Transcript Causal sets:

An introduction to Causal sets
1: Discreteness without symmetry
breaking
Joe Henson: Causal sets
Discovery : a case study
How did we discover the properties of matter at the atomic
scale?
Lord Kelvin’s approach:
Atoms really are vortices in some aether.
“For the only pretext seeming to justify the monstrous assumption of
infinitely strong and infinitely rigid pieces of matter, the existence of which
is asserted as a probable hypothesis by some of the greatest modern chemists
in their rashly-worded introductory statements, is that urged by Lucretius and
adopted by Newton—that it seems necessary to account for the unalterable
distinguishing qualities of different kinds of matter.”
•
•
•
•
•
Hydrodynamics was the best developed understanding of matter;
Mathematically natural from Kelvin’s perspective;
A conservative generalisation of current theories;
Attractive properties on paper;
Wrong.
Joe Henson: Causal sets
Discovery : a case study
Lord Rayleigh’s approach:
Make a simple modelling assumption and look for observable
consequences.
Modelling atoms as small perfectly reflective spheres gives:
By noting that Kanchenjunga could be seen “fairly bright” from
Darjeeling, Rayleigh estimated 1/β to be 160km and thus obtained a
value for Avagadro’s constant as around 4 x 10^23 !
• Minimal (but not totally generic) assumption.
• Natural physically, not a mathematical generalisation of current theory
or what seems natural based on mathematics.
• Allows calculation.
• Leads towards correct theory.
Main points for this series
• It is possible – but not easy – to discretize
spacetime while preserving symmetries in the
continuum approximation.
• This provides an interesting foil for arguments
about “generic phenomenology of QG”.
• It also led to the only successful prediction that
has ever come out of QG.
• Investigations of possible dynamics for causal
sets present unique challenges but also give
some reasons for hope.
Joe Henson: Causal sets
Planck Scale Problems
Only a particle of mass greater than m can probe distances less than
. But it is only at scales >> that the effects of the particle on
spacetime are negligible.
Possibilities: at the Planck scale…
•geometrical properties become fuzzy or uncertain;
•Geometrical concepts are inadequate;
•Spacetime is discrete.
Several “clues” from current theory (infinities in GR, QFT, BH
entropy) suggest that the replacement should be discrete. Are
there well-motivated ways to model the effects of spacetime
discreteness?
Joe Henson: Causal sets
Further Considerations
• GR is most naturally treated as a spacetime
rather than space + time theory, but
cannonical quantisation requires the latter,
leading to considerable problems.
• Many types of discretisation also break the
symmetries of GR.
• A Lorentzian Gromov-Hausdorff type distance
is hard to define.
• With these facts in mind we consider
discretising spacetime: is there a long list of
possibilities, or are models restricted by
simple principles like symmetry preservation?
Joe Henson: Causal sets
Continua as approximations
E.g. consider a classical non-relativistic particle. If we had
reason to believe that time was discrete, how could we recover
continuous trajectories from discrete?
Here, we might say that C
approximates A if C is within
some given distance of the
linear interpolation B of A.
A
B
Discrete
Structures
g
C
Discrete/continuum
correspondence ~
Continua
~ must be surjective;
If g ~ g1’ and g ~ g2’ then g1’ and g2’ must be physically
indistinguishable.
Joe Henson: Causal sets
g’
Quantum complications?
If the underlying dynamics is quantum, shouldn’t we be concerned with
approximations between classical continua and quantum sums of
histories?
But note: in a semi-classical state, the path integral is dominated by
paths that approximate the classical path being “tracked”:
Measurements in accordance with the classical
approximation are very “course-grained” and, by
definition, do not measurably disturb the state.
If you have a sum of many paths that don’t
approximate the classical one, you can’t magically get
back the right classical measurement results!
Conclusion: some of the individual paths in the history
space must approximate to continua, so we still need a
discrete/continuum correspondence.
Joe Henson: Causal sets
Spacetime as approximation
Along these lines we might imagine many discretisations of spacetime +
fields.
Euclidean geometry could be
approximated by a piecewiselinear version as before.
But we live in a Lorentzian world!
Technical problem: define distance between Lorentzian manifolds?
Killer problem: No way any such approximation of Minkowski can
be Lorentz invariant!
Joe Henson: Causal sets
An intriguing result
We can define a relation between points in a Lorentzian
manifold such that x y if x is to the past of y. The causal
structure of a manifold is this relation up to symmetries.
Given the causal structure and the conformal factor
of a manifold with Lorentzian metric, one can recover
the dimension, differential structure, topology and
metric of that manifold.
Causal structure + volumes = geometry
Taking this causal structure as fundamental, we
arrive at a simple way of discretising spacetime.
Joe Henson: Causal sets
Causal sets
The causal order of a spacetime is a partial order < on the
set of points C, meaning:
To get a discrete version of this, we add:
Order  Causal structure
Number  Volume
Joe Henson: Causal sets
Continua as approximations
Discrete
Structures
Continua
g’
g
Discrete/continuum
correspondence
Binary relation g ~ g ’
~ must be surjective;
If g ~ g1’ and g ~ g2’ then g1’ and g2’ must
be physically indistinguishable.
Joe Henson: Causal sets
Spacetime as approximation
We must recover the spacetimes of GR as approximations to
some of these causal sets. When does a spacetime (M,g)
approximate to (C, )?
If (C, ) is the partial order on some set of points in M
which is the order induced by the
causal order of (M,g), we say that
(C, ) is “embeddable” in (M,g).
It is “faithfully embeddable” if that
set of points could have arisen, with
relatively high probability, from
“sprinkling”:
This ensures that, for large
regions, n ≈ ρV
This defines the discrete/continuum approximation.
Joe Henson: Causal sets
Recovering Geometry
Given a causal set, how do we work out the properties of its
continuum approximation (and if it has one)?
E.g: If (M,g) ~ (C, ) , what dimension does M have?
In an interval of Minkowski, the fraction of pairs of points that are
causally related is a function of the dimension. E.g. in 2D half of all
pairs of points are related, and in 3D it’s less. Reversing this relation
gives a dimension estimator:
“Manifoldlike” causal sets have integer valued, matching dimension
estimators.
Similar results for lengths, topology, etc…
Joe Henson: Causal sets
Lorentz invariance?
A fertile ground for phenomenology, and a problem for most
discrete structures. Does discreteness imply Lorentz violation?
“Is this discrete structure Lorentz invariant?”
Bad question: can only talk about continuum symmetries when
there is a continuum! What we rally want to know is:
“Does the discrete structure, in and of itself, serve to pick out a
preferred direction in the approximating continuum?”
Similar example: in the continuum approximation, a sphere of glass
is rotationally invariant but a sphere of crystal is not. (NB: it is not
the transformations of the microscopic configuration we are worried
about here).
Joe Henson: Causal sets
A hint for Lorentz invariance
The Poisson process under boosts
Joe Henson: Causal sets
Causal sets
So, does the discrete structure, in and of itself, serve to pick
out a preferred direction in the approximating continuum?
1)
This distribution is invariant under all volume preserving
transformations. In Minkowski this includes Lorentz
transformations.
2)
3)
Causal information is Lorentz invariant.
Outcomes of the sprinkling process do not pick a direction.
Theorem: There is no “equivariant” map between outcomes of
sprinklings and directions in Minkowski.
Joe Henson: Causal sets
Lorentz Invariance: a theorem
But there is no uniform probability distribution on this non-compact
group, so that can be no such map D. So a sprinkling picks out no
direction.
Joe Henson: Causal sets
Illustrating the theorem
In Euclidean space, sprinkling are rotationally invariant,
but a direction from the marked point can be defined to
the nearest sprinkled point:
In Lorentzian space, for any finite distance
from the marked point, there is an infinite
volume closer to the marked point, which must
contain sprinkled elements.
Corrolaries:
• No direction from a sprinkling without marked point;
• No set of directions from a sprinkling;
• No finite graph from a sprinkling.
Joe Henson: Causal sets
Lorentz invariance!
Random discreteness saves symmetry
Joe Henson: Causal sets
Conclusion
• Causal set discreteness is the only known
way to make a “fuzzy” structure that
approximates to Minkowksi space at large
scales in a fully Lorentz-invariant way.
• Good symmetry properties are hard to
come by and so this is a restrictive, simple
principle to build from, like Rayleigh’s.
• Next: what can this tell us about possible
deviations from standard theory?
Joe Henson: Causal sets
An introduction to Causal sets
2: consequences of spacetime
discreteness
Joe Henson: Causal sets
Applying Causal Sets
• Causal set discreteness provides the only known way to
“fuzz” spacetime at small scales while preserving
symmetries at the continuum level. This impacts on
important questions…
• Are there generic signals of Planck-scale spacetime
fuzziness? Does Lorentz Invariant discreteness have
specific signals? Atomic Matter : attenuation of light,
dispersion, scattering, defects…
• Other compelling, general expectations of what a
quantum theory of discrete spacetime would predict?
Predicting the cosmological constant.
Joe Henson: Causal sets
Fields on causal sets
Is discreteness/fuzziness consistent with observation?
It has been suggested that any fuzziness in distance
measurements will cause loss of coherence of light from distant
sources.
But this line of reasoning does not accord with Lorentz
symmetry. Can we put a field on a causal set to test this?
The problem is also relevant to dynamics. How do
we recover effective locality from causal sets?
Joe Henson: Causal sets
Scalar fields on Minkowski
We need to make some approximation to the local, Lorentz
invariant D’Alembertian operator. A lattice provides an easy
way to recover locality, but breaks Lorentz invariance. On the
other hand, the Lorentz invariant causal set discretisation
makes it more difficult to recover locality.
On a light-cone lattice:
A weighted sum of field values at a finite set of “near
neighbours”.
But in a truly Lorentzian discretisation, there can be no such
finite set.
E.g.: how many “links” to a given element are there in a
sprinkling of Minkowski?
Joe Henson: Causal sets
Lorentz invariance or locality: a
choice
x
D
Consider a sprinkling of Minkowski.
If there is a non-zero probability of
a near neighbour of x being
sprinkled into region D…
There is an equal
probability in D’.
Thus there must be an
infinite amount of near
neighbours, however they
are defined.
D’
Joe Henson: Causal sets
Approximating Green’s functions
Equivalently to the d’Alembertian, the field theory
can be defined by the Green’s function of the
d’Alembertian:
In 4-dimensional Minkowski space, the Retarded
Greens’ function is given by
A delta function on the future light-cone of x.
This function is defined using purely causal information.
Joe Henson: Causal sets
Approximating Green’s functions
We have seen that the links from one element “hug the lightcone”. Consider following function on pairs of causal set
elements:
In the limit of dense sprinkling (with suitable normalisation)
this function goes to the delta-function on the future lightcone, G(x,y).
This can be used to define the propagation of a scalar field
on the causal set:
This method has some problems, but can be used to give a model of a
scalar field propagating from source to detector. This helps us to see
whether causal set discreteness is consistent with the coherence of light
travelling over long distances, and gives an example of Lorentz invariant
discrete dynamics.
Joe Henson: Causal sets
A Model of Propagation
We can model propagation from a small source to a distant
detector and compare the standard model with the causal
set model. We define the signal F as follows:
In the continuum we are finding the measure of the set of pairs of points
(one in source, one in detector) that are null related.
Joe Henson: Causal sets
A Model of Propagation
Source
Discrete version:
Joe Henson: Causal sets
R
Detector
A Model of Propagation
In the causal set case, to find the detector signal we counted the number
of links between the source and detector region for a typical causal set
approximating to Minkowski space. The result is the same, with negligible
corrections.
The signal varies with the strength of the source just as in the continuum.
No significant random or systematic effects come in, e.g. to change the
phase of a propagating wave.
I.e. no Lorentz violation, no loss of coherence.
Spacetime “fluctuations” → loss of coherence
Joe Henson: Causal sets
The cosmological constant
problem
The cosmological constant:
• If the cosmological constant comes from the
zero-point energy of QFT, shouldn’t it be 1?
• In other words, why is there an approximately flat
manifold at all?
• If it’s not 1 why isn’t it 0?
• Is it a coincidence that L has only just become
significant?
Joe Henson: Causal sets
The cosmological constant
problem
A hint for a possible solution:
Is the cosmological constant a product of
discreteness and random/quantum fluctuations?
Joe Henson: Causal sets
A condensed state analogy
Consider a (square) membrane embedded in 3D space with
metric g, extrinsic curvature K and intrinsic curvature H:
Thermal fluctuations will impart a an energy of ~ T to each mode
on the membrane up to molecular cutoff , and thus we expect a
surface tension of order 1 in dimensionless units.
BUT: not all actually existing membranes have such high surface
tension!
Some have a low, fluctuating surface tension with 0 mean value of
!
Joe Henson: Causal sets
A condensed state analogy
Fluid membranes are made of amphiphilic molecules (like soap), which
have low solubility. At high enough concentration, the surfacereaches a
critical density of molecules and wrinkles rather than taking on a higher
density. The free energy density on the membrane has a minimum:
And the surface tension (conjugate to the total area) is therefore zero.
Joe Henson: Causal sets
Unimodular gravity and L
Unimodular gravity is a gravity theory in which the volume element is held
constant but the rest of the diffeos – the unimodular group – are allowed.
Einstein
Unimodular
Classically this is equivalent to GR: for any co-ordinate patch there is a
co-ordinate systems for which |g|=1, and then the actions agree.
In invariant language, in unimodular gravity the total volume is a
physical constant. Implementing this as a constraint on the variation of g,
the cosmological constant is now a Lagrange multiplier:
Joe Henson: Causal sets
The argument
Ingredient 1 (unimodular gravity): in QG, L and V will be conjugate
variables like E and t are in standard QM. So there is an uncertainty
relation:
Ingredient 1 (causal sets): there is an intrinsic uncertainty in
continuum volumes because they are not fundamental:
So we have an expression for the uncertainty in L:
Joe Henson: Causal sets
THIS IS THE ONLY
SUCCESSFUL PREDICTION
FROM QUANTUM GRAVITY.
EVER.
What scientists do: take note of successful heuristic predictions
and develop on them!
Joe Henson: Causal sets
Modelling everpresent L
Can we further test these ideas?
We don’t know a QG theory in which L and V are conjugate and V
has “sqrt(N)” fluctuations.
Stochastic toy version of argument:
Try implementing a stochastically fluctuating L in
finite difference approx:
Joe Henson: Causal sets
Modelling everpresent L
Consider a homogeneous, isotropic universe: how do we add the
fluctuating L?
We could throw away the acceleration equation and substitute a
fluctuating L, or try a linear combination of the two equations. This is
not GR but perhaps may model the causal set idea of an uncertain L.
Does the toy argument pan out?
Joe Henson: Causal sets
Modelling Everpresent L
The cosmological constant does track the matter
energy density in this mode as expected (this result
is independent of the exact ansatz used).
Joe Henson: Causal sets
Modelling Everpresent L
Joe Henson: Causal sets
Modelling Everpresent L
This work only scratches the surface of event the toy model.
• What effect does the addition of anisotropy and
inhomogeneity have?
• Still need to model the actual observations of L
accord with what we see, within this model.
• Without a quantum gravity theory, is there other
hueristic reasoning that can be used for modelling
beyond the stochastic level?
• Condensed state analogs?
• Develop quantum causal set dynamics.
Joe Henson: Causal sets
Conclusions
THIS IS THE ONLY
SUCCESSFUL PREDICTION
FROM QUANTUM GRAVITY.
EVER.
However it has not received the attention it deserves, and as a
result there are lots of avenues for interesting research and
chances for further predictions.
Joe Henson: Causal sets
An introduction to Causal sets
3: Indications for causal set
dynamics
Joe Henson: Causal sets
A formidable task
• Aim: a theory with QFT on curved
spacetime and GR as limits.
• Two ideas: formal generalisation or theory
construction from physical principles.
• For causal sets, there are many problems
to tackle either way (non-manifoldlike
causal sets, recovering locality, no Wick
rotation…)
Joe Henson: Causal sets
An entropic problem?
Almost all causal sets are KR orders:
Their number
This looks nothing like a manifold according to our discrete/continuum
correspondence, sprinkling. A local dynamics could not suppress such a
large entropy.
But: the number of possible relations scales like this too; an action on
causal sets cannot be local in the sense used above.
Questions:
•Do we always see these kinds of posets dominating in toy dynamics?
•sprinklings make sense physically – are they also natural
mathematically?
Joe Henson: Causal sets
Classical Sequential Growth
A quantum SOH can be seen as a generalization of a stochastic
theory; we can test the principle approach in a stochastic setting.
Sequential growth: the causal set, starting from one element, is
“grown” by randomly adding elements to the future (or spacelike)
to existing elements:
Defining all the “transition
probabilities” gives a probability
measure on infinite causal sets.
“Percolation” is a well studied model
of this type in which the new element
is related to any given past element
with probability p before transitive
closure is taken.
Joe Henson: Causal sets
2
2
1
0
1
2
Classical Sequential Growth
Physical Principles:
• general covariance= labeling invariance, e.g.:
3 4
4 3
P(
1
2
) = P(
1
2
)
• Bell causality: the ratios of probabilties of two
transitions does not depend on elements
spacelike to the two possible new elements.
Percolation obeys these rules. In fact (almost) all
processes that obey these rules have infinitely many
elements with no spacelike elements, and “flow
towards” percolation over many bounces.
Typical percolated posets look nothing like KR orders. They also have
a continuum limit in a sense, but not the right one.
Joe Henson: Causal sets
Defining the path integral
Instead we might try to learn something from a generalised path
integral
1. How to do analytical continuation?
2. What is the action? How do we approximate any local operator
on a sprinkled causal set?
3. How to suppress non-manifoldlike causal sets? Limit set of
causal sets summed over/dynamics?
Joe Henson: Causal sets
A simple 2D example
“Quantum gravity” in 2D is a trivial theory.
Consider a path integral over all spacetime
intervals, of trivial topology, in 2D. The path
integral reduces to
Z
iS (g )
D

(
g
)
e

g 
1
S
LV
8G
Reducing again to the case of fixed volume
(like the “unimodular restriction” in 4D), all
that matters is the measure. We now
discretise this path integral.
A simple 2D example
Consider partial orders on some set of objects, for example the
integers.
3
1
2

1
2
3
1
=
2
3
The intersection of two partial orders on the same set has only the
relations shared by both. The intersection of two linear orders is
called a 2D order.
Claim: the set of 1D and 2D orders contains all of the causal sets
that can be embedded into Minkowski space, which includes
everything that can be embedded in any topologically trivial 2D
causal interval.
A simple 2D example
5
6
4
5
3
6
4
5
6
4
3
1
2
1
3
2
2
1
To see this, think of
the two liner orders
and light-cone coordinates. A point in
the in causal future
of another in 2D
Minkowski if both its
co-ordinates are
greater Same rule
as 2D orders.
In 2D all of the metric
is in the conformal
factor, so varying the
density of embedding
gives faithful
embeddings for
every metric.
A simple 2D example
Any 2D order can be embedded in Minkowski
space, and conversely any finite number of
points that can be embedded in Minkowski
space, with their causal order, are a 2D order.
So all finite sprinklings into 2D Minkowski are
included in this set. Also, in 2D, all metrics on
the interval are diffeomophic to a flat metric
multiplied by a position dependent factor:
Sprinklings in these spaces are also included,
so we have discretisation s of all 2D intervals
of trivial topology.
A simple 2D example
This gives an natural way to define a limited class of causal sets that
contains all 2D intervals.
Z~
g D (g )

Z~
1
2D orders
As the original histories were diffeomophism classes, the discrete
ones are relabelling classes, or “unlabelled causets”.
Is this easily-defined restriction enough? There are still nonmanifoldlike causets in the sum. What kind of causal set dominates?
A little technology
The uniform measure on unlabelled causal sets (i.e. isomorphism
classes of causal sets) of size N is called U(N).
Another measure of interest is the measure on (labelled) causal sets
is P(N). This comes from putting uniform measure on two linear order
(i.e. two permuntations of {0,…,N}) and intersecting them.
Also, it is not hard to see that the random causet obtained
by sprinkling N points into an Interval of Minkowski space is
the same as that obtained from P(N).
Putting these two facts together, we see that causets that
are faithfully embeddable into Minkowski dominate the sum
as N -> infinity.
Joe Henson: Causal sets
2D model: Open questions
Intriguing result, but in a very limited model.
Hard to compare to other 2D quantum geometry models. Can
we extend this to a path integral between an initial and final
closed boundary? Characterise causal sets that can be
embedded on cylinders?
Study the magnitudes of fluctuations about flat space. No
fluctuations if we send “Planck scale” to 0.
Is something similar true in higher dimension? Conjecture:
faithful embeddings (with flat metric?) are always more
numerous than general embeddings, for any Lorentzian
manifold.
In general
In higher dimensions, we need an expression fro the Einstein
Hilbert action.
How do we approximate any local operator at all?
Joe Henson: Causal sets
A non-local, causal
“d’Alembertian”
Back to scale fields on sprinkled causal sets.
In the lattice we had a weighted sum of the field values at the
nearest neighbours to define d’Alembertian. Can be “average this
out” over the hyperboloid without divergences? E.g. in 2D:
L_i : all elements y such that there are only i elements causally
between x and y.
Averaging over sprinklings, this will define an approximation
to the D’Alembertian as a weighted integral over the past
light cone. Remarkably, it does not diverge.
Joe Henson: Causal sets
A potential problem
Take e.g. the expression for a constant field. The approximation is
a weighted integral over the whole past light cone, and the
integrand has significant non-zero values over an infinite volume.
Can we avoid divergances due to contributions from “far down the
light-cone”?
We can rewrite this integral:
Crucially, some “large”
terms in I do not
contribute to J:
Joe Henson: Causal sets
A potential problem
Let us assume that the field is of compact support, and
slowly varying in some frame (for now), and work in that
frame. We are in 2D, so in light-cone co-ords:
Work in units where k >> 1.
Using the previous equations, which get rid of terms of order k^ -1 and k^-2, we
see that the contribution to J(k) is small -- of order k^-3. This shows how the
above relations help to regain approximate locality. Thus the only significant
contribution is from region 3:
Joe Henson: Causal sets
Curved space
Consider a sprinkling into curved space where the appears on a
scale larger than the support of the field (which is >> the
discreteness scale). We can apply the same operator. If we
recover a local operator at all, dimensional considerations give
A straightforward but tedious calculation using Riemann normal coordinates shows the constant to be a half for dimensions < 8.
By applying this operator to the constant field we can then read off
the curvature, and thus the EH action:
Where N_i is the number of intervals containing i+1 elements .
Joe Henson: Causal sets
Current directions
The action is a sum of the number of certain sizes of intervals with
some parameters. If we analytically continue these we can get a
real path integral to play with.
Monte Carlo simulations with causal sets are being developed and
we now see that thermalisation is practical.
Joe Henson: Causal sets
Conclusions
• The causal set offers a discretisation of
spacetime consistent with the symmetries we
observe.
• It led to a prediction which deserves to be
followed up.
• While there are intriguing hints, a good
dynamics has yet to be found.
• There are lots of open problems in kinematics,
dynamics and phenomenology.
Joe Henson: Causal sets
WhY?
Joe Henson: Causal sets
HOW?
Joe Henson: Causal sets
Where?
Joe Henson: Causal sets
HOW?
Joe Henson: Causal sets
A potential problem
We need to expand the field:
The constant term gives a constant contribution. Let us examine the uv
term
Joe Henson: Causal sets
A potential problem
Joe Henson: Causal sets
What is quantum theory?
Probability for a sequence of
measurement outcomes :
Can be thought of as a function
from subsets of a history space
Ω to the reals.
What is essential?
Consider the probability for a an outcome of a
sequence of measurements, e.g. double slit:
A = “Particle went through slit 1 and ended up at point x”.
B = “Particle went through slit 2 and ended up at point x”.
A or B = “Particle ended up at point x.”
However,
A non-local, causal
“d’Alembertian”
For a sprinklings in flat space:
Definition depends only on the causal set structure. Applying it to
sprinklings into curved spacetimes, a calculation shows that
Joe Henson: Causal sets
A potential problem
Take e.g. the expression for a constant field. The approximation is
a weighted integral over the whole past light cone, and the
integrand has significant non-zero values over an infinite volume.
Can we avoid divergances due to contributions from “far down the
light-cone”?
We can rewrite this integral:
Crucially, some “large”
terms in I do not
contribute to J:
Joe Henson: Causal sets
The state of the art
This gives an ansatz for a “path integral”
Analytically continuing parameters a,b,c,d can give a real path
integral.
Problems:
• Solve analytically?
• Can we efficiently sample space of causal sets?
• If so how can be have any confidence in analytically continuing
back?
Joe Henson: Causal sets
The Problems
• Our criterion for manifoldlikeness is
physically nice, but can I be made to arise
naturally in some theory?
• We lack a natural way to characterise
manifoldlikeness.
• Manifoldlike causets form a vanishingly
small subset of causal sets.
• For a “traditional” – local – dynamics this
represents a problem.
• Similar problems for any theory of QG –
getting configurations to dominate that look
smooth at large scales.