Some persistent puzzles in background independent approaches to quantum gravity Lee Smolin Perimeter Institute for Theoretical Physics and UW Work by and.

Some persistent puzzles in background independent approaches to quantum gravity

Lee SmolinPerimeter 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) What has loop quantum gravity accomplished for sure?2) What are the persistent hard problems?3) Non-locality: problem or opportunity?4) Particle physics from non-local edges. 5) A bimetric low energy limit

Some persistent puzzles in background independent approaches to quantum gravity

and a possible remedy to them

Lee SmolinPerimeter 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) What has loop quantum gravity accomplished for sure?2) What are the persistent hard problems?3) Non-locality: problem or opportunity?4) Particle physics from non-local edges. 5) 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 diffeomorphisminvariant 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 andthe 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 inthe 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 andnot 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 propagatorMarkopoulou et al 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 emergesare 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 atomicdynamics remotely right to understand why the air in the room isuniform, or understand why metals form at low temperature.

We then need a general, thermodynamic type argument.

Also, phenomenology predictions, low energy symmetry shouldbe 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 theHamiltonian quantum theory? Do we have to choose between them?

Claim: expansion moves are necessary for generating long distancecorrelations, 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 graphcausal structure of a single spin foam history.

Macrolocality: nearby in the classical metric that emerges

Issues: Semiclassical states may involve superpositionsof large numbers of graphs. Their notions of localitymay not agree. Which notion of locality emergesas macrolocality? Similar issue for histories.

Are there states contributing to a semiclassical state for a classical metric qab

whose connectivity is non-local with respectto qab?

Weaves: Spherically symmetric caseMetric :

Consider a set of N spherical spheres, between which there are shells.This gives rise to a coarse grained geometryIn the form of a list: g= {Ai, Vi }.

|g > is a weave state that matches this

But there are non-local weaves that equallywell satisfy these conditions

Local weave: all linkscross only one sphere.

A= {6,8, 10}V= {3,4,5,6}

A= {6,8, 10}V= {3,4,5,6}

The conditions areequally well satisfiedby non-local weaves

So the weave conditions do not imply locality.

There seems nothing that guarantees that microscopiclocality defined by the connectivity of a given spinnet goes over into locality of a semi-classical or coherent statefrom which classial geometry would emerge.

Furthermore, there is a problem suppressing non-locallinks, as there are potentially so many more of them.

This is the inverse problem.

The inverse problem is a general problem for backgroundindependent approaches to quantum gravity:

Its easy to approximate smooth fields with discrete structures.

The inverse problem is a general problem for backgroundIndependent 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 thenclassical 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:

1/2

1/2

1/2

1/21

1/2

1/2

Perform a 2 to 2 move:

1/2

1/2

1/2

1/21

1/2

1/2

1/2

1/2

1

1/21/2

1/21/2

Exchange moves can increase the non-local edges.

Perform a 2 to 2 move:

1/2

1/2

1/2

1/21

1/2

1/2

1/2

1/2

1

1/21/2

1/21/2

Exchange moves can increase the non-local edges.

The two left and two right edgescan now evolveaway from each other, leading totwo non-localedges.

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 rednonlocal magenta

Added: local blacknonlocal green

R=1 blowup

R=1 blowup

Initial: local rednonlocal magenta

Added: local blacknonlocal green

R=1 blowup

Initial: local rednonlocal magenta

Added: local blacknonlocal green

Expansion dominated phase:

spiky, not a random sampling of any manifold

R=100

R=100 blowup

Initial: local rednonlocal magenta

Added: local blacknonlocal green

Initial: local rednonlocal magenta

Added: local blacknonlocal green

Initial: local rednonlocal magenta

Added: local blacknonlocal 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 ofnon-local links (locality defects)??

What is the effect of a small proportion of non-local edgesin a regular lattice field theory?

If this room had a small proportion of non-local link, with no twonodes in the room connected, but instead connectingto 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 Curietemperature increases slightly.

Tentative conclusions: To a certain point, the effect of non-localdefects 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. thereis an action, natural in the discrete setting, that forcesthe discrete system to condense to approximate alow 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. thereis an action, natural in the discrete setting, that forcesthe discrete system to condense to approximate alow 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. thereis an action, natural in the discrete setting, that forcesthe discrete system to condense to approximate alow 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 ofnonlocality in semiclassical states:

1. matter from non-locality

2. large macroscopic corrections to the low energylimit (MOND-like effects)

3. Cosmological implications

4. Hidden variables theories of quantum mechanicsgr-qc/0311059 PRD 04

We have been studying the effects of small amounts ofnonlocality in semiclassical states:

1. matter from non-locality

2. large macroscopic corrections to the low energylimit (MOND-like effects)

3. Cosmological implications

4. Hidden variables theories of quantum mechanicsDiscussed 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 SUSYin 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 dynamicsof 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 amplitudei

j

k

lj

i

l

k

Aijn klm m

n

Let the i=1/2 line be non-local

i

j

k

lj l

A1/2jn klm m

n

This is a propagation amplitude for a fermion

j

k

lj

A1/2jn klm m

n

k

k

Lets look at this in detail:

1/2

1

1

A1/2 1/2 1/2 111 1

1

1/2

1/2

1

The standard LQG fermion amplitude has the form:

j

1/2 1F[1]

1

1

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

Interactions come from moves that are local microscopically,but non local macroscopically:

A spin-1 boson:

1

1/2

1/2

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

1/2

1/2

1/2

1/2

B1

1/2

1/2

1/2

1/21

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

1/2

1/2

B1

1/2

1/2

1/2

1/21

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

1/2

1/2

B1

1/2

1/2

1/2

1/21

1/2

1/2

1/2

1/2

11

1/21/2

1/21/2

Interactions come from moves that are local microscopically,but non local macroscopically:

Locally this looks like:

1

1/2

1/2

1/2

1/2

B1

1/2

1/2

1/2

1/21

1/2

1/2

1/2

1/2

11

1/21/2

1/21/2

1

1/2

1/2

Interactions come from moves that are local microscopically,but non local macroscopically:

Locally this looks like:

1

1/2

1/2

1/2

1/2

B

1

1/2

1/2

i

j

k

lj

i

l

k

Aijn klm m

n

So if the pure gravity amplitude is:

The amplitude for matter interactioncomes from the pure gravity evolutionamplitude.

Amp B -> = 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 likeRegge trajectories in string theory.

•This gives a unification in which fermions appear in fundamental representations of gauge groups-unlike SUSY where they appear in adjoint reps-but 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 aslittle as possible, and we get out thestandard model, as observed?

But we still have to input the gauge group.

Could there be a version where we input aslittle as possible, and we get out thestandard 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 aslittle as possible, and we get out thestandard 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, twistedribbon 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 ina 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 Right braid: SU(2)R

+ 0 + - 0 -

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:

0 0 0 0 0 0

L R

Spin 1/2 states

q =0:

q= +1:

0 0 0 0 0 0

L R

+ + + + + + - - - - - -

Spin 1/2 states

q =0:

q= +1:

0 0 0 0 0 0

L R

+ + + + + +

e+L e+

R

- - - - - -

e-L e-

R

Spin 1/2 states

q =0:

q= +1:

q= +2/3

0 0 0 0 0 0

L R

+ + + + + +

e+L e+

R

- - - - - -

e-L e-

R

+ + 0

uL

+ 0 + 0 + + + + 0

uR

+ 0 + 0 + +

Spin 1/2 states

q =0:

q= +1:

q= +2/3

q= + 1/3

0 0 0 0 0 0

L R

+ + + + + +

e+L e+

R

- - - - - -

e-L e-

R

+ + 0

uL

+ 0 + 0 + + + + 0

uR

+ 0 + 0 + +

- 0 0 0 - 0 0 0 - - 0 0

dR

0 - 0 0 0 -

dL

0 0 0

0 0 0L R

+ + + + + + e+

L e+R

- - - - - - e-

L e-R

+ + 0uL

+ 0 + 0 + + + + 0uR + 0 + 0 + +

- 0 0 0 - 0 0 0 - - 0 0 dR 0 - 0 0 0 -

dL

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

There is a general flavor/colour index

R=1,...,15

Fermion =| helicity, R>

Assume that the physical states live in a subspace that satisfies the additional rule for non-coincident edges:

= q q-1

But not necessarily the other recoupling rules:

To get statistics and independence of left and right stateswe need another rule:

As a result under physical braiding (not coincident), ends of ribbons in 2d surfaces behave as anyons.

= q9 +…

q= ei/9

= q +…

So for triplets: If q9 = -1 the triplets braidas fermions, so they can moveas particles in 3d.

This means that individualribbons could never behaveas relativistic particles in3+1 dimensions.

So projected onto braids:

= -

Under the rules assumed. the left and right braids,in all their twisted states, are independent

LeftRight

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 allends of non-local links. So the probability of a link beingnon-local is at least

1080/10180 ~ 10-100

There could be many more non-local links and we could stillbe in a very sparse domain. The effects of non-locality mayonly become apparent when one looks out to cosmologicalscales.

Could there be macroscopic non-local effects that onlyappear on cosmological scales?

These would be effects that are characterized by the cosmologicalconstant scale L = -1/2

•There are anomalies in the CMB data at the scale L = -1/2:

One interpretation: no power on scaleslarger than -1/2

•Neutrino masses are at the scale L: m ~ 1/4 ~ lP-1/2

1/4~ .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 byan 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 accelerationholding low in the acceleration limit

Newtonian gravitational acceleration: aN = - GM/r2

Milgram’s Law:

aN >a0 a=aN aN <a0 a=-(aNa0)1/2

a0= 1.2 10-8 cm/sec2~ c2/6

This calls for non-locality as the force falls slower than 1/r2

Galaxy rotation curves:

Fits to data:

•The MOND formula does embarrassingly well!

•Could MOND be a consequence of quantum gravity?

•In particular since it suggests non-locality, could itbe 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 byother degrees of freedom:

gab = 2 (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 macroand microlocality.

Bi-metric theories as the low energy limit of quantum gravity Markopoulou, Premont-Schwarz, ls to appear

From a given quantum gravity state | > we extract two metrics:

Micro-metric: matches geometry operators: < V| and < |

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 }.

|g > is a weave state that matches this

But there are non-local weaves that equallywell satisfy these conditions

The macro metric counts local and non-local edges differently:

Rules: 1) Macro and micro metrics agree for local weaves2) 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 mappingr and 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/

which tells us: r02= 2GML

> r for > 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 ( ) dnumber of outgoing non-local links crossing the shell at

D ( ) dnumber of ingoing non-local links crossing the shell at

a0 area deficit from a non-local edge

v0 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 withgeometry 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, mightbe able to account for observed astrophysical deviationsfrom Newton’s laws.

THE END