Download Unsolved Questions in String Theory

Unsolved Questions in String Theory
1 Historical overview : 45 years of string theory
2 Unsolved questions
3 Remarks on string theory in the Rindler coordinates
and space-time uncertainty
4 Conclusion
Tamiaki Yoneya
KEK workshop
2014 Feb. 18
To the memory of Bunji Sakita (1930-2002)
and Keiji Kikkawa (1935-2013)
1 Historical (and personal) overview : 45 years of string theory
1969 ~ 1979 Initial developments
Nambu-Goto action
Light-cone quantization, no-ghost theorem, critical dimensions, ...
Ultraviolet finiteness (modular invariance)
Neveu-Schwarz-Ramond model
Space-time supersymmetry
Developments related to field-theory/string connection
Fishnet diagram interpretation, Nielsen-Olesen vortex
Derivation of gauge theory, general relativity and supergravity from strings in the
zero-slope limit
unification including gravity
Construction of various supersymmetric gauge and gravity theories
String picture from strong-coupling lattice gauge theory
t` Hooft’s large N expansion
String field theories (light-cone)
1984~1989 First revolution
Green-Schwarz anomaly cancellation
Five consistent perturbative superstring vacua in 10D
Compactifications, various connections to mathematics
CFT technique, renormalization group interpretation
1990~1994 “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 (BFSS, IKKT), supermembranes, M(atrix) theory conjecture, .....
AdS/CFT correspondence, GKPW relation, ....
Current main stream:
general idea of gauge-gravity correspondence
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
Spiral stair case: review talks in JPS meetings
1 Dual model and field theory 1975.3
2 Condensation of monopoles and
quark confinement 1979.3
3 Hamiltonian quantum gravity 1985.4
4 Strings and gravity 1986.10
5 Lower-dimensional quantum gravity
and string theory 1993.9
6 Toward non-perturbative string theory 1997.3
7 General relativity and elementary-particle theory 2005.9
8 What is string theory? : reminiscences and outlook 2010.3
Guggenheim museum (New York)
The Physical Society of Japan (JPS)
abstract of the JPS review talk in 1975
Dual model and field theory
Main message of that talk was:
Is it possible to resolve the following dichotomic duality relations ?
(1) local field theories
strings
non-perturbative and higher order effects
fishnet diagrams, 1/N-expansions
lattice gauge theories, etc
(2) strings
local field theories
zero-slope limits
local gauge symmetry from world-sheet conformal invariance
closed strings
general relativity
open strings
gauge theory
“Interaction between field theory and string theory”
How could general relativity be contained in gauge theory?
Cambridge Univ. Press, 2012
a preliminary version
arXiv:0911.1624
What we have achieved:
Gravity and and gauge forces emerge from quantum mechanics of
relativistic strings, in loop expansions, in such a way that
no ultraviolet divergence in loop corrections
unitarity is preserved
external degrees of freedom (or parameters) are not allowed
deformations of backgrounds from flat space-time can be,
at least infinitesimally, described using only the
degrees of quantum strings (condensation of string fields)
derivations of black hole entropy in some special cases
new understanding about dual connections between gravity
and gauge theory
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
All these suggest that string theory is quite promising as a
unified theory of all natural interactions, the structure of
matter and space-time.
All these suggest that string theory is quite promising as a
unified theory of all natural interactions, the structure of
matter and space-time.
an artistic image of string unification ?
Gunas
Isamu Noguchi (1904-1988)
Guṇa (Sanskrit: ग"ण) means 'string' or 'a single thread or
strand of a cord or twine'. (from Wikipedia)
All these suggest that string theory is quite promising as a
unified theory of all natural interactions, the structure of
matter and space-time.
an artistic image of string unification ?
Gunas
Isamu Noguchi (1904-1988)
Guṇa (Sanskrit: ग"ण) means 'string' or 'a single thread or
strand of a cord or twine'. (from Wikipedia)
But, leaving aside the ultimate question of
phenomenological validity,
there are many fundamental questions,
which seem almost insurmountable at the present time,
in both technical and conceptual aspects.
2 Unsolved questions
principles or any hidden (geometrical) symmetries behind
Why strings and branes ?
Why matrices or gauge theories?
non-perturbative formulation
Could gauge/gravity correspondence be the non-perturbative
definition of string theory?
exact formulation of strings and branes in curved space-time
No string theory in nontrivial curved space-times
which are understood at the same level as in flat space-time
background independent formulation
What are the primordial degrees of freedom of string theory?
Could holographic principle be formulated in the
background-independent fashion?
Where are the degrees of freedom for deforming
background space-time in “boundary” theory?
What is the non-perturbative formulation of
open/closed string duality?
stringy description of space-time geometry
To what extent, space-time horizons and space-time
singularities meaningful?
How to formulate information puzzle in stringy language?
Is it possible to formulate “compactification” dynamically?
observables other than the S-matrix elements
What are the invariants or symmetries characterizing stringy
observables?
Is there any theory of measurement for quantum string theory?
Some of these questions motivated various new approaches
appeared, especially in the period of second revolution
(1995-1999)
M-theory conjecture, matrix models (BFSS, IKKT, ...), and
general conjecture of gauge-gravity correspondence
Progress has been made. But it seems fair to say that we are still
far and away in answering any of these fundamental questions.
Some of these questions motivated various new approaches
appeared, especially in the period of second revolution
(1995-1999)
M-theory conjecture, matrix models (BFSS, IKKT, ...), and
general conjecture of gauge-gravity correspondence
Progress has been made. But it seems fair to say that we are still
far and away in answering any of these fundamental questions.
In the following, I would like to recall
some of my past attempts motivated by these questions.
Sole purpose is to stimulate you to think further.
33
φ
φ suggestion
φ
An earliest
3
toward background independence:
conjecture of purely
cubic action
1 131 33
1 1
1 311 33
= !ψ
→
SS=!ψ
→
SSWitten
! cψ̃ψ
ψ̃ψ
ψ̃ +
+
Witten
ccψ̃
S=
"!ψ
→""S
= ! =ψ̃ψ
ψ̃ +
ψ̃ "ψ̃ψ̃ ""
Witten
6 66
2 2
6 66
2 2 = 0,
2 ψ
= 0,ψ
ψc ψ=c c0,
ψ
+
ψ̃
c
ψ
+
ψ̃
=ψψψ==
+
ψ̃
c
c
1
1 3
" → S = ! ψ̃ψc ψ̃ + ψ̃ "
2
6
0 Q
Q− Q−
−Q
Q
−
Q
D
φ =φφV==VV
φ =φφ0==0Q
QD Q
−
Q
QD −
D QDD
talk “Approaches to string field theory”
ψ=
ψ
+
ψ̃
c
in ICOBAN’86 International conference on
grand unification (site near Kamiokande)
Q
S#
0
Q
S#
See
alsoQ
related works
0 S#0 R
|E|
=
V
=
|E|d
Q
=
CV
C
=
|E||E|
= =T.Y. PRL
V0 =V |E|d
Q =QCV
C =C = d R R
55(1985)1828,
S#
=
|E|d
=
CV
d
PLB
197(1987)76
0S#
= 0 Q S#
−
Q
Q
−
Q
d
D
D
0
After the conference, interesting attempts toward possible realization of this
conjecture have been made in Hata-Ito-Kugo-Kunitomo-Ogawa, PL 175B, 138(1986)
Horowitz-Lykken-Rohm-Strominger, PRL 57, 283(1986)
In view of the present status of superstring field theory, this idea
was a bit too naive. But in any case,
String fields, or (if any) other possible degrees of freedom
for describing background independence, themselves
should be regarded as a fundamental geometric entity of
“string geometry”.
Classical geometry must be an emergent phenomena, or
any property of space-time must be defined by physical
processes of strings themselves.
Another unfinished project: non-perturbative understanding
of open-closed string duality
motivation:
gravitational degrees of freedom in gauge-theory
description of D branes
(and hence in gauge-gravity correspondence)
are hidden in the whole quantum configuration space of
gauge theories
For instance, the correct 3-point interactions
of gravitons are obtained only as a loop effect in
D0 susy gauge theory.
(Y. Okawa and T. Y., 1998)
an analogy : Mandelstam duality in 2D field theories
massive Thirring model
D-particle field theory
sine-Gordon model
closed string field theory
Project of D-brane field theories
open-closed duality
open-string field theories
effective Yang-Mills theories
closed-string field theories
bosonization
or
Mandelstam duality
first quantization
or
second quantization
D-brane field theories
T.Y., side:
arXiv:0705.1960[hep-th]
135 (2007)
An analogy on the right-hand
soliton operator ↔(PTP118,
Dirac field
T.Y., arXiv:0804:0297[hep-th]
(IJMPA23,
2343(2008)
in the duality between sine-Gordon
model and massive Thirring
model
exp(π(x) ± iφ(x)) ↔ ψ(x),
$µν ∂ν φ ↔ ψγµ ψ(x) etc
1

 (3) 
 (4) 
x(2) 
 · 
·

 · 


 z2 
z
2



2 




 (4) 

i · 
(3) 
 · 
 · 

 z3 

x




3 


Sbh
·

 (4) 
∼ $γ
Ψ
,
δ
Ψ
∼
$
Ẋ
·
·
·
(3)
(4)
(2)
(1) +
ab
susy
ab
ab



,
z
=
,
z
=
,
z
=
z
=


Fock
space
of
D
particle
gauge
theories
with
different




·
∼
e
 · 
· 
xN

 · 

4 








 XN ×N

 · 
 · 
 · 
 · 






 XN ×N 
 · 
 · 



·
· 
√
 ∆X


∆Th→ 0 ⇒
→
∞
h
·
·
·
·
 · 
·
S
=
2π
n
+
−
bh
1


Gauge
symmetry
· together with their complex conjugates. The dots indicate inifinitely many dummy com1
·
λ(
= quantum
ofspatial
D-particle
space
 X1×1  ponents.
Of course, bothstatistics
X and (z, z̄) are
vectors, withHilbert
the corresponding
indices
gµν (x) → e


 X2×2  being suppressed.
+ 
−Thus the D-particle fields creating and annihilating a D particle must be defined
+ 
−

φ
φ  X
φφ
g
⇒
g
+
 3×3  conceptually
µν
µν
as
Gauge invariants
 · 
+
−


(1) (1)
 · 
z ]|0! → φ+ [z (2)fields
, z (2) ]φ+ [z (1) , z (1) ]|0! → · · · ,
φ+ : |0! → φ+ [z
,D-particle
=
bi-linears
of

+ −
+
−
φ F
φ · φ 
F
φ


φ− : 0 ← |0! ← φ+ [z (1) , z (1) ]|0! ← φ+ [z (2) , z (2) ]φ+ [z (1) , z (1) ]|0! ← · · · .
 XN ×N 
 The manner in which the degrees of freedom are added (or subtracted) is illustrated in

 · 
“c
G
h”
Creation and annihilation of D particles and associated open
strings
Fig. 1. As we explain below, the multiplication rules of the D-particle field operators are
·
φ
φ
φ Fφ
26
!
Gh
=created 3
connecting them, respectively. The real lines are open-string degrees of freedom which $
have
P been
before the latest operation of the creation field operator, while the dotted lines indicate those created by c
A Difficulty：
necessity of non-associative structure
the last operation. The arrows indicate the operation of creation (from left to right) and annihilation
Fig. 1: The D-particle coordinates and the open strings mediating them are denoted by blobs and lines
(from right to left) of D-particles.
actually not associative, nor commutative, and hence we need some special notation for
+
−
i
!φ
,
∂
φ
" = 0,
ulas.
z
equation
nhrödinger
order to simplify
formulas. equation
S = A/4G
3.3theSchrödinger
N
A = 4πR
2
(3.31)
R = 2G
N
in order to simplify the formulas.
It
is
possible
toSchrödinger
rewrite
the entire
of Yang-Mills
mechanics
We
can
now rewrite
the content
Schrödinger
equation
terms of operators.
these
bilinearInope
rewrite
the
equation
in terms
of thesein quantum
bilinear
3.3now
Schrödinger
equation
Sbh
for D particles
with all different
as
ane extended quantum field theory.
N
∼
configuration
space,
we
have
ration
space,
we have
Schrödinger
equation
hrödinger
equation
inthe
terms
of these equation
bilinear operators.
We
can3.3
now
rewrite
Schrödinger
in terms ofInthese bilinear operators. In
# g #√ ∂ ∂ $ 1
$
∂
#
s s
i
j 2
onfiguration
have thegSchrödinger
∂
1
∂space,
S
=
2π
n
n
n
We can
now we
rewrite
equation
in
terms
of
these
bilinear
In
Ψ[X]
=
−Tr
+
[X
,
X
] operators.
Ψ[X]
s #is ∂
bh
5
i1
j i2 p
i
5
= −Tr
+
X ] Ψ[X]
∂t$ i
25 [X
∂X , ∂X
4gs #s
# g # i∂∂t Ψ[X]
i
#2gj 2#∂X∂ ∂X∂
$
4g
#
∂ ∂ space,
1 we ihave
s
configuration
s s
s
1
s s
i
j 2
−Tr
+
[X
, X ] "Ψ[X]
λ(x)
%
Ψ[X]
=
−Tr
+
[X
,
X
]
Ψ[X]
i
i
5
1
∂
∂
1
i ∂X(x)
i
5 e
2
∂X
g
→
igµν
j (x)
i j%$
i⇒
i j jψ(x) →
∂t 4gs #s = − 2Tr ∂X
4g
#
µν
#
"
s
s
g
#
+
(X
X
X
X
−
X
X
X X ) Ψ[X].
s gss #s i∂
∂
1
1
∂ ∂ ∂
1
i
5
i∂Xj ∂X
i j g
ji , Xjj ]2 Ψ[X]
2 −Tr(X
#s i X i[X
sX
Ψ[X]
=
+
i
%
=
−
Tr
g
#
+
X
X
X
−
X
X
)
Ψ[X].
(3.32)
s s"
%
i
i
5
∂
12
i
i
5
2 i ∂X
∂X
4g
# j j
1iHere
∂ X i∂suppressed
j ∂X
i ∂t
j ∂X
i gsj # 1j
j i j (3.32)
i s i swhich
s
+
(X
X
X
X
−
X
X
X
)
Ψ[X].
we
have
the
fermionic
part,
isΨ[X].
treated in (3.32)
the next secti
=
−
Tr
g
#
+
(X
X
X
X
−
X
X
X
X
)
g
⇒
g
+
a
s
s
i
i
5
µν
µν
X ∂X
gs # s 2
∂X" i ∂X i gs #5s µν
%
e have suppressed
the
fermionic
part,
which
is
treated
in
the
next
section.
In the
1
∂
∂
1
second-quantized
form,
this
is
expressed
i j i asj
i i j j
= − isthe
Tr
gfermionic
+ which
(XisXtreated
X the
X in
−X
Xnext
X Xsection.
) Ψ[X].
(3.32)
e fermionic
part,
which
treated
next
section.
In
s #s in ithe
Here
we have
suppressed
part,
the
In
the
i
5
∂X ∂X
quantized form, this 2is expressed
as gs #s
is
expressed as form, this is expressed as
econd-quantized
H|Ψ"
= 0, in the next section. In the
Here we have suppressed the fermionic part, which
is treated
∂X i
second-quantized
expressed
as
=
0,
(3.33)
H|Ψ" = 0, form, this isH|Ψ"
(3.33)
+
−
H|Ψ"
=
0,
(3.33)
“c G h”
H = i(4!φ , φ " + 1)∂t +
$
#
=, ∂0,i · ∂ i φ− " + 3!φ+ , ∂ i φ− " · !φ+ , ∂ i φ(3.33)
−
++ +− −−H|Ψ" +
H = i(4!φ+ , φ− " + 1)∂t +H
"
2g=
#
(!φ
,
φ
"
+
1)!φ
i(4!φ
+z̄ z
i(4!φ, φ
, φ ""+
+ 1)∂
1)∂tt+
sHs =
z̄
z
$ !
$ "$
#
#
1
+
−
+
− +
− +
+
−+, ∂ i φ
−
i" +j 1)∂
2−−+ i + +
j
j− −
i
−
−
+
−
+
+
− ,3!φ
+− "++·, ,∂
iφ
H
=
i(4!φ
,
φ
φ
"
"
+
!φ
,
∂
·(!φ
∂sz#isφ
+ 1)!φ2g
, ∂#+z̄i2g
(4!φ
φ
"
+
1)(!φ
φ
"
+
1)!φ
,
(z̄
·
z
)
−
(z̄
·
z
)(z̄
·
z
)
φ
".
(3.34)
i
i
i
i
t
φ
"
"
+
3!φ
,
∂
φ
"
·
!φ
,
∂
·
∂
(!φ
1)!φ
z̄
z
s s 2gs !5 , φ " + 1)!φ , ∂z̄ i z̄· ∂z izφ " + 3!φ , ∂z̄z̄ i φ29" · !φ ,z ∂z i φ "
!
s
$
#
−
+
−
−
+
+
−
+
In the large N2glimit
and
in
the
center-of-mass
frame,
this
is
simplified
to
i
i
i
i
φ
"
φ
"
+
3!φ
,
∂
φ
"
·
!φ
,
∂
·
∂
#
(!φ
,
φ
"
+
1)!φ
,
∂
s s
z̄
z
z
z̄
Gh
−33
29
29
(with
some
minor
constraints)
# g!
!
"
$
+
−
$
=
∼
10
cm
29
P
!φ
,
φ
"
s s
i∂ |Ψ" = −
!φ+ , ∂ i · ∂ i φ− " −
!φ+ ,c3(z̄ i · z j )2 − (z̄ i · z j )(z̄ j · z i ) φ− " |Ψ".
3 Remarks on string theory in the Rindler coordinates
and space-time uncertainties
Another crucial unsolved problem is “information paradox” of
black hole. But majority of recent works discussing this question
ignore stringy nature of gravity, basing on low-energy effective
(field) theory
(except perhaps for holographic arguments and “fuzz-ball” conjecture)
However, by definition, the black hole and space-time horizon
involves arbitrarily high energy (short distance) physics,
corresponding to infinite red shift associated with the horizon.
3 Remarks on string theory in the Rindler coordinates
and space-time uncertainties
Another crucial unsolved problem is “information paradox” of
black hole. But majority of recent works discussing this question
ignore stringy nature of gravity, basing on low-energy effective
(field) theory
(except perhaps for holographic arguments and “fuzz-ball” conjecture)
However, by definition, the black hole and space-time horizon
involves arbitrarily high energy (short distance) physics,
corresponding to infinite red shift associated with the horizon.
It seems of vital importance,
from the viewpoint of physics in the bulk space-time,
to take into account the non-local nature of string theory.
µν
µν
µν
What is the appropriate“c
of non-locality of strings?
ccharacterization
G
Gh
h”
!
This is also an unsolved
question.
Gh
!
−33 if we set two of classical
In any theory
gravity,
$P of
= quantum
∼
10
cm
3
c
Gh
−33
fundamental
constants
to
unity,
$ =
∼ 10 cm
P
c3
⇒
G=c=1
$P =
√
h
√
Once we take into
gravity,
$s account
=
α!
G
=
c
=
1
⇒
$
=
h
P
√
quantization=introduction
of fundamental length
$P = h = f (Φ)$s
In string theory, the role of fundamental length is played
f (Φ)
= exp$(2φ/(D
− 2))
by the string
length
s
φ
gs = e
!
α =
24
2
$s
pc = 4 g = 0 g = 1 g = 2
2g
µν
µν
µν
What is the appropriate“c
of non-locality of strings?
ccharacterization
G
Gh
h”
!
This is also an unsolved
question.
Gh
!
−33 if we set two of classical
In any theory
gravity,
$P of
= quantum
∼
10
cm
3
c
Gh
−33
fundamental
constants
to
unity,
$ =
∼ 10 cm
P
c3
G=c=1
⇒
$P =
√
h
√
Once we take into
gravity,
$s account
=
α!
G
=
c
=
1
⇒
$
=
h
P
√
quantization=introduction
of fundamental length
$P = h = f (Φ)$s
In string theory, the role of fundamental length is played
f (Φ)
= exp$(2φ/(D
− 2))
by the string
length
s
φ
The space-time
uncertainty
relation was originally
gs = e
proposed with this in mind. In the absence of any
!
2
definite mathematical
formalism and axioms, only
α = $s
24
way was to adopt a qualitative approach.
pc = 4 g = 0 g = 1 g = 2
2g
(written in 1986, and published in “Wandering in the fields” , Festschrift for
Prof. K. Nishijima on the occasion of his sixtieth birthday, World Scientific,
1987)
!P = h = f (Φ)!s
∆E∆t argument?
!h
he difference, in string theory, regarding this general
Actuwe consider
the high-energy
regime,
ber of the allowed states with a large Ifenergy
uncertainty
∆E
behaves
√
k!s /∆t with some positive coefficient k, and " ∝
# being the string
α
s
h
Φ) = exp (2φ/(D
−
2))
∆E ∼ ∆X This
∆t
= ∆T of the
2
nt, where α# is the traditional slope parameter.
increase
!s
much faster than that in local field theories. The crucial difference
This
was derived
by
quantizing
strings
φ
d theories,
however,
is
that
the
dominant
string
states
among
these
exgs = e
the time-like gauge.
egenerate states are not the states withinlarge
center-of-mass momenta,
e massive states with higher excitation modes along strings. The excitamodes
α! along
= !2s strings contributes to the large spatial extension of string
ms reasonable to expect that this effect completely cancels the short diswith respect to the center-of-mass coordinates of strings, provided that
gmodes
= 0 contribute
g = 1 g =appreciably
2
2g to physical processes. Since the order of
the
spatial extension
corresponding
to a large energy uncertainty ∆E
Time-energy
uncertainty
relation
behave as ∆X ∼ "2s ∆E, we are led to a remarkably simple relation for
is
reinterpreted
as
the
time-space
Ω1 + Ω2 , ∆X
Ω1 ∩for
Ω2 fluctuations
=∅
magnitude
along spatial directions of string states
relation.
within the time interval ∆T = ∆t of interactions:
∆E∆t ! h
2
∆X∆T >
"
∼ s.
(2.2)
25
to call this relation the ‘space-time uncertainty relation’.
It should be
h
the property
of this how
path integral.
The absence
ofdistance
the ultravioleto
eart
language.
by
briefly
recalling
to
define
the
on characteristic
the basis space-time
of conformal
of
theofworldny
property
ofinvariance
the
string
amplitudes
can
be
understood
from
string
theory
from
this
point
view
is
a
consequence
of
thesurface
modular
uncertainty
relation
on
the
basis
of
conformal
invarian
briefly
recalling
how
to
define
the
distance
on
a
Riemann
19) This derivation seems to support
19)(2This
.2) can
an This
oldof
work.
operty
this
path
integral.
The
absence
ofuncertainty
ultraviolet
in
ally
invariant
manner.
Fordirectly
athean
given
me
see
that
thederived
space-time
relation
be regarde
relation
can
also
be
andwork.
inRiemannian
adivergences
more
sheetwill
string
dynamics,
following
old
derivation
#
ariant
manner.
For
a
given
Riemannian
metric
ds
=
ρ(z,
z)|dz|,
theory
from
this
point
of
view
is
a
consequence
of
the
modular
invariance.
We
uncertainty
relation
should
be
valid
universally
in
generalization
of
the
modular
invariance
for
arbitrary
string
amplitu
#
our proposal that the space-time
uncertainty relation should
be va
the
Riemann
surface
has
length
L(γ,
ρ)
=
ρ|dz|.
.2) can
universal
way
using
world-sheet
conformal
invariance.
emann
that
the
space-time
uncertainty
relation
(2
be regarded
as
a natural
the
direct
space-time
language.
mits.
surface
has
length
L(γ,
ρ)
=
ρ|dz|.
This
length
γis, howboth short-time and long-time limits.
γ
of(T.Y.,
the in
modular
invariance
forbriefly
arbitrary
string
amplitudes
in
of on a R
Let
us
start
by
recalling
how
to define
theterms
distance
eization
formulated
terms
of
path
integrals
as
weighted
MPL
1989
:
for
an
extensive
review,
see
Prog.
Theor.
Phys.
103:
1081--1125(2000))
the
choice
of
the
function
ρ.
If
we
consider
some
finite
All
themetric
stringof
amplitudes
are formulated
in terms
of path
inte
ent
on
the
choice
the
metric
function
ρ.
If
we
ect
language.
in a conformally
invariant manner.
For a given Riemannian
metric d
ble space-time
Riemannmappings
surfaces
tofrom
a target
space-time.
There#
the
set
of allthe
possible
Riemann
surfaces
to
a target
sp
f
arcs
defined
on
Ω,
the
following
definition,
called
the
‘extremal
t
us
start
by
briefly
recalling
how
to
define
distance
on
a
Riemann
surface
an
arc
γ
on
the
Riemann
surface
has
length
L(γ,
ρ)
=
ρ|dz|.
This
quadrilaterals
on
thefollowing
world sheet, definition,
weγ have
d ofa the
setFor
ofarbitrary
arcs
defined
on
Ω,
the
string
amplitudes
can
be
understood
from
fore, ever,
any
characteristic
property
ofthe
the
string
amplitudes
canconsid
be
27)
dependent
on
the
choice
of
metric
function
ρ.
If
we
nformally
invariant
manner.
For
a
given
Riemannian
metric
ds
=
ρ(z,
z)|dz|,
tical
literature,
is ultraviolet
known
to
give
a# conformally
invariant defial. The
absence
of
the
divergences
in
conformal
invariants,
called
the
extremal
length
27)
the property
of length
this
path
The
absence
of
the
ultraviol
athematical
literature,
is
to
give
a
confor
region
Ω has
and
a set of
arcsintegral.
defined
on
Ω,
the
following
definition,
called
γ on the Riemann
surface
L(γ,
ρ)
= known
ρ|dz|.
This
length
is,
howγ
iew
a which
consequence
of thefrom
modular
invariance.
We
hependent
of isthe
set
Γ
of
arcs:
27)
corresponds
to
proper
time
of
trajectory
string
theory
this
point
of
view
is
consequence
modula
length’ofinthe
mathematical
literature,
known
tosome
giveof
a the
conformally
on the choice
metric function
ρ. Ifparticle
weaisconsider
finite
.2) can
relation
(2
be
regarded
as
a
natural
eertainty
length
of
the
set
Γ
of
arcs:
.2) can be rega
nition
for
the
length
of
the
set
Γ
of
arcs:
that on
theΩ,space-time
uncertainty
relation
(2
Ω and a set ofwill
arcssee
defined
the following
definition,
called
the
‘extremal
2
variance
for arbitrary
string27)amplitudes
ingive
terms
of
L(Γ,
ρ)
in mathematical
literature,
is
known
to
a
conformally
invariant
defi- . ampli
generalization
of
the
modular
invariance
for
arbitrary
string
2
(2 7)
λΩ (Γ ) = sup
L(Γ, ρ)
for the lengththe
of the
set Γ
of ρarcs:A(Ω,
2
λΩ (Γ ) = sup
direct
space-time
language.
ρ)
ρ A(Ω, ρ)
ng how to define the distance on a Riemann surface arbitrarily
chosen
world-sheet
Let us start by briefly
recalling
how
to
define
the
distance
on
a
2
L(Γ,
ρ)
Ω
er. For a given Riemannian
metric
ds
=
ρ(z, z)|dz|, metric
.7)
#
(2
λ
(Γ
)
=
sup
in a with
conformally
invariant manner.
For
a
given
Riemannian
metric
Ω
$
$
ρ
# 2
ρ) is, howρ A(Ω,
has length L(γ, ρ) = γ ρ|dz|. This
length
L(Γ, surface
ρ) = inf has
L(γ,
ρ), A(Ω,
dzdz. Th
an arc γ on the Riemann
L(γ,ρ)ρ)== γρ ρ|dz|.
2 length
L(Γ,
= inf
L(γ,ρ.ρ),If weA(Ω,
ρ) =
ρ dzdz.
f the ρ)
metric
function
consider
someγ∈Γfinite
Ω
ever,
dependent
on
the
choice
of
the
metric
function
ρ. If we con
$
γ∈Γ
Ω
d on Ω, the following definition, called the ‘extremal
2
L(Γ,
ρ)
=
inf
L(γ,
ρ),
A(Ω,
ρ)
=
ρ
region Ω and a set of arcs defined ondzdz.
Ω, the following$definition, ca
27)
e,
is known to giveγ∈Γ
a conformally invariantΩdefi27) is known to give a conforma
2
length’
in
mathematical
literature,
of arcs:
nition for the γ∈Γ
length of the set Γ of arcs:
Ω
2
L(Γ, ρ)
(2.7)
L(Γ, ρ)2
Ω (Γ ) = sup
L(Γ, ρ)
λ (Γ ) = sup
A(Ω, ρ)
L(Γ, ρ) = inf L(γ, ρ),
A(Ω, ρ) =
ρ dzdz
pc =satisfy
4 g =the
0 composition
g = 1 g = law,
2
2g
The extremal lengths
which
partially justifies the
naming “extremal length”: Suppose that Ω1 and Ω2 are disjoint but adjacent open
Ω = Ω1 surface.
+ Ω2 , Ω
= ∅ Γ2 consist of arcs in Ω1 and
composition
theorem
: Riemann
1 ∩ΓΩ
regions on an
arbitrary
Let
1 2and
Ω2 , respectively. Let Ω be the union Ω1 + Ω2 , and let Γ be a set of arcs on Ω.
1. If every γ ∈ Γ contains a γ1 ∈ Γ1 and γ2 ∈ Γ2 , then
24
λΩ (Γ ) ≥ λΩ1 (ΓT.
λΩ2 (Γ2 ).
1 ) +Yoneya
T. T.
Yoneya
Yoneya
12
2. If every γT.
Γ1T.
and Yoneya
γ2 ∈ Γ2 contains a γ ∈ Γ , then
1 ∈ Yoneya
12
T. Yoneya
nn
surface
corresponding
to
a
string
amplitude
can
be
decomposed
T. surface
Yoneya
ann
surface
corresponding
to
a
string
amplitude
can
be
decomposed
Since
any
Riemann
corresponding
to
a
string
amplitude
can
1/λ
(Γ
)
≥
1/λ
(Γ
)
+
1/λ
(Γ
).
1
2
Ω
Ω
Ω
1
2
mann surface corresponding to a string amplitude can be decomposed
iemann
surface
corresponding
tocorresponding
a string
amplitude
can
betwisting
decomposed
adrilaterals
pasted
along
the
boundaries
(with
some
twisting
operSince
any
Riemann
surface
to
a
string
amplitude
can
be decomposed
adrilaterals
pasted
along
the
boundaries
(with
some
operinto
a
set
of
quadrilaterals
pasted
along
the
boundaries
(with
som
quadrilaterals
pasted
along
the boundaries
(with
some
twisting
operThese
two
cases
correspond
to
two
different
types
of
compositions
of
open
regions,
12the
T.
Yoneya
reciprocity
theorem
:
Yoneya
ny
Riemann
surface
corresponding
to
a T.
string
amplitude
canfor
be
decomposed
al),
it
is
sufficient
to
consider
the
extremal
length
an
arbitrary
f12
quadrilaterals
pasted
along
boundaries
(with
some
twisting
oper- operinto
a
set
of
quadrilaterals
pasted
along
the
boundaries
(with
some
twisting
eral),
it
is
sufficient
to
consider
the
extremal
length
for
an
arbitrary
al),
it
is sufficient
to along
consider
the Ω
extremal
length
for
an
arbitrary
ations,
in on
general),
it side
is sufficient
toΩconsider
the
extremal
length
depending
whether
the
where
are
joined
does
not divide
the
sides
1 and
2 some
!
!
set
of
quadrilaterals
pasted
the
boundaries
(with
twisting
oper!
!
ations,
in
general),
it
is
sufficient
to
consider
the
extremal
length
for
an
arbitrary
ment Ω.
the
twopairs
pairs
of
opposite
sides
of α,
ΩOne
beand
α, β,
αfor
β . composition
eneral),
itLet
isγ sufficient
to of
consider
the
extremal
length
arbitrary
segment
Ω.
Let
the
two
opposite
sides
of Ω be
α
βand
. anβ,
!
! .Ω ! be
which
∈
Γ
connects,
or
do
divide,
respectively.
consequence
of
the
!
gment
Ω.
Let
the
two
pairs
of
opposite
sides
of
Ω
be
α,
α
and
β,
β
quadrilateral
segment
Ω.
Let
the
two
pairs
of
opposite
sides
of
in
general),
it
is
sufficient
to
consider
the
extremal
length
for
an
arbitrary
Since
any
Riemann
surface
corresponding
to
a
string
amp
!
quadrilateral
segment
Ω.
Let
the
two
pairs
of
opposite
sides
of
Ω
be
α,
α
and
β,
β
.
!
!
!
12
T.
Yoneya
Since
any
Riemann
surface
corresponding
to
a
string
amplitude
can
be
decomposed
eet
set
all
connected
set
of
arcs
joining
α
and
α
.
We
also
define
the
ofofall
connected
set
of
arcs
joining
α
and
α
.
We
also
define
the
lateral
segment
Ω.
Let
the
two
pairs
of
opposite
sides
of
Ω
be
α,
α
and
β,
β
.
! region
lawTake
is that
thethe
extremal
length
from
a point
to
any
finite
is! .infinite
and! the
the
!β,
!
:
segment
Ω.
Let
the
two
pairs
of
opposite
sides
of
Ω
be
α,
α
and
β
into
a
set
of
quadrilaterals
pasted
along
the
boundaries
Γ
be
set
of
all
connected
set
of
arcs
joining
α
and
α
.
We
also
define
!. α
Take
Γ
be
the
set
of
all
connected
set
of
arcs
joining
α
and
α
.
We
set
of
all
connected
arcs
joining
and
α
.
We
also
define
the
∗ ∗ be
! . then
nto
a
set
of
quadrilaterals
pasted
along
the
boundaries
(with
some
twisting
oper!
of
arcs
Γ
be
the
set
of
arcs
joining
β
and
β
We
have
two
!
arcs
Γ
the
set
of
arcs
joining
β
and
β
We
then
have
two
conjugate
length
is
zero.
This
todefine
the
that
the
vertex
he
of
all
connected
set
of
arcs
joining
α corresponds
and
α
. also
We
also
define
the
∗ arcs
! . fact
be set
thecorresponding
setconjugate
ofSince
all
connected
set
of
joining
α
and
α
.
We
the
ations,
in
general),
it
is
sufficient
to
consider
the
extrem
set
of
arcs
Γ
be
the
set
of
arcs
joining
β
and
β
We
then
have
two
∗
!
∗
!
∗
any
Riemann
surface
corresponding
to
a
string
amplitude
can
be
decomposed
12
T.
Yoneya
ances,
λ
(Γ
)
and
λ
(Γ
).
The
important
property
of
the
extremal
ations,
in
general),
it
is
sufficient
to
consider
the
extremal
length
for
an
arbitrary
∗
conjugate
set
of
arcs
Γ
be
the
set
of
arcs
joining
β
and
β
.
We
fces,
arcs
Γ
be
the
set
of
arcs
joining
β
and
β
.
We
then
have
two
∗
!
Ω
Ω (Γ ).
∗describe
! . We
:λ
operators
the
on-shell
asymptotic
states
whose
coefficients
are
represented
∗
λarcs
(Γ
)
and
The
important
property
of
the
extremal
ateof
set
of
arcs
Γ
be
the
set
arcs
joining
β
and
β
then
have
two
et
Γ
be
the
set
of
arcs
joining
β
and
β
.
We
then
have
two
Ω extremal
Ω of quadrilaterals
λΩ (Γ
) andpasted
λΩ (Γ along
). The
important
property
of the
extremal
quadrilateral
segment
Ω.
Let the
two pairs
oftwisting
opposite
sid
into a distances,
set
the
boundaries
(with
some
oper!
!.
∗
∗
s
the
reciprocity
∗
quadrilateral
segment
Ω.
Let
the
two
pairs
of
opposite
sides
of
Ω
be
α,
α
and
β,
β
by
local
external
fields
in
space-time.
We
also
recall
that
the
moduli
parameters
of
extremal
distances,
λitTake
(Γsufficient
)important
and
λset
(Γofproperty
).the
The
important
property
ces,
λΩ (Γ
)λations,
λisΩ
(Γ
).∗).).
The
important
property
of
the
extremal
al
(Γ
) us
and
λ
(Γ
The
property
ofextremal
theof
extremal
Ω
Ω
the
reciprocity
Ωand
Ω(Γ
hedistances,
reciprocity
Γ
be
the
all
connected
set
of
arcs
joining
αa
stances,
λlength
)for
and
λ
The
important
the
extremal
in
general),
is
to
consider
length
for
an
arbitrary
Ω (Γ
Ω
Since
any
Riemann
surface
corresponding
to
a
string
amplitude
can
be
!
∗
Take
Γ
be
the
set
of
all
connected
setthe
ofset
arcs
joining
and
αset
. ofWe
also
define
the
world-sheet
Riemann
surfaces
are
nothing
but
aofΓ
set
of(2
extremal
lengths
with
some
! and
!.
.8)sides
∗ α
for
us
is
the
reciprocity
λ
(Γ
)λ
(Γ
)
=
1.
quadrilateral
segment
Ω.
Let
two
pairs
opposite
Ω
be
α,
α
β,
β
Ω
Ω
length
for
us
is
the
reciprocity
∗
the
reciprocity
conjugate
of
arcs
be
the
of
arcs
joining
β
an
.
s is theassociated
reciprocity
into
a)λ
set (Γ
of ∗quadrilaterals
along
the boundaries
(with
some
λΩ (Γ
)λpasted
(Γ
)=
1.operations,
(2
8) tw
∗
!
Ω
.
angle
variables,
associated
with
twisting
which
are
necessary
!
λ
(Γ
)
=
1.
(2
8)
conjugate setTake
of arcs
Γ Ωset
be ofthe
set
ofdistances,
arcs
joining
β) and
βλand
. (Γ
We
then
have
two
∗.).We
∗connected
ΓΩbe the
all
set
of
arcs
joining
α
α
also
define
the
extremal
λ
(Γ
and
The
important
.
Ω
Ω
λ
(Γ
)λ
(Γ
)
=
1.
(2
8)
ations,
in
general),
it
is
sufficient
to
consider
the
extremal
length
for
a
Ω mutually
Ω ∗∗of
s impliesin that
one
of
the two
conjugate
extremal
lengths
is !
∗
∗
∗
order
to
specify
the
joining
the
boundaries
of
quadrilaterals.
.
.then
that this
implies
that
of).
the
two
mutually
conjugate
lengths
extremal Note
distances,
λΩ
(Γ
))λ
and
λΩone
(Γ
The
important
property
of(2the
extremal
conjugate
set
of
arcs
Γ(Γ
be
the
set
of
arcs
joining
β
and1.β extremal
. We
have is
two
λ
(Γ
)λ
(Γ
)
=
λ
(Γ
(Γ
)
=
1.
(2
8)
Ω
λ
(Γ
)λ
)
=
1.
8)
length
for
us
is
the
reciprocity
Ω
Ω
Ω
Ω
Ω
quadrilateral
segment
Ω.
Let
the
two
pairs
of
opposite
sides
of
Ω
be
α,
mplies
that
one
of
the
two
mutually
conjugate
extremal
lengths
is
∗
Conformal
invariance
allows
us
to
conformally
map
any
quadrilateral
to
a
recthat
this
implies
that
one
of
the
two
mutually
conjugate
extremal
lengths
is
larger
than
1.
extremal
distances, λΩ (Γ ) and λΩ (Γ ). The important property of the! extremal
ength
for
us
is
the
reciprocity
! )∗and
! )als
mal
lengths
satisfy
the
composition
law,
which
partially
justifies
the
Take
Γ
be
the
set
of
all
connected
set
of
arcs
joining
α
and
αis.1.
We
λ
(Γ
)λ
(Γ
)
=
angle
on
the
Gauss
plane.
Let
the
Euclidean
lengths
of
the
sides
(α,
α
(β,
β
The
extremal
lengths
satisfy
the
composition
law,
which
partially
justifies
the
than
1.
Ω
Ω
his
implies
that
one
of
the
two
mutually
conjugate
extremal
lengths
length
for
us
is
the
reciprocity
Note
this
implies
that
one
of
the
two
mutually
conjugate
ext
mplies
that
one
of
the
two
mutually
conjugate
extremal
lengths
is
∗
!
∗
emal
length”:
Suppose
thatthe
Ωlength”:
and
are)λ
but
adjacent
open
conjugate
ofΩ
arcs
Γdisjoint
bewhich
the
set
of
arcs
joining
βthe
and
β . (2
We
.8) the
1set
2 (Γ
be
a
and
b,
respectively.
Then,
the
extremal
lengths
are
given
by
the
ratios
naming
“extremal
Suppose
that
Ω
and
Ω
are
disjoint
but
adjacent
open
λ
(Γ
)
=
1.
e
extremal
lengths
satisfy
composition
law,
partially
justifies
the
1
2
Ω
Ω
al
lengths
satisfy
the
composition
law,
which
partially
justifies
∗ that
1.arbitrary
.8)
∗in
larger
than
1.
Note
that
this
implies
one
of
the
two
mutually
cont
λ
(Γ
)λ
(Γ
)
=
1.
(2
Riemann
surface.
Let
Γ
and
Γ
consist
of
arcs
Ω
and
Ω
Ω
extremal
distances,
λ
(Γ
)
and
λ
(Γ
).
The
important
property
of
1
2
1
Ω
Ω
regions
on
an
arbitrary
Riemann
surface.
Let
Γ
and
Γ
consist
of
arcs
in
Ω
and
gal“extremal
length”:
Suppose
that
Ω1 and
Ωare
disjoint
but adjacent
open
2
1.
2 are
∗ 1but
length”:
Suppose
that
Ω
and
Ω
disjoint
adjacent
open
1
2
remal
lengths
satisfy
the
law,
which
partially
justifies
λ(Γ
)than
= be
a/b,
λ(Γ
) = b/a.
(2 9) is
larger
1.
ly. Letthat
Ω
bethis
the implies
union
Ω that
+ Ωcomposition
, and
Γ
a set
of
arcs
on
Ω.
Note
one
ofletthe
two
mutually
conjugate
extremalthe
lengths
Ω (Γ )
 b, respectively. Then, the extremal lengths are give
≥ 1/λΩ1 (Γ1 ) + 1/λΩ1 2 (Γ
2 ). 2be a and

 One
·
nects, or do divide, respectively.
consequence
of the composition


∗
two The
different
types of compositions
of
open
regions,
geometrical
properties
of
target
space-time
are
λ(Γ
)
=
a/b,
λ(Γ
) are
= b/a.


The
boundary
conditions
chos
X
xtremal length from a point
to
any
finite
region
is
infinite
and
the
N
×N
 the sides27)

where
Ω
and
Ω
are
joined
does
not
divide
1
2
µRef.
µ =the
related
through
these
conformal
invariants
 ∂tosee
corresponds
For
a proof,
njugate
length
is
zero.
This
the
fact
that
·
x
·
∂
x
0 vertex
in the conformal
1
2
ide, respectively. One consequence of the
composition
Let uscoefficients
now consider
how
the extremal length is reflec
be
the
on-shell
asymptotic
states
whose
are
represented
·is infinite
h from a pointRiemann
to any
finite regionstructure
and
the
Space-Time
path
integral
then
contains
fa
sheet
probed
by
general string
amplitudes.
Thethe
euclide
space-time.
Wetoalso
recall
thatthe
the
moduli parameters of
1 !
h fields
is zero.inThis
corresponds
the
fact
that
vertex
conformal
gauge is Principle
essentially governed by the action "2 !Ω1
Space-Time
Uncertainty
s
mann
surfaces
are whose
nothing
but a setare
of represented
extremal lengths with some
asymptotic
states
coefficients
rectangular region as above and the boundary conditions (z
exp
−
+modulioperations,
−
me
Uncertainty
Principle
13
variables,
withthe
twisting
which
are
necessary
e-time.
We associated
also recall
that
parameters
of
φ
φ
µ
µ
µ2
x
(0,
ξ
)
=
x
(a,
ξ
)
=
δ
Bξ2 /b,
oundary
conditions
areextremal
chosenof
such
that
kinematical
constrain
2 momentum
2
a joining
fy
quadrilaterals.
arethe
nothing
butofathe
set boundaries
of
lengths
withthe
some
µ
µ
µ1 ∗)
µ = 0 in the conformal gauge is satisfied for xthe
(ξ
,
0)
=
x
(ξ
,
b)
=
δ
Aξ1 /a.Th
1
1
∂
x
classical
solution.
nvariance
allows
us
to
conformally
map
any
quadrilateral
to
a
rectociated
with
twisting
operations,
which
are
necessary
Space-Time
Uncertainty
Principle
2
A
n such that the kinematical momentum
constraint
This
indicates
that
the
square
ro
!
!
of the
boundaries
quadrilaterals.
uss
plane.
Letcontains
theoffor
Euclidean
lengths
of the sides
α ) and (β, β )
∗) (α,
ntegral
then
theclassical
factor
+
−solution.
auge
is
satisfied
the
The
φ
F
φ
of
the
length
probed
by
strings
in
ws
us to conformally
map
any
quadrilateral
to
a
rectectively.
Then,
the
extremal
lengths
are
given
by
the
ratios
!
#$
"
b
2
2 are chosen
The
boundary
conditions
such that the kinematical m
tor
!
!
B
1
A
the Euclidean lengths of exp
the sides
(α,
α
)
and
(β,
β
)
natural,
as
suggested
from
the
de
∗
+
−
.
µ
µ
.
#$
"
λ(Γ
) 2·=∂a/b,
λ(Γ
) 2=
b/a.
9)
∗ )gauge is (2
∂
x
x
=
0
in
the
conformal
satisfied
for the clas
2
1
2
the
extremal
lengths
are
given
by
the
ratios
"
λ(Γ
)
λ(Γ
B
1
A
s
%
%
∆T
∼∗scale
along
the
longitudinal
directions
+
.
path
integral
then
contains
the
factor
∗
2 # measur
.
Γ
)
=
a/b,
λ(Γ
)
=
b/a.
(2
9)
"ndicates
λ(Γ
)
λ(Γ
)
that
the
square
root
of
extremal
length
can
be
used
as
the
"A
∼
λ
∆A
=
s
B
!by the
#$
" space-time
consider
how
the
extremal
length
is
reflected
2
2
length
probedlength
by strings
in space-time.
The
appearance
of the
B square root
1
A
t
of
extremal
can
be
used
as
the
measure
d by general
string amplitudes.
The euclidean
path-integral
in+the
exp
−
.
contribution
to the amplitude
: In
particular,
this
implies
that
p
.
!
2
∗
l,
as
suggested
from
the
definition
(2
7):
the
extremal
length
is
reflected
by
the
space-time
1
∆XThe
∼
scale
along
theof
transverse
directions
"
)
λ(Γ
)
µ . Take
µ ∂ λ(Γ
space-time.
appearance
the square
root
is
s
dzdz
x
a
is essentially
governed
by
the
action
∂
x
z
z
Ωin the
"2smultaneously
%
%
%
% always restricted
ring amplitudes.
The
euclidean
path-integral
is
.
!
nition
(2∆A
7):
2
2)# as
1
on
as
above
and
the
boundary
conditions
(z
=
ξ
+
iξ
µ
µ,∂ square
This
indicates
that
the
root
of
extremal
can be
1
2
"A #"2∼Ω dzdz
λ(Γ
∆B
=
"B
∼
λ(Γ )"
.
= action
x
.
Take
a
governed by the
∂)"
x
2
s
slength
z
z
%
length,
∆A∆B
∼
"
.
In
Minkow
%
s
'
s
µof the
µ
µ2
length
probed
by
strings
in
space-time.
The
appearance
dΓ the
boundary
conditions
(z
=
ξ
+
iξ
)
as
∗
2
1
2
x
(0,
ξ
)
=
x
(a,
ξ
)
=
δ
Bξ
/b,
2 "B #that
)#ss.2 short
)"s , ∆B
∼ 2 probing
λ(Γ )"
ticular,
this =implies
distances
along
both
directions
s
the
other
is
space-like,
as
required
µ suggested
µ1 from the definition (2.7):
natural,
as
µ xµ
µ2
(ξ
, 0)δ =restricted
x 2(ξ
δ the
Aξ1 /a.
.8) of the extrem
ξ2 ) = x (a,
ξalways
Bξ
/b,1 , b) =by
2 )1=
neously
is
reciprocity
property
(2
the space-time
uncertainty
relati
robing µshort distances
along both
%
% directions
% si%
µ1
2
= x (ξ1∼
, b) "= .δ In
AξMinkowski
,, 0)
∆A∆B
metric, one of the directions is time-like an
1 /a.
2 27)
Ref.
In black hole space-times;
For remote observers outside black holes, a finite length
of time corresponds to an infinitesimally short time on
the horizon.
as we can derive in the Minkowski coordinates.
Space-time uncertainty
implies
that∆T
the, the uncertainty w
Thus inrelation
the limit
of small
longitudinal extension of strings in the near horizon region is
2
arbitrary large.
!s
∆X1 ∼
→ ∞
∆T
Essentially the same relation is noticed later by Susskind (1994)
who also the
emphasized
its relevance
black
hole physics.
Therefore
Hilbert
spaceinof
string
states can
never
Then there
is almost
no meaning
considering
space
corresponding
to in
a single
wedge.horizons
Both wedges
using local field approximations. Information puzzle must
This
means
that
there
is
no
thermalization.
It
should
be formulated by taking due account of non-local nature
of strings.the definition of space-time distances itself must b
the physical properties of quantum strings. Thus th
D−1
!
2 ρ−τD−1
2
2
2
2
ρ+τ )
1
The
Rinder
space
!
ds
=
−X
(dτ
+
(dX
)
+
(dX
)
U
=
∓e
,
V
=
±e
(4)
1 is well known,
i
As
the space-time
1 2
Penrose
space-time
2
2 diagram of Schwarzschild
) 1+ (dX
+
(dXi )
(2)
The1 )Rinder
space
i=2
1accelerated
The Rinder space
i=2
uniformly
observers
th
As
is
well
known,
the
space-time
coordinates
which
are
approp
which gives
Note
by=
Tamiaki
Yoneya
A
of space-time
an
accelerated
observer
is given
by defined
X
Astrajectory
is well known,
the
coordinates
which
are
appropriate
for
describing
the1 =b
accelerated
observer
is
given
by
X1 =observers
R,
∂τ R
0,Rindler
and
the
proper
uniformly
accelerated
the
coordinates
1
0
AsX
is well
known,
thexspace-time
coor
x
=
cosh
τ,
=
X
sin
2
2
1
1
time
is
τ
=
Rτ
.
The
range
of
the
Rindler
time
τ
is
from
−
ds
=
−R
dU
dV
(5)
e range
of
the
Rindler
time
τ
is
from
−∞
to
+∞.
The
±sign
of
X
1
uniformly accelerated observers
the Rindler coordinates
defined
by
2014
1
0
uniformly
accelerated
observers
Rin
x = X1 cosh τ,
x = X1 sinh
τ,
x
(i
= 2, . .the
.,D
−
i = Xi
( I ) and left ( IIto
) Rindler
respectively.
If )
weRindler
restrict ourselves
corresponds
right wedge,
( I ) and
left
( II
wedge,
respec
In
terms
of
these,
the
metric
is
1
0
The Rindler
horizons
correspond
to
U
V
=
0.
x
=
X
cosh
τ,
x
=
X
sinh
τ,
x
=
X
(i
=
2,
.
.
.
,
D
−
1)
(1)
1
0
1
1 ρ (R= a
i positive
i
ρ
wedges,
we
can
set
X
=
±Re
constant)
and
then
the
x
=
X
cosh
τ,
x
= X1 sinh
In
terms
of
these,
the
metric
is
1
1
to one of these two wedges, we can set X1 = ±Re (R=
a τ,
p
As +∞,
is1wellThe
known,
this metric
can becorresponds
regarded as an to
approximation
of the
near-horizon
Rinder
space
∞ to
the former
of which
the
horizon,
and
the
2
2
2
2
D−1
!
In
terms
of
these,
the
metric
is
range of ρ is from −∞
to
+∞,
the
former
of
which
corresp
ds
=
−X
(dτ
)
+
(dX
)
+
1
1 these,
the
metric
is
2
2
2
2 In terms of
2
region
→ 2GM
of the Schwarzschild
dsmetric,
= −X1spacet-time.
(dτ ) + (dX1 ) +
(dXi )
to
the rusual
Minkowski
%
$
metric is conformal
to D−1
the usual Minkowski
metric,
i=2
!
D−1
!
2
2
2
2
2
2GM
1
As
is
well
known,
the
space-time
coordinates
which
are
appropriat
#
2
2
2
2
2
2
2
2
2
ds ==−−X
(dτA) trajectory
+ (dX
(dX
) +trajectory
(2)
1 )++
iA
2ds
2 11 −
of
an
accelerated
ds
=
−X
(dτ
)
+
(dX
)
+R,ob
(d
dr
r
d
Ω
(6)
dt
1
1
of
an
accelerated
observer
is
given
by
X
=
∂
−(dτ
)
+
(dρ)
(3)
1
τ
2GM
"
#
ate change is2
1−
i=2 2
2 r 2ρ
2
i=2
r + (dρ)
ds
=
R
e
−(dτ
)
time
is
τ
=
Rτ
.
The
range
of
the
uniformly
accelerated
observers
theofRindler
coordinates
byto
time
is $
τ = Rτ
. The range
the Rindler
time τ isdefined
from −∞
"
#
A trajectorytoof define,
an accelerated
observer
is right
given by
X1 =
R, ∂τ R = of
0, an
andaccelerated
the properobserve
A
trajectory
convenient
in
the
left
and
wedge,
respectively
2GM
t corresponds
corresponds
to right
(
) Rindler
respectively
1 I ) and left ( IIto
right wedge,
( I ) and
left (
− range of, the
=time
τ τ to
(7)to +∞.
t time
isX1also
sometimes
convenient
define,
in
the
left
and
rig
ρ
11The
0
is=τ 4GM
= Rτ
.
Rindler
is
from
−∞
The
±sign
of
X
time
is
τ
=
Rτ
.
The
range
of
the
1 Rind
r
4GM
to
one
of
these
two
wedges,
we
can
set
X
=
±Re
(R=
a
positive
1
x =toX1one
sinhof
τ, these
xi = two
Xi (i
= 2,(4)
. . . ,we
D−
1)
wedges,
can
V = ±eρ+τ t x = X1 cosh τ,
to right
( I Xof
0)
( IIto
X1+∞,
< 0) the
Rindler
wedge,
respectively.
corresponds
to
( I corresponds
X1 > If
0) and
range
ρ (7)
is and
fromleft
−∞
former
of right
which
1 >
, corresponds
τ=
range
of
ρ
is
from
−∞
to
+∞,
th
ρ−τ
ρ+τ
U 4GM
= ∓emetric
, is conformal
V = ±eto the usual Minkowski metric,
ρ
ofwe
therestrict
Schwarzschild
metric
is
transformed
into
the
two-dimensional
we
restrict
ourselves
of these tw
ourselves
to
one
of
these
two
wedges,
we
can
set
X
=
±Re
(R=to
a one
positive
1
In terms of these, the metric
is
metric
is conformal to
the
usual M
" −∞ to
#
ic
constant)
then the
range of ρ is fro
constant)
and
then
the
range
of
ρ
is
of which
corresponds
2 approximation
2 from
2ρ
2 +∞, the
2 former and
near
horizon
ds
=
R
e
−(dτ
)
+
(dρ)
which
gives
"
D−1
V
(5)
ransformed
into the two-dimensional
2
2 2ρ
2
2
!
ds
=
R
e
−(dτ
)
+
(dρ)
to
the
horizon,
and
the
metric
is
confor
to the horizon,
and
to2the usual Minkowski
metric,
2 the metric
2 is conformal
2
2
1
2
2
2
2
2
ds "=
−X
(dτ
)
+
(dX
)
+
(dX
)
1
i
1
)s correspond
+
(dr)
−X
(dτ
)
+
(dX
)
(8)
It
is
also
sometimes
convenient
to
define,
in the" left and right
we
2
2
1
1dU
2GM =
#
to
U
V
=
0.
Rindler
wedges
of
Minkowski
space-time
ds
−R
dV
"
#
1 − 2r
2
2 2ρ
2
2
2 2ρ
2
2
i=2
ds
=
R
e
−(dτ
)
+
(dρ)
It
is
also
sometimes
convenient
to
ds = R e −(dτ ) + (dρ) ρ−τ
(3)
ρ+τ
2
2
U = ∓e
,
V = ±e
(dX
) trajectory
1A
of an (8)
accelerated
observer
isValso
given
∂toτ R
=
The
Rindler
horizons
correspond
to It
Uis
=
0.by Xconvenient
1 = R,
sometimes
defin
ρ−τ
ρ+τ
e+
It is also sometimes convenient to define, in the leftUand
=right
∓ewedge,
, respectively
V = ±e
x = X1 cosh τ, x = X1 sinh τ, xi = Xi (i = 2, . . . , D − 1)
previous
discussions
toobservers
strings.the
The
action
is
uniformly
accelerated
Rindler
coordinate
2014
propagator
using
the
Minkowski
vacuum,
using
the
1
0
The
Rinder
space
x
=
X
cosh
τ,
x
=
X
τ,
x
=
X
(i
=
2,
.
.
.
,
D
−
1)
1
1 sinh
i 1
icoordinates
uniformly
accelerated
observers
the
Rindler
defined
by
#
#
In
terms
of
these,
the
metric
is
µ
ν
String
theory
in
the
Rindler
coordinates
.
1
0
$
1
∂x
∂x
x ab= X1 sinh τ, xi = Xi (i =
1 in
The
space x = X
onic string
the Rinder
Rindler coordinates
2 1 cosh τ,
Notw
Sstring [x(τ, σ)] = −
d
ξ
−γ(ξ)γ
(ξ)g
(x)
µν
D−1
As
is
well
known,
the
space-time
coordinates
!
"
a ∂ξ
Note
onb quantum s
4πα
∂ξ
2
2
2
2
2
1
0
In terms
ofisX
these,
the
metric
is
ds
=sinh
−Xof
) +uniformly
(dX
(dX
As=
well
known,
the
space-time
coordinates
which
are
for coor
des
1) +
i )=
1
The
x
cosh
τ,
x
=
X
τ,
x
=
X
(i
2,
.appropriate
. .Rinder
, Dthe
−Rindler
1)space
1 (dτ
In
terms
these,
the
metric
is
1
1
i
i
accelerated
observers
us
extend the previous discussions to strings. The action is
r coordinates
i=2
uniformly
accelerated
observers
the
Rindler
coordinates
defined
by
# metric
D−1
D−1
where γab (ξ)1is #the
for
two-dimensional
world
sheet,
and
g
(x)
is
th
µ
ν
As
is
well
known,
the
sp
µν
$
!
!
∂x
∂x
1
0
2of an accelerated
2 τ,
A trajectory
is)2given
by=(dX
XX1 1=sinh
R,
∂τ R x=i =
0, X
ani
x2 observer
=X
cosh
τ,+ x(67)
2= −
2
22 ξ
2ab (ξ)g
212 (dτ )
[x(τ,
σ)]
d
−γ(ξ)γ
(x)
1 uniformly
ds
=
−X
+
(dX
)
ing
µν
1
i
accelerated
obs
1
0
ds
=
−X
(dτ
)
+
(dX
)
+
(dX
)
"
a
b
sions
to strings.
The
action
isτ, 1x is= X1 sinh
iτ,∂ξ xi = Xi (i = 2, . . . , D − 1)
In
terms
of
these,
the
metric
1 X
4πα
∂ξ
x
=
cosh
1
space-time. In the Rindler
space,
time is
τ = i=2
Rτ .we
Thehave
range of the Rindler time τi=2
is from −∞ to +∞. The
1=The
Rinder
space
µ
ν
$
x
X
cosh
τThe
,
x
In
terms
of
these,
the
metric
is
∂x
∂x
1
R
(ξ) is theIn
metric
for
two-dimensional
world
sheet,
and
g
(x)
is
that
of
the
target
%
#
#
µν
ab terms of these, the
A
trajectory
of
an
accelerated
observer
is
given
by
X
corresponds
to
right
(
I
)
and
left
(
II
)
Rindler
wedge,
respectively.
If
we
rest
metric
is
D−1
−γ(ξ)γ (ξ)gµν (x) a1 b
(67)
$
!
∂τ
∂X
∂X
1=∂X
1and
As
is well
known,
the space-tim
abis given
2 ∂τ
∂ξ
∂ξ
A In
trajectory
of
an
accelerated
observer
by
X
=
R,
∂
R
0,
the
ρ
D−1
In
terms
of
these,
the
me
me.
theSRindler
space,
we
have
2
2
2
2
2
1of
τ
!
time
is
τ
=
Rτ
.
The
range
the
Rindler
time
τ
is
frok
to
one
of
these
two
wedges,
we
can
set
X
=
±Re
(R=
a
positive
constant)
=
−
−γ(ξ)γ
(ξ)
−X
+
+
D−1
1
As
is
well
string
ds =
−X
(dτ
)
+
(dX
)
+
(dX
)
!
1
2
2
2
2
2
uniformly
accelerated
observers
1
i
"
a
b
a
b
a
2 1 4πα
2
2
2
2 =∂ξ
ds
−X
(dτ
)
+
(dX
)
+
(dX
)
%
&
# #ds
∂ξ
∂ξ
∂ξ
∂ξ
1
i
1
−Xg1 (dτ
) is
+that
(dX
+∂X
(dX
1 )from
i )and
$ =and
nsional world
sheet,
(x)range
the
target
corresponds
to right
left
wedge, respective
uniformly
of
is
−∞
to +∞,
the former
of which corresponds
to the hora
1
∂τρ of
∂τ
∂X
∂X
∂XRindler
oordinates
right wedge.
Now let inustheextend
the
1
0
1
time is τ = Rτ . The ab
range
of 2the Rindler
+∞.
The ±sign
µν
2
2
2
1time
1 τ is from −∞
i=2i=2
dsto
=
−Xi=2
+(
1 (dτ )
ing
=−
4πα"
−γ(ξ)γ (ξ) −X1
+
+
·
(68)
x1 = X1 cosh τ,
x 0 = X1
a ∂ξ b
a ∂ξ b
a
b
∂ξ
∂ξ
∂ξ
∂ξ
of
these
two
wedges,
we
can
set
X1 = ±Reρ (R= a posi
metric
is
conformal
to
the
usual
Minkowski
metric,
∂
have
1
corresponds
to right
( I an
)∂x
and
left0( and
II ) observer
Rindler
wedge,
respectively.
If
we
restrict
ou
In
the
limit
→
γ
=
0,
this
reduces
to
the
previous
x
=
X
A
trajectory
of
an
accelerated
observer
is
given
Aparticle
trajectory
of
accelerated
is
given
by
X
=
R,
∂
R
=
0,
and
1
τ
a1
In
terms
of
these,
the
metric
is
A trajectory
1
A %trajectory
of an accelerated
observer
is
given
by
X1 =ofR,
∂of
= accele
0, an
&
τ Ran
∂
of
ρ
is
from
−∞
to
+∞,
the
former
which
correspon
"
#
article limit
and
γ. a1The
= 0,∂X
this2 ∂X
reduces
the
particle
action
time
is→τ 0=
Rτ
range
of 2the
Rindler
time
τtime
isThe
from
to Rindler
+∞.
The
±
ρ
2ρ to
2 previous
2 = Rτ
time
is
T
.
range
of
the
time
∂τ
∂τ
∂X
∂X
∂x
is
τ−∞
=
Rτ
. The
ran
1
1
1
2
ds
=
R
e
−(dτ
)
+
(dρ)
to
one
of
these
two
wedges,
we
can
set
X
=
±Re
(R=
a
positive
constant)
and
th
1to
2
with
m
=
0.
The
equations
of
motion
and
constraints
are
ξ)
−X
+
+
·
(68)
ds2 = −X12In
(dτ )terms
+ (dX1of
)2
is
conformal
the
usual
Minkowski
metric,
1
b
a right
b
aofleft
time iscorresponds
τ a= Rτ
. The
range
thebRindler
Rindler
timerespectively.
τ is from −∞
+∞. ours
Th
to
and
wedge,
If weto
restrict
= 0. The equations
motion
∂ξ ∂ξ of ∂ξ
∂ξ and constraints
∂ξ
∂ξ are
to right
( I res
) a
corresponds tocorresponds
right and left Rindler
wedge,
range of
is from
−∞
to It+∞,
the
former
of
which
corresponds
to
the
horizon,
a
ρ
to
one
ofA these
two
wedge
"
#
of∂ρthese
two
wedges,
we
can
set
X
=
±Re
(R=
a
positive
constant)
and
the
2
ρ
1
is
also
sometimes
convenient
to
define,
in
the
left
and
right
wedge,
respec
trajectory
of
an
accelerated
2 of
2ρ these two2wedges, we
2 can set X1 = ±Re ds
=
∂τ) and leftds(2 II
(R=
√
∂τ√ to right
=
R
e
−(dτ
)
+
(dρ)
corresponds
(
I
)
Rindler
wedge,
respectively.
If
we
res
ab
2
ab
2
=−γγ
0, this
reduces
to−∞
the to
previous
particle
action
range of
ρ is
isτ =
from
X
=
0
(69)
time
Rτ . The−∞
rangeand
ofto
th
−γγ
X
=
0
of
ρ
is
from
+∞,
the
former
of
which
corresponds
to
the
horizon,
1
b
1
There
are
a
large
number
of
previous
works
studying
string
metric ∂ξ
is conformal
to
the
usual
Minkowski
metric,
a
b
∂ξ
of ρ ρ+τ
is from −∞
to +∞,iscorresponds
the
former
of which
co
∂ξ
ρ−τ
metric
conformal
to
th
to
right
and
left
Ri
ρ
U
=
∓e
,
V
=
±e
is
conformal
to
the
usual
Minkowski
metric,
A
trajector
theory
in
the
Rindler
coordinates,
but
they
are
unsatisfactory
and
constraints
are
to
one
of
these
two
wedges,
we
can
set
X
=
±Re
(R=
a
positive
constant)
√
√
∂τ ∂τ It is also sometimes
1 conformal
convenient
to define,
inwedges,
the
left
an
ab ∂X1
ab
of
these
two
we
can
set
is
to
the
usual
Minkowski
metric,
"
√
√
−γγ
+
−γγ
X
=
0
(70)
∂
∂X
∂τ
∂τ
"
#
1
2
2
2ρ
1
in dealing
with
the
Virasoro
constraints
or
in
using
is T =
a
b
dsof ρ=
R −∞
e time
−(dτ
)
"2∂ξ
2b
2 2 2ρ ab 2 ∂ξ
2 ab X2 #
∂ξ
is
from
to
+∞,
the
for
2ρ
2
−γγ
+
−γγ
=
0
ds
= is
R
e = ItR
−(dτ
) −(dτ
+ +∞,
(dρ)
1(orρ−τ
dsfrom
einbgeneral
)gives
+ the
(dρ) former
which
range
of
ρ
−∞
to
which
the
ho
ρ+τcorresponds
"
# tocorresponds
a
a ,∂ξof
bV 2=
unjustifiable
gauges.
is
impossible
U
=
∓e
±e
√
(69)
2
2ρ
2
2
∂ξ
∂ξ
∂ξ
∂X
is
conformal
to
the
usual
Minko
ab
ds = R It
e is−(dτ
) +
(dρ)
also
sometimes
conv
−γγ
=
0
(71)
inconsistent)
to
satisfy
the
momentum
constraint
if
one
b
of" these
two
∂ξ
2
2to define, in
√
2
2 2ρ
2
∂
∂X
metric
is
conformal
to
the
usual
Minkowski
metric,
It
is
also
sometimes
convenient
the
left
and
right
wedge,
respect
ds = R e −(dτ ) + (dρ
ds = −R
dUdV
ab
τIt∂τ
restricts
oneself
to
incomplete
space-like
surfaces.
ρ−τ
which
gives
is also
sometimes
convenient
to
define,
in
the
left
and
right
wedge,
respectively
U
=
∓e
,of ρ V
=
is
from
−γγ
=
0
=
0
(70)
It
is
also
sometimes
convenient
to
define,
in
the
a
b
a
b
∂ξ
∂ξ
ξ ∂ξ
ρ−τ
ρ+τ
U
=
∓e
,
V
=
±e
∂τ
∂τ
∂X
∂X
∂X
∂X
"
#
1
1
2
The Rindler horizons
correspond
to UVwhich
= 0.
2
2
X1
+2
2 2ρ
+
·
2
ds2 = −R dU dV
ρ−τ
It is also sometimes
convenient t
is conforma
gives
ρ+τ
γ11σ 2+ 2π for finite
the
!γ we can solve
!gauge
odicity
σ
→
all function
dynamical
variables
Pτ = −κ − X1
∂
γ ∂X
∂
γ
10
11
1 function
00
appropriately
choosing the periodic transformation
f
(σ
γ00
0
−
−
−
is
no
problem
in
this
gauge
choice.
!
Physically, the
most
natural gauge choice∂τ
is the time-like
γ
∂τ
∂σ
γ
00
11
γ
∂X
finite
function
γ
we
can
solve
the
gauge
condition
perturbative
11
1
10
! gauge freedom is!then
The
residual
√ ar
P
=
κ
−
1
and orthogonal
gauge.
∂
γ11 ∂X
∂1 = κ −γγ
γ00
√
P
γ∂X
∂τ
1
a0is
00
no problem in this gauge choice.
− gauge, −
− r
P1 = κ −γγ
!
σ.
In
this
time-like
the
momenta
a
∂τ
γ00 ∂τ
∂σ √ γ11a
0
∂ξ∂X
γ11
01 freedom is then!arbitrary timePindepende
= κ −γγ
The residual gauge
P√
=κ −
γ
11
2 "
∂τ
a0γ∂X
00
P
=
−κ
−
X
τmomenta reduce 1to
γ11
P residual
= κ −γγ
σ. 1In
this
time-like
gauge,
the
freedom=
time-independent
reparametrization
of
a
Note
that,
if
we
set
κ
−
=
1/e
γ
where
00
∂ξ
γ
00
!
!
The total Hamiltonian which generates
of τγ11is∂X1
γ11 2 translation
1 to
equations
precisely
correspond
κ=
P1 = κ −
re canonical momenta Pτ = −κ − γ X1
!
"
2πα
γ
∂τ
00
00
γ11
2
!
!
(82) shows that
−
X
is inde
H = −2πPτ
!
!
!
1
γ
For
comparison
00
γ
∂X
γ111∂X1 ∂
11
1
∂
γ
γ
∂X
γ
11 ∂X
00
1
11
P
=
κ
−
P
=
κ
−
1
−
−
−
+
−
X
=
0
κ =eqs. of motion
!
! ch
1possible,
as
we
γ
∂τ
γ
∂τ
!
reparametrization
of
σ,
γ
(τ,
σ
)=
00
00
∂τ
γ00 ∂τ
∂σ
γ11 ∂σ
γ00
11
2πα
The equations of motion
now take the!form
! !
of arbitrary transf
γ
∂X
11
∂
γ
∂X
∂
γ
∂X
!
11
00 Hamiltonian which generates tr
The
total
!
P
=
κ
−
−
− also making
=γ11
0 2 physical
For comparison
with
theγ11−
particle
case
and
picture
assuming
periodici
∂ ∂τ
∂
γ
∂τ
00
γ
∂τ
∂σ
γ
∂σ
2
00 = 0
11κ
−
X
=
e
01
Pτ = −κ
− X
1
1
10
0
anyleaves
τ . The residua
γ
00
∂τ
∂τ
γ
possible, we choose
the
time-like
gauge
ξ
=
τ
.
This
still
us t
H
=
−2πP
00
τ
"
The total Hamiltonian which
generates
translation
of
τ
is
We
can further
ass
Note that, if we set κ − γγ11
=
1/e
where
e
is
the
auxiliary
variab
constant
energy
density
1
00
rbitrary transformation of spatial coordinate
ξ
≡ the
σ. positive
Weinfinitesimal
consider
cloo
with
e
being
and
consta
form
11
Thetoequations
of motion
now take
the
equations
the particle
case. Note
also tha
H =precisely
−2πPτ correspond
"
! Thus,
uming periodicity
in σ with periodicity
σ
→
σ
+
2π
for
all
dynamical
v
(closed string:
)
of
motion
above.
the
spatia
γ11
2
∂of time,
γ11 and
∂
= ∂τ f (σ,
2δγ10
hence,
u
10
τ (82) shows that − γ00 X1 is independent
P
=
−κ
−
X
=
0
τ
1
The
equations
of
motion
now
take
the
form
#
$
τ . The residual arbitrariness of the σ reparametrization
must
respect
the
∂τ
∂τ
γ
dσ 2e ∂X1 00
!
!
reparametrization of !
σ, γ11 (τ, σ ) =
γ
we
can
choos
which
can
always
m
11 (τ, σ),
!
P
=
dσ
1
2
∂
γ11 2condition
∂ the orthogonality
X
∂τ
can further assume that
γ
=
0
is
satisfied.
01
1
appropriately
choo
!P = −κ
− X =0
sible, we choose the time-like gauge ξ =
he σ and
reparametrization
must respect
periodic
case
also making physical
picture the
as clearly
itrary
transformation
of
spatial
coordinate
hogonality condition γ = 0 is satisfied. Indeed
gauge ξ = τ . This still leaves us the freedom
ing
periodicity
in
σ
with
periodicity
σ
→
trization
δσ ξ= f≡(σ,
= fconsider
(σ + 2π,closed
τ ) is strings
coordinate
σ.τ )We
Theσresidual
arbitrariness
of thevariables
σ reparam
icity
→ σ + 2π
for all dynamical
at
(
reparametrization
must the
respect
the periodicity.con
nσfurther
assume that
orthogonality
gonality
condition
γ
=
0
is
satisfied.
Indeed
the
but arbitrary periodic function δγ (σ, τ ) to zero
esimal form of the σ reparametrization δσ =
zation
δσ
=
f
(σ,
τ
)
=
f
(σ
+
2π,
τ
)
is
transformation function f (σ, τ ) = f (σ + 2π, τ ).
δγ
=
∂
f
(σ,
τ
)
auge condition perturbatively. So in principle
(77) th
∂X1
∂X
2
2
2
2
2
2
e = X1 P1 + P + κ
+κ
Virasoro conditions
#
$ ∂σ %2
$ ∂σ %2 &
∂X1
∂X
2
2
2
2
2
2
1 The+Rinder
space
e = the
X1 momentum
P1 + P +constraint
κ
κ
Similarly,
is
∂σ
∂σ
Note by Tami
2014
As is well known, the space-time coordinates
∂X1
∂X
Similarly,
momentum
is accelerated observers the Rindler co
P1 the +
P·
=constraint
0
uniformly
∂σ Note on∂σquantum string theory in the Rindler spacetime
∂X1
∂X
x1 = X1 cosh τ, x0 = X1 sinh τ, xi =
P1us first
+ study
P · the=equation
0
Let
for X1 in comparison with the case o
∂σ
∂σ
These
complicated-looking
(nonlinear)
by Tamiaki
Yoneya
Incase
terms
of time
these,
is also w ≡ eiσ
the same variable as in the latterNote
for
zthe
=metric
eτ and
2014
system Let
of equations
can inthe
principle
us first study
equation
for
X
in
comparison
with
the
case
o
1
$
%
D−1
2
2
2
2
!
∂
X
∂X
2z
∂X
∂
X
∂X
2
2
2
2
2
be exactly solvable,
classically.
1
1
1
1+
1
2
2
τ
iσ
ds
=
−X
(dτ
)
+
(dX
)
(dX
)
1
i
1 κz = e
the same
as
in
the
latter
case
for
time
and
also
w
≡
e
z 1 variable
+
z
−
+
X
−
=
0
1
The
Rinder
space
2
∂z
X1 $ ∂z %
∂σ e2 ∂σ i=2
But exact quantization
is∂zvery difficult.
2
2
2
2
∂
X
∂X
2z
∂X
∂
X
A
trajectory
of
an
accelerated
is giv
1 the space-time 1coordinates which2 are appropriate
1 ∂X1observer
2 As is 1well known,
for
describi
following
ansatz
for
small +
z behavior.
z a the
+ z approach
−
X1 − κ
=0
Let Assume
us take
qualitative
and
2
2
time
is T =coordinates
Rτ . The
∂z
∂z
X1
∂z
∂σrange
e of∂σ
uniformly
accelerated
observers
the
Rindler
defined
bythe Rindler tim
derive the space-time uncertainty corresponds to right and left Rindler wedge, r
η
$ 0
1 (1 + O(z
X
∼
az
>
0τ, zxi behavior.
Assume
the
following
ansatz
small
x viewpoint.
= X1 cosh τ, ))x =for
X$1 sinh
= Xi (i = 2, . . . , D − 1)
1 this
relation from
of these two wedges,
we can set X1 = ±Reρ (R
In termsη of these, the$ metric is of ρ is from −∞ to +∞, the former of which
In theXleading
(1 + O(z ))
1 ∼ az order,
$>
0
is conformal to the usual Minkowski metric,
ds2 = −X12 (dτ )2 + (dX1 )2 +
D−1
!
(dXi )2
2
"
#
κ
d
2
2 2ρ
2 da 3η 2
η
η
2η
η
2
= R−e
−(dτa) + (dρ)
In the⇒
leading
aη(η −order,
1)z + aηz − 2aηzi=2 ds
+ az
z =0
2
e dσ dσ
κ 2 The energy constraint shows
In the regions of the worldc(σ,
sheetτ )where
small,
= X1onis the
world
sheet
vanishes.
e
∂X1
the world
From
the
e
velocity of longitudinal mode in theontarget
space sheet vanishes.
∼
X
(σ,
τ
)
σ
→
1
approximately
by
∂τ
Let us denoteapproximately
the local horizon
by a function
by
velocity of propagating transverse oscillationsIn this small X
κ 1 region
on
2
c(σ,
κ 2τ ) = X1
along the world sheet
c(σ, τ ) = X1
e
and
X1 (σ0 , τ ) = 0, neglected,
σ0 =
) hence its beha
e F (τ
Let
us
denote
the
local
The Rindler horizon leads to horizons
in
world
sheet,
inthe
mind
that
thewhich
form
fu
Let us denote
the
local
horizon
by horizo
a of
func
are only dynamically
determined
self-consistently
The energy
constraint
shows for
consistently
byeach
solving the eq
classical solution.
τ 0) =
= F0,(τ ) σ0 = F
X1 (σ0 , τ )X
=1 (σ
0, 0 , σ
In order to satisfy the mom
∂X
1
∼ X1 (σ, τ )directions
σ → Fmust
(τ ) be excited, a
The energy
constraint
shows
∂τ
The
energy
constraint
shows
We cannot divide the Hilbert space of (1st quantized) strings into
must
be
orthogonal
to
the
ta
∂X
1
left and right wedges.
nd quantized)
fields
must
be
defined
∂X
1
∼
X
(σ,
τ
)
σworld
→ F (τ )sheet,
In (this
small Xstring
region
on
the
1
1
∼X
τ) σ →
F
∂τ
1 (σ,lengths
the
average
wave
of
using both Rindler regions simultaneously. Hence, the Hilbert space
2
∂τ
neglected,
and
hence
its Xbehavior
isthe
almost
psh
of string fields cannot
be decomposed
In either.
this small
region
on
world
1 frequency is of order
average
In this small X1 region on t
in mind thatneglected,
the form
function
F cannot
andof
hence
its behavior
is almo
κnbeha
2
neglected,
and
hence
its
ν
∼
c/λ
∼
nc
=
X
1c
in mind
thatthe
theequations
form of function
F
consistently by
solving
of motio
e
in mind that the form of fu
∂τ Then
from the Hamiltonian constraint
average
frequency is of order
2
the
momentum
X
∼
1/κ
using
the
Hamilto
wave
lengths
of
such
transverse
excitations
along
th
small
X
region
on
the
world
sheet,
the
terms
involving
σ-derivatives
1
1
space-time uncertainty relation
2
2 (106) κn
along
the
world
sheets,
and
hence
for
th
e
!
n
ed,
and
hence
its
behavior
is
almost
partilce-like
for
each
fixed
σ
near
σ
potential
energies
are
of
the
2
0s
.uency
If weis
consider
ν∼
∼ nc =
X1
ofc/λ
order
d that the form of function Fe cannot given
externally,
being determine
always
has
extendedness
by
along
the
world
sheets,
and
hence
for
the
ving
σ-derivatives
can
beexcitation number
κXalong
!
∆ν
1 ∆X
: average
of 1transverse
dently
sheet
is
n,
its
the
world
sheets,
and
he
by
solving
the
equations
of
motion
and
the
constraints.
κn
which
is of
course
satisfied
not
only
in
the
Rindler
coor
2
Then
from
the
Hamiltonian
c
The
order
of
the
energy
constant
can
be
estimated
by
consid
oscillations
along
the
world
sheet
ch
fixed
σ
near
σ
.
Keep
/λ
∼
nc
=
0X1
order to satisfy the
momentum
constraint,
at1 least
one component
of non-lo
the tra
κ|X
|∆X
!
∆ν
:
average
energy
density
This
relation
originates
from
the
1
e
2
case.
Thus
the
average
of
the
wave
satisfies
alongfrequency
the world sheets,
and
hence
for
the
uncertainty
κX
∆X
!
∆ν
y, X
being
determined
self∼
1/κ
using
the
Hamiltonian
constraint
with
the
assu
1
1
along
the
world
sheets,
and
hence
for
the
1
2
2
ons must be excited,
as
is
obvious
from
its
intuitive
meaning
that
the
mom
Hamiltonian constraint requires
e !n
relation ∆ν∆τ ∼ 1
κX1 ∆X
! ∆νorder.by
ints.
1 originates
the
energy
constant
can
be
estimated
considering
potential
energies
are
of
the
same
This
assumption
This
relation
the
e orthogonal to2 the tangent of the profile of strings at from
each
fixed
τnon-loca
. If we th
c
This
relation
originates
from
t
κX
!
ν
κ|X
!
2
1 |∆X
1 ! ∆ν
mponent
of
the
transverse
1
This
relation
originates
from
the
non-locality
of sheet
strings
ais
X
∆X
∆τ
!
2πα
∼
&
erage
wave
lengths
of
such
transverse
excitations
along
the
world
which
is
of
course
satisfied
no
1
1
relation
∆ν∆τ
∼ 1withoscillations
s
always
extendedness
by
zero-point
of
ord
using
thehas
Hamiltonian
constraint
the
assumption
relation
∆ν∆τ ∼ 1
relation
∆ν∆τ
∼
1
that the
ening
frequency
is ofmomentum
order
case.
Thus the
average
freque
Then
from
the
Hamiltonian
constraint
This
relation
originates
from
the
non-loc
! time
2 T is
! proper
2
In
terms
of
the
=
X
τ
ergies
are
of
the
same
order.
This
assumption
reas
1
X
∆X
∆τ
!
2πα
∼
&
|X
|∆X
∆τ
!
2πα
∼
&
1
1
ch
fixed
τ
.
If
we
consider
16
!
2
s
1
1
of strings where
s
κn 2
X
∆X
∆τ
!
2πα
∼
&
1
1
√
s
∼ c/λ ∼ nc =
X1
relation
∆ν∆τ
∼
1
2
relation
Fluctuations
of
and
also
e
the world2 sheet2isby
n, its
νof
In terms of the
properκX
time
T =
X1 τorder
, we arrive at α
th
extendedness
zero-point
oscillations
1 !
e !
n and In terms of the proper time T = X τ ,
to uncertainties,
thatcontribute
kinetic
relation
1
In
terms
of
the
proper
time
are either
of the samecan
order
der ofbut
thethey
energy
constant
be estimated by considering
the state
of stringsT
2
!
2
!
&
the Hamiltonian
|X
|∆X
!
1 ∆T
s2πα ∼ &s
1∆X
1 ∆τ
or of non-leading order. constraint
2
relation
∆X1 ∆T
! &the
s
1/κ
using
the
Hamiltonian
constraint
with
that coordin
that kine
because
strings
relation
which is of course satisfied not only inassumption
the Rindler
aswewe
can
derive
in the
Minkowski
coord
al
This
assumption
is reasonable
as
can
derive
in the
Minkowski
coordinates.
In order.
terms
of
the
proper
time
T =because
X
τ,
2 energies are of the same
2
NoThe
thermalization
!
energy constraint shows
This is as it should be:
∂X
∼ X (σ, τ ) σ → F (τ )
∂τ
We cannot localize the string fields in finite space-like regions
There is no allowed invariants (observables) on the world-sheets
1
which can be defined in restricted localized
1 regions of
if we assume the validity of the space-time uncertainty relation
with respect to target space-time. Again no1 legitimate observables
restricted space-like regions, other than S-matrices,
In this small X region on the w
and seems to support
‘black holeand
complementarity’.
neglected,
hence its
behavior
in mind that the form of functio
consistently by solving the equatio
In order to satisfy the momentu
NoThe
thermalization
!
energy constraint shows
This is as it should be:
∂X
∼ X (σ, τ ) σ → F (τ )
∂τ
We cannot localize the string fields in finite space-like regions
There is no allowed invariants (observables) on the world-sheets
1
which can be defined in restricted localized
1 regions of
if we assume the validity of the space-time uncertainty relation
with respect to target space-time. Again no1 legitimate observables
restricted space-like regions, other than S-matrices,
In this small X region on the w
and seems to support
‘black holeand
complementarity’.
neglected,
hence its
behavior
in mind that the form of functio
entangled pure states
thermal mixed state
consistently
by
solving
the
equatio
cannot happen, at least in association with the existence
of space-time
horizons,
in
string
theory
In order to satisfy the
momentu
This strongly indicates that the notion of space-time horizons is
meaningless in string theory.
Of course, concrete resolution of the information puzzle is yet a big
open question. Its final resolution would require the construction of
genuinely stringy geometry of space-times. Note that, for instance, any
non-linear sigma models for describing curved space-times intrinsically
rely, still, upon classical geometry.
Also, our claim does not mean that the idea of thermalization is
completely devoid of meaning.
It must be useful as an approximate concept.
But we have to keep in mind that there is a serious limit on this idea,
as we can infer, for instance, also from the existence of limiting
temperature (Hagedorn temperature) in string theory.
4 Concluding remark
No doubt, string theory is promising toward a final and complete
unification. Unfortunately, however, there are many fundamental
unsolved questions. They seem to be too difficult and almost
insurmountable at this time.
But I believe that there must be different and entirely new
ways of looking at these questions. Perhaps, and hopefully,
steady efforts of pursuing what we can, such as various
computer simulations and studies of simple models and so on,
may somehow open up doors to new unexpected directions and
angles of resolving these questions.
4 Concluding remark
No doubt, string theory is promising toward a final and complete
unification. Unfortunately, however, there are many fundamental
unsolved questions. They seem to be too difficult and almost
insurmountable at this time.
But I believe that there must be different and entirely new
ways of looking at these questions. Perhaps, and hopefully,
steady efforts of pursuing what we can, such as various
computer simulations and studies of simple models and so on,
may somehow open up doors to new unexpected directions and
angles of resolving these questions.
The next year 2015 is the centenary of General Relativity.
Pray for the next (3rd) revolution of string theory
in the not so distant future !