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Some persistent puzzles in background independent
approaches to quantum gravity
Lee Smolin
Perimeter Institute for Theoretical Physics and UW
Work by and with Fotini Markopoulou,
Mohammad Ansari, Sundance O. Bilson-Thompson, Hal Finkel,
Jacob Foster, Isabeau Premont-Schwarz, Yidun Wan
1)
2)
3)
4)
5)
What has loop quantum gravity accomplished for sure?
What are the persistent hard problems?
Non-locality: problem or opportunity?
Particle physics from non-local edges.
A bimetric low energy limit
Some persistent puzzles in background independent
approaches to quantum gravity
and a possible remedy to them
Lee Smolin
Perimeter Institute for Theoretical Physics and UW
Work by and with Fotini Markopoulou,
Mohammad Ansari, Sundance O. Bilson-Thompson, Hal Finkel,
Jacob Foster, Isabeau Premont-Schwarz, Yidun Wan
1)
2)
3)
4)
5)
What has loop quantum gravity accomplished for sure?
What are the persistent hard problems?
Non-locality: problem or opportunity?
Particle physics from non-local edges.
A bimetric low energy limit
What has loop quantum gravity accomplished for sure!
A Quantum geometries: spin networks for algebra A
B. Quantum spacetimes: spin nets evolve by local rules.
C. Derivations of A and B from classical diffeomorphism
invariant theories.
D. Applications: black holes, cosmology, phenomenology, etc.
There is lots of good news. Steady progress.
But there are also several persistent unsolved problems.
Unification:
We claim we can include any standard matter, SUSY and
the theory stays finite consistent.
BUT what about anomalous chiral gauge theories???
Is there fermion doubling as there is in lattice gauge theory?
In that case can LQG really include the standard model?
Is there a spin statistics theorem?
There are many LQG/spin foam models. Are there criteria to pick
out one that should describe nature?
If LQG is even roughly right, the right version should have
implications for the problem of unification.
Interpretation of quantum cosmology:
Several claims, but still open.
Evolving constants of motion: YES, but how to implement in
the real theory??
Relational quantum theory (Crane, Rovelli, et al):
Sounds good in principle, but what determines the boundaries
between different domains?
Quantum and algebraic causal histories (Markopoulou et al)
Also sounds good, but requires a fixed causal structure.
(Terno reports on some developments)
What is an event when we sum over causal histories?
Maybe quantum theory should come from quantum gravity and
not the other way around??
The emergence of classical spacetime geometry.
-We can assume ansatz’s for semi-classical, coherent or weave
states and derive predictions from them. But we can’t know if these
are predictions of the theory unless we can find the ground state
and show that classical geometry emerges.
There are new approaches to this problem to be discussed here.
Rovelli et al
Markopoulou et al
propagator
particles as decoherence free subspaces
Why can’t we find the Hamiltonian operator for asymptotically
flat b.c. and show that it is positive definite on physical states?
Why is this so hard?
What if the quantum hamiltonian is not positive definite?
Three possibilities:
0
The theory is wrong
1
Those spin foam models from which classical spacetime emerges
are very special. This is a criteria to pick out good theories.
(Perhaps they are supersymmetric, and underlie string theory….)
2
The emergence of spacetime is generic.
Shouldn’t 2 be right? You don’t need to get the details of atomic
dynamics remotely right to understand why the air in the room is
uniform, or understand why metals form at low temperature.
We then need a general, thermodynamic type argument.
Also, phenomenology predictions, low energy symmetry should
be generic.
(But what about theories with the “wrong” Immirzi parameter?)
There is one issue which matters: the two types of moves:
Expansion moves:
Exchange moves:
•Hamiltonian constraint gives only expansion moves.
•Spin foams give both (finite evolution, crossing symmetry)
How then could spin foam models be precisely derived from the
Hamiltonian quantum theory? Do we have to choose between them?
Claim: expansion moves are necessary for generating long distance
correlations, hence, emergence of spacetime.
Possible ways out: regulate in space and time, master constraint???
The problem of non-locality
Two kinds of locality:
Microlocality:
connectivity of a single spin net graph
causal structure of a single spin foam history.
Macrolocality:
nearby in the classical metric that emerges
Issues:
Semiclassical states may involve superpositions
of large numbers of graphs. Their notions of locality
may not agree. Which notion of locality emerges
as macrolocality? Similar issue for histories.
Are there states contributing to a
semiclassical state for a classical metric qab
whose connectivity is non-local with respect
to qab?
Weaves: Spherically symmetric case
Metric :
Consider a set of N spherical spheres, between which there are shells.
This gives rise to a coarse grained geometry
In the form of a list: g= {Ai, Vi }.
|Gg > is a weave state that matches this
But there are non-local weaves that equally
well satisfy these conditions
Local weave: all links
cross only one sphere.
A= {6,8, 10}
V= {3,4,5,6}
The conditions are
equally well satisfied
by non-local weaves
A= {6,8, 10}
V= {3,4,5,6}
So the weave conditions do not imply locality.
There seems nothing that guarantees that microscopic
locality defined by the connectivity of a given spinnet
goes over into locality of a semi-classical or coherent state
from which classial geometry would emerge.
Furthermore, there is a problem suppressing non-local
links, as there are potentially so many more of them.
This is the inverse problem.
The inverse problem is a general problem for background
independent approaches to quantum gravity:
Its easy to approximate smooth fields with discrete structures.
The inverse problem is a general problem for background
Independent approaches to quantum gravity:
Its easy to approximate smooth fields with combinatoric structures.
But generic graphs do not embed in manifolds of low dimension,
preserving even approximate distances.
?
Those that do satisfy constraints unnatural in
the discrete context,
One reason for worry:
We believe the universe starts in a non-classical state and then
classical spacetime emerges as it evolves. So the initial states should
not approximate any classical geometry.
The evolution is by local moves. Will these generate local spacetime?
Local moves are unlikely to remove non-local edges.
So once there in the initial state, they are defects, trapped in!
Combinatorial definition of non-local edge:
smallest cycle containing the edge is very large.
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move:
1
1/2
1/2
1/2
1/2
1/2
1/2
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move:
1
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1
1/2
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move:
1
1/2
1/2
1/2
1/2
1/2
The two left and
two right edges
can now evolve
away from each
other, leading to
two non-local
edges.
1/2
1/2
1/2
1/2
1/2
1/2
1
1/2
LQG cosmological scenario
•Universe starts with a random spinnet
•Expands by a combination of expansion and exchange moves
•Becomes local (low valence nodes) decorated with a small number
of the original links, which are now non-local.
•What really happens?
Hal Finkel will report on a series of numerical experiments
using stochastic evolution with various mixes of evolution moves
•No quantum mechanics
•No labels, only graphs
•Random start ~200 nodes
•Grow to ~5,000 nodes
•Vary R= exchange moves/expansion moves
R=1
Initial: local red
nonlocal magenta
Added: local black
nonlocal green
R=1 blowup
Initial: local red
nonlocal magenta
R=1 blowup
Added: local black
nonlocal green
Initial: local red
nonlocal magenta
R=1 blowup
Added: local black
nonlocal green
Expansion dominated phase:
spiky, not a random sampling of any manifold
R=100
wup
Initial: local red
nonlocal magenta
Added: local black
nonlocal green
Initial: local red
nonlocal magenta
Added: local black
nonlocal green
Initial: local red
nonlocal magenta
Added: local black
nonlocal green
Exchange dominated phase:
Well mixed, spikiness eliminated.
Lots of non-locality created by local exchange moves!!!
For details see Hal Finkel’s talk
Compare R=100 to R=1
R=100
R=1
Tentative conclusion: dominance by exchange moves is needed to
recover macro-geometry
Is this a problem for Hamiltonian evolution??
Suppose the ground state is contaminated by a small proportion of
non-local links (locality defects)??
What is the effect of a small proportion of non-local edges
in a regular lattice field theory?
If this room had a small proportion of non-local link, with no two
nodes in the room connected, but instead connecting
to nodes at cosmological distances, could we tell?
Yidun Wan studied the Ising
model on a lattice contaminated by
random non-local links.
R=non-local links/local links
= 20/800=1/40
The critical phenomena is the same, but the Curie
temperature increases slightly.
Tentative conclusions: To a certain point, the effect of non-local
defects on the lattice is just to raise the critical temperature,
Correlation functions alone apparently cannot detect small amounts
of non-locality, at least away from Tc.
For details, see Yidun Wan’s talk.
What are we to do about the inverse problem and
the locality problem?
What are we to do about the inverse problem and
the locality problem?
1. Hope that the problem is solved by dynamics, i.e. there
is an action, natural in the discrete setting, that forces
the discrete system to condense to approximate a
low dimensional spacetime.
Little evidence of this so far
What are we to do about the inverse problem and
the locality problem?
1. Hope that the problem is solved by dynamics, i.e. there
is an action, natural in the discrete setting, that forces
the discrete system to condense to approximate a
low dimensional spacetime.
Little evidence of this so far
2. The theories are wrong.
But these appear to be generic problems!!!
What are we to do about the inverse problem and
the locality problem?
1. Hope that the problem is solved by dynamics, i.e. there
is an action, natural in the discrete setting, that forces
the discrete system to condense to approximate a
low dimensional spacetime.
Little evidence of this so far
2. The theories are wrong.
But these appear to be generic problems!!!
3. Assume a sparse distribution of non-local links are
locked in from the early universe and hence connect to
cosmological scales. See what this implies for physics.
We have been studying the effects of small amounts of
nonlocality in semiclassical states:
1. matter from non-locality
2. large macroscopic corrections to the low energy
limit (MOND-like effects)
3. Cosmological implications
4. Hidden variables theories of quantum mechanics
gr-qc/0311059 PRD 04
We have been studying the effects of small amounts of
nonlocality in semiclassical states:
1. matter from non-locality
2. large macroscopic corrections to the low energy
limit (MOND-like effects)
3. Cosmological implications
4. Hidden variables theories of quantum mechanics
Discussed at Marseille
gr-qc/0311059 PRD 04
Consider LQG coupled to Yang-Mills with gauge group G
A network with a non-local link labeled (j=1/2, r= fundamental)
looks to a local observer like a spin 1/2 particle in the
fundamental rep. of G.
(1/2,N)
So we naturally get fermions, and unlike SUSY
in the fundamental representation of any gauge fields.
So a small amount of non-locality is nothing to be afraid of.
A spinnet w/ non-local links looks just like a local spinnet
with particles.
So a small amount of non-locality is nothing to be afraid of.
A spinnet w/ non-local links looks just like a local spinnet
with particles.
But this implies that the dynamics and interactions of
matter fields are already determined by the dynamics
of the gravity and gauge fields.
Could this work?
Model:
trivalent spinnets (2+1) with local moves.
fm gr-qc/9704013
Relation between fermion and gravity dynamics:
pure gravity amplitude
i
k
m
j
i
k
Aijn klm
l
n
j
l
Let the i=1/2 line be non-local
k
i
k
m
j
A1/2jn klm
l
n
j
l
This is a propagation amplitude for a fermion
k
Y
j
m
l
Y
A1/2jn klm
j
k
n
Lets look at this in detail:
1
Y
1
1/2
Y
A1/2 1/2 1/2 111
1
1
1/2
1/2
1
The standard LQG fermion amplitude has the form:
1
Y
1/2
F[1]
1
1
Y
1/2
1
1
We have to do this twice to reproduce the pure gravity move:
F[1]2 = A1/2 1/2 1/2 111
j
Interactions come from moves that are local microscopically,
but non local macroscopically:
A spin-1 boson:
1/2
1/2
1
1/2
1/2
B
Interactions come from moves that are local microscopically,
but non local macroscopically:
A spin-1 boson as a non-local link w/ j=1
1/2
1/2
1
1/2
1/2
1
1/2
1/2
B
1/2
1
1/2
1/2
1/2
Interactions come from moves that are local microscopically,
but non local macroscopically:
Perform a 2 to 2 move:
1/2
1/2
1
1/2
1/2
1
1/2
1/2
B
1/2
1
1/2
1/2
1/2
Interactions come from moves that are local microscopically,
but non local macroscopically:
Perform a 2 to 2 move:
1
1/2
1/2
1
1/2
1/2
1/2
B
1/2
1
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1
1/2
1/2
1
1/2
Interactions come from moves that are local microscopically,
but non local macroscopically:
Locally this looks like:
1
1/2
1/2
1
1/2
1/2
B
1/2
1
1/2
1/2
1/2
1/2
1/2
1/2
y+
1
y
1/2
1/2
1/2
1
1/2
1/2
1/2
1
1/2
Interactions come from moves that are local microscopically,
but non local macroscopically:
Locally this looks like:
So if the pure gravity amplitude is:
1/2
1/2
1
1/2
1/2
1/2
y+
1
y
1/2
B
i
i
j
k
m
l
Aijn klm
j
k
n
l
The amplitude for matter interaction
comes from the pure gravity evolution
amplitude.
Amp B -> y+ y = A1/2 1/2/1 1/2 1//2 1
“Matter without matter”
J A Wheeler
•Works also when coupling to gauge fields are included.
Just label edges by reps of SU(2) X G.
•Pair creation possibly implies spin-statistics connection.
Dowker, Sorkin, Balachandran.....
•CPTgravity
CPTmatter
same for CP, T etc
•Does CP breaking in matter imply CP breaking in gravity?
•We get a tower or particles of increasing spin, just like
Regge trajectories in string theory.
•This gives a unification in which fermions appear in
fundamental representations of gauge groupsunlike SUSY where they appear in adjoint repsbut like nature.
But we still have to input the gauge group.
But we still have to input the gauge group.
Could there be a version where we input as
little as possible, and we get out the
standard model, as observed?
But we still have to input the gauge group.
Could there be a version where we input as
little as possible, and we get out the
standard model, as observed?
Minimal model: no labels, just graphs...
too simple...
But we still have to input the gauge group.
Could there be a version where we input as
little as possible, and we get out the
standard model, as observed?
Minimal model: no labels, just graphs...
too simple...
Next simplest model:
Ribbon graphs
Let’s play a simple game:
(Bilson-Thompson)
Basis States: Oriented, twisted
ribbon graphs, embedded in S3
topology, up to topological class.
There is a label, which is twisting:
t=0
t=+1
t=-1
Rule 1: Twist number is conserved at nodes.
We will be interested in states with triplets of edges:
Some possible topologies for triplets:
unbraid
Left braid
Right braid
Each strand also can be twisted:
+ +0
Two more rules:
Rule 2: Conservation of braiding number
across nodes.
Rule 3: No states with both + and - twists in
a single triplet.
Topological embeddings of ribbon graphs modulo
these rules span a Hilbert space Hedges
Discrete symmetries:
C:
twist
-twist
P:
Left
Right:
T: Reverse orientation:
CPT=I
Left braid: SU(2)L
+ 0 +
- 0 -
Right braid: SU(2)R
Assume this theory has a low energy limit, defined in
terms of an emergent 3+1 dimensional spacetime metric.
Assume that in the low energy limit the resulting
effective dynamics is Poincare invariant.
What do the twisted braid states look like?
Classification of braided states:
(Bilson-Thompson hep-ph/0503213)
Interpretation: Twist = charge in units of e/3
Braiding = left and right fermion number
Spin 1/2 states
q =0:
0 0 0
0 0 0
Spin 1/2 states
q =0:
nL
nR
0 0 0
0 0 0
Spin 1/2 states
q =0:
nL
nR
0 0 0
0 0 0
q= +1:
+++
+++
- - -
- - -
Spin 1/2 states
q =0:
nL
nR
0 0 0
0 0 0
q= +1:
e+L
e+R
+++
e-L
+++
e-R
- - -
- - -
Spin 1/2 states
q =0:
nL
nR
0 0 0
0 0 0
q= +1:
e+L
e+R
+++
e-L
+++
q= +2/3 uL
e-R
- - -
- - -
uR
++0
+0+
0++
++0
+0+
0++
Spin 1/2 states
q =0:
nL
nR
0 0 0
0 0 0
q= +1:
e+L
e+R
+++
e-L
+++
e-R
- - -
q= +2/3 uL
- - -
uR
++0
+0+
0++
q= + 1/3 dL
++0
+0+
0++
dR
-00
0- 0
00 -
-00
0 -0
0 0 -
The 30 fermion states of the first generation are all here.
•Color is naturally explained as place in the braid. It is clear
why only the charge 1/3 and 2/3 states have color. 24 states
•There are only two neutral states, one left, one right handed.
•There are four q=+ 1 states n
L
There is a general
flavor/colour index
R=1,...,15
0 0 0
0 0
e+L 0
e+R
e-L
e-R
+++ +++
- - - - -
uL
dL
Fermion =| helicity, R>
nR
uR
++0+0+ 0++
dR
-00 0- 0 00 -
++0+0+ 0++
-00 0 -0 0 0 -
To get statistics and independence of left and right states
we need another rule:
Assume that the physical states live in a subspace that
satisfies the additional rule for non-coincident edges:
Y[
] = q Y[
]+
-1
q
Y[
But not necessarily the other recoupling rules:
]
As a result under physical braiding (not coincident),
ends of ribbons in 2d surfaces behave as anyons.
=q
+…
This means that individual
ribbons could never behave
as relativistic particles in
3+1 dimensions.
So for triplets:
=
9
q
+…
If q9 = -1
the triplets braid
as fermions, so
they can move
as particles in
3d.
q= eip/9
So projected onto braids:
=-
+…
Under the rules assumed. the left and right braids,
in all their twisted states, are independent
Y[ Left] = Y[ Right]
Single ribbons cannot behave as particles in 3d.
They are anyons. They can live as ribbons in 3+1 but as
particles only in 2+1 But triplets can!
P: projection operator onto triplet states:
P
=
-
Suppose this works, so that the observed fermions are all
ends of non-local links. So the probability of a link being
non-local is at least
1080/10180 ~ 10-100
There could be many more non-local links and we could still
be in a very sparse domain. The effects of non-locality may
only become apparent when one looks out to cosmological
scales.
Could there be macroscopic non-local effects that only
appear on cosmological scales?
These would be effects that are characterized by the cosmological
constant scale
L = L-1/2
•There are anomalies in the CMB data at the scale
L = L -1/2:
One interpretation:
no power on scales
larger than L-1/2
•Neutrino masses are at the scale L:
m ~ r1/4 ~ lP-1/2 L1/4~ .1 eV
•We should expect anomalies at the acceleration scale given by
ac = c2/L ~ 10-8 cm/sec2
•The Pioneer Anomaly is at the scale ac:
a is approximately 8 10-8 cm/sec2
astro-ph/0104064, 0208046
•The anomalous galaxy rotation curves are characterized by
an acceleration scale near ac:
The Tully Fischer Relation:
•Galaxies have flat rotation curves, with velocity V astro-ph/0204521
k Ga0 M= V4
a0= 1.2 10-8 cm/sec2 ~ ac
k= mass/luminosity ratio
The MOND phenomenological law accounts for this:
A modification of Newton’s law of gravitational acceleration
holding low in the acceleration limit
Newtonian gravitational acceleration:
aN = - GM/r2
Milgram’s Law:
aN >a0
aN <a0
a=aN
a=-(aNa0)1/2
a0= 1.2 10-8 cm/sec2~  L c2/6
This calls for non-locality as the force falls slower than 1/r2
Fits to data:
Galaxy rotation curves:
•The MOND formula does embarrassingly well!
•Could MOND be a consequence of quantum gravity?
•In particular since it suggests non-locality, could it
be non-locality from quantum gravity?
Basic idea:
The low energy limit of quantum gravity is
a bi-metric theory
(Markopoulou)
Bi-metric theories as the low energy limit of quantum gravity
Usual bimetric theories have two classical metrics, differing by
other degrees of freedom:
gab = f2 (qab + Ba Bb )
qab satisfies something like einstein eqs
propagation of matter is determined by gab.
The proposal is that the difference arises from mismatch of macro
and microlocality.
Bi-metric theories as the low energy limit of quantum gravity
Markopoulou, Premont-Schwarz, ls to appear
From a given quantum gravity state |Y > we extract two metrics:
Micro-metric:
matches geometry operators:
< Y| V| Y > and < Y|A | Y >
treats non-local links the same as local links
satisfies approximate Einstein equations
Macro-metric:

derived from propagation of matter
ignores non-local links beyond matter scale.
Recall: Spherically symmetric, static weave
Consider a set of N spherical spheres, between which there are shells.
This gives rise to a list of areas g= {Ai, Vi }.
|Gg > is a weave state that matches this
But there are non-local weaves that equally
well satisfy these conditions
The macro metric counts local and non-local edges differently:
Rules:
1) Macro and micro metrics agree for local weaves
2) Macro metric gives less weight in areas for non-local edges
and less weight in volume for ends of non-local edges.
3) Both are static.
The disagreement about areas leads to a mapping
r and r refer to the same physical surface, given different areas by
the two metrics
The non-locality does not affect the other components, so the lapse is:
The macrometric determines orbits of stars according to:
To reproduce the observations (MOND law) we need
But the micrometric must be approx Schwarzchild, n2= 1 - 2 GR/r
2= 2GML
r
0
which tells us:
r > r for r > r0
Can the distribution of non-local links be chosen to reproduce this,
keeping the macro-spatial geometry flat to zero’th order in GM/r?
YES
(Note an upper cutoff r< R= er0 )
C ( r ) dr:
D ( r ) dr:
number of outgoing non-local links crossing
the shell at r.
number of ingoing non-local links crossing
the shell at r.
Da0
area deficit from a non-local edge
Dv0
volume deficit from a non-local edge
Conclusions:
Non-locality does not necessarily kill a theory, it may be
hard to observe directly.
Non-local links leads to a new unification of matter with
geometry and forces.
Maybe it is hard to derive classical GR as the low energy
limit because the low energy limit is a bimetric theory??
Bimetric theory can roughly account for effects of non-local
links in semiclassical or weave states.
Non-locality, modeled by such a bimetric theory, might
be able to account for observed astrophysical deviations
from Newton’s laws.
THE END