Some Remarks on the Semi-Classical Limit of Quantum Gravity

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by  Etera R. Livine

Some Remarks on the Semi-Classical Limit of Quantum Gravity
Etera R. Livine∗
Perimeter Institute, 31 Caroline Street North, Waterloo, ON, Canada N2L 2Y5
(Dated: January 12nd 2005)
Oneofthemostimportantissuesinquantumgravityistoidentifyitssemi-classicalregime. First
the issue is to define for we mean by a semi-classical theory of quantum gravity, then we would
like to use it to extract physical predictions. Writing an effective theory on a flat background
is a way to address this problem and I explain how the non-commutative spacetime of deformed
specialrelativity isthenaturalarenaforsuchconsiderations. Ontheotherhand,Idiscusshowthe
definition of the semi-classical regime can be formulated in a background independent fashion in
terms of quantuminformation and renormalisation of geometry.
PACSnumbers:
5
0 Building a theory of Quantum Gravity is one of the where G is the Newton coupling constant for gravity, c
0 biggest problems in theoretical physics. In this context, the speed of light or highest speed of any signal in flat
2 the conceptofsemi-classicalityisparticularlyrelevantin spacetime, ~the Planckconstantofthequantumtheory.

n ordertomakeitaphysicaltheory. Indeed,onapractical Itisusuallybelievedthatgravitationaleffects(blackhole
a level, building an effective theory would allow to extract formation)impliesthatthePlancklengthl isaminimal
P
J
physics and experimental predictions for quantum grav- distance and that the geometry is quantized in quanta

5
ity. In fact, it would even allow us to say something on of size l . The relevance of the Planck mass m , or
P P
2 quantum gravity without having to build the full consis- PlanckenergyE =m c2, is more subtle. It cannotbe
P P
tent theory. This is particularly interesting since experi- a maximal mass since macroscopic systems have a rest
1
v ments like GLAST, AUGER and others, should be able energy much larger than EP. Obviously, it can not be a
6 to measure effects due to a quantum gravity regime. minimalmasseither. Aneasywaytocharacterizeitisas
7 Now, in order to define a semi-classical domain, one the maximal energy or mass content of a cell of size l .
P
0 shouldfirstunderstandthenotionofobserversinaquan- Moreprecisely,let us considera systemofcharacteris-
1
tum gravity context. Then one could construct an effec- tic size L. Then black holes in general relativity imply a
0
tive theory for an observer who can define a regime in maximal bound on the mass of the system:
5
which geometry doesn’t need to be described as quan-
0 c2L
/ tum anymore. First, I will describe the different limits M M (L) . (2)
c of quantum gravity and what I call an effective theory. ≤ max ≡ 2G
q
Then I will show that the non-commutative space-time
- On the other hand, quantum effect implies mass fluctu-
r ofdeformedspecialrelativityprovidesaneffectivetheory
g ations of the system of order:
for quantum gravity. And I will define the semi-classical
:
v regimeofthe theory. I willdiscussthe necessityofintro- ~
Xi ducing coherent states in a non-commutative spacetime δM(L)≡ cL. (3)
in order to describe a semi-classical notion of spacetime
r Assumingthatgeneralrelativityandquantummechanics
a points. Section IV will discuss the interplay of quan-
stillholdatsuchscale,thePlanckscalel canbedefined
tum information and quantum gravity. Finally, I will P
as the scale where quantum fluctuations automatically
discuss the tools necessary in order to understand the
saturate the gravitational bound: M (l ) = δM(l ).
semi-classical regime, or more generally a macroscopic max P P
Then the mass bound is m .
regime,in full backgroundindependent quantumgravity P
Furthermore, there is another quantity relevant to
such as loop quantum gravity.
quantum gravity. It is the cosmologicalconstant Λ, who
provides in more local terms an energy density or a lo-
cal mean curvature. It provides us with a second length
I. EFFECTIVE THEORY AND
scale whichwe canuse to obtaina dimensionless param-
SEMI-CLASSICALITY
eter: z l √Λ. z is usually the quantum deforma-
P
≡
tionparameter in the quantumgroupapproachto quan-
Quantum Gravity is usually defined as the regime
tum gravity. In the full theory, we expect l to impose
where the Planck quantities become relevant: P
a minimal bound on distances while the ”cosmological”
2G~ ~c length lC 1/√Λ defines a maximal scale. Note that,
lP =r c3 , mP =r2G, (1) evenstarti≡ngwith a vanishing cosmologicalconstant, we
wouldnecessarilyget Λ=0 under a coarse-grainingflow
6
or more generally a renormalisationgroup flow.
The differentlimits ofquantumgravitycanbe defined
∗Electronicaddress: [email protected] as ~ 0 (the classical limit), c (the Newtonian
→ → ∞
2
limit), G 0 (the no-gravitylimit orquantumfieldthe- withconstantcurvatureΛ,seenbyanobserverwillfinite
→
ory limit), Λ 0 (which I call the flat spacetime limit). resolution δl as a flat spacetime.
→
This is usually translated in Planck units into l 0, Now,assuming the observeris workingin a flatspace-
P
→
m ,z 0. Howevergenericallyonecanhavemore time, smallmovements aroundhim cannot be described
P
→∞ →
subtlemathematicallimitshavingsomePlanckunitsrun- infinitesimally but are necessarily larger than δl. This
ning to 0 while other ones remain finite. finite displacements are embedded in the originalcurved
spacetime: the”tangentspace”tooureffectiveflatspace-
I define an effective theory in a generic regime when
time is curved. Thus we get a curved momentum space
the considered scales are much larger compared to the
on a flat spacetime. The curvature of the momentum
quantum gravity scales. This still allows quantum ef-
space is related to a maximal moment norm, which is
fects and general relativistic effects. More operationally,
givenbytheminimallengthscaleδl ontheoriginalman-
I consider an observer, provided with a resolution δl .
obs ifold. Ontheotherhand,themaximallengthscalel on
C
Gravity imposes a minimal bound on the observer’s res-
the original spacetime defines a minimal moment scale.
olution, δl & l , but the observer doesn’t necessarily
P Putting the right units where they belong, we start with
saturate it. Then there is the scale L of the observed
a curved spacetime with curvature Λ seen with a resolu-
phenomena. As explained above, the two relevant mass
tion δl provided with the usual flat tangent space, and
(or energy) scales relevant here are a lower bound im-
we show it is equivalent to a flat spacetime but provided
posedbyquantumeffectsandrelatedtotheresolutionδl with a curved moment space with curvature (δl/~)2 and
and a maximal bound imposed by general relativity and
a resolution δp ~√Λ.
related to the system scale L: ≡
Considering that δl l , we can work in the limit
C
≪
~ 1 c2 Λ 0 , or precisely z 0. Then we are the mathe-
→ →
δM = M L=Mmax. (4) matical structure of a smooth curved ”tangent space”.
c δl ≤ ≤ 2G
The coordinates on the flat spacetime need to be recon-
The natural requirement that δM M is equivalent structed as the translation generators on the moment
max
to the assumption Lδl l2. In th≤e limit case that L = space. Now the moment space is curved, so the coor-
δl = l , we recover the≥PlPanck regime δM = M = dinate becomes non-commutative:
P max
mP. i~2
2
My first assumption for an effective theory of quan- [Xµ,Xν]=−κ2 Jµν =−i(δl) Jµν, (5)
tum gravity only concerns the observer and will be that
δl l . Then,usingtheequivalenceprincipleofgeneral where we have defined the quantum deformationparam-
C
rel≪ativity, I translate the fact that all manifolds are lo- eter κ = ~/δl. Mathematically, the moment space is
cally flat in an second assumption that the observer will identified as the hyperboloid SO(4,1)/SO(3,1) and the
describe the spacetime as being flat. The goal of the ef- coordinate operators are the DeSitter translation gener-
fective theory is to describe quantum gravity effects as ators. And we recover the mathematical set-up of de-
on that flat spacetime arena. formed special relativity (DSR).
A semi-classical regime is naturally defined such Diagonalizing the coordinate operators, we get a dis-
that quantum fluctuation are not relevant: δM(δl) crete space structure and a continuous time. Another
≪
M (L), so that we would be far from a strong quan- choicewouldbetoconsideranAntiDeSitterspace,which
max
tum gravity regime and black hole formation would be wouldleadto the reversestructure ofacontinuousspace
improbable. From this point of view, working out an and a discrete time. However, one can always question
effective theory doesn’t need to work in a semi-classical the experimental relevance of such spectra. Indeed, a
limit,butthatlimitwillstillneedtobeconsideredinthe quantized time would necessary implied phenomenologi-
derived effective framework. callyadiscretespacestructuresince wetraditionallyuse
Iproposethiscriteriatoreplacedifferentotherchoices time-of-flightexperimentstomeasurespacelikedistances.
of definition for semi-classicality such as ~ 0, G 0, Moreover, any clock used to measure time intervals will
coarse-grained limit L δl, flat limit g → η →, en- have discrete ticks and time will alwaysbe operationally
µν µν
ergy scales very large c≫ompared to m , inte∼raction en- quantized.
P
ergy scales much smaller than mP. The (squared) length is still expressed as L2 =XµXµ
and is a Lorentz-invariant: we are truly working in a
flatspacetimewithatrivialmetric. Theresultinguncer-
II. DEFORMED SPECIAL RELATIVITY tainty relations read as:
2
δX .δX =(δl) J ,
µ ν µν
Iwouldliketoshowthatdeformedspecialrelativity[1, h i
2]providesanarenainwhichtobuildaneffectivetheory so that δX δl, which is consistent with the starting
∼
for quantum gravity. These results are explained in de- pointthat the observerhas a finite resolutionδl onmea-
tailsin[3]. IconsiderthecosmologicalconstantΛascom- surements on the space-time coordinates.
ing from quantum gravity corrections to the flat space- It is possible to generalize this construction to an ar-
time. Thenwearetryingtodescribeacurvedspacetime, bitrary initial metric, which gets translated into an ar-
3
bitrary metric on the moment space, and thus take into III. LOCALIZING POINTS IN
account some dynamical effects of quantum gravity [3]. NON-COMMUTATIVE SPACES
Furthermore,onedoesn’tneedtotakethelimitΛ 0.
→
Thenone needs to takeinto accountthe finite resolution As we have seen in the previous section, non-
ofthemomentumspacewhichgetstranslatedintoacom- commutative geometries are a natural set-up for effec-
mutation relation: tive theories of quantum gravity. In fact, already special
relativitycanbeconsideredasanon-commutativegeom-
2
[pµ,pν]= i(δp) Jµν. (6) etry. Indeed the curved space of velocities induces the
−
non-commutativityofthe spacecoordinatesifwelookat
This leads to a doubly deformed special relativity. Al-
special relativity from a purely 3d space point of view.
thoughitisknownthatthedeformedspecialrelativityal-
Moreprecisely,foragivenmassivesystem,[4]derivesthe
gebraisisomorphicto the κ-deformationofthe Poincar´e
following commutation relation:
group,itisonlyconjecturedthatthealgebrabehindthis
extension to Λ = 0 is isomorphic to the quantum defor- 2
[x ,x ] = il J ,
mation so (4,1)6 with parameter q =e−z. i j c ij
q
wherel =~/mcistheComptonlengthassociatedtothe
Now that we have shown how the non-commutative c
system.
spacetime ofdeformedspecialrelativityprovidesa moti-
More generally, as argued in [6], requiring Lorentz in-
vatedarenato study effective quantumgravity,the issue
variance at the same time as some discrete spacetime
is to make sense of the DSR physics and check whether
structure leads naturally to geometry (length and time)
we can formulate a consistent kinematics. The main ob-
been defined as (quantum) operators. In such a frame-
stacle to a straightforward interpretation of DSR is the
work, a necessary step towards talking about a semi-
”soccer ball” problem: κ defines a mass scale which ac-
classical limit with a smooth spacetime manifold is to
tually becomes a maximalmasswhen lookingat the dis-
introduce a notion of localized spacetime points: it is
persion relation on the theory. Intuitively, it is the same
necessarytobuildcoherentstatesofgeometrydescribing
thing as when we deformed the flat velocity space into
a notion of ”semi-classicalpoints”.
the mass-shell hyperboloid in special relativity in order
to introduce a maximal speed c [4]: deforming the mo- Let us look at the simplest case: we consider 3d space
ment space into a 4-hyperboloidintroduces the maximal coordinates forming a su(2) algebra. So we define xi =
rest mass κ. However the existence of a maximal rest lP Ji with [Ji,Jj] = iǫijkJk. This can actually be seen
mass is not consistent with the existence of macroscopic as 3d Loop Quantum Gravity. The length spectrum is
system. Indeed in the limit case where δl = l , κ is the obviously derived:
P
Planckmassandasoccerballweightsmorethanm for
instance. P ˆl2 = lP2 J~2 ⇒ l2 = lP2 j(j+1). (7)
Thesolutionistoconsiderthatonecopyofthealgebra
describesa”fundamental”systemorone-particlesystem Thenonenaturallydefinesthespreadofgeometrystates
for the observer. To describe a two-particle states, we as the uncertainty (δl)2 = ~x2 ~x 2 . Evaluatingit on
|h i−h i |
would need two copies of the DSR algebras. Then the the Jz eigenvectors jm , we have:
| i
algebra describing the kinematics of the center of mass
would be the diagonal part of the direct sum of these (δl)|jmi = lP j(j+1) m2. (8)
−
two algebras. It turns out that if the two original alge- p
bras have κ as deformation parameter, then the center It is easy to show that the minimal spread is obtained
ofthe massalgebrahas2κasdeformationparameter[5]. forthe maximalweightvectorm=j,forwhichwehave:
Thus, let us start at the Planck length with the Planck (δl)min = lP√j. These are the usual SU(2) coherent
mass bound: consideringsystems largerthan the Planck states, as defined by Peremolov[7], and provide the no-
length, i.e. many-particle states, will require to renor- tion of semi-classical points with minimal spread. Let
malize the mass bound κ from the Planck mass m to us conclude this example by pointing out that we derive
P
the actualmaximal mass M (L) corresponding to the here the uncertainty relation:
max
length scale of the system.
Following this logic, [5] proposes a law of addition of δl = ll , (9)
P
momentausingafive-componentenergy-momentumand p
a generalized relativity principle which states that this which matches the prediction from distance measure-
new extended 5-momentum is conserved during scatter- mentuncertaintyasderivedfromtheholographicprinci-
ing processes in any reference frames. The deformation ple in three spacetime dimensions (see [6] and references
parameter κ then becomes an intrinsic property of the therein).
considered system, in some sense the maximal mass for We hope thatsuch a study ofsemi-classicalproperties
the system could have. In this effective theory context, of non-commutative spacetime can be extended to the
the semi-classical limit is then defined as the κ four spacetime dimension case, in which the holographic
limit, when DSR effects become negligible. → ∞ principle would require (δl)3 = ll2.
P
4
IV. QUANTUM GEOMETRY FROM in a diffeomorphism invariant context, the only notion
QUANTUM INFORMATION? of distance in a purely geometrical theory which can be
naturalisgivenbythesimplecriteria: iftwosystemsare
close,thentheyarestronglycorrelated;thefurthestthey
A field which becomes more and more relevant to the
are, the weaker the correlations are. More precisely, we
study of quantum gravity is quantum information. It
coulddefineadistanceintermsofcorrelationandentan-
has been often proposed that the quantum geometry be
glement. I propose to talk about a semi-classical geom-
reconstructed in terms of flow of information at the fun-
etry state when this notion of distance is well-behaved
damental level, especially since the holographic princi-
and continuous.
ple has been put forward as a fundamental principle for
To conclude this section, we see that there is a lot
quantum gravity. Let us start by a simple remark about
theprecisionofmeasurements. Ifonewouldlikeincrease of room for an interesting interplay between quantum
informationandresearchinquantumgravity[8]. All the
the precision of the measurement of a property of a sys-
tem, one would need to use a larger and more massive previousremarkscanbeappliedtoloopquantumgravity,
which is the main candidate to a theory realizing the
measurement device in order to achieve the macroscopic
amplifications of the smaller fluctuations. More exam- quantization of general relativity.
ple, the precision bound on a length measurement goes
as δl ~/cM. But changing the measurement device to
alarg∼eronewillmodify the geometryofthe surrounding V. SEMI-CLASSICAL (LOOP) QUANTUM
and perturb the measured system. Of course, one could GRAVITY
argue that we could place the measurement device fur-
ther in order to lessen the perturbation of the geometry Loop quantum gravity[9, 10, 11] is a canonical quan-
that it causes. Nevertheless, ultimately, the information tization of general relativity. It describes the quantum
itselfwhichweextractfromthemeasuredsystemshould states of space geometry and their evolution in time.
carrysomeenergyandsomemasswhichperturbitselfthe Thesearethespinnetworks,graphdecoratedwithSU(2)
geometryandthe system. Thus,they shouldexistafun- spin representations on their edges (a half-integer num-
damental uncertainty principle between the information ber j N) and with SU(2) invariant tensors at their
∈
extracted on a system (or the geometry of a space-time vertices (Clebsh-Gordan coefficients). The main result
region) and the reliability (or truth) of this information of this quantization is the derivation of a discrete spec-
(or equivalently the perturbationof the system or geom- trum for the area and volume operators. More precisely,
etry). Thesemi-classicalregimecouldthenbe definedas a spin network can be understood as representing a dis-
minimizing such an uncertainty relation. crete manifold. Vertices represent chunks of 3d volume.
A quantum information concept of particularly inter- The graph links are dual to the surfaces boundary of
esting for quantum geometry is the notion of quantum these pieces of volume. An elementary surface intersect-
referenceframes. Indeed,thesearequantumsystemswho ing only one link of the spin network labeled by a spin j
state spaces approximates a classical symmetry group. has an area jl2. Actually, there is an ambiguity in the
P
When the symmetry group is the 3d rotation group or areaspectrum,whichcouldalsobe j(j+1)orj+1/2,
theLorentzgroup,thenwehaveaquantumsystemwhich but these details do not matter inpthe regime of large
can be used to localize a direction. Imagining a system spins j. A generic surface is decomposed in elementary
composedofmanysuchquantumreferenceframeswould patches, each of them intersecting the spin network only
makeareasonablequantumgeometrybackground. More once. Thetotalareaofthe surfaceisthesumofthe area
precisely,the main difficulty indiffeomorphism invariant of all its elementary patches. More precisely, noting Γ
theories such as general relativity is to construct rele- the spin network and the surface, the area is:
S
vant physical observables. The practical solution is to
use relational observables, built as physical correlations ( ) l2 a(M), (10)
A S ≡ P
between two systems A and B: we are dealing with the MX∈S∩Γ
probability amplitude of A being in some state when B
is in such or such a state. Choosing B to be given by where a(M) is the elementary area, j or j(j+1), as-
one of these quantum reference frame would allow us to sociated to the intersected spin network edpge.
”localize”thesystemA,withrespecttothereferencesys- In this context, I could talk about the construction of
tem B. Imagining a quantum geometry region made of quantum reference frames as discussed in [8]. But I pre-
many quantum reference frames would allow us to have fer to discuss the renormalisation of geometrical quanti-
a notion of ”where”, which is necessary to talk about a ties. Let us consider the area of a surface. The actual
semi-classical geometry. measure area will depend on the resolution of the ob-
Finally,inabackgroundindependentquantumgravity server doing the measurement. Just like for fractals, the
theory, there is no natural notion of distance. Consider- measured macroscopic area will depend on the way the
ing a state of quantum geometry of, we should first be surface is folded in space. The surface is made of ele-
abletodefineasubsystemofthisgeometry,whichwould mentary patches, each of them is attached a SU(2) spin
correspondtothenotionofaregionofspace(time). Then representation. Thewaythesurfaceisfoldedisdescribed
5
through a tensor between these SU(2) representations. surface since we can not localize it, we can at least talk
If the surface is closed, it should be an invariant tensor about the renormalisation flow of its geometrical prop-
-an intertwiner. This tensor or intertwiner defines the erties. We can push this further and directly define a
way the surface will get coarse-grained. And ultimately, quantum geometry state as the renormalisation flow of
one could coarse-grain it to a single elementary patch, theareasinloopquantumgravity. Moreprecisely,asthe
which represents the surface macroscopically. One can probabilityamplitudedistributionofthetensorsorinter-
prove that the most probable macroscopic area goes as twiners describingthe waythe surfaceis folded andthus
the squareroot of the microscopic area when we have no the way it gets coarse-grained. From this point of view,
knowledge of the way the surface is folded: each quantum geometry state contains the information
which defines its continuum limit or semi-classical limit.
trivial geometry state l .
macro P micro
⇒A ∼ A
p To conclude, the notion of semi-classicality is an im-
Knowledge will influence the coarse-graining flow of the portant issue in quantum gravity. In a background in-
surface. Nowoneshouldunderstandthatthatknowledge dependent theory, either one could introduce effective
of the tensor or intertwiner between the surface elemen- description of quantum gravity depending explicitly on
tarypatchesisactuallygivenbythespinnetwork,which the observer and taking place in a (non-commutative)
defines the waythe surfaceif embeddedin the surround- flat spacetime, or one should introduce new ways to de-
ing volume. fine the semi-classical regime in a background indepen-
Reversing this logic, describing the renormalization dent fashion. I proposed to address the latter alterna-
flowoftheareaofsurfacescanbeconsideredasproviding tive using tools of quantum information and developing
thewholequantumstateofgeometry. Alreadyatanaive the mathematical relationbetween the quantum state of
level,onecanseethatthesignofthecurvatureisrelated geometryandthe correspondingrenormalisation/coarse-
to the increasing or decreasing of the flow. It should graining of geometrical objects.
neverthelessbe investigatedin details whether one could
actuallyreconstructtheunderlyingspinnetworkstateor
not.
Acknowledgments
To sumup, we cantrade the knowledgeofthe state of
quantumgeometryforthe informationaboutthe coarse-
grainingflowofareasofsurfacesinloopquantumgravity. I am grateful to Laurent Freidel, Florian Girelli and
This actually makes sense in such a diffeomorphism in- Daniele Oriti for plenty of discussions on quantum grav-
variant theory: as one needs to talk about an abstract ity and deformed special relativity.
[1] G Amelino-Camelia, The three perspectives on the [7] A. Perelomov, Generalized Coherent States and Their
quantum-gravity problem and their implications for the Applications, Springer-Verlag (1986).
fate of Lorentz symmetry, gr-qc/0309054. [8] F.Girelli, E.R.Livine, Reconstructing Quantum Geome-
[2] JKowalski-Glikman, Introduction to DoublySpecial Rel- try from Quantum Information: Spin Networks as Har-
ativity, hep-th/0405273. monic Oscillators, in preparation; Talk ”The Harmonic
[3] F. Girelli, E.R. Livine, D. Oriti, Deformed Special Rela- Oscillator Representation of Loop QuantumGravity”at
tivity as an effective flat limit of quantum gravity, Nucl. theVIMexicanSchoolonGravitationandMathematical
Phys.B, in press, gr-qc/0406100. Physics ”Approaches to Quantum Gravity”(11/2004).
[4] F. Girelli, E.R. Livine, Special Relativity as a non com- [9] C Rovelli, Loop Quantum Gravity, Living Rev.Rel. 1
mutative geometry, lessons for DSR, gr-qc/0407098. (1998) 1, gr-qc/9710008.
[5] F. Girelli, E.R. Livine, Physics of Deformed Special [10] T Thiemann, Lectures on Loop Quantum Gravity,
Relativity: Relativity Principle revisited, gr-qc/0412004; Lect.Notes Phys.631 (2003) 41-135, gr-qc/0210094.
Physics of Deformed Special Relativity, gr-qc/0412079. [11] A Ashtekar, J Lewandowski, Background Independent
[6] E.R. Livine, D. Oriti, About Lorentz invariance in Quantum Gravity: A Status Report, Class.Quant.Grav.
a discrete quantum setting, JHEP 0406 (2004) 050, 21 (2004) R53, gr-qc/0404018.
gr-qc/0405085.