Transcript Document

The Quest for Quantum Gravity
Joseph Polchinski
KITP, UC Santa Barbara
Daniel Heineman Prize Lecture
APS, Jacksonville, April 16, 2007
Discovery of D-branes, and their role:
Details of history
Discovery vs. invention
Discovery is often messy: you often don’t get
to where you expected. The history of string
theory is full of unexpected twists and turns.
The Veneziano formula:
Semi-phenomenological formula for meson-meson
scattering,
t
s
Veneziano amplitude (1968)
meson = infinite tower of harmonic oscillators
= string
(Nambu, Susskind, Nielsen, 1969-70)
Veneziano
string
supersymmetry
extra dimensions
gravity
duality
supergravity
D-branes
M theory
braneworlds
and warping
the landscape
black hole entropy
matrix theory
gauge/gravity duality
Veneziano
string
supersymmetry
extra dimensions
gravity
duality
supergravity
D-branes
M theory
braneworlds
and warping
the landscape
black hole entropy
matrix theory
gauge/gravity duality
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.
Some of the tools of theoretical discovery:
Thought experiments
Einstein
Galileo
Maxwell
Black holes: Bekenstein, Hawking and
many others
Consistency
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)
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.
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!
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
Instant of time:
.
point
loop
(or strand
)
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.
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
2pR
•
For a point particle the effect is to quantize the
momentum, pcompact = n/R with integer n. This is a contribution to the effective mass seen by a lowerdimensional 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).
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.
The zero R limit is the same as the infinite R limit!
• R = √a’ 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.
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.
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?
No, it’s a D-brane*:
*Dirichlet membrane
Phase diagram of highly supersymmetric string
backgrounds (today):
M theory
Type IIA
heterotic E8 x E8
Type IIB
heterotic SO(32)
Type I
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).
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.
D-branes provide a simple description of the halfBPS objects required by the dualities:
M theory
Type IIA
heterotic E8 x E8
Type IIB
heterotic SO(32)
Type I
(Almost obvious: T-duality preserves SUSY, and SUSY
of Type I is half that of Type II).
Unexpected consequences:
• New connections to mathematics (K theory, noncommutative 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 Juan’s talks).
• New ideas about the nature of `string’ theory:
strings are present only in certain classical limits, Dbranes seem to come closer to the fundamental
degrees of freedom.
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
.i 2
H = M ∑i,a,b (x ab) + M 3 ∑i,j,a,c |x iab x jbc - x jab x ibc |2
1
2
Nonrelativistic kinetic energy plus potential energy
for commutator.
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 longrange force between
the particles:
aa
bb
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.
Veneziano
string
supersymmetry
extra dimensions
gravity
duality
supergravity
D-branes
M theory
braneworlds
and warping
the landscape
black hole entropy
matrix theory
gauge/gravity duality
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.
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 compactifications, and especially for cosmology.
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…