which share the given hinge (for a complete characterization see 11,12)). Given a hinge one can define the deficit angle relative to it by bh=21r-S6, (2 .2) .Ch where a are the four dimensional simplices which possess h as an hinge. Regge [12) postulated the following action 2 E Ah6h h
(2.3)
where the sum is extended to all hinges . The relation of action (2 .3) to the continuum theory has been examined in several papers 1131 . Regge's action turns out to be unique in the following sense: in we consider a sequence of smooth manifolds approaching the piecewise flat manifold we have
f V9- d"a R -+ 2 Eh Ahbh
(2 .4)
(see references 113) for the exact meaning of "approachin,,' ). On the other hand it is not unique in the following sense: if we consider a sequence of piecewise flat manifolds approaching a given smooth manifold there are actions which differ from Regge's (e.g . by adding higher powers of bh) which in the limit converge to the usual continuum Einstein's action . There is no problem in extending Regge's action to include the cosmological term a& V® . Hamber and Williams [141 gave the transcription of terms quadratic in R,,, .\, on the Regge skeleton . To do this it is necessary to associate to each (two dimensional) hinge a four dimensional volume Vh . This can be done in a variety of ways . The simplest higher derivative term is h AVhh
-~
4
f
r,- d"a R,x,R" xP .
(2 .5)
This association is easily understood as the vanishing of all bh imposes to the Regge manifold to be flat, in the same way as the identical vanishing of
31
the Riemann tensor implies the flatness of the four dimensional space. Hamber and Williams 114) gave also the translation on the Regge skeleton of the other invariants quadratic in the Riemann tens. They associate to each four dimensional simplex s the Riemann tensor [R,.%,] .
Ab
= E w.,h[ V U, .UAP)h hC .
(2 .6)
with U, , functions of the links 1(.) 1(b) which specify the hinge h U,.~
_ _1 lx e,. aP (a)l(b) . 2A
(2 .7)
A, b, V are the area, the deficit angle and the volume associated to the hinge h and Vh N.IhV.
(2 .3)
with N.1h the number of simplices per hinge. In all cases the transcriptions are not unique and they become equivalent in the continuum limit i.e . when the maximum link itngth goes to zero relative to some characteristic radius of curvature of the continuum manifold . 2. Gauge approach . Here one exploits the (incomplete) analogy which gravity shares with gauge theories . The attitude is akin to considering gravity as a field theory in Minkowski space (or euclidean space) whose lagrangian is invariant under a group of local transformations (reparametrization transformations) . Several peoples 1151 introduced along these lines discretized versions of gravity. I shall here breefly describe the formulation due to Smolin 115) which can be considered as the prototype of these gauge formulations. Essentially Smolin's formulation is the discretized version of the MacDowell - Mansouri [161 gauge formulation of De Sitter gravity. Let us consider the De Sitter group 0(4,1) which goes over to 0(5) in the euclidean, and introduce the usual hypercubic lattice familiar to gauge theories . Associate now to each link of the hypercubic lattice a finite element of 0(5)
U(n, n + p) = exp( 1aJbcwb + aPce')
(2.9)
with Jab and Pa obeying the following 0(5) commutation relations
P. Menotti/Non perturbative quantum gravity
32
[-7&bv Jed] = &Jad - 6ac .7bd - bbd-lac + bad.lbc
gauge transformations
V.6, PC] = Sbl- P. - b, [Pa , P6] = The action is
(2.10)
-J b -
ritten oyapU
here a =
hw--,hw+k,,f +Lf, .
4
3~â
(n)Ua,D(n)£ABCDS
(2 .11)
and
Ur (n) = UAC (n, n + j&)UCD (n UDE (n + I& + v, n + v)UEB(n
+,u9 n + jL + v, n)
+ v)
(2 .12)
is the plaquette in the vector representation . The action is obviously invariant under local O(4) transformations, but not under O(5) transformations . This is the result of the incomplete analogy between gravity and gauge theories. Going over to the formal continuum limit one gets the ac Dowell - ansouri action z [det -r (m pR - :+) é1~Ap
$-
XpE
(2 .13)
38a which is the usual Einstein's action in presence of the cosmological term to which a Gauss - Bonnet term has been added. The discrete action is invariant under reparametrization only in the formal continuum limit. Extension of this treatment to the Poincaré group can be easily given [17] . The written discrete action is obviously non symmetric with respect to reflections because it favors the sitive directions on the lattice. One has to perform the symmetrization with respect to directions otherwise one ends up with propagators containin complex terms of the form exp(iak,,) -1, when expanding e.g. the Poincaré version around a flat background . When such an expansion is performed using the symmetrized lagrangian one obtains at the quadratic level the quadratic continuum Einstein's lagrangian 2kAkahm~(-k)~~(k) - 2kak~h~~(-k)hv~(k) +kake"ha~"(®k)hw(-k) ° kAkVha~(°k)hev(k)
(2 .14)
here however the continuum momentum k,, is replaced by k,, -- sin(ak,,)/a. Such a quadratic la rangian is invariant under the following linear
(2 .15)
However the appearance of k, = sin(ak,,)/a and not of 2sin(ak,,/2)/a implies the phenomenon of graviton doubling (16 gravitons !). H. Nielsen [18] suggested us that if a point is reached where reparametrization invariance is recovered only one graviton should remain massless while the others should get a mass divergent with the cut - off, because reparametrization invariance protects the mass of one and not of 16 particles. I shall come back to this problem in the next section. We have given a euclidean formulation and thus we should care about reflection positivity which is the property which assures unitarity in the reconstruction of the theory with Minkowski signature [19] . Symmetrization as already noticed, is necessary but it is not sufficient for having reflection positivity. In fact the written action is pseudoscalar in the internal space ar-d thus not equivalent to the original action . We have that 2 ,J
Rab n r` A rdeabcd
=
f det r R d4x
(2 .16)
is not equal to
f ~,fj_ R d°x = f
1
det-rIR d4a
(2 .17)
unless we multiply the lagrangian in (2.16) by sign(det r) . In order to do so we must extract from the fundamental link variables the local vierbeins,
i.e . vierbeins which transforms locally under O(4) . Such vierbeins are also necessary to write down the Poincaré formulation of gravity. These local vierbeins in the le Sitter case can be extracted through the formula [17] T,;(n)
_ -j atr[HP°U(n, n + p)HU(n + Ftv)] (2 .18)
where lip = diag(1,1,1,1, -1). After introduction of sign(det T) in (2 .11) reflection positivity can be proven for all O(4) invariant abservables (which are the only physical ones). Moreover given the local vierbeins one can easily translate all RI terms of the continuum on the lattice and again reflection positivity proven [17] . An extension to N = 1 supergravity has also been given 1201, but here no reflection positivity result is yet available. The gauge approach is strictly bound to the
P. Menotti/Non perturbative quantum gravity program outlined by Weinberg of finding a fixed point around which to renormalize the theory. The Re e approach can also be used to investigate this problem . On the other hand' the Regge approach has been proposed [211 as a way to introduce a fundamental length in physics by putting an upper bound on the number of sites of the random lattice in a given four dimensional voulme . We shall come back to this issue in section 5.
3. The problem o the measure There have been several proposal for the non perturbative integration measure for quantum gravity on the continuum . They are all based of formal arguments, which explains the variety of the obtained results. I shall limit myself here in outlining the origin of a few of them without any pretence of completeness . De Witt [221 looks for a definition of distance between two nearby geometries whose difference is characterized by bg,,r, which is invariant under reparametrizations and ultralocal, i.e function only of 9,," and not of its derivatives . The expected result, whose uniqueness is proven by De Witt is given 1169""II' where
= 1&"z d"y
G"AP(x, y) is
G"A"(x, y)
b9""(x)G ""AP(x, y)69AP(y)
(3.1)
the local bitensor density
9`9A°WZ-y) = V9_(9"A9"P+9"P9"A+-/ (3 .2)
and y is an arbitrary constant. This is also the starting point in the treatment given by Polyakov in string theory [231. One defines as usual the volume element as the square root of the determinant of the metric ; the determinant of the infinite continuous matrix G""AP(x, y) can be easily computed at the formal level to get
G(x)
det GI"P(x, y) = with
G(x) = const . 9(x)
(D-®1(D+1)
(3.3) (3.4)
and thus for the measure in 1) dimensions we have const.
.1(D+o1 41 9(x)(D`
11 /sÇ"
d9""(x)
(3 .5)
33
which in four dimensions is simply a constant . Other reasonings [241 are based on the imposition of invariance under local confamal tranformation noticing that ;7, , under confamal tranformation behaves formally like a dimension 2 object 89w(z) := é(x)(x - 8 + 2)9,,,(x) .
(3.6)
Starting from this remark De Witt's result is again obtained; using on the other hand the vierbeins a similar reasoning [241 gives for the measure in four dimensions dr, (x) (3.7) which is not the simple jacobian transformation of (3.5), due to the fact that under a dilatation the density of points itself varies . The integration over vierbeins is preferable at the non perturbative level because as pointed out by Hawking [251 it assures the correct signature of 9,u, . Faddeev and Popov [261 using the hamiltonian approach obtain 9(x)6/=
JUL
d9w,(z).
(3 .6)
"<"
Other results re obtained in the hamiltonian approach by Leutwyler and by Fradkin and Vilkovisky [271. (3.5) is not a measure in the Lebesgue sense, and it is an interesting question whether a Lebesgue measure for Riemann geometries can e defined at all. Gromov 1251 was able to give a rigorous definition of measure on the space of geometries; his main result is that it is possible to define a distance between geometries and that according to such a distance the set of eometries of bounded diameter and with Ricci tensor bounded from below, R,, , > -cg. is (pre-)compact thus allowing, going over to the dual of continuous functions the introduction of an infinity of true Lebesgue measures. Such ideas are very profound but they do not solve the problem of which is the correct measure for quantum gravity, a problem again handled over to physicists. to discrete formulations the mea Coming n sure employed in actual Re e calculus computations is the so called scale invariant measure
inspired by the Fadd v - Popov result (as many
P Menotti/Non perturbative quantum gravity
34
rs of 9 in the numerator as many in the denominator) . Hamber [29] has tried the measure (3 .9) on the two dimensional model getting within 10% the exact critical index -fme derived by ICnizhnik, Polyakov and Zamolodchikov [30] . This lends to the measure (3 .9); however such a measup s'_-!re is not immune from critics. I report here a typical one by Jevicki and Ninomiya [31] . Their point is that the link lengths are invariants which characterize completly the geometry of space; due to reparametrization invariance there are many 9,", which describe the same geometry. Thus given the {k-) one can construct in some fixed gauge some g,(a ; {h)) to which a reparametrization transformation g,, can be applied, thus obtaining the whole family of equivalent gw(z ; {h), &,). In the variation bg. one should consider both the one arising from the variation of the geometry i.e . of the {1s) and that arising from the variation of the ~,,. Starting from De Witt result (3.X) one can rewrite it as
(b9, b9) _
bl; K;; bl; + ff, 0)" b&, (3.l0)
where G(t) is easily proven from (3 .2) on the continuum to be
G(E) _ -4vfg-(9"V a + Rl')6'(z - y) = -4,1g-(-L" + 2R")b4 (z - y).
(3.11)
Rew is the Ricci tensor and LIv the Lichnerowicz
operator
LI'
=
-(g"'®'
-
R,`~),
(3 .12)
with Va the vector covariant laplacian. Going over to the lattice the Lichnerowicz operator has a simple translation in the form of L = b d + d b [32], being d and b the boundary and coboundary operations. Also as we saw above, Rw can be translated on the discrete and thus dl; (det K)=(det G( 4)) 12'
(3 .13)
where det K turns out to be an essentially local object while det G(E) is a non local object of the Faddeev - Popov type . Turning over to the gauge formulation of gravity we saw how on the continuum the proposed measures in the 9, , or in the vierbeins ra, are of the type (det r)M . To throw some light on the measure
in presence of the already described lattice cut - off an explicit one loop computation of the SlavnovTaylor identities was performed and the local measure determined such as to satisfy for a --4, 0 the above mentioned identities [33] . The calculation can be performed in completeley analytical way and it gives for the local measure (det r)N
f(E r1,rl) a
3.14
where the O(4) invariant function f(E. T~asr) has to be allowed as it happens in Yang - Mills theory, and if one likes, can be absorbed in the definition of the vierbeins themselves. The result is N= 3+3t with t = 2
f
dz [e - ZIo (z)] 4 = 0.619 . ..
(3 .15) The result is rather disappointing as a flat integer value would have been more appealing. Higher loop can introduce corrections to the result and from the structure of the calculation one also gets the impression that the obtained value is strictly regularization dependent; using e.g . a different lattice would have given a different result . On the other hand the dependence of the measure on the regularization is not a new fact . Imposition of the Slavnov - Taylor identities is equivalent to imposing that the (0, 0, 0, 0) graviton does not acquire a mass. On the other hand the three graviton vertex which can be worked out completly [33] is not invariant under the transformation ak,, --+ 7r-ak,, as it contains terms with sin(ak,,/2) and cos(ak,/2) . Thus in agreement with Nielsen's suggestion the other 15 gravitons should take a mass of order 1/(mp as), a calculation on the way of being performed. The measure in lattice gravity is a major problem and some attention should be devoted to it either by putting on rigorous grounds existing formal arguments or by performing experiments on exactly known results.
4 . Numerical simulations for lattice gravIn four dimensions we have two numerical simulations for lattice gravity in the Regge formulation, one by Berg [34] and the other by Hamber and Williams 1351 and one simulation by Caracciolo and Pelissetto [36] for De Sitter lattice gravity in the gauge formulation . We shall give here a short re-
P Menotti/Non perturbative quantum gravity
view of their result : referring to the original paper for full details. Berg [341 considers the pure Einstein case where the partition function is given by dl
1
e
2rc2
r h
Ahbh
(4.1)
The simulation is performed at fixed total four dimensional volume which corresponds to introducinga kind of microcanonical cosmological constant. Such a constraint is necessary to get a meaningful functional integral. We recall that a bare cosmological constant is necessary [371 even in perturbative calculations [331 to kill the quartic divergence which arises already to one loop (except in the dimensional regularization scheme where it is put to zero by hand). The meaning of the pure number rc2 in the numerical simulation is that the value of the total volume is 0 Vro/mp if during the simulation the total volume is kept equal to the pure number Yo. A random Regge lattice i.e. with arbitrary incidence matrices appears at present too complex for a numerical computation . Thus Berg starts from a regular hypercubic lattice with positive diagonals or with a kind of centered hypercubic lattice and considers variations of the link lengths. It appears that such structures provide a good approximation to the random lattice even if here the topology is kept fixed. The simulation is performed on 2s and 34 lattices which already have a large number of link variables (15 for each hypercube in the hypercubic model with the positive diagonals) . Berg computes the average values of several quantities : the average action density S = (2bhAh), the average area of the hinges (Ah) and the average deficit angle (bh). First the simulation is performed putting rc= = 0, thus one measures here only the effect of the entropy of the configurations. The main result here is that the average action density is small and negative S = -0.056 f 0.006 . The simulation is then repeated increasing rc2 away from zero up to the value 0.3. The main result found by Berg is a sudden transition in all the above listed average values at a critical value rc~ estimated in the range 0.02 <; rc, G 0.04 . For values below rcc' i.e. in the so called entropy dominated region the situation is very similar to the pure entropy case, while across the transition S changes sign and becomes S " 102 i.e.orders of magnitude larger than below rc2 . More-
35
over one ends up into non reproducible situations thus pointing to the existence of many metastable states for r62 > r .,2 similar to a spin glass model. A similar discontinuous behavior is experienced by the other quantities like (bh) and (Ah). A notable fact in the region rc2 > rc;2 is a positive value of S coupled with a negative value of (bh) which shows a correlation between the deficit angles and the area of the hinges. Hamber and Williams [351 consider gravity in the Regge formulation in presence of terms quadratic in the Riemann tensor. The general structure of the action they use in the numerical simulation is the lattice equivalent of J
~ dz'' [mpR - A - 4R, ,i,R"-% p]
(4.2)
thus setting a = 4b in (1.8) . Due to the scale invariance of the adopted measure all resiAt r are functions of the adimensional constant mp%a and a = 4b. Thus in Hamber and Williams' simulation the cosmological constant is explicitly introduced ; the term R.a,R'--% P is expected to stabilize the functional integral. They consider the following adimensional quantities
and
R = ( 2 Eh Ahbh) (12)
(4 .3)
R2 = (4 Eh 461/Îrh) (12)2 .
(4.4)
(Eh Vh)
(Eh Vh)
Again putting all couplings to zero Hamber and Williams find a negative average curvature R -10 and 7i.2 se 58200. Thus we have a large adimensional curvature with large fluctuations . For non zero action two typical cases are explored the first with a small value of the R2 term i.e. a = 0.005 and the second with a = mp/A i.e equal to the other adimensional constant. Fixing a = 0.005 they find, varying mp/A, the following results for R and R2 R R2 mpla
0.125 0.167 0.250 0 .500
-0.612 8900 -0.420 8660 132 134000 123 143000
Thus in presence of a small R$ term again one finds for small mP/a an average curvature which is small
P. Menotti/Non perturbative quantum gravity
36
and negative and with large roughness Rs/R >> 1. Above the critical value m4/,X = 0.2 f 0.04 one has a sudden transition to large postive average curvature and the jump is so large that appears to indicative of a discontinuous transition . Values of R and Ra much smaller than 1 would signal a transition toward a continuum limit; from the a ve we see that for small a (a = 0.005) such a situation is not achieved. For a sizable R2 term i.e. for a = rnp/a the situation changes considerably
mP/A
0.167 0.250 0.500
R -0.044 -0-031 -0-030
Ra 496 432 208
e see n that R is consistently small though with large roughness. Thus with regard to a continuum limit the situation has definitely improved even though cannot say having reached the continuum limit. I come now to the simulation performed by Caracciolo and Pelissetto 1361 of the De Sitter model in the gauge formulation. They made ex tensive simulations on an 8s lattice . The reflection positive action is given by
4
sign(det T)E~A°tIjIIAPD EABCD5
(4 .5)
M4
3 where Q = 32 T and one sums over positive and negative directions . The factor sign(det T) is rather time consuming and thus the first simulation by Caracciolo and Pelissetto has been performed in absence of such a factor i.e. in the pure Smolin model . With regard to the measure they adopted the general local form (det T)N and the results were examined in a large range of N (0 < N < 150). Due to the compact nature of the group SO(5) and the presence of the lattice cut - off the action is bounded and thus the theory must exhibit a well defined ground state. For small Q the vacuum is similar to the QCD vacuum in the strong coupling region . The large Q region is characterized by the vanishing of the vierbein so that the action becomes dominated by the topological term . An interesting order parameter is given by P~
RaaV6 Ra6 pv
(4.6)
whose distribution is peaked around zero in the small Q region and around the extremes ±1 in the large ,6 region. A sudden transition is observed between the two phases at Q, = 0.08 f 0.01 . Around such a value all measured quantities like the mean value of the action per site, the trace of the metric, the O(4) curvature and P, , exhibit very large hysteresis cycles thus showing that the transition is first order. Moreover they also performed simulation with mixed phases showing definitively the first order nature of the transition. For Q > & the system ends up in non reproducible states showing the existence of many metastable states above & recalling a situation similar to a spin glass. The existence of a dense set of topologically metastable states is due to the dominance of the Gauss - Bonnet term in the action for Q > Q. and renders the thermalization of the system above Q. problematic even if the existence of the thermal average is assured by the boundedness of the action. Preliminary results [38] of simulations with the full action (4.5) i.e. with the term sign(det T) included show still greater problem in the thermalization above the critical coupling and appear not to alter the nature of the transition. 5. Conclusions
It is clear from the above review that the computational approach to quantum gravity is still in its infancy. We have only three numerical simulation on relatively small lattices . One conclusion that can be drawn is that despite the difference in the formulations (Regge calculus, gauge formulation) the different treatment of the cosmological constant and the uncertainty in the measure, the three simulations agree on the absence of a second order phase transition for the pure Einstein plus cosmological term model. If this result is going to be confimed as we expect, the above described treatements are unable to define a continuum limit for the pure Einstein gravity at the critical coupling constant . Terms quadratic in the Riemann tensor can be added and as we saw the investigation of their effect has already been started by Hamber and William in the Regge formulation . As we mentioned in sec.(2) the Regge approach has been suggested as a way to introduce in physics a fundamental length [21]. If one accepts this viewpoint the theory becomes more elastic and different
P. Menotti/Non perturibative quantum gravity questions must be addressed. For example Berg [2] suggests to couple gravity to a matter field e.g . to a Yang - Mills field (we know alread how to write down a Yang - Mills field on a random lattice [39]) and to look for a point in the parameter space where the mass gap goes to zero compared to the Planck mass . An other possibility as suggested by Hamber
and William [35] is to extract the renormalized Newton constant by computing correlation func tions at geodesic distances (here one needs somewhat larger lattices ) and look for a point in the parameter space where the renormalized cosmological constant, defined in terms of the average curvature of the universe vanishes compared to the square of the inverse of Newton's constant . The study of correlation functions at geodesic distances appear particularly significant in quantum gravity and correspond to the introduction of the "radial" gauge [40] . Another crucial question in the Regge formulation in presence of terms quadratic in the Riemann tensor is checking the unitarity of the theory a problem about which very little is known.
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