The Quest for Quantum Gravity

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The Quest for Quantum Gravity

• Joseph Polchinski
• KITP, UC Santa Barbara
• Daniel Heineman Prize Lecture
• APS, Jacksonville, April 16, 2007

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Discovery of D-branes, and their role
Discovery vs. invention
Discovery is often messy you often dont get to
where you expected. The history of string theory
is full of unexpected twists and turns.
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The Veneziano formula
Semi-phenomenological formula for meson-meson
scattering,
t
s
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Veneziano amplitude (1968)
meson infinite tower of harmonic oscillators
string
(Nambu, Susskind, Nielsen, 1969-70)
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Veneziano
string
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A messy history, but the underlying object seems
to be uniquely determined. We still do not have
the complete form of the theory there are more
surprises in store.
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Some of the tools of theoretical discovery
Thought experiments
Einstein Galileo Maxwell Black holes Bekenstein,
Hawking and many others
Consistency
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An example the weak interaction
The Fermi of a local interaction (plus V-A),
m
ne
nm
e
works well to a point, but gives uncontrollable
divergences at higher order. (High precision or
high energy)
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The solution is to resolve the local interaction
into the exchange of an intermediate vector boson
ne
m
nm
W
e
This IVB must have very specific properties it
must originate from a spontaneously broken
Yang-Mills theory.
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A correct theory must be
Renormalizable Unitary Lorentz invariant
It is very hard to satisfy all three (e.g., a
spatial smearing Lorentz invariance would imply
smearing in time as well, and loss of causality).
(There are many solutions if we impose only two
of the three).
Spontaneously broken gauge symmetry works because
there are gauges satisfying any two (Coulomb,
Feynman, unitary), but all are equivalent!
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For gravity it is harder. It seems that the
infinities of quantum gravity are not cured
simply by adding new particles, but by changing
the nature of the particles or even the nature of
spacetime. One solution
graviton
electron
.
Instant of time
point
loop
(or strand
)
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Is there one theory of quantum gravity or many?
If one imposes only two of the three
consistency conditions, one can find many
theories of quantum gravity. Many attempts
give up Lorentz invariance at the start, and it
has even been argued that this is a necessary
feature of quantum gravity. It is hard to see
how the successes of Special Relativity can then
be maintained. E.g., the Standard Model would
have 20 extra parameters (different speeds of
light for every particle), and even Planck-scale
breaking will feed down into the low energy
Lagrangian.
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Towards D-branes T-duality
An interesting thought experiment is to put
strings in a periodic space, length 2pR, and then
take R to be very small (Kikkawa Yamasaki,
1984, Sakai Senda, 1986)
2pR
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2pR

For a point particle the effect is to quantize
the momentum, pcompact n/R with integer n.
This is a con-tribution to the effective mass
seen by a lower-dimensional observer -pm pm M
2 implies
Meff 2 -(pm pm)noncomp. M 2 (pcompact)2
M 2 n2/R2
As R goes to zero, any state with momentum in the
extra dimension becomes infinitely massive, so we
just lose a dimension. (The zero radius limit of
a cylinder is a line).
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For strings there is an additional effect -- they
can wind
2pR
w 0
w 1
w 3
It costs energy to wind around the cylinder
Meff 2 M 2 n2/R2 w2R2/a (a string
length-scale2)
As R goes to infinity, the winding states get
massive and the momentum states form a
continuum. As R goes to zero, the momentum
states get massive and the winding states form a
continuum. There is a symmetry, T-duality
n w, R a/R
The zero R limit is the same as the infinite R
limit.
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The zero R limit is the same as the infinite R
limit!
R va is an effective minimum radius.
This is a simple example of stringy geometry,
the fact that strings seem spacetime differently
than point particles. Further applications lead
to topology change and the resolution of some
spacetime singularities. This is a simple
example of emergent spacetime the dimension that
reemerges in the R to zero limit is not manifest.
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This is for closed strings. Open strings do not
have a conserved winding number
There is no quantum number w, so the mass formula
is just
Meff 2 M 2 n2/R2
As R goes to zero, there is no continuum the
cylinder effectively becomes a line.
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In a theory with open and closed strings, if we
start with D dimensions, compactify, and take R
to zero, the closed strings live in D dimensions
and the open strings live in D-1 dimensions. Is
this an inconsistency?
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Phase diagram of highly supersymmetric string
backgrounds (today)
M theory
heterotic E8 x E8
Type IIA
Type IIB
heterotic SO(32)
Type I
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Phase diagram of highly supersymmetric string
backgrounds (in 1989)
heterotic E8 x E8
Type IIA
Dine, Huet, Seiberg Dai, Leigh, JP
Narain
Type IIB
heterotic SO(32)
Dai, Leigh, JP Horava
Type I
(all perturbative dualities).
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Phase diagram of highly supersymmetric string
backgrounds (in 1989)
heterotic E8 x E8
Type IIA
Dine, Huet, Seiberg Dai, Leigh, JP
Narain
Type IIB
heterotic SO(32)
Dai, Leigh, JP Horava
Type I
Weak-strong dualities connect the two sides, and
lead to new phases.
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D-branes provide a simple description of the
half-BPS objects required by the dualities
M theory
heterotic E8 x E8
Type IIA
Type IIB
heterotic SO(32)
Type I
(Almost obvious T-duality preserves SUSY, and
SUSY of Type I is half that of Type II).
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Unexpected consequences
New connections to mathematics (K theory,
non-commutative geometry, topological string
theory ). New possibilities for phenomenology
(braneworlds, warped spaces, brane inflation).
The first counting of black hole entropy.
Gauge/gravity duality (see Stan and Juans
talks). New ideas about the nature of string
theory strings are present only in certain
classical limits, D-branes seem to come closer to
the fundamental degrees of freedom.
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One example of new degrees of freedom
Let us try something seemingly completely
different, starting with the idea that a minumum
length scale could arise from noncommutative
coordinates,
x i, xj ? 0.
To implement this, let us make them matrices in
some N-dimensional space, x iab. We want to
recover the usual commutative coordinates at low
energy, but have noncommutativity at high energy.
We can do this with
.
H M ?i,a,b (x iab)2 M 3 ?i,j,a,c x iab x
jbc - x jab x ibc 2
Nonrelativistic kinetic energy plus potential
energy for commutator.
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Quantum corrections destroy the hierarchy, so add
supersymmetry,
M 2?i,a,b y ab g i y bc x ica
(The most supersymmetry theory, 16 supercharges,
has 9 spatial directions i).
At low energy, virtual noncommutative matrices
give a long-range force between the particles
aa
bb
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This can be interpreted as a theory of quantum
gravity, in light-cone gauge. In fact, it is
the full Hamiltonian for M theory, with boundary
condition of flat spacetime with a null circle
(DLCQ). (de Wit, Hoppe, Nicolai Banks,
Fischler, Shenker, Susskind). The connection
comes because it is the low-energy Hamiltonian
for N D0-branes.
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In contrast to string perturbation theory, which
does not converge,
Matrix Theory is a complete nonperturbative
description, which could even be simulated on a
computer. It describes, e.g. black hole
formation and decay, spacetime topology change,
trans-Planckian scattering, and resolution of
some spacetime singularities.
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Where to from here?
Matrix theory defines string/M theory with a
certain simple boundary condition it is almost
background independent. AdS/CFT defines it for
other boundary conditions. Quantum gravity is a
holographic theory, so is very sensitive to its
boundary conditions.
Our current understanding allows us to answer
many questions, but there are many others that
remain to be answered the nonperturbative
formulation for realistic compactifi-cations, and
especially for cosmology.
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In cosmology, the only natural boundaries are in
the past and future, rather than in spatial
directions. Apparently we need to extend our
understanding of emergent space to emergent time
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