The world as a hologram: News from string theory

The world as a hologram:
News from string theory
Prof. Dr. Jan Plefka
Humboldt-Universität zu Berlin
Integrability in Gauge and String Theory 2015
Public Lecture
The problem
Two keystones of fundamental physics:
1. Einstein’s theory of gravity
[1915]
2. Quantum theory: [1920-1930]
⇒
Standard model of elementary particle physics
Electromagnetic, weak and strong forces
Not known how to combine the two!
The most promising ansatz:
⇒
String theory
[1950-75]
“unified theory”
[since 1984]
Both keystones are intimately connected
⇒
Holographic principle of quantum gravity
[1/25]
[since 1997]
The world as a holgram
[2/25]
Die Allgemeine Relativitätstheorie
Das Prinzip:
Die Raumzeit bestimmt die Bewegung der Materie, die Materie
Gravity
bestimmt die Krümmung
der Raumzeit.
Einsteinsche Feldgleichungen:
Rµν − 12 gµν R + gµν Λ =
8π GN
c4
Tµν
Geometrie (Mathematik) = Materie (Physik)
Krümmung R
Kosmologische Konstante Λ
Energie-Impuls-Tensor Tµν
Gravity according to Newton
Sir Isaac Newton, 1642-1727
Two bodies moving through empty space in the absence of forces
FA = mA
d2 x
=0
dt2
⇒ x(t) = vt+c
Asbodies
have masses mA und mB there is an attractive gravitational force
acting between them
F = GN
mA · mB
(xA − xB )2
Newton’s law of gravity
Time is absolute: t “ticks” the same for A and B.
Forces are mediated instantaneously
Space is flat and infinitely extended (Euclidean): R3
[3/25]
Gravity according to Einstein
Albert Einstein, 1879-1955
The attraction of two masses arises through the curvature of space (and time)
Bodies A and B continue to move on straight
paths albeit in a curved space! There is no force
acting between them.
Gravity is a fictitious force similar to the centrifugal force known from everyday
life.
Space-time is dynamic: Gets curved by the matter moving through it and
influences the motion of matter via its curvature. A highly coupled system.
[4/25]
A Gedankenexperiment
Let us remove the sun from our solar system!
Newton’s World: The gravitational force acts instantaneous:
The “removal” of the sun is immediately felt by the earth
Einstein’s World: Changes in the curvature of space and time propagate with c
the velocity of light
Gravitational waves =
ˆ ripples of space-time!
Predicted alrady in 1915, presently searched for in graviational wave detectors.
[5/25]
A Gedankenexperiment
Let us remove the sun from our solar system!
Newton’s World: The gravitational force acts instantaneous:
The “removal” of the sun is immediately felt by the earth
Einstein’s World: Changes in the curvature of space and time propagate with c
the velocity of light
Gravitational waves =
ˆ ripples of space-time!
Predicted alrady in 1915, presently searched for in graviational wave detectors.
[5/25]
The theory of general relativity
The principle:
Space-time dictates the movement of matter, matter dictates the
curvature of space-time.
Einstein’s field equations:
Rµν − 12 gµν R + gµν Λ =
8π GN
c4
Tµν
Geometry (Math) = Matter (Physics)
Curvature R
Cosmological constant Λ
[6/25]
Energy-momentum tensor Tµν
The theory of general relativity
The principle:
Space-time dictates the movement of matter, matter dictates the
curvature of space-time.
Einstein’s field equations:
Rµν − 12 gµν R + gµν Λ =
8π GN
c4
Tµν
Geometry (Math) = Matter (Physics)
Curvature R
Cosmological constant Λ
[6/25]
Energy-momentum tensor Tµν
Beispiel Quantenelektrodynamik:
Elementarteilchenph
Physikal. Theorie der Elektronen und P
(g:“Kopplungskonstante”)
• Beschrieben durch Quantenfeldtheor
Quantum mechanics
e−
e−
=
e−
Zeit
e−
g2 ·
Ph
Streuung von
Elektronen
Streuprozesse
Störun
Renormierung: g → g(E)
• Renormierung: g → g(E)
Drei Naturkräfte beschrieben durch Ei
Quantum mechanics
M.Planck
W.Heisenberg
E.Schrödinger
A particle starts it motion from point A at time tA . Will it reach the point B at tb ?
Classical physics:
Prediction (from Newton’s law): Depending on its intitial velocity and the
forces acting upon it, the particle will either reach B at tb or not:
Deterministic prediction: yes/no
Quantum physics:
There is no definite answer to this question! Only the prediction for a probability
of observing the particle at space-time point B at the time tB is possible.
W(A,ta )→(B,tb ) = 0.73
[7/25]
Quantum mechanics
M.Planck
W.Heisenberg
E.Schrödinger
A particle starts it motion from point A at time tA . Will it reach the point B at tb ?
Classical physics:
Prediction (from Newton’s law): Depending on its intitial velocity and the
forces acting upon it, the particle will either reach B at tb or not:
Deterministic prediction: yes/no
Quantum physics:
There is no definite answer to this question! Only the prediction for a probability
of observing the particle at space-time point B at the time tB is possible.
W(A,ta )→(B,tb ) = 0.73
[7/25]
Feynman’s path integral
Richard Feynman, 1918-1988
Feynman’s approach to quantum mechanics allows for the computation of this
probability:
Consider all possible paths from A to B. Every path is weigthed by a factor
(the action) and the total probability follows from the sum of all possible paths
X
2
W(A,ta )→(B,tb ) = eaction/~ all paths
The classical path is the path with minimal action ⇒ typically gives a dominant
contribution to W .
Time is still an absolute quantity here ⇒ “non-relativistic” quantum mechanics
[8/25]
Quantum field theory or relativistic quantum mechanics
Absolutely surprising effect of the constancy of the speed of light:
γ
Creation and annihilation of particles
γ
Photon
e−
Elektron
Photon
e
−
e−
Elektron
Photon radiation
e−
e−
e−
Forces are transmitted via the exchange of elementary particles
e−
e−
Carrier particle of the electro−
e−
magnetic force:ePhoton (“light”)
Photon
e−
Photon
e−
[9/25]
[1950s-70s]
Quantum field theory
Example: Quantum electrodynamics:
Elementarteilchenphysik (ohne Gravitation)
Theoretical description of electrons and photons and their interactions
(g: charge or “coupling constant”)
• Beschrieben durch Quantenfeldtheorie: (hier QED)
e−
e−
−
−
=
e
Zeit
e
g2 ·
Photon
+ g4 ·
[1950-1975]
+ g 6(. . .) + . . .
ScatteringStreuprozesse
of electrons
Perturbative
in g 1
Störungsreihe
in g �series
1 g:“Kopplungskonstante”
Renormalization: g → g(E)
• Renormierung:
→ g(E)
But
what happens gwhen
g ∼ 1? ⇒ non-perturbative quantum field theory
Three fundamental forces described via gauge field theory
• Drei Naturkräfte beschrieben durch Eichfeldtheorien
• Was passiert bei g ∼ 1? ⇒ nichtperturbative Quantenfeldtheorie
[10/25]
[1955,1971]
[1955,1971]
Standard model of particle physics
Forces: Gauge bosons
SU(N) gauge fields N × N matrices
Matter: Quarks & leptons
Higgs boson: Scalar particle
Standard Model of Elementary Particles
18.02.15 09:56
[11/25]
Photon
The strong force: Quantum
chromodynamicse
Gluon
−
e−
e−
Photon
SU(3) gauge theory: gluons and quarks (q)
q
q
q
g
Interaction:
e−
q
e−
g
Responsible for the staq
bility of the proton and
nuclei
Gluon
e−
q
Gluon
q
q
g
Gluons have selfg interactions
g
g
g
Gluon
g
q
g
g
g
g
g2 ∼
g∼
g
Strong force:
q
g
(at LHC energies)
g
g
g
1000 = 10g3 times stronger thatg electromagnetic
force
Gluon
Gluon
100000 = 105 times stronger than the weak force
10000 . . . 000 = 1038 times stronger than gravity!!
[12/25]
g
Quantum chromodynamics: The inverse giant
Coupling strength depends on energy
[Gross, Wilczeck, Pollitzer]
g → g(E)
g
E
Low Energies (g 1): Confinement
No free quarks and gluons are being observed.
Instead: Bound states (hadrons)
Mesons:
q̄
q
L
q̄q-potential:
V =
“Jim Knopf”, M. Ende
[13/25]
1
2
lS
L
“color flux tube”
The strong force as a string theory
Color flux tube reminds of a microscopic string:
Hadron
String picture
Meson (e.g. pion)
=
ˆ
Glueball
=
ˆ
q̄
q
q
Baryon (e.g. Proton)
[’t Hooft, 1974]
q
=
ˆ
q
Vision: Strings are adequate description for strongly coupled gauge fields.
But: Strings describe quantum gravity and the strong force does not contain
gravity!
Puzzle resolved by holographic principle...
[14/25]
!
Quantum
gravity andeiner
string
theorySchwerpunktsb
• Quantenmechanik
Quantum
mechanics
of a“Saite”:
relativistic
string
• Quantenmechanik einer “Saite”: Schwerpunk
Graviton
Gauge bosonS
Stringwechselwirkungen:
Vibrationsspektrum
Spektrum Eichteilche
der
Graviton =
Oscillation
spectrum
ˆ =ˆspectrum
of “Elemen
“elem
• Verallgemeinerung
Teilchengraphen:
[198
Vibrationsspektrum
=
ˆ Spektrum
der “Elem
Extended
structurevon’softens’
divergences:
Gravitation ?
Gravity as a quantum theory?
Treat Einstein’s theory of gravity b y the rules of quantum field theory:
• Behandle Einsteins Gravitationstheorie als Quantenfeldtheorie:
Gravitons: Small curvature fluctuations about a given space-time structure
Gravitonstreuung: gScattering
κ · hµν (x)
Gedankenexperiment:
of+gravitons
µν (x) = ηµν
h
h
=
h
(Gedankenexperiment)
Zeit
κ2 ·
Graviton
+ κ4 ·
+ ...
h
=∞
NOT “renormalizable” ⇒ Forced
to extend Einstein’s theory
NICHT “renormierbar” ⇒ Quantenfeldtheorie der Gravitation existiert nicht !
Quantum field theory of gravity requires measurements of infinitely
many empirically determined parameters
Kopplungskonstante der Gravitation: [κ] = M 1
=
ˆ 10−33cm
Planck
Very limited predictability . . .
4
√
Coupling constant κ = GN of gravitons extremely weak:
Becomes relevant only at length scales 10−33 cm
[15/25]
String theory
How can we reach predictability within
quantum gravity?
Stringtheorie
• Simple
Idee: Ersetze
Teilchen
durch by
ausgedehntes
Objekt:
“String”
idea: Replace
particles
extended 1d1d
object:
“String”
o�
−33
lS ∼
cm cm
lS 10
∼ 10−33
Quantum mechanics of a relativistic string:
• Quantenmechanik einer “Saite”: Schwerpunktsbewegung + Eigenschwingung
Centre of mass movement + internal oscillations:
Graviton
Eichteilchen
Materieteilchen
Vibrationsspektrum =
ˆ Spektrum der “Elementarteilchen”
[16/25]
7
String interactions: Perturbative series
Stringwechselwirkungen: Störungsreihe
Generalization of particles interactions:
[1984-1995]
• Verallgemeinerung von Teilchengraphen:
+
gS2 ·
[1984-1995]
+
gS4 ·
+ gS6 · (. . .) + . . .
Zeit
gS : Stringkopplungskonstante
gS : string coupling constant
Es treten keine Divergenzen mehr auf! Wechselwirkung ist “weich”
There are no divergencies! Interaction is “soft” as not localized to a point
• Gravitonstreuung:
Graviton scattering:
gS2 ·
=
ˆ Einsteins Gravitationstheorie
=
ˆ Quantenkorrekturen zu Einsteins Theorie
=
ˆ Einstein’s theory of gravity
7
gS2
·
=
ˆ Quantum corrections to Einstein’s theory
[17/25]
Properties and predictions of string theory
Higher dimensions:
As a quantum theory strings are only consistent in 9 space dimensions!
⇒ 6 extra space dimensions
or
Graviton
Materieteilchen
Geometry of hidden 6 dimensions predicts particle spectrum 3 dimensional world
Prediction of supersymmetry: bosons ⇔ fermions
Puzzle: There exist a gigantic number of possible compactifications from 9 → 3
[18/25]
The world as a hologram
Source: Scientific american
Quantum gravity in negatively curved space-times
Since 1997 revolutionary progress in our understanding of quantum gravity in
anti-de-Sitter space (AdSd ) =
ˆ constant neg. curvature [Willem de Sitter, 1872-1934]
AdS5 is (4+1)-dimensional space-time with a boundary: R3 × time
“Gravity in a box”
String theory well defined on AdS5 × M5 , e.g. choose M5 = S 5 5d-sphere.
[19/25]
Maldacena’s String-Gauge Duality
[1997]
Holographic principle: Strings in the bulk of space-time (Anti-de-Sitter space),
quantum particles (gluons) on the boundary
String theory in higher
dimensional space
Gauge field theory on 4d boundary
Two dual descriptions of one physical entity: Gauge theory =
ˆ String theory in AdS
[20/25]
The world as a hologram: Resolution of dimensions
4d particle theory ⇔ (5+5)d gravity theory ⇔ 2d string theory
[21/25]
Integrability in gauge and string theory
Novel insights discussed at the IGST Conference allow computation of exact
quantum properties in both theories.
⇒ No approximation in coupling constant g any longer!!
Probability for the scattering of two gluons into two gluons in
supersymemtric QCD
Example 1:
g
g
=
?
g
g
(
73 6
4 6
2
8
8g 2 − 38 π 2 g 4 + 88
45 π g − 16( 630 π + 4 ζ(3) ) g + . . .
=
27 ζ(3) 1
2
K
1
4g − 3 log
− 4π
2 g−3 log 2/4π − 29 π 3 g 2 − . . .
π
for
for
g 1o
g1
× (Simple functions of the involved momenta)
Exactly known function of g and gluon-momenta
[22/25]
[Beisert,Eden,Staudacher]
Stringtheorie
Theory as a consistent
theory of quantum gravity
pling asymptotics. For this purpose Stringtheorie
we solve numerically the integral form of the
em equations for the exact energies of excited states proposed by us and A. Kozak.
Integrability in gauge and string theory
ee:
Ersetze
ausgedehntes1d1d
Objekt:
“String”
Idee:
ErsetzeTeilchen
Teilchen durch
durch ausgedehntes
Objekt:
“String”
: Example
Replace2: particle
by extended
1d object:
string
Exact internal
excitation energy
of a closed
string in AdS-space.
!!
−33 −33
lS ∼
lS 10
∼ 10cm
cm
UCTION
ills theories are at the heart
cs, describing all fundamenty. Nevertheless, in spite of
First excitation mode of the
most
40mechanics
years, we stilleiner
don’t
Quantenmechanik
Schwerpunktsbewegung
+ Eigenschwingung:
ntum
of a“Saite”:
relativistic
string in flat space-time:
string
uantenmechanik
einer
tive
description of the
most“Saite”: Schwerpunktsbewegung + Eigenschwingung:
h as QCD, in the region of
plings. The low energy quanis mostly known only from
7
ice YM theories. A few imned the topological, BPS secGraviton
Gauge bosonStörungsreihe
Matter particle
tained.
Stringwechselwirkungen:
on complete
exact 4D soluVibrationsspektrum
Spektrum
der
“Elementarteilchen”
Figure 1:
Numerical
solutionstring
of exact finite
size integral YGraviton
Eichteilchen
Materieteilchen
∆K=
=
Energy
stored
in vibrating
lation
spectrum
ˆ(λ)=ˆspectrum
of
“elementary
particles”
uantities
given
by nontrivial
system equations for the Konishi dimension
∆K (λ) in a wide
4 to the asymptotic
o start waning, N=4 superrange
of ’tder
Hooft
couplings
λ, compared
Curvature
radius
of AdS-space
allgemeinerung
von Teilchengraphen:
[1984-1995]
Vibrationsspektrum
=
ˆλ =
Spektrum
“Elementarteilchen”
5
g 2 λ asymptotics
y gave structure
us serious hopes
for
nded
’softens’
divergences:
Bethe
ansatz curve and to the predicted =
large
String
length
1/4
1/4
∆K (λ) ≃ 2λ
+ 2/λ
obtained by fit.
he dynamics of strongly inDue to the AdS/CFT cor[Gromov,Kazakov,Vieira]
2
4
6
o the quantum integrability
+
g ·
+
[23/25]
g ·
+ g · (. . .) + . . .
Summary
Quantum gravity (=
ˆ string theory) in (d+1)-dimensions is equivalent to
quantum particle theory (gauge field theory) in d-dimensions w/o gravity
Novel tools for solution of the quantum particle theory in 3 space dimensions
[24/25]
Outlook
Prospect: Use (2+1)d particle theory to describe quantum gravity in (3+1)d
Black holes! Our universe?
[25/25]
Thank you for your
attention