Quantum gravity and the origin of quantum theory Lee Smolin Perimeter Institute for Theoretical Phys

1
Quantum gravity and the origin of quantum
theoryLee SmolinPerimeter Institute for
Theoretical Physics

• 1 Questions and Motivations
• 2 Non-locality in quantum gravity
• Particle model
• Particles from non-locality
• Lattice model (if time)
• Non-locality on astrophysical scales (if time)
• 7 Conclusions
• Work with Fotini Markopoulou
• gr-qc/0311059 and in preparation

2
Prelude
3
Prelude Anton says that quantum
unpredictability reflects events which happen
without a sufficient reason. Indeed, no reason
can be found within a nucleus to tell us when it
will decay, or within a photon to tell us
whether or not it will pass a polarizer.
4
Prelude Anton says that quantum
unpredictability reflects events which happen
without a sufficient reason. Indeed, no reason
can be found within a nucleus to tell us when it
will decay, or within a photon to tell us
whether or not it will pass a polarizer. If we
still demand a sufficient reason, we must
look outside the system, to a more detailed
description of its relations with the rest of the
universe.
5
Prelude Anton says that quantum
unpredictability reflects events which happen
without a sufficient reason. Indeed, no reason
can be found within a nucleus to tell us when it
will decay, or within a photon to tell us
whether or not it will pass a polarizer. If we
still demand a sufficient reason, we must
look outside the system, to a more detailed
description of its relations with the rest of the
universe. Perhaps there we will find Chriss
ZING.
6
We have made progress in quantum gravity, but
the foundational problems are unsolved

• The measurement problem is still unsolved after
  80 years.
• It gets worse when the observer is inside the
  system

7
We have made progress in quantum gravity, but
the foundational problems are unsolved

• The measurement problem is still unsolved after
  80 years.
• It gets worse when the observer is inside the
  system
• Attempts to make sense of quantum cosmology are
  not
• convincing, to us.

8
We have made progress in quantum gravity, but
the foundational problems are unsolved

• The measurement problem is still unsolved after
  80 years.
• It gets worse when the observer is inside the
  system
• Attempts to make sense of quantum cosmology are
  not
• convincing, to us.
• In addition there is the problem of constructing
  real 4d
• observables measurable by observers in the
  universe.

9
LQG has not resolved the foundational problems of
quantum cosmology

• The measurement problem is still unsolved after
  80 years.
• It gets worse when the observer is inside the
  system
• Attempts to make sense of quantum cosmology are
  not
• convincing, to us.
• In addition there is the problem of constructing
  real 4d
• observables measurable by observers in the
  universe.
• And there is the problem of proving that general
  relativity
• emerges as the low energy limit. There are
  indications,
• but no proof.

10
Technical issues with the low energy limit

• There are problems defining the sum over spin
  foams
• Sums over complex numbers seldom converge.
• In QM and QFT on backgrounds, this is resolved by
  making use
• of Euclideanization to make sums convergent. But
  without a
• background metric any Euclideanization
  prescription is arbitrary
• and may break diffeomorphism invariance.
• Any renormalization group must renormalize sums
  over spin
• foams, not just single spin foams

11

• There are also difficulties recovering smooth
  manifolds
• as a low energy limit of spin foam models.
• Originally spinnets and spin foams were embedded.
• It was observed that the formulation is much
  simpler if
• we drop the embedding.
• Fotinis causal spinnet histories
• CLKRs reduction of spin foams to group field
  theory.
• Crane et al Without manifold conditions, dual
  triangulation
• gives rise to manifold with conical
  singularities.
• Perhaps these are the matter degrees of freedom?
• Inverse locality problem local in the coarse
  graining derived
• from averaging over spin foams may not be
  strictly local in
• the microscopic level.

12
The inverse problem for discrete spacetimes
Its easy to approximate smooth fields with
discrete structures.
13
The inverse problem for discrete spacetimes
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,
14
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
15
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the
embedding?
16
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the embedding? d N
ways that dont.
17
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the embedding? d N
ways that dont. N2 ways that do. Thus, if the
low energy definition of locality comes from a
coarse graining of a combinatorial graph, it will
be easily violated in fluctuations.
18
New hypothesis The conceptual and technical
obstacles to understanding the low energy limit
are related
Perhaps quantum theory cannot be fully made sense
of because it is an approximation to a deeper
theory. Perhaps the quantum sums over spin foams
cannot be defined because quantum mechanics
emerges from spin foams rather than the reverse.
But, by the experimental disproof of the Bell
inequalities Any deeper theory must be
non-local.
Perhaps the non-local theory is based on
combinatorial histories like spin foams. This
idea is the subject of this talk.
19

• The basic idea
• The fundamental theory is combinatorial and
  deterministic or stochastic. It is based on an
  evolving graph like a spin network.
• The low energy theory recovers spacetime as an
  averaged description, with the graph embedded in
  it.
• Because the coarse grained notion of locality
  incompletely represents the fundamental notion,
  there are stray edges, connecting nodes that are
  far away in the coarse grained notion of
  locality.
• Statistical mechanics for the whole system, plus
  reasonable conditions, implies quantum mechanics
  for subsystems.
• The stray non-local links are the missing
• hidden variables.

20
We have been studying models of non-locality in
discrete spacetime models such as LQG A
regular lattice or weave with a random
distribution of non-local links. P(n,m)
probability that nodes n and m are connected.
21
We have been studying models of non-locality in
discrete spacetime models such as LQG A
regular lattice or spinnet with a random
distribution of non-local links. P(n,m)
probability that nodes n and m are connected.
22
We have been studying models of non-locality in
discrete spacetime models such as LQG A
regular lattice or spinnet with a random
distribution of non-local links. P(n,m)
probability that nodes n and m are
connected. Pltlt1
These models have conflicting macro and micro
notions of locality
23

• We have found so far four applications of such a
  conflict
• between micro and macro locality
• Hidden variables theories of quantum mechanics
• gr-qc/0311059 PRD 04
• matter fields from gauge fields non-locality
• large macroscopic corrections to the low energy
• limit (MOND-like effects)
• Cosmological implications (microscopic
• derivation of bi-metric or VSL theories)

24

• We have found so far four applications of such a
  conflict
• between micro and macro locality
• Hidden variables theories of quantum mechanics
• gr-qc/0311059 PRD 04
• matter fields from gauge fields non-locality
• large macroscopic corrections to the low energy
• limit (MOND-like effects)
• Cosmological implications (microscopic
• derivation of bi-metric or VSL theories)

25

• Quantum mechanics has been understood as a low
  energy approximation to the statistical mechanics
  of non-local systems.
• Approaches that use matrix models
• Steve Adler
• Artem Starodubtsev
• ls
• Local variables are invariants, such as
  eigenvalues
• Off diagonal elements carry non-local information
• Quantum mechanics for unitary invariants is
  derived from
• certain limits of statistical mechanics for
  matrix models.
• Approach based on stochastic differential
  equations Nelson
• Brownian motion on configuration space
• Conservation of an average energy
• gives solutions to Schrödinger's equation.

26
In this talk we make use of Nelsons
method. Matrix methods are more powerful,
and are under investigation.
27
The basics of Brownian motion on configuration
space. probability density probability
current Conservation Evolution Forward Bac
kward Current velocity Osmotic velocity
28
Nelsons assumptions 1) There is a
universal Brownian motion. 2) Energy is
conserved, on the average
H, the average energy,is a function only of, r,
va and position. H is positive definite. H
is invariant under time reversal invariance. H
is invariant under rotations H is local, so it
is of the form H contains only those terms
that dominate in the low velocity and long long
wavelength limit. The only density in the
problem is r, hence Under time
reversal Hence
29
lt Energy gt conservation also implies the
Nelson/Newtons law
where
3d assumption vanishing vorticity Thus S
exists such that
30
From Nelsons assumptions 1) There is a
universal Brownian motion. 2) Energy is
conserved, on the average 3) Vanishing
vorticity We have the conservation of
Now choose

The conservation of H implies
31

• Problem Nelsons proof is not constructive.
• We provide constructive, but approximate.
• realizations of Nelsons conditions.
• 1 Particle mechanics
• Lattice gauge theory (in progress)
• These are inspired by the structures in LQG

32
The particle model
33

• Assume that the low, low (non-relativistic) limit
  of QG satisfies
• There is a flat space of dimension d.
• There are N particles in that space.
• They satisfy Newtons laws in the limit hG l2 -gt
  0 with some V.
• To leading order in hG there are corrections that
  come from
• the failure of the low energy approximation to
  coincide with
• locality in the underlying theory.
• These residual non-localities are
• described by a graph G on N nodes.
• More detailed assumptions
• The average distance between particles, Lgtgtl
• n, the valence of G. 1 ltlt n ltlt N.
• The graph G is randomly distributed in space.

34
Matrices Q adjacency matrix of G. Qija
Qij 1 if i and j connected, 0 if
not Xa diagonal Xaij dijxa Combine the
matrices
35
Matrices Q adjacency matrix of G. Qija
Qij 1 if i and j connected, 0 if
not Xa diagonal Xaij dijxa Combine the
matrices

• The basic result
• Consider the eigenvalues lai of Ma.
• Assume that some regions of the universe are
  hot
• so that for those degrees of freedom nx does not
  vanish.
• Assume the average nx is large, so nnx l4/L4 is
  O(1).
• n will be the number of connections to hot nodes.
• Then the evolution of the probability
  distribution for the ls
• is given to leading order by the Schrödinger
  equation..

36
How Nelsons conditions are satisfied Condition
1 Every degree of freedom feels a universal
Brownian motion.
Dysons theorem If some elements of a matrix
undergo Brownian motion, so do the eigenvalues
In our case The off diagonal elements are
constant. Some xas fluctuate, with average
diffusion constant nx Consider a l whose x is
cold (nx0).
37
How Condition 2 is satisfied The condition
is Compute where From the assumption that
Newtons laws are satisfied to leading order we
know that Because to zeroth order the ls trace
the xs The stochastic derivatives are defined
by
38
The correction due to non-local terms is
The ratio of the correction to the classical
acceleration is order
Under reasonable physical conditions the is
O(1). Hence, to leading order we have shown,
39

• Showing condition 3
• Assume that the probability currents for the xs
• are curl free.
• A similar calculation shows that to leading order
  the
• same is true of the probability currents for the
  ls

40

• Open issues
• What happens to higher order? Are there
• non-linear corrections?
• Does coherence decay in time?
• Nelsons dynamics is not quite equivalent to
  Schrödinger for configuration spaces with
• non-trivial topology. (Because eiS is not
• guaranteed to be a phase.)
• But averaged energy is still conserved. Do we
• then get mixed states?

41
But, If LQG really unifies gravity and QM,
shouldnt it automatically tell us about
unifying the rest of physics?
F. Markopoulou, ls, hep-th/05???
42
A spin network with a non-local link
1/2
43
A network with a non-local link Add a loop in
the fundamental rep, N, of G.
(1/2,N)
Couple to Yang-Mills means add labels, a rep r,
of gauge group G SU(N) on each link,
similarly for nodes.
44
It looks to a local observer like a spin
1/2 particle in the fundamental rep. of SU(N).
A network with a non-local link labeled (j1/2,
r fundamental)
(1/2,N)
45
It looks to a local observer like a spin
1/2 particle in the fundamental rep.
A network with a non-local link labeled (j1/2,
r fundamental)
(1/2,N)
So we naturally get fermions, and unlike SUSY in
the fundamental representation of any gauge
fields.
46
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.
47
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 (21)
with local moves.
Markopoulou gr-qc/9704013
48
Relation between fermion and gravity
dynamics pure gravity amplitude
i
k
i
k
Aijn klm
n
m
j
l
j
l
Let the i1/2 line be non-local
k
i
k
A1/2jn klm
n
m
j
l
j
l
This is a propagation amplitude for a fermion
k
Y
k
A1/2jn klm
Y
n
m
j
l
j
49
Lets look at this in detail
1
Y
1
A1/2 1/2 1/2 111
Y
1/2
1
1/2
1
1/2
1
The standard LQG fermion amplitude has the form
1
1
Y
Y
F1
1/2
1/2
1
1
1
We have to do this twice to reproduce the pure
gravity move
F12 A1/2 1/2 1/2 111
j
50
Interactions come from moves that are local
microscopically, but non local macroscopically
A spin-1 boson
1/2
1/2
B
1
1/2
1/2
51
Interactions come from moves that are local
microscopically, but non local macroscopically
A spin-1 boson as a non-local link w/ j1
1
1/2
1/2
1/2
1/2
1/2
B
1
1
1/2
1/2
1/2
1/2
1/2
52
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
1/2
B
1
1
1/2
1/2
1/2
1/2
1/2
53
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
1/2
B
1
1
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1
1
1/2
1/2
1/2
54
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
1/2
B
1
1
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
y
1
y
1
1
1/2
1/2
1/2
1/2
55
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
i
k
B
1
i
k
1/2
Aijn klm
n
m
1/2
j
l
j
l
1/2
The amplitude for matter interaction comes from
the pure gravity evolution amplitude. Amp B -gt
y y A1/2 1/2/1 1/2 1//2 1
y
1
y
1/2
56

• 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 groups-
• unlike SUSY where they appear in adjoint reps-
• but like nature.

57
The lattice model (in progress)
58

• Regular lattice R. Nd sites in d dimensions.
  Lattice spacing a
• Decorate it with K links between random sites.
  The result is L
• Site n has a set of neighbors N(n)
• Its non-local neighbors, not in R are F(n)
• On L, define a lattice scalar field theory

A
The hamiltonian is
B
We divide the lattice into two parts A and B.
We assume each is much larger than the
correlation length, and that we make all
measurements in A. The degrees of freedom in B
will then be averaged over.
59
We block spin renormalize so the lattice spaceing
a -gt Pa. After the canonical field rescaling, the
Hamiltonian becomes
g ltlt1 because the density of non-local links is
low.

• To show Nelsons conditions
• 1) Because of the non-local interactions the
  fields
• in A are subject to a universal Brownian motion.
• 2) The average energy in A is conserved for small
  g
• because in the local partition function ltfngt0
• 3) Under reasonable conditions the probability
• currents are curl free.

60
Nelsons condition 1 Universal Brownian
motion. We consider the one dimensional case,
and for clarity separate A and B.
A
B
Mass term
Equation of motion
Wc/a
Fn is a noise term coming from the non-local links
61
We study Fourier modes in A
We have harmonic oscillators plus noise
With classical frequencies
The noise term is random
Thus, each mode in A is subject to a random
noise, uncorrelated with the other modes in A.
Thus, the non-local links cause a universal
Brownian motion of the field modes.
62
Nelsons condition 2 the average energy in A is
conserved. We divide the Hamiltonian into three
parts
HA contains the degrees of freedom we can measure
HB is similar, but we dont measure it. The
local interactions between A and B are absent
or negligable. HNL has the non-local interactions
63
The averaged energy is the sum
The averages satisfy
Lcorrelation length
Assume that the non-local links are few enough
that correlations are not established by them.
Then, for n in A and m in B we have
Hence
Hence the averaged energy in A is conserved.
64
Nelsons third condition Set up the initial
statistical state so that in A dVn/dfm- dVm/dfn
0 Then there is an Sfm such that Vn d
Sfm /dfn Then, as before, we can invent a y
Br eiS that approximately satisfies the
Schrodinger equation.
65

• Conclusions
• There are obstructions to recovering smooth, low
  dimensional manifolds as approximations to
  combinatorial structures such as spinnets and
  spinfoams.
• Can we get some physics from them?
• In particular locality is unstable. Hard to get
  low energy and microscopic notions of locality to
  coincide.
• Can we use these for non-local hidden variables
  and recover quantum mechanics of subsystems from
  statistical mechanics of spin nets or foams?
• If we can, we can resolve the foundational
  problems of quantum cosmology, by making use of
  an obstacle to spin foams having a good low
  energy limit.
• The first indications are positive, but much to
  do.

66

• Possible Evidence for non-local effects in very
  low energy
• astrophysics
• The Tully Fischer Relation
• Galaxies have flat rotation curves, with velocity
  V.
• Total luminosity L astro-ph/0204521
• C L Va a3.9? 0.2
• K L/M (M-total mass)
• CK M V4
• CK should be prop to G
• CK Ga0
• a0 1.2 10-8 cm/sec2
• ? L c2/6

67
MOND proposal There is a critical acceleration
a0 1.2 10-8 cm/sec2 c2/L agta0
aaN aN GM/r2 alta0 a(aNa0)1/2
L cosmological radius
68
MOND proposal There is a critical acceleration
a0 1.2 10-8 cm/sec2 c2/L agta0
aaN aN GM/r2 alta0 a(aNa0)1/2
critical radius where aa0
rgt rc v2/r (GMa0 )1/2 /r
v4 GM a0
L cosmological radius
69
MOND proposal There is a critical acceleration
a0 1.2 10-8 cm/sec2 c2/L agta0
aaN aN GM/r2 alta0 a(aNa0)1/2
critical radius where aa0
rgt rc v2/r (GMa0 )1/2 /r
v4 GM a0
The MOND potential
70

• Explains flat rotation curves and
  Tully-Fischer, but...
• How well does this explain the details of galaxy
• rotation curves?
• How well does this compare with the hypothesized
• dark matter?

astro-ph/020452
71
Galaxy rotation curves
72
(No Transcript)
73
(No Transcript)
74
Galaxies not fit by MOND none Galaxies for
which MOND fit is dubious 11 Galaxy
problem NGC 2841 distance discrepant from
Hubble flow value NGC 2915 distance
uncertain DDO 154 last few points dropping (no
Newtonian fit, either) IC 1613 very uncertain
inclination and asymmetric drift F565-V2
inclination very uncertain UGC 5750 inclination
very uncertain UGC 6446 distance uncertain UGC
6818 interaction? UGC 6973 very dusty - does
light trace mass? Ursa Minor very sensitive to
Milky Way parameters Draco sensitive to Milky Way
parameters
Galaxies well fit by MOND 84 listed at
present UGC 2885 NGC 5533 NGC 6674 NGC 7331 NGC
5907 NGC 2998 NGC 801 NGC 5371 NGC 5033 NGC 2903
NGC 3521 NGC 2683 NGC 3198 NGC 6946 NGC 2403 NGC
6503 NGC 1003 NGC 247 NGC 7739 NGC 300 NGC 5585
NGC 55 NGC 1560 NGC 3109 UGC 128 UGC 2259 M 33 IC
2574 DDO 170 DDO 168 NGC 3726 NGC 3769 NGC 3877
NGC 3893 NGC 3917 NGC 3949 NGC 3953 NGC 3972 NGC
3992 NGC 4010 NGC 4013 NGC 4051 NGC 4085 NGC
4088 NGC 4100 NGC 4138 NGC 4157 NGC 4183 NGC 4217
NGC 4389 UGC 6399 UGC 6446 UGC 6667 UGC 6818 UGC
6917 UGC 6923 UGC 6930 UGC 6973 UGC 6983 UGC
7089 NGC 1024 NGC 3593 NGC 4698 NGC 5879 IC 724
F563-1 F563-V2 F568-1 F568-3 F568-V1 F571-V1
F574-1 F583-1 F583-4 UGC 1230 UGC 5005 UGC 5999
Carina Fornax Leo I Leo II Sculptor Sextans Sgr
75

• The MOND formula does embarrassingly well!
• Dark matter calculations do not do nearly as
  well
• Dont account for Tully-Fischer
• Have cusps, dark matter should dominate in the
  centers of galaxies, but in the data they dont
• Doesnt explain the occurrence of an acceleration
  scale as the threshold for breakdown of Newtons
  laws with visible matter.
• Doesnt explain why it involves L.

76

• The MOND formula does embarrassingly well!
• Dark matter calculations do not do nearly as
  well
• Dont account for Tully-Fischer
• Have cusps, dark matter should dominate in the
  centers of galaxies, but in the data they dont
• Doesnt explain the occurrence of an acceleration
  scale as the threshold for breakdown of Newtons
  laws with visible matter.
• Doesnt explain why it involves L.

A force that falls off like 1/r for d3 implies
non-locality.
77

• Let us go back to our model
• A regular weave for a flat metric qab with a
  random distribution
• of non-local links.
• P(x) probability that nodes n
• and m are connected if they
• are a distance x apart in the
• metric qab.

78

• Bimetrics for graphs
• rnm distance between n and m in metric qab.
• The weave can be chosen so the graph metric
  matches rnm
• P(r) probability that nodes n and m are
  connected if they are a distance r apart in the
  metric qab
• d (r) minimal expected graph distance
  between two nodes r apart in qab.

79

• Bimetrics for graphs
• rnm distance between n and m in metric qab.
• The weave can be chosen so the graph metric
  matches rnm
• P(r) probability that nodes n and m are
  connected if they are a distance r apart in the
  metric qab
• d (r) minimal expected graph distance
  between two nodes r apart in qab.

Physical ansatz The grav potential from a source
of mass M j(r) -GM/d(r)
80
The express train problem You are trapped in a
very complicated metro system with local and
express trains. You know only that probability
that a station is an express station. What is
the expected time to go from n to m? The random
wormhole problem There is a random distribution
of wormholes in spacetime. You know the
probability that two points are connected by a
wormhole. What is the expected distance using
the wormholes between two points a distance r
apart?
Slide 47
Slide 47
81
Prob of path w one jump We want z st the prob
from a region around n and m are so connected.
This means This gives
zltr w r
w
z
r
z
m
n
When this is true
This tells us the relationship between P(r) and
d(r)
82

• Our physical ansatz requires that jMOND(r)
  -GM/d(r)
• Mond
• Our calculation found
• These imply

So there is a wormhole distribution that leads to
MOND N1, P 10-216