1. No category

What is String Theory ?
- present status from a historical perspective -
NTU, 2011
Prologue：What is physics?
“The endeavor of exploring,
on the basis of empirical facts,
the most fundamental laws
which exist behind all kinds
of phenomena surrounding us.”
from What is physics ?, Shin-ichiro Tomonaga, 1979
As the understanding of basic physical laws were
developed deeper, our horizons have been broadened,
both in macroscopic and microscopic directions.
“More is different”
(P. W. Anderson, 1972)
and
Deeper is more !
What is the most fundamental
problem remaining unsolved
in the 20th century physics?
量子力学
相対性理論
Conflict between two central
pillars of modern physics,
quantum theory
and
general relativity
相対性理論
量子力学
General Relativity
LHS: ‘made of marble‘
Geometry of space-time
RH: ‘made of mere wood’
Quantum theory of matter
according to Einstein himself.
The principles of general relativity are governed
entirely by classical concepts of particles and fields.
No uncertainties, and no fluctuations at all.
“The real goal of my research has always been the
simplification and unification of the system
of theoretical physics.
I attained this goal satisfactorily
for macroscopic phenomena,
but not for the phenomena
of quanta and atomic structure.
..................
I believe that despite considerable success,
the modern quantum theory is still far from a
satisfactory solution of the latter group of problems.”
A. Einstein, 1936
Quantum Theory :
characterized by fluctuations of physical states
superposition principle
uncertain relation
The constitution and the
stability of matter can never
be explained without
quantum theory.
Even the vacuum and the geometry
of space-time itself are microscopically
fluctuating and superposing
among themselves,
by pair creation/annihilation
of all sorts of elementary particles.
Quantum Gravity:
quantum theory of
space-time geometry
creation/annihilation
of gravitons
If general relativity is combined naively to the ordinary rules
! we encounter serious difficulties.
of quantization,
P/π
Q/2π
$
"#
Non-renormalizable
ultra-violet
2n
κ
divergences, due to space-timefn
(D−2)/2
fluctuations:
a
n "#
n
κ
a(D−2)/2
$2n
fn
Information puzzle in
(approximate) quantum
mechanics around black-holes:
This is related to the ultraviolet difficulty,
since the wave lengths of signals reaching
infinity must be infinitely short at the spacetime horizon. (Time lapse becomes infinitely
slow as seen from an outside observer.)
a→0
cut-off length scale at
small distances
a→0
This conflict indicates that
General Relativity is not a fundamental theory, but is
・a low-energy effective theory
which is valid only at sufficiently long distances,
but breaks down microscopically.
historical analogy of low-energy effective theory:
- four-fermi theory of weak interactions
Weinberg-Salam theory
- sigma model for low-lying hadrons
QCD
Quantum theory and general relativistic
theory of gravity must be truly unified,
by replacing general relativity by a more
fundamental microscopic theory of gravity
But, the conceptual framework of these
theories are quite different :
・the concept of “states” in quantum theory is
defined globally on the basis of
linear superposition principle.
・space-time and states in general relativity are
defined on the basis of patching works of
local events .
Unfortunately, we do know any valid direct
approaches, so far, to the unification of these two
different conceptual frameworks in the framework of
(quantum) local field theory.
And also, old attempts towards non-local field theories
suffered from either violation of quantum mechanical
unitarity and or Lorentz invariance.
String Theory
suggests new intrinsic mechanisms
for a complete elimination of ultraviolet infinities
and
for unifying all fundamental interactions
including gravity,
within the general framework of quantum theory
(at least in the perturbation theory).
But, unfortunately, is still immature and is in the
“making” stage.
What is string ?
fundamental string (‘ F1’)
only has 1 dimensional extension
・mass density :
・
strictly constant
string theory
the unique fundamental constant of
string length constant
closed string ： generate gravitational field
open string ： generate gauge fields
・closed string interacts with all kinds of strings
in a universal manner which is consistent with
general relativity at long distances.
・open strings interacts among themselves
in a universal manner which is consistent with
gauge theory at long distances.
classical images of strings in motion
time
space
graviton
dilaton
In string theory, gravity and gauge interactions are
“emergent” phenomena arising from a single
dynamical framework of quantum mechanics of
relativistic strings :
quantum fluctuations of strings
quantum fluctuations of space-time and gauge fields
and
the condensation of various modes of strings
induces space-time curvatures, coupling constant,
and all allowed `external` fields:
the structure to be expected for a backgroundindependent theory.
Furthermore, closed strings can also be regarded as being induced
from quantum fluctuations (or loop effects) of open strings.
related to gauge/gravity correspondence, later
“String theory is, at a new level, the realization
of old ideas concerning induced gravitation!
I cannot refrain from feeling proud on this point!”
General Relativity and Gravitation, Vol. 32, No. 2, 2000
A. D. Sakharov (1921-89), 1985
Vacuum Quantum Fluctuations in Curved Space
and the Theory of Gravitation†
Academician A. D. Sakharov
Translated from Doklady Akademii Nauk SSSR, vol. 177, No. 1,
pp. 70–71, November 1967. Original article submitted August 28, 1967.
In Einstein’s theory of gravitation one postulates that the action of spacetime depends on the curvature (R is the invariant of the Ricci tensor):
1
S(R) = −
16πG
!
√
(dx) −g R.
(1)
Nobel peace prize 1975
If we neglect dilation of violin strings,
the degrees of freedom of vibration at each point are ２.
2=3+1-(1+1)
space
dimensions
time
The fundamental string live in (critical) 10 (=1+9)
dimensional space-time, and the vibrational
degrees of freedom are only from transverse directions
8=9+1-(1+1) This is why graviton and gauge fields can emerge.
Potentially, the critical dimension suggests that string theory might be able to
explain why our space-time is 4 dimensional, provided that we find appropriate
dynamical mechanisms of reducing 10 dimensions to 4.
n=0
n=0
sion, of the string sented
per unit b
2.1 Discovery of relativistic strings
sion,
of
the
s
2.1 (first
Discovery
of
relativistic
strings
integrals,
we
have
to
spec
String
theory
evolved
from
a
proposal
made
in
the
late
equality,
‘t-channel’
description) or through forma- (first
index
0
equalit
Brief
history
of
string
theory
w
to is
thethe
initi
evolved
from
a proposal
made
the
late integrals,
60s forString
ation
particular
2↔2
scattering
amplitude,
called
the in corresponding
oftheory
resonance-like
states
(second
equality,
‘s-channel’
tion of reson
2
suppressed
incorrespondi
the present s
‘Veneziano
of mesons
which satisfies
a special called
60s description).
for formula’,
a particular
2↔2
scattering
amplitude,
the
description)
Here,
s
and
t
are
Lorentz
invariant
combinational
fig
1968
Veneziano
model
S
is
an
integral
over
a
2
symmetry
requirement
calledofs-tmesons
‘channel’
duality.
The
Σ
i
‘Veneziano
formula’,
which
satisfies
a
special
2
2suppressed
tions
of
ener
tions ofthat
energy-momenta
s=
−(p1 +p-of2Channel
)ele,t =
−(p
+p3form
) , essary fi
2the
(s-t)
duality
takes
latter demands
the amplitude is
composed
#
is
an
in
symmetry
duality.
The
# requirement called s-t ‘channel’
3S!
Σ
and
α
and
Regge
behavior
and
α
and
α
are
two
parameters.
It
soon
turned
out
grees
of
0
ments such as the formula
2
- narrow-resonance
d ξthis
L(X,
takes
the
fo
=
that
am
latter
demands
that
the
amplitude
is
composed
of
ele!
1
that this amplitude
and its various
generalizations can Σ The con
approximation
!
!
−α s−α
−α t−α0 −1
0 −1
be interpret
ments
as
the
formula
dx
x
(1
−
x)
(1)
V (s,be
t) such
=
interpreted
in
terms
of
the
dynamics
of
relativistic
fies
the
with
! 1
0
open strings
!
! provided α
open
strings
propagating
space-time,
ent
top
0 = 1. L
−α
s−α
−1
−α
t−α0 −1
0
=
g
(X)∂
The
analogo
which can
equally
be
described
by
exchanges
of
particles
µν
dx x
(1)
V (s, t) =
4 (1 − x)
with
analogous
amplitudes which correspond
to
closed
Rieman
betweenThe
two interacting
strings
0 particles
where (ξ1 , ξ2 ) with z were
= ξ1
poles at
strings
were
dlesexam
and
∞also constructed.
∞
For
"
"
dimensional
coordinates
pa
which
can equally
be
described
by
exchanges
of
particles
rn (s)
rn (t)
#
2
V
(s, t)
=
=
(2)
(n
−
1)/α
a
For
example,
the
pole
singularities
at
s
or
t
=
m
=
χ(Σ)
≡
face
Σ.
The
space-time
co
n
2
2
between two #interacting
particles
t
−
m
s − mn
where
n
µ (ξ1 , ξ
n=0
n=0
of
strings
sented states
by fields X
(ξ) (µw
(n − 1)/α are interpreted
as
representing
possible
Rieman
∞
∞
"
"
dimensional
2
number
of d
r
(s)
r
(t)
(first equality,
‘t-channel’
description)
or
through
formaindex
0
being
the
time
n
n
of strings
with
definite
(mass)
.
There
are
an
infinite
mation
V (s, t)
= (second equality,
= ‘s-channel’
(2)
face
Σ.
The
and
rotation
tion of resonance-like
states
is
the
metric
tensor
of t
2
2
t
−
m
s
−
m
spectrum
ofn=0
relativistic
open
strings
number
of them
corresponding
to
various
vibrational
inserted
n
n
n=0
that forψ
description). Here, s and t are Lorentz invariant combina- tional field variable
infi
sented
bycom
and
rotational
modes
of
strings.
Actually,
spective
Nambu,
Susskind,
.....
2Nielsen,
2 it also turned
string are
tions(first
of energy-momenta
s = −(p1description)
+p2 ) , t = −(por
essary
fields, index
which
us
2 +p
3) ,
equality, ‘t-channel’
through
forma0theor
bei
3
thatα0for
completely
consistent
formulations
of quantum
ing
infin
andtion
α# and
are
two
parameters.
It
soon
turned
out
grees
of
freedom,
such
as s
sions
must
of resonance-like
states
(second
equality,
‘s-channel’
is
the
metr
5
string
theory,
necessary
that the space-time
dimen- sions),
strip
that this amplitude and it
itsisvarious
generalizations
can The constant
gs , of
called
st
26,
Similar
formula
(Virasoro,
Shapiro),
description).
Here, s and
t are Lorentz
invariant combina- tional fieldor
be interpreted
in terms
of the
dynamics
of relativistic
fies the
of final)
Riemann
sions must
be at
some
particular
value
(critical
dimenex
fermi
2
2weightsheet)
of energy-momenta
s = −(p
essary
fields
corresponding
to
closed
strings
1 +p2 )α0, t==1.−(pent
2 +ptopologies.
3) ,
opentions
strings
propagating
space-time,
provided
It
is
well o
can
elimina
sions),
26,
or
if
we
want
to
include
space-time
(and
worldtively,
Y. Nambu
(1921~ )
#
3
4
and
α
and
α
are
two
parameters.
It
soon
turned
out
grees
of2 clas
free
0
The analogous
amplitudes
Riemann
surfaces
are
6 which correspond to closed
(mass)
with
sheet) fermions consistently, at 10. In the latter case we
This
In the 1970s, the string theory made its development
from the S-matrix formula to a dynamical theory of
relativistic strings. Various ideas which would become
the basis for later developments was emerging from
the studies of its connection to local field theory.
1
Gravity from strings: personal reminiscences
of early developments
Tamiaki Yoneya
Institute of Physics, University of Tokyo
Komaba, Meguro-ku, Tokyo 153-8902, Japan
Abstract
I discuss the early developments of string theory with respect to its connection with gauge theory and general relativity from my own perspective. The
period covered is mainly from 1969 to 1974, during which I became involved
in research on dual string models as a graduate student. My thinking towards the recognition of string theory as an extended quantum theory of
gravity is described. Some retrospective remarks on my later works related
to this subject are also given.
(arXiv:0911.1624,)
to appear in
The Birth of String Theory,
Cambridge Univ. Press.
1970 ~ 1978 Initial developments
(models for hadronic interactions)
Nambu-Goto action
Light-cone quantization, no-ghost theorem, critical dimensions (26 or 10)
Ultraviolet finiteness (modular invariance)
Neveu-Schwarz-Ramond model (inclusion of “G”-partiry and fermionic degrees of freedom)
Space-time supersymmetry
Developments related to field-theory /string connection (‘70s)
`Fishnet’ diagram interpretation, Nielsen-Olesen vortex
Derivation of gauge theory, general relativity and supergravity from strings in the zeroslope limit
gravity and unification
Construction of various supersymmetric gauge and gravity theories
String picture from strong-coupling lattice gauge theory
t` Hooft’s large N limit
1984~1989 First revolution
Green-Schwarz anomaly cancelation
Five consistent perturbative string vacua (I, IIA, IIB, 2xHetro) in 10D
Compactifications, new connections to mathematics
CFT technique, renormalization group interpretation
1990~1994 Development of “old” matrix models
Double scaling limit
c=1 strings, 2D gravity, ‘non-critical’ strings
topological field theories and strings
1995~1999 Second revolution
discovery of D-branes
statistical interpretation of black-hole entropy in the BPS or near-BPS limits
conjecture of M-theory
New matrix models, supermembranes, M(atrix) theory conjecture, .....
AdS/CFT correspondence, .......
The most important development after 1998 :
General idea of gauge-gravity correspondence
has been flourishing.
starting from Maldacena’s paper in 1997
unification of two old ideas on strings from the 70s ?
hadronic strings for quark confinement from
gauge theory
string theory for ultimate unification as an
extension of general relativity
Gauge/Gravity correspondence
The fact that gravity (general relativity) and gauge
theory are united in a single framework leads to an
entirely new connection between them,
giving nontrivial predictions to the gaugetheory side from gravity side (and vice versa)
a realization of
`holography’:
quantum gravity in the bulk
must be formulated using only
the degrees of freedom
on the boundary.
(‘t Hooft, Susskind, .....)
open-closed string duality in string perturbation
theory (simplest one-loop case)
D-branes
creation and annihilation
of open strings
exchange of closed strings
(including graviton exchange)
analytic continuation
effective theory
=gauge theory
effective theory
=gravity
description on
the boundary
description
in the bulk
ators with the exponents η = 1 + 4!/5
Gravity
Correspondence
!
Example:
1999) on"the correlation functions for a strong-coupling
1 (Sekino-T.Y.
+ij Predictions
Str
F
X
·
·
·
X
≡
ij
i
il
gaugeJltheory
dimensional
super
Yang-Mills
theory, the so-called `Matrix
1
,i1 ,··· ,i(0+1
1. theory’) has
l
N
Overview:
recently been checked through Monte Carlo simulations
Matrix
theory
2.
D-partilce
interactions
0.01
(!O(t)O(t " )# ∼ (|t − t " |−η ))
Three typical examples:
0.0001
!
3. Strings
as wrapped
membranes
J + -type operators with the exponents η = 1 + 4!/5
1e-006
+ij
Jl ,i1 ,··· ,il
+
J1 , N=3, !=12
-4 1.8
3.15*10
/t
+
J2 , N=3, !=12
-5 2.6
1.68*10
/t
+
J3 , N=3, !=16
-6 3.4
1.24*10
/t
+
J4 , N=3, !=16
5.16*10-8
+/t4.2
1e-008
!4. Strong"
1
Fij Xi1 · · · Xil
≡ Str coupling
behavior of
N
Matrix
theory
" )# ∼
" |−η ))
5.
"
"
−
η
Three
typical
examples:
(!O(t)O(t
(|t
−
t
Three typical examples:
(!O(t)O(t )# ∼ (|t − t |Concluding
))
remarks:
0.01
! J -type operators with the exponentscan
ηwe= 1 + 4!/5
1e-012
! J + -type operators
with the exponents η = 1 + 4!/5
1e-010
0.1
1
t
make
progress
towards
M-theory?
0.0001
!
1
+ij
!≡ Str F X"
1
+ij
J
1e-006
ij i1
=J1, 2, 3, 4)≡are
plotted
(in
l ,i1 ,···
,il Fij X
Str
·
·
·
X
il
N i1
l ,i1 ,··· ,il
· · · Xil
point functions !Ji+ (t)Ji+ (0)# (i
N
for N = 3, β = 4, Λ = 12
for i = 1, 2 and1e-008
Λ=
16Rev.
forLett. J104
Hanada-Nishimura-Sekino-T.Y..
Phys.
(2010)151601
, N=3,
!=12
3.15*10
/t
−1/3
see
also
arXiv:1108.5153[hep-th]
,
N=3,
!=12
J
.
nit of time length is λ
1.68*10 /t
1
2
+
-4 1.8
+
-5 2.6
+
J , N=3, !=16
"
The meaning of string theory
・encompasses almost all relevant ideas and/or methods
devised in the past towards unification
・provides an entirely new scheme of unifying
all interactions including gravity
(motion or spectra and interaction are completely unified)
・provides a microscopic explanation of black hole
entropy in terms of quantum statistical language
using D-branes
(albeit in some special cases)
・resolves non-renormalizable divergences in terms of
intrinsic non-locality in a way which is consistent
with unitarity

 X2×
cance

tates. It seems reasonable to expect that this effect completely

1
 X3×
The
characterization
of
short
distance
structure
of

 X1×1

ance effect with respect to the center-of-mass coordinates
of
string
 ·

p
=
4
g
=
0
g
=
1
g
=
2
2g
c


X
space-time
embodied
in
string
theory
relates
to
one
of
2×2
hese
appreciably to physical processes.
  Si

4 ghigher
= 0 gmodes
= 1 contribute
g=2
2g
 X3×3   ·
 ·

mostof
fundamental
questionscorresponding
of quantum gravity.
magnitude
the spatial extension
to a large
energy
 · 
  XN ×

2
  sim

·
∆E,
we
are
led
to
a
remarkably
s expected to
behave
as
∆X
∼
"
 ·

Ω = Ω1 + Ω2 , Ω1 ∩ Ωs2 = ∅
 · 
a
possible
expression
:

 directions
·
Ω
Ω1 +of
Ω2magnitude
, Ω1 ∩ Ω2∆X
= ∅ for fluctuations along spatial
he=order
 XN ×N 


uncertainty
relation
of
space
and
time
 · 
participating within the time interval ∆T = ∆t of interactions:
·
∆E∆t ! h
φ+
2
>
∆X∆T
"
∆E∆t ! h
∼ s.
φ+
φ−
φ+ F
h
∼ ∆X
t is natural ∆E
to call
this2 relation
∆t =the
∆T ‘space-time uncertainty relatio
!
s
h this relation
+
mphasized that
is not a modification of the φusual
F φ− unce
∆E ∼ ∆X 2
∆t = ∆T
∆T ∼ scale along the lo
∗)
s be
・This!can
interpreted
the we use units in which ¯
Throughout
the
present as
paper,
h = 1, c = 1.
space-time representation of the
world-sheet conformal symmetry
which governs the quantum dynamics
of strings and is responsible for the
reduction of degrees of freedom both
in UV and IR regions.
∆T ∼ scale along the longitudinal
∆X ∼ scale along the
∆X ∼ scale along the transverse d
To summarize:
string theory enables us to derive general relativity and
gauge interaction naturally from quantum mechanics of
fundamental strings. In this sense, string theory provides
the ingredients which are necessary for achieving the final
unification. (This is already a grand prediction!)
At the present stage of development, however,
it is still an immature ‘theory’.
・we only have a perturbative definition for constructing
scattering amplitudes.
・we do not know the primordial principles behind it.
・we cannot yet make any definite predictions to the real
world.
The present immaturity of string theory does not mean that
the theory cannot make any predictions in principle.
There is no other competing theory, exhibiting such a tight
structure of self-consistency and hence in principle
of highest predictability at least potentially.
The present status of string theory is somewhat
similar to that of early quantum theory before 1924.
We do not yet have true languages for describing
string theory.
It is unclear to what extent various recent ideas
(say, the “string landscape”, “world-brane scenarios”, ...)
for model building survive
when we would finally find such true languages.
We need non-perturbative definition of string theory
such that its whole apparatus could be put on computers
(remember how lattice gauge theory has been benefitting QCD).
We have to be patient and to learn more,
in view of such a grandiose synthesis we are looking for !
On the experimental side, hopefully, various increasingly
precise data from cosmological observations
(background fluctuations, dark matter, dark energy, etc)
would provide us key basis for the future development.
Epilogue: Historical remarks
・ Dirac’s anticipation of strings in 1955
“The lines would then be the elementary
concept in terms of which the whole theory
of electrons and the electromagnetic field
would have to be built up. Closed lines
would be interpreted as photons and open
lines would have their ends interpreted as
electrons or positrons. ...”
P. A. M. Dirac
(1902-1984)
GAUGE-INVARIANT FORMULATION OF
QUANTUM ELECTRODYNAMICS1
Canadian Journal of Physics, 33(1955) 650
ABSTRACT
Electrodynamics is formulated so as t o be manifestly invariant under general
gauge transformations, through being built up entirely in terms of gauge-invariant
dynamical variables. The quantization of the theory can be carried out by the
usual rules and meets with the usual difficulties.
I t is found that the gauge-invariant operation of creation of an electron
involves the simultaneous creation of an electron and of the Coulomb field
around it. The requirement of manifest gauge invariance preventsonefrom using
the concept of a n electron separated from its Coulomb field.
H. Yukawa (1907-1981) was also one of main
advocates of non-local field theory from the late 1940s.
「素粒子を点と思っていたのでは、 てん で話にならない」
It is pointless to conceive elementary
particles as mere points.
Einstein: General
Relativity
Quantum
Mechanics
Kaluza
Klein
Supergravity
Weyl
Gauge
Principle
Super
Symmetry
Standard
Model
Quantum Field
Theory
Black hole
Unitarity puzzle
(information problem)
Yang-Mills
Theory
UV problem
Nonrenormalizability
Superstring
M-theory
Gauge/string
Correspondence
‘holography’
Web of Unification
Perhaps, we are at the dawn
of a new era of physics.
The Night sky in
Zermatt
©T.Y. 1983
What would be the appearance of
string theory after the next 30 years?
Thanks !
Matterhorn
©T.Y., 1983