unsolved questions in string - 東京大学hep1.c.u-tokyo.ac.jp/~tam/kek2014.pdfunsolved questions...

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 YoneyaKEK workshop2014 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

Developments related to field-theory/string connection

Fishnet diagram interpretation, Nielsen-Olesen vortexDerivation of gauge theory, general relativity and supergravity from strings in the zero-slope limit unification including gravityConstruction of various supersymmetric gauge and gravity theoriesString picture from strong-coupling lattice gauge theoryt` Hooft’s large N expansionString field theories (light-cone)

1969 ~ 1979 Initial developments Nambu-Goto actionLight-cone quantization, no-ghost theorem, critical dimensions, ...Ultraviolet finiteness (modular invariance)Neveu-Schwarz-Ramond modelSpace-time supersymmetry

Green-Schwarz anomaly cancellationFive consistent perturbative superstring vacua in 10DCompactifications, various connections to mathematicsCFT technique, renormalization group interpretation

1984~1989 First revolution

1990~1994 “Old” matrix modelsDouble scaling limitc=1 strings, 2D gravity, ‘non-critical’ stringstopological field theories and strings

1995~1999 Second revolutiondiscovery of D-branesstatistical interpretation of black-hole entropy in the BPS or near-BPS limitsconjecture of M-theoryNew 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

Guggenheim museum (New York)

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

The Physical Society of Japan (JPS)

NII-Electronic Library Service

abstract of the JPS review talk in 1975

Dual model and field theory

(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

Main message of that talk was: Is it possible to resolve the following dichotomic duality relations ?

“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

Weyl

KaluzaKlein

Quantum FieldTheory

Gauge Principle

Quantum Mechanics

Super Symmetry

Supergravity

Yang-Mills Theory

SuperstringM-theory

StandardModel

Black hole Unitarity puzzle

(information problem)

UV problemNonrenormalizability

Gauge/stringCorrespondence

‘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.


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.


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)

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.

An earliest suggestion

toward background independence: conjecture of purely cubic action

talk “Approaches to string field theory”in ICOBAN’86 International conference on grand unification (site near Kamiokande)

!3

S =1

6!"3" # SWitten = !1

2""c" +

1

6"3"

"2c = 0, " = "c + "

! = V ! = 0 Q $ Q QD $ QD

|E| =Q

S#0V = |E|d Q = CV C =

S#0

dR

! b

a

E · dx =

= 0 $ 0 =

E % B

c v c $ v

N = 3NA R = kBNA NA = 6.02 % 1023 mol!1

N = 2 N = &

35

!3

S =1

6!"3" # SWitten = !1

2""c" +

1

6"3"

"2c = 0, " = "c + "

! = V ! = 0 Q $ Q QD $ QD

|E| =Q

S#0V = |E|d Q = CV C =

S#0

dR

! b

a

E · dx =

= 0 $ 0 =

E % B

c v c $ v

N = 3NA R = kBNA NA = 6.02 % 1023 mol!1

N = 2 N = &

35

!3

S =1

6!"3" # SWitten = !1

2""c" +

1

6"3"

"2c = 0, " = "c + "

! = V ! = 0 Q $ Q QD $ QD

|E| =Q

S#0V = |E|d Q = CV C =

S#0

dR

! b

a

E · dx =

= 0 $ 0 =

E % B

c v c $ v

N = 3NA R = kBNA NA = 6.02 % 1023 mol!1

N = 2 N = &

35

!3

S =1

6!"3" # S = !1

2""c" +

1

6"3"

"2c = 0, " = "c + "

! = V ! = 0 Q $ Q QD $ QD

|E| =Q

S#0V = |E|d Q = CV C =

S#0

dR

! b

a

E · dx =

= 0 $ 0 =

E % B

c v c $ v

N = 3NA R = kBNA NA = 6.02 % 1023 mol!1

N = 2 N = &

35

See also related works T.Y. PRL55(1985)1828, PLB197(1987)76

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.

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,

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)

4

Project of D-brane field theories

open-string field theories

effective Yang-Mills theories

D-brane field theories

closed-string field theories

open-closed duality

first quantization

or

second quantization

bosonization

or

Mandelstam duality

An analogy on the right-hand side: soliton operator ! Dirac fieldin the duality between sine-Gordon model and massive Thirring model

exp(!(x) ± i"(x)) ! #(x), $µ%&%" ! #'µ#(x) etc

an analogy : Mandelstam duality in 2D field theories

massive Thirring model

sine-Gordon model

D-particle field theory

closed string field theory

T.Y., arXiv:0705.1960[hep-th] (PTP118, 135 (2007)T.Y., arXiv:0804:0297[hep-th] (IJMPA23, 2343(2008)

Sextrebek = 2!

!Q1Q5Np

P =Np

R

d(N,Q1, Q5) ! exp 2!!

cN/6 ! exp 2!!

Q1Q3NP c = 4Q1Q5"3/2 N = NP

N ="

{ni}

niNni

gsQ ! g2YMQ # 1

g2YMQ $ 1, Q # 1

Rcurvature # "s

Seff =

#dt (

v2

2"sgs+a1

r

"2s

+a2"22v

2

r3+ · · · + b1

"6sv

4

r7+ b2

g2s"

13s v6

r14+ · · · )

v2

2gs"s

"

q=0, p=!1

cq,p

$gs"3

s

r3

%q $"4sv

2

r4

%p

p = q p < q p > q

#susyXiab ! $%i!ab , #susy!ab ! $Xab + · · ·

"Th % 0 & "Xh % '&

''''''''''''''(

1X1"1

X2"2

X3"3

···

XN"N

··

)

**************+

26

!

""""""""""""""#

1X1!1

X2!2

X3!3

···

XN!N

··

$

%%%%%%%%%%%%%%&

!+ !"

!+F!"

27

!

""""""""""""""#

1X1!1

X2!2

X3!3

···

XN!N

··

$

%%%%%%%%%%%%%%&

!+ !"

!+F!"

27

Fock space of D particle gauge theories with different

!

""""""""""""""#

1X1!1

X2!2

X3!3

···

XN!N

··

$

%%%%%%%%%%%%%%&

!+ !"

!+F!"

27

A Difficulty： necessity of non-associative structure

Gauge symmetry = quantum statistics of D-particle Hilbert space

Gauge invariants = bi-linears of D-particle fields

complex vectors,

z(1) =

!

""""""""#

x(1)1

······

$

%%%%%%%%&

, z(2) =

!

""""""""#

z(2)1

x(2)2

·····

$

%%%%%%%%&

, z(3) =

!

"""""""""#

z(3)1

z(3)2

x(3)3

····

$

%%%%%%%%%&

, z(4) =

!

"""""""""#

z(4)1

z(4)2

z(4)3

x(4)4

···

$

%%%%%%%%%&

together with their complex conjugates. The dots indicate inifinitely many dummy com-

ponents. Of course, both X and (z, z) are spatial vectors, with the corresponding indices

being suppressed.

Thus the D-particle fields creating and annihilating a D particle must be defined

conceptually as

!+ : |0! " !+[z(1), z(1)]|0! " !+[z(2), z(2)]!+[z(1), z(1)]|0! " · · · ,

!! : 0 # |0! # !+[z(1), z(1)]|0! # !+[z(2), z(2)]!+[z(1), z(1)]|0! # · · · .

The manner in which the degrees of freedom are added (or subtracted) is illustrated in

Fig. 1. As we explain below, the multiplication rules of the D-particle field operators are

Fig. 1: The D-particle coordinates and the open strings mediating them are denoted by blobs and linesconnecting them, respectively. The real lines are open-string degrees of freedom which have been createdbefore the latest operation of the creation field operator, while the dotted lines indicate those created bythe last operation. The arrows indicate the operation of creation (from left to right) and annihilation(from right to left) of D-particles.

actually not associative, nor commutative, and hence we need some special notation for

expressing N -body states with N $ 3. But for the moment, let us avoid such complica-

tions for the purpose of explaining only the key ideas in the case of simple 1-body and

2-body states.

9

Creation and annihilation of D particles and associated open strings

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

“cGh”

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P =#

h

$s =#

!!

30

It is possible to rewrite the entire content of Yang-Mills quantum mechanics for D particles with all different as an extended quantum field theory.

We also have

!!+,"

"zi!!"!!+,

"

"zi!!"|!" =

1

4

!ddX]

[dU(N)]

"Tr

# "

"X i

$Tr

# "

"X i

$![X]

%|N [X]". (3.28)

This relation holds because, for the N -body states, only the term !!+, "xn!!"2/4 con-

tributes, as seen from (3.21) and (3.22).

Collecting all these results, we have derived the identity

(4!!+,!!" + 1)

![ddX]

[dU(N)]

"Tr

# "

"X i

"

"X i

$![X]

%|N [X]"

= 4"(!!+,!!" + 1)!!+,

"

"zi· "

"zi!!" + 3!!+,

"

"zi!!"!!+,

"

"zi!!"|

%!". (3.29)

We also note that in our case of a flat spacetime, we can assume that the total center-of-

mass momentum vanishes, and hence we can set

Tr# "

"X i

$![X] = 0, (3.30)

or equivalently,

!!+, "zi!!" = 0, (3.31)

in order to simplify the formulas.

3.3 Schrodinger equation

We can now rewrite the Schrodinger equation in terms of these bilinear operators. In

configuration space, we have

i"

"t![X] = #Tr

#gs#s

2

"

"X i

"

"X i+

1

4gs#5s

[X i, Xj]2$![X]

= #1

2Tr

"gs#s

"

"X i

"

"X i+

1

gs#5s

(X iXjX iXj # X iX iXjXj)%![X]. (3.32)

Here we have suppressed the fermionic part, which is treated in the next section. In the

second-quantized form, this is expressed as

H|!" = 0, (3.33)

H = i(4!!+,!!" + 1)"t+

2gs#s

#(!!+,!!" + 1)!!+, "zi · "zi!!" + 3!!+, "zi!!" · !!+, "zi!!"

$

29

We also have

!!+,"

"zi!!"!!+,

"

"zi!!"|!" =

1

4

!ddX]

[dU(N)]

"Tr

# "

"X i

$Tr

# "

"X i

$![X]

%|N [X]". (3.28)




(4!!+,!!" + 1)

![ddX]

[dU(N)]

"Tr

# "

"X i

"

"X i

$![X]

%|N [X]"

= 4"(!!+,!!" + 1)!!+,

"

"zi· "

"zi!!" + 3!!+,

"

"zi!!"!!+,

"

"zi!!"|

%!". (3.29)



Tr# "

"X i

$![X] = 0, (3.30)

or equivalently,

!!+, "zi!!" = 0, (3.31)





i"

"t![X] = #Tr

#gs#s

2

"

"X i

"

"X i+

1

4gs#5s

[X i, Xj]2$![X]

= #1

2Tr

"gs#s

"

"X i

"

"X i+

1

gs#5s




H|!" = 0, (3.33)

H = i(4!!+,!!" + 1)"t+

2gs#s

#(!!+,!!" + 1)!!+, "zi · "zi!!" + 3!!+, "zi!!" · !!+, "zi!!"

$

29

We also have

!!+,"

"zi!!"!!+,

"

"zi!!"|!" =

1

4

!ddX]

[dU(N)]

"Tr

# "

"X i

$Tr

# "

"X i

$![X]

%|N [X]". (3.28)




(4!!+,!!" + 1)

![ddX]

[dU(N)]

"Tr

# "

"X i

"

"X i

$![X]

%|N [X]"

= 4"(!!+,!!" + 1)!!+,

"

"zi· "

"zi!!" + 3!!+,

"

"zi!!"!!+,

"

"zi!!"|

%!". (3.29)



Tr# "

"X i

$![X] = 0, (3.30)

or equivalently,

!!+, "zi!!" = 0, (3.31)





i"

"t![X] = #Tr

#gs#s

2

"

"X i

"

"X i+

1

4gs#5s

[X i, Xj]2$![X]

= #1

2Tr

"gs#s

"

"X i

"

"X i+

1

gs#5s




H|!" = 0, (3.33)

H = i(4!!+,!!" + 1)"t+

2gs#s

#(!!+,!!" + 1)!!+, "zi · "zi!!" + 3!!+, "zi!!" · !!+, "zi!!"

$

29

We also have

!!+,"

"zi!!"!!+,

"

"zi!!"|!" =

1

4

!ddX]

[dU(N)]

"Tr

# "

"X i

$Tr

# "

"X i

$![X]

%|N [X]". (3.28)




(4!!+,!!" + 1)

![ddX]

[dU(N)]

"Tr

# "

"X i

"

"X i

$![X]

%|N [X]"

= 4"(!!+,!!" + 1)!!+,

"

"zi· "

"zi!!" + 3!!+,

"

"zi!!"!!+,

"

"zi!!"|

%!". (3.29)



Tr# "

"X i

$![X] = 0, (3.30)

or equivalently,

!!+, "zi!!" = 0, (3.31)





i"

"t![X] = #Tr

#gs#s

2

"

"X i

"

"X i+

1

4gs#5s

[X i, Xj]2$![X]

= #1

2Tr

"gs#s

"

"X i

"

"X i+

1

gs#5s




H|!" = 0, (3.33)

H = i(4!!+,!!" + 1)"t+

2gs#s

#(!!+,!!" + 1)!!+, "zi · "zi!!" + 3!!+, "zi!!" · !!+, "zi!!"

$

29

We also have

!!+,"

"zi!!"!!+,

"

"zi!!"|!" =

1

4

!ddX]

[dU(N)]

"Tr

# "

"X i

$Tr

# "

"X i

$![X]

%|N [X]". (3.28)




(4!!+,!!" + 1)

![ddX]

[dU(N)]

"Tr

# "

"X i

"

"X i

$![X]

%|N [X]"

= 4"(!!+,!!" + 1)!!+,

"

"zi· "

"zi!!" + 3!!+,

"

"zi!!"!!+,

"

"zi!!"|

%!". (3.29)



Tr# "

"X i

$![X] = 0, (3.30)

or equivalently,

!!+, "zi!!" = 0, (3.31)





i"

"t![X] = #Tr

#gs#s

2

"

"X i

"

"X i+

1

4gs#5s

[X i, Xj]2$![X]

= #1

2Tr

"gs#s

"

"X i

"

"X i+

1

gs#5s




H|!" = 0, (3.33)

H = i(4!!+,!!" + 1)"t+

2gs#s

#(!!+,!!" + 1)!!+, "zi · "zi!!" + 3!!+, "zi!!" · !!+, "zi!!"

$

29+1

2gs!5s

(4!"+,"!" + 1)(!"+,"!" + 1)!"+,!(zi · zj)2 # (zi · zj)(zj · zi)

""!". (3.34)

In the large N limit and in the center-of-mass frame, this is simplified to

i#t|!" =## gs!s

2!"+, #zi · #zi"!" # !"+,"!"

2gs!5s

!"+,!(zi · zj)2 # (zi · zj)(zj · zi)

""!"

$|!".

Obviously, the Schrodinger equation commutes with the number operator !"+,"!"and with the translation operator !"+,

%"n=1

!!xi

n"!", according to the above discussion

of the global translation in our infinite-dimensional base space. We expect that after

appropriate ‘bosonization’ the former would be interpreted as a consequence of the gauge

symmetry associated with the RR gauge fields in the bulk. We also note the following.

As emphasized in some previous works [15] by the present author, Yang-Mills quantum

mechanics embodies the characteristic scales of D-particle dynamics in the form of scaling

invariance under

X i $ $X i, t $ $!1t, gs $ $3gs,

conforming to the space-time uncertainty relation "t"X ! !2s [16] as a qualitative char-

acterization of the non-locality of string theory. In the present field-theory formulation,

the same scaling symmetry is satisfied under

(zi, zi) $ $(zi, zi) t $ $!1t, gs $ $3gs,

provided that the field operators possess scaling such that the combination&

[d2dz]"±[z, z]

has zero scaling dimension. It is very desirable to clarify the geometrical meanings of all

these symmetries in terms of some intrinsic language which is suitable for our infinite-

dimensional base space.

4. Supercoordinates

In this section, we briefly describe the extension of our formalism to fermionic matrix

variables. They can be taken into account by introducing infinite-dimensional super-

coordinates for D-particle fields, through a straightforward extension of what we have

presented so far. Let us first recall the fermionic Hamiltonian in the configuration space,

Hf =1

2Tr

!#$i[X i, #]

", (4.1)

where the components of the spinor matrices, #, are Hermitian (in the sense of Majorana-

Weyl) and Grassmannian matrix coordinates, and the quantities $i are the SO(9) Dirac

30

(with some minor constraints)

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

“cGh”

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P =#

h

$s =#

!!

30

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.

What is the appropriate characterization of non-locality of strings?

In any theory of quantum gravity, if we set two of classical fundamental constants to unity,

Once we take into account gravity,

quantization=introduction of fundamental length

In string theory, the role of fundamental length is played

by the string length

g4o g2

s

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

cGh

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P = h

$s

24

g4o g2

s

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

“cGh”

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P =#

h

$s =#

!!

$P =#

h = f(!)$s

f(!) = exp (2%/(D & 2))

gs = e#

!! = $2s

pc = 4 g = 0 g = 1 g = 2 2g

24

This is also an unsolved question.

What is the appropriate characterization of non-locality of strings?

In any theory of quantum gravity, if we set two of classical fundamental constants to unity,

Once we take into account gravity,

quantization=introduction of fundamental length

In string theory, the role of fundamental length is played

by the string length

g4o g2

s

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

cGh

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P = h

$s

24

g4o g2

s

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

“cGh”

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P =#

h

$s =#

!!

$P =#

h = f(!)$s

f(!) = exp (2%/(D & 2))

gs = e#

!! = $2s

pc = 4 g = 0 g = 1 g = 2 2g

24

This is also an unsolved question.

The space-time uncertainty relation was originally

proposed with this in mind. In the absence of any

definite mathematical formalism and axioms, only

way was to adopt a qualitative approach.

(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)

Time-energy uncertainty relation

is reinterpreted as the time-space relation.

!P =!

h = f(!)!s

f(!) = exp (2"/(D " 2))

gs = e!

#! = !2s

pc = 4 g = 0 g = 1 g = 2 2g

" = "1 + "2, "1 # "2 = $

#E#t ! h

#E % #Xh

!2s

#t = #T

25

!P =!

h = f(!)!s

f(!) = exp (2"/(D " 2))

gs = e!

#! = !2s

pc = 4 g = 0 g = 1 g = 2 2g

" = "1 + "2, "1 # "2 = $

#E#t ! h

#E % #Xh

!2s

#t = #T

25

Space-Time Uncertainty Principle 5

the violation of unitarity and/or locality. However, string perturbation theory isperfectly consistent with (perturbative) unitarity, preserving all the important ax-ioms for a physically acceptable S-matrix, including the analyticity property of theS-matrix. It should be recalled that the analyticity of the S-matrix is customarilyattributed to locality, in addition to unitarity, of quantum field theories. However,locality is usually not expected to be valid in theories with extended objects. Fromthis point of view, it is not at all trivial to understand why string theory is free fromthe ultraviolet di!culty, and it is important to give correct interpretations to itsmechanism.

2.1. A reinterpretation of energy-time uncertainty relation in terms of stringsThe approach which was proposed in Ref. 18) is to reinterpret the ordinary

energy-time uncertainty relation in terms of the space-time extension of strings:!)

!E!t >! 1. (2.1)

The basic reason why we have ultraviolet divergencies in local quantum field theoriesis that in the short time region, !t " 0, the uncertainty with respect to energyincreases indefinitely: !E ! 1/!t " #. This in turn induces a large uncertaintyin momentum !p ! !E. The large uncertainty in the momentum implies thatthe number of particles states allowed in the short distance region !x ! 1/!pgrows indefinitely as (!E)D"1 in D-dimensional space-time. In ordinary local fieldtheories, where there is no cuto" built-in, all these states are expected to contributeto amplitudes with equal strengths. This consequently leads to UV infinities.

What is the di"erence, in string theory, regarding this general argument? Actu-ally, the number of the allowed states with a large energy uncertainty !E behavesas ek!s"E ! ek!s/"t with some positive coe!cient k, and "s $

%## being the string

length constant, where ## is the traditional slope parameter. This increase of thedegeneracy is much faster than that in local field theories. The crucial di"erencewith local field theories, however, is that the dominant string states among these ex-ponentially degenerate states are not the states with large center-of-mass momenta,but can be the massive states with higher excitation modes along strings. The excita-tion of higher modes along strings contributes to the large spatial extension of stringstates. It seems reasonable to expect that this e"ect completely cancels the short dis-tance e"ect with respect to the center-of-mass coordinates of strings, provided thatthese higher modes contribute appreciably to physical processes. Since the order ofmagnitude of the spatial extension corresponding to a large energy uncertainty !Eis expected to behave as !X ! "2

s!E, we are led to a remarkably simple relation forthe order of magnitude !X for fluctuations along spatial directions of string statesparticipating within the time interval !T = !t of interactions:

!X!T >! "2s. (2.2)

It is natural to call this relation the ‘space-time uncertainty relation’. It should beemphasized that this relation is not a modification of the usual uncertainty relation,

!) Throughout the present paper, we use units in which h = 1, c = 1.

This was derived by quantizing strings in the time-like gauge.

If we consider the high-energy regime,


and space components, we have

!!"[X0, Xi]2# =

"12!"[Xi, Xj ]2# + !4

s>$ !2

s. (2.6)

This shows that the inequality (2.2) of the space-time uncertainty relation can inprinciple be consistent with Lorentz invariance, conforming to the first argument.This also suggests a possible definition of the proper frame such that the noncom-mutativity of space-like operators is minimized. To avoid a possible misconception,however, it should be noted that the present formal argument is not meant to implythat the author is proposing that the operator constraint (2.5) is the right way forrealizing the space-time uncertainty principle. In particular, it is not at all obviouswhether the uncertainties can be defined using Lorentz vectors, since they are notlocal fields. Here it is only used for an illustrative purpose to show schematically thecompatibility of the space-time uncertainty relation with Lorentz invariance. Theremight be better way of formulating the principle in a manifestly Lorentz invari-ant manner. We will later present a related discussion (§5) from the viewpoint ofnoncommutative geometric quantization of strings based on the Schild action.

2.3. Conformal symmetry and the space-time uncertainty relationFor the sake of completeness, we now explain an independent derivation of the

space-time uncertainty relation on the basis of conformal invariance of the world-sheet string dynamics, following an old work. 19) This derivation seems to supportour proposal that the space-time uncertainty relation should be valid universally inboth short-time and long-time limits.

All the string amplitudes are formulated in terms of path integrals as weightedmappings from the set of all possible Riemann surfaces to a target space-time. There-fore, any characteristic property of the string amplitudes can be understood fromthe property of this path integral. The absence of the ultraviolet divergences instring theory from this point of view is a consequence of the modular invariance. Wewill see that the space-time uncertainty relation (2.2) can be regarded as a naturalgeneralization of the modular invariance for arbitrary string amplitudes in terms ofthe direct space-time language.

Let us start by briefly recalling how to define the distance on a Riemann surfacein a conformally invariant manner. For a given Riemannian metric ds = "(z, z)|dz|,an arc # on the Riemann surface has length L(#, ") =

#! "|dz|. This length is, how-

ever, dependent on the choice of the metric function ". If we consider some finiteregion $ and a set of arcs defined on $, the following definition, called the ‘extremallength’ in mathematical literature, 27) is known to give a conformally invariant defi-nition for the length of the set % of arcs:

&"(% ) = sup#

L(%, ")2

A($, ")(2.7)

withL(%, ") = inf

!!$L(#, "), A($, ") =

$

""2dzdz.



!!"[X0, Xi]2# =

"12!"[Xi, Xj ]2# + !4

s>$ !2

s. (2.6)








&"(% ) = sup#

L(%, ")2

A($, ")(2.7)

withL(%, ") = inf

!!$L(#, "), A($, ") =

$

""2dzdz.



!!"[X0, Xi]2# =

"12!"[Xi, Xj ]2# + !4

s>$ !2

s. (2.6)








&"(% ) = sup#

L(%, ")2

A($, ")(2.7)

withL(%, ") = inf

!!$L(#, "), A($, ") =

$

""2dzdz.



!!"[X0, Xi]2# =

"12!"[Xi, Xj ]2# + !4

s>$ !2

s. (2.6)








&"(% ) = sup#

L(%, ")2

A($, ")(2.7)

withL(%, ") = inf

!!$L(#, "), A($, ") =

$

""2dzdz.



!!"[X0, Xi]2# =

"12!"[Xi, Xj ]2# + !4

s>$ !2

s. (2.6)








&"(% ) = sup#

L(%, ")2

A($, ")(2.7)

withL(%, ") = inf

!!$L(#, "), A($, ") =

$

""2dzdz.

arbitrarily chosen world-sheet metric



!!"[X0, Xi]2# =

"12!"[Xi, Xj ]2# + !4

s>$ !2

s. (2.6)








&"(% ) = sup#

L(%, ")2

A($, ")(2.7)

withL(%, ") = inf

!!$L(#, "), A($, ") =

$

""2dzdz.

This relation can also be derived directly and in a more universal way using world-sheet conformal invariance. (T.Y., MPL 1989 : for an extensive review, see Prog. Theor. Phys. 103: 1081--1125(2000))

For arbitrary quadrilaterals on the world sheet, we have conformal invariants, called the extremal length

which corresponds to proper time of particle trajectory

12 T. Yoneya

Since any Riemann surface corresponding to a string amplitude can be decomposedinto a set of quadrilaterals pasted along the boundaries (with some twisting oper-ations, in general), it is su!cient to consider the extremal length for an arbitraryquadrilateral segment !. Let the two pairs of opposite sides of ! be ","! and #,#!.Take $ be the set of all connected set of arcs joining " and "!. We also define theconjugate set of arcs $ " be the set of arcs joining # and #!. We then have twoextremal distances, %!($ ) and %!($ "). The important property of the extremallength for us is the reciprocity

%!($ )%!($ ") = 1. (2.8)

Note that this implies that one of the two mutually conjugate extremal lengths islarger than 1.

The extremal lengths satisfy the composition law, which partially justifies thenaming “extremal length”: Suppose that !1 and !2 are disjoint but adjacent openregions on an arbitrary Riemann surface. Let $1 and $2 consist of arcs in !1 and!2, respectively. Let ! be the union !1 + !2, and let $ be a set of arcs on !.

1. If every & ! $ contains a &1 ! $1 and &2 ! $2, then

%!($ ) " %!1($1) + %!2($2).

2. If every &1 ! $1 and &2 ! $2 contains a & ! $ , then

1/%!($ ) " 1/%!1($1) + 1/%!2($2).

These two cases correspond to two di"erent types of compositions of open regions,depending on whether the side where !1 and !2 are joined does not divide the sideswhich & ! $ connects, or do divide, respectively. One consequence of the compositionlaw is that the extremal length from a point to any finite region is infinite and thecorresponding conjugate length is zero. This corresponds to the fact that the vertexoperators describe the on-shell asymptotic states whose coe!cients are representedby local external fields in space-time. We also recall that the moduli parameters ofworld-sheet Riemann surfaces are nothing but a set of extremal lengths with someassociated angle variables, associated with twisting operations, which are necessaryin order to specify the joining of the boundaries of quadrilaterals.

Conformal invariance allows us to conformally map any quadrilateral to a rect-angle on the Gauss plane. Let the Euclidean lengths of the sides (","!) and (#,#!)be a and b, respectively. Then, the extremal lengths are given by the ratios

%($ ) = a/b, %($ ") = b/a. (2.9)

For a proof, see Ref. 27)

Let us now consider how the extremal length is reflected by the space-timestructure probed by general string amplitudes. The euclidean path-integral in theconformal gauge is essentially governed by the action 1

"2s

!! dzdz 'zxµ'zxµ. Take a

rectangular region as above and the boundary conditions (z = (1 + i(2) as

xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

composition theorem :

g4o g2

s

eH = y

z =L2

r

J =1

2!!E2

g2YMN ! 1

S = A/4GN A = 4"R2 R = 2GNM

N " eSbh

Sbh = 2"#

n1n5np

gµ!(x) $ e"(x)gµ!(x) % #(x) $ ei"(x)#(x)

gµ! % gµ! + aµ!

“cGh”

$P =

!Gh

c3" 10"33 cm

G = c = 1 % $P =#

h

$s =#

!!

$P =#

h = f(!)$s

f(!) = exp (2%/(D & 2))

gs = e#

!! = $2s

pc = 4 g = 0 g = 1 g = 2 2g

" = "1 + "2, "1 ' "2 = (

24

reciprocity theorem :

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

:

:

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

The geometrical properties of target space-time are related through these conformal invariants

a

b

A

B

Riemann sheet Space-Time

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

12 T. Yoneya


%!($ )%!($ ") = 1. (2.8)




%!($ ) " %!1($1) + %!2($2).


1/%!($ ) " 1/%!1($1) + 1/%!2($2).



%($ ) = a/b, %($ ") = b/a. (2.9)



"2s



xµ(0, (2) = xµ(a, (2) = )µ2B(2/b,

xµ((1, 0) = xµ((1, b) = )µ1A(1/a.

contribution to the amplitude :


The boundary conditions are chosen such that the kinematical momentum constraint!1xµ · !2xµ = 0 in the conformal gauge is satisfied for the classical solution.!) Thepath integral then contains the factor

exp!! 1

"2s

"A2

#($ )+

B2

#($ !)

#$.

This indicates that the square root of extremal length can be used as the measureof the length probed by strings in space-time. The appearance of the square root isnatural, as suggested from the definition (2.7):

%A =%"A2# $

%#($ )"s, %B =

%"B2# $

%#($ )"s.

In particular, this implies that probing short distances along both directions si-multaneously is always restricted by the reciprocity property (2.8) of the extremallength, %A%B $ "2

s. In Minkowski metric, one of the directions is time-like andthe other is space-like, as required by the momentum constraint. This conforms tothe space-time uncertainty relation, as derived in the previous subsection from avery naive argument. Also note that the present discussion clearly shows that thespace-time uncertainty relation is a natural generalization of modular invariance, orof open-closed string duality, exhibited by the string loop amplitudes.

Since our derivation relies on conformal symmetry and is applicable to arbi-trary quadrilaterals on arbitrary Riemann surfaces, which in turn can always beconstructed by pasting quadrilaterals appropriately, we expect that the space-timeuncertainty relation is robust with respect to possible corrections to the simple setupof the present argument. In particular, the relation, being independent of the stringcoupling, is expected to be universally valid to all orders of string perturbation the-ory. Since the string coupling cannot be regarded as the fundamental parameter ofnonperturbative string theory, it is natural to expect that any universal principleshould be formulated independently of the string coupling.

We have assumed a smooth boundary condition at the boundary of the rectan-gle. This leads to a saturation of the inequality of the uncertainty relation. If weallow more complicated ‘zigzag’ shapes for boundaries, it is not possible to establishsuch a simple relation as that above between the extremal distance and the space-time uncertainties. However, we can expect that the mean values of the space-timedistances measured along the boundaries of complicated shapes in general increase,due to the entropy e!ect, in comparison with the case of smooth boundaries (namelythe zero mode) obtained as the average of given zigzag curves. Although there is nogeneral proof, any reasonable definition of the expectation value of the space-timedistances conforms to this expectation, since the fluctuations contribute positivelyto the expectation value. Thus the inequality (2.2) should be the general expressionof the reciprocity relation (2.8) in a direct space-time picture. Since the relationis symmetric under the interchange $ % $ !, it is reasonable to suppose that thespace-time uncertainty relation is meaningful in both limits %T & 0 and %T & ',as we proposed in the previous subsection.

!) The Hamilton constraint (!1x)2 = (!2x)2 is satisfied after integrating over the moduli pa-rameter, which in the present case of a rectangle is the extremal length itself.



exp!! 1

"2s

"A2

#($ )+

B2

#($ !)

#$.


%A =%"A2# $

%#($ )"s, %B =

%"B2# $

%#($ )"s.








exp!! 1

"2s

"A2

#($ )+

B2

#($ !)

#$.


%A =%"A2# $

%#($ )"s, %B =

%"B2# $

%#($ )"s.








exp!! 1

"2s

"A2

#($ )+

B2

#($ !)

#$.


%A =%"A2# $

%#($ )"s, %B =

%"B2# $

%#($ )"s.






!

""""""""""""""#

1X1!1

X2!2

X3!3

···

XN!N

··

$

%%%%%%%%%%%%%%&

!+ !"

!+F!"

!T ! scale along the longitudinal directions

!X ! scale along the transverse directions

'"("#)#s

27

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.

Space-time uncertainty relation implies that the longitudinal extension of strings in the near horizon region is arbitrary large.

Then there is almost no meaning in considering horizons using local field approximations. Information puzzle must be formulated by taking due account of non-local nature of strings.

as we can derive in the Minkowski coordinates.

Thus in the limit of small !T , the uncertainty with respect to X1 is arbitrarily large.

!X1 !!2s

!T" #

Therefore the Hilbert space of string states can never be decomposed into separate Hilbert

space corresponding to a single wedge. Both wedges must be treated on an equal footing.

This means that there is no thermalization. It should be emphasized that in string theory,

the definition of space-time distances itself must be reconstructed (or reemerged) from

the physical properties of quantum strings. Thus the concept of horizon is at best some

kind of approximate concept which can only be used in low-energy e"ective theory whose

validity is restricted only to the description of space-times observed macroscopically from

remote distances.

For comparison with this, let us again consider the case of particles. Although there

is no intrinsic spatial extension, we still have uncertainties with respect to its position.

Does this destroy the concept of horizon? Answer seems No. The basic reason for this is

that the e"ective mass of a particle becomes arbitrarily large as X1 " 0. The momentum

and position uncertainty relation

!X1!P1 ! 1 (107)

reduces, for a given energy P!

!X1 ! 1

!P1! X2

1

!P!V1(108)

where V1 is the velocity V1 = dX1d! , if we assume that the uncertainty of P1 is due to that

of energy P! . Thus even if the uncertainty of energy is small the uncertainty of X1 can

become arbitrarily small as particle approach the horizon. The energy-time uncertainty

relation, on the other hand,

!P!!" ! 1 (109)

Thus when the resolution in " increases, the uncertainty increases as

!P! ! 1

!"

17

Essentially the same relation is noticed later by Susskind (1994)

who also emphasized its relevance in black hole physics.

Note on quantum string theory in the Rindler spacetime

Note by Tamiaki Yoneya

2014

1 The Rinder space

As is well known, the space-time coordinates which are appropriate for describing the

uniformly accelerated observers the Rindler coordinates defined by

x1 = X1 cosh !, x0 = X1 sinh !, xi = Xi (i = 2, . . . , D ! 1) (1)

In terms of these, the metric is

ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)

A trajectory of an accelerated observer is given by X1 = R, "!R = 0, and the proper

time is ! = R! . The range of the Rindler time ! is from !" to +". The ±sign of X1

corresponds to right ( I ) and left ( II ) Rindler wedge, respectively. If we restrict ourselves

to one of these two wedges, we can set X1 = ±Re" (R= a positive constant) and then the

range of # is from !" to +", the former of which corresponds to the horizon, and the

metric is conformal to the usual Minkowski metric,

ds2 = R2e2""!(d!)2 + (d#)2

#(3)

It is also sometimes convenient to define, in the left and right wedge, respectively

U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)

The Rindler horizons correspond to UV = 0.

1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1

Penrose diagram of Schwarzschild space-time

Rindler wedges of Minkowski space-time



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


As is well known, this metric can be regarded as an approximation of the near-horizon

region r $ 2GM of the Schwarzschild spacet-time.

ds2 = !$

1 ! 2GM

r

%dt2 +

1

1 ! 2GMr

dr2 + r2d2! (6)

1

The appropriate coordinate change is

dr!1 ! 2GM

r

= dX1,

"X1 = 4GM

#1 ! 2GM

r

$,

t

4GM= ! (7)

the two-dimensional part of the Schwarzschild metric is transformed into the two-dimensional

part of the Rindler metric

!%

1 ! 2GM

r

&(dt)2 +

1

1 ! 2GMr

(dr)2 " !X21 (d!)2 + (dX1)

2 (8)

2 Quantum Particle

Let us start from the reparametrization invariant relativistic action (e(") ='

#(") is the

einbein auxiliary variable).

S[x("), #(")] =

(d"

1

2

'#(")

)#(")!1

*ds

d"

+2! m2

,

=

(d"

1

2

-!e(")!1X2

1

%d!

d"

&2

+ e(")!1

%dX1

d"

&2

+ e(")!1.

i

%dXi

d"

&2

! m2e(")

/

(9)

Thus the equations of motion and the constraint are

d

d"

%X2

1

e(")

d!

d"

&= 0 (10)

d

d"

%1

e(")

dX1

d"

&+ e(")!1X1

%d!

d"

&2

= 0 (11)

d

d"

%1

e(")

dXi

d"

&= 0 (i = 2, . . . , D ! 1) (12)

X21

%d!

d"

&2

!%

dX1

d"

&2

!.

i

%dXi

d"

&2

= m2e(")2 (13)

The conjugate momenta are

P! = !e!1X21

d!

d"(14)

P1 = e!1dX1

d"(15)

P = e!1dX

d"(16)

2

The appropriate coordinate change is

dr!1 ! 2GM

r

= dX1,

"X1 = 4GM

#1 ! 2GM

r

$, ! =

t

4GM(7)

the two-dimensional part of the Schwarzschild metric is transformed into the two-dimensional

part of the Rindler metric

!%

1 ! 2GM

r

&(dt)2 +

1

1 ! 2GMr

(dr)2 " !X21 (d!)2 + (dX1)

2 (8)

2 Quantum Particle

Let us start from the reparametrization invariant relativistic action (e(") ='

#(") is the

einbein auxiliary variable).

S[x("), #(")] =

(d"

1

2

'#(")

)#(")!1

*ds

d"

+2! m2

,

=

(d"

1

2

-!e(")!1X2

1

%d!

d"

&2

+ e(")!1

%dX1

d"

&2

+ e(")!1.

i

%dXi

d"

&2

! m2e(")

/

(9)

Thus the equations of motion and the constraint are

d

d"

%X2

1

e(")

d!

d"

&= 0 (10)

d

d"

%1

e(")

dX1

d"

&+ e(")!1X1

%d!

d"

&2

= 0 (11)

d

d"

%1

e(")

dXi

d"

&= 0 (i = 2, . . . , D ! 1) (12)

X21

%d!

d"

&2

!%

dX1

d"

&2

!.

i

%dXi

d"

&2

= m2e(")2 (13)

The conjugate momenta are

P! = !e!1X21

d!

d"(14)

P1 = e!1dX1

d"(15)

P = e!1dX

d"(16)

2



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)




ds2 = !$

1 ! 2GM

r

%dt2 +

1

1 ! 2GMr

dr2 + r2d2! (6)

1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)



corresponds to right ( I X1 > 0) and left ( II X1 < 0) Rindler wedge, respectively. If

we restrict ourselves to one of these two wedges, we can set X1 = ±Re" (R= a positive

constant) and then the range of # is from !" to +", the former of which corresponds

to the horizon, and the metric is conformal to the usual Minkowski metric,

ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)




ds2 = !$

1 ! 2GM

r

%dt2 +

1

1 ! 2GMr

dr2 + r2d2! (6)

1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)



corresponds to right ( I X1 > 0) and left ( II X1 < 0) Rindler wedge, respectively. If

we restrict ourselves to one of these two wedges, we can set X1 = ±Re" (R= a positive

constant) and then the range of # is from !" to +", the former of which corresponds

to the horizon, and the metric is conformal to the usual Minkowski metric,

ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)




ds2 = !$

1 ! 2GM

r

%dt2 +

1

1 ! 2GMr

dr2 + r2d2! (6)

1

near horizon approximation

String theory in the Rindler coordinates



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)

A trajectory of an accelerated observer is given by X1 = R, "T R = 0, and the proper

time is T = R! . The range of the Rindler time T is from !" to +". The ±sign of X1

corresponds to right and left Rindler wedge, respectively. If we restrict ourselves to one

of these two wedges, we can set X1 = ±Re! (R= a positive constant) and then the range

of # is from !" to +", the former of which corresponds to the horizon, and the metric

is conformal to the usual Minkowski metric,

ds2 = R2e2!"!(d!)2 + (d#)2

#(3)


U = #e!!" , V = ±e!+" (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2!"!(d!)2 + (d#)2

#(3)


U = #e!!" , V = ±e!+" (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)




of these two wedges, we can set X1 = ±Re" (R= a positive constant) and then the range



ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2""!(d!)2 + (d#)2

#(3)


U = #e"!! , V = ±e"+! (4)

which gives

ds2 = !R2dUdV (5)


1

This implies that any observable restricted in a single wedge is expressed as statistical

average

!0M|O±|0M" = Tr±!e!2!"b±!,k b±†

!,!kO±"

(66)

which corresponds to temperature T = 1/2!, with respect to the energy scale of the

Rindler time " .

Now let us consider the Feynman propagator using the Minkowski vacuum, using the

Rindler coordinates in the right wedge.

3 Bosonic string in the Rindler coordinates

Now let us extend the previous discussions to strings. The action is

Sstring[x(",#)] = # 1

4!$"

##d2%

$#&(%)&ab(%)gµ#(x)

'xµ

'%a

'x#

'%b(67)

where &ab(%) is the metric for two-dimensional world sheet, and gµ#(x) is that of the target

space-time. In the Rindler space, we have

Sstring = # 1

4!$"

## $#&(%)&ab(%)

%#X2

1

'"

'%a

'"

'%b+

'X1

'%a

'X1

'%b+

'X

'%a· 'X

'%b

&(68)

In the particle limit $$x1

$ 0 and &a1 = 0, this reduces to the previous particle action

with m = 0. The equations of motion and constraints are

'

'%a

%#&&abX2

1

'"

'%b= 0 (69)

'

'%a

%#&&ab 'X1

'%b+%#&&abX1

'"

'%a

'"

'%b= 0 (70)

'

'%a

%#&&ab 'X

'%b= 0 (71)

#X21

'"

'%a

'"

'%b+

'X1

'%a

'X1

'%b+

'X

'%a· 'X

'%b

=1

2&ab&

cd

%#X2

1

'"

'%c

'"

'%d+

'X1

'%c

'X1

'%d+

'X

'%c· 'X

'%d

&(72)

There are only two independent constraints, due to the world-sheet Weyl invariance under

&ab(%) $ ((%)&ab(%). The canonical momenta are

P% = #)%#&&a0X2

1

'"

'%a(73)

10


average

!0M|O±|0M" = Tr±!e!2!"b±!,k b±†

!,!kO±"

(66)


Rindler time " .





Sstring[x(",#)] = # 1

4!$"

##d2%

$#&(%)&ab(%)gµ#(x)

'xµ

'%a

'x#

'%b(67)



Sstring = # 1

4!$"

## $#&(%)&ab(%)

%#X2

1

'"

'%a

'"

'%b+

'X1

'%a

'X1

'%b+

'X

'%a· 'X

'%b

&(68)




'

'%a

%#&&abX2

1

'"

'%b= 0 (69)

'

'%a

%#&&ab 'X1

'%b+%#&&abX1

'"

'%a

'"

'%b= 0 (70)

'

'%a

%#&&ab 'X

'%b= 0 (71)

#X21

'"

'%a

'"

'%b+

'X1

'%a

'X1

'%b+

'X

'%a· 'X

'%b

=1

2&ab&

cd

%#X2

1

'"

'%c

'"

'%d+

'X1

'%c

'X1

'%d+

'X

'%c· 'X

'%d

&(72)



P% = #)%#&&a0X2

1

'"

'%a(73)

10


average

!0M|O±|0M" = Tr±!e!2!"b±!,k b±†

!,!kO±"

(66)


Rindler time " .





Sstring[x(",#)] = # 1

4!$"

##d2%

$#&(%)&ab(%)gµ#(x)

'xµ

'%a

'x#

'%b(67)



Sstring = # 1

4!$"

## $#&(%)&ab(%)

%#X2

1

'"

'%a

'"

'%b+

'X1

'%a

'X1

'%b+

'X

'%a· 'X

'%b

&(68)




'

'%a

%#&&abX2

1

'"

'%b= 0 (69)

'

'%a

%#&&ab 'X1

'%b+%#&&abX1

'"

'%a

'"

'%b= 0 (70)

'

'%a

%#&&ab 'X

'%b= 0 (71)

#X21

'"

'%a

'"

'%b+

'X1

'%a

'X1

'%b+

'X

'%a· 'X

'%b

=1

2&ab&

cd

%#X2

1

'"

'%c

'"

'%d+

'X1

'%c

'X1

'%d+

'X

'%c· 'X

'%d

&(72)



P% = #)%#&&a0X2

1

'"

'%a(73)

10

There are a large number of previous works studying string

theory in the Rindler coordinates, but they are unsatisfactory

in dealing with the Virasoro constraints or in using

unjustifiable gauges. It is in general impossible (or inconsistent) to satisfy the momentum constraint if one restricts oneself to incomplete space-like surfaces.

Physically, the most natural gauge choice is the time-like and orthogonal gauge.

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)

For comparison with the particle case and also making physical picture as clearly

as possible, we choose the time-like gauge $0 = ' . This still leaves us the freedom

of arbitrary transformation of spatial coordinate $1 # (. We consider closed strings

assuming periodicity in ( with periodicity ( $ ( + 2% for all dynamical variables at

any ' . The residual arbitrariness of the ( reparametrization must respect the periodicity.

We can further assume that the orthogonality condition "01 = 0 is satisfied. Indeed the

infinitesimal form of the ( reparametrization )( = f((, ') = f(( + 2%, ') is

)"10 = #!f((, ') (77)

which can always make infinitesimal but arbitrary periodic function )"10((, ') to zero by

appropriately choosing the periodic transformation function f((, ') = f(( + 2%, '). For

finite function "10 we can solve the gauge condition perturbatively. So in principle there

is no problem in this gauge choice.

The residual gauge freedom is then arbitrary time independent re-parametrization of

(. In this time-like gauge, the momenta reduce to

P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)

The total Hamiltonian which generates translation of ' is

H = "2%P! (81)

The equations of motion now take the form

#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

residual freedom= time-independent reparametrization of

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

canonical momenta

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

eqs. of motion

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

!

!"

!!#11

#00

!X1

!"! !

!$

!!#00

#11

!X1

!$+

!!#11

#00X1 = 0 (83)

!

!"

!!#11

#00

!X

!"! !

!$

!!#00

#11

!X

!$= 0 (84)

Note that, if we set %"!!11

!00= 1/e where e is the auxiliary variable of particle case, these

equations precisely correspond to the particle case. Note also that #00 < 0, #11 > 0. Now,

(82) shows that"!!11

!00X2

1 is independent of time, and hence, using a time-independent

reparametrization of $, #!11(",$

!) =#

d"d"!

$2#11(",$), we can choose $ such that

%

!!#11

#00X2

1 = e (85)

with e being the positive and constant energy density, which is consistent with the equation

of motion above. Thus, the spatial components of momenta are

P1 =e

X21

!X1

!"(86)

P =e

X21

!X

!"(87)

The equations of motion for other variables become

1

%

!

!"

e

X21

!X1

!"! %

!

!$

X21

e

!X1

!$+

1

%

e

X21

X1 = 0 (88)

1

%

!

!"

e

X21

!X

!"! %

!

!$

X21

e

!X

!$= 0 (89)

On the other hand, the Hamilton constraint is

! X21 +

!X1

!"

!X1

!"+

!X

!"· !X

!"

=1

2#00#

00

%!X2

1 +!X1

!"

!X1

!"+

!X

!"· !X

!"

&

+1

2#00#

11

%!X1

!$

!X1

!$+

!X

!$· !X

!$

&

=1

2

%!X2

1 +!X1

!"

!X1

!"+

!X

!"· !X

!"

&! %2

2

X41

e2

%!X1

!$

!X1

!$+

!X

!$· !X

!$

&

which is

!X21 +

!X1

!"

!X1

!"+

!X

!"· !X

!"= !%2X4

1

e2

%!X1

!$

!X1

!$+

!X

!$· !X

!$

&(90)

12

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

!

!"

!!#11

#00

!X1

!"! !

!$

!!#00

#11

!X1

!$+

!!#11

#00X1 = 0 (83)

!

!"

!!#11

#00

!X

!"! !

!$

!!#00

#11

!X

!$= 0 (84)

Note that, if we set %"!!11

!00= 1/e where e is the auxiliary variable of particle case, these

equations precisely correspond to the particle case. Note also that #00 < 0, #11 > 0. Now,

(82) shows that"!!11

!00X2

1 is independent of time, and hence, using a time-independent

reparametrization of $, #!11(",$

!) =#

d"d"!

$2#11(",$), we can choose $ such that

%

!!#11

#00X2

1 = e (85)

with e being the positive and constant energy density, which is consistent with the equation

of motion above. Thus, the spatial components of momenta are

P1 =e

X21

!X1

!"(86)

P =e

X21

!X

!"(87)

The equations of motion for other variables become

1

%

!

!"

e

X21

!X1

!"! %

!

!$

X21

e

!X1

!$+

1

%

e

X21

X1 = 0 (88)

1

%

!

!"

e

X21

!X

!"! %

!

!$

X21

e

!X

!$= 0 (89)

On the other hand, the Hamilton constraint is

! X21 +

!X1

!"

!X1

!"+

!X

!"· !X

!"

=1

2#00#

00

%!X2

1 +!X1

!"

!X1

!"+

!X

!"· !X

!"

&

+1

2#00#

11

%!X1

!$

!X1

!$+

!X

!$· !X

!$

&

=1

2

%!X2

1 +!X1

!"

!X1

!"+

!X

!"· !X

!"

&! %2

2

X41

e2

%!X1

!$

!X1

!$+

!X

!$· !X

!$

&

which is

!X21 +

!X1

!"

!X1

!"+

!X

!"· !X

!"= !%2X4

1

e2

%!X1

!$

!X1

!$+

!X

!$· !X

!$

&(90)

12

constant energy density

P1 = !!"""a0#X1

#$a(74)

P = !!"""a0#X

#$a(75)

where

! =1

2%&! (76)








)"10 = #!f((, ') (77)







P! = "!

!""11

"00X2

1 (78)

P1 = !

!""11

"00

#X1

#'(79)

P = !

!""11

"00

#X

#'(80)


H = "2%P! (81)


#

#'P! = "!

#

#'

!""11

"00X2

1 = 0 (82)

11

(closed string: )

Virasoro conditions

!

X21 =

X41

e2

!P 2

1 + P 2 + !2"X1

"#

"X1

"#+ !2"X

"#· "X

"#

"(91)

!

e2 = X21

#P 2

1 + P 2 + !2

$"X1

"#

%2

+ !2

$"X

"#

%2&

(92)

Similarly, the momentum constraint is

P1"X1

"#+ P · "X

"#= 0 (93)

Let us first study the equation for X1 in comparison with the case of particle. We use

the same variable as in the latter case for time z = e! and also w " ei".

z2"2X1

"z2+ z

"X1

"z# 2z2

X1

$"X1

"z

%2

+ X1 # !2 "

"#

X21

e2

"X1

"#= 0

Assume the following ansatz for small z behavior.

X1 $ az#(1 + O(z$)) $ > 0

In the leading order,

% a%(% # 1)z# + a%z# # 2a%z2# + az# # !2

e2

d

d#a2 da

d#z3# = 0

which is only allowed with % = 0 if a satisfies

a # !2

e2

d

d#a2 da

d#= 0 & w

d

dwa2w

d

dwa # e2

!2a = 0

& d2a

dw2+

2

a

$da

dw

%2

+1

w

da

dw# e2

!2w2a= 0 (94)

To investigate other possibilities, let us take into account the Hamilton constraint. It

suggests that, for small X1, the contribution of'

%X1%"

(2can be as large as 1

X21. Then the

last term of the equation of motion can have significant e!ect

# "

"#X2

1

"X1

"#= #2X1

$"X1

"#

%2

+ X21

"2X1

"#2$ # 2

X1

13

!

X21 =

X41

e2

!P 2

1 + P 2 + !2"X1

"#

"X1

"#+ !2"X

"#· "X

"#

"(91)

!

e2 = X21

#P 2

1 + P 2 + !2

$"X1

"#

%2

+ !2

$"X

"#

%2&

(92)

Similarly, the momentum constraint is

P1"X1

"#+ P · "X

"#= 0 (93)

Let us first study the equation for X1 in comparison with the case of particle. We use

the same variable as in the latter case for time z = e! and also w " ei".

z2"2X1

"z2+ z

"X1

"z# 2z2

X1

$"X1

"z

%2

+ X1 # !2 "

"#

X21

e2

"X1

"#= 0

Assume the following ansatz for small z behavior.

X1 $ az#(1 + O(z$)) $ > 0

In the leading order,

% a%(% # 1)z# + a%z# # 2a%z2# + az# # !2

e2

d

d#a2 da

d#z3# = 0

which is only allowed with % = 0 if a satisfies

a # !2

e2

d

d#a2 da

d#= 0 & w

d

dwa2w

d

dwa # e2

!2a = 0

& d2a

dw2+

2

a

$da

dw

%2

+1

w

da

dw# e2

!2w2a= 0 (94)

To investigate other possibilities, let us take into account the Hamilton constraint. It

suggests that, for small X1, the contribution of'

%X1%"

(2can be as large as 1

X21. Then the

last term of the equation of motion can have significant e!ect

# "

"#X2

1

"X1

"#= #2X1

$"X1

"#

%2

+ X21

"2X1

"#2$ # 2

X1

13

These complicated-looking (nonlinear) system of equations can in principle be exactly solvable, classically. But exact quantization is very difficult.

Let us take a qualitative approach and

derive the space-time uncertainty relation from this viewpoint.



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2!"!(d!)2 + (d#)2

#(3)


U = #e!!" , V = ±e!+" (4)

which gives

ds2 = !R2dUdV (5)


1



2014

1 The Rinder space





ds2 = !X21 (d!)2 + (dX1)

2 +D!1!

i=2

(dXi)2 (2)







ds2 = R2e2!"!(d!)2 + (d#)2

#(3)


U = #e!!" , V = ±e!+" (4)

which gives

ds2 = !R2dUdV (5)


1

In the regions of the world sheet where is small,

on the world sheet vanishes. From the equations of motion, we can define this velocity

approximately by

c(!, ") =#

eX2

1 (104)

Let us denote the local horizon by a function

X1(!0, ") = 0, !0 = F (") (105)

The energy constraint shows

$X1

$"! X1(!, ") ! " F (") (106)

In this small X1 region on the world sheet, the terms involving !-derivatives can be

neglected, and hence its behavior is almost partilce-like for each fixed ! near !0. Keep

in mind that the form of function F cannot given externally, being determined self-

consistently by solving the equations of motion and the constraints.

In order to satisfy the momentum constraint, at least one component of the transverse

directions must be excited, as is obvious from its intuitive meaning that the momentum

must be orthogonal to the tangent of the profile of strings at each fixed " . If we consider

the average wave lengths of such transverse excitations along the world sheet is n, its

average frequency is of order

% ! c/& ! nc =#n

eX2

1

The order of the energy constant can be estimated by considering the state of strings where

X21 ! 1/# using the Hamiltonian constraint with the assumption that that kinetic and

potential energies are of the same order. This assumption is reasonable because strings

always has extendedness by zero-point oscillations of order#

'! for every component.

Then from the Hamiltonian constraint

e2 ! n2

which is of course satisfied not only in the Rindler coordinates but also in the Minkowski

case. Thus the average frequency of the wave satisfies

#X21 ! % (107)

16

velocity of propagating transverse oscillations along the world sheet


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

velocity of longitudinal mode in the target space

The Rindler horizon leads to horizons in the world sheet, which

are only dynamically determined self-consistently for each

classical solution.


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

We cannot divide the Hilbert space of (1st quantized) strings into

left and right wedges. (2nd quantized) string fields must be defined

using both Rindler regions simultaneously. Hence, the Hilbert space of string fields cannot be decomposed either.

space-time uncertainty relation


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

: average excitation number of transverse oscillations along the world sheet


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

Hamiltonian constraint requires


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

along the world sheets, and hence for the uncertainty

!X1!X1 ! !"

This relation originates from the non-locality of strings and implies through the classical

relation !"!# ! 1

X1!X1!# ! 2$%! ! &2s (108)

In terms of the proper time T = X1# , we arrive at the same space-time uncertainty

relation

!X1!T ! &2s (109)



!X1 !&2s

!T" #








remote distances.






!X1!P1 ! 1 (110)


!X1 ! 1

!P1! X2

1

!P!V1(111)

17


!X1!X1 ! !"


relation !"!# ! 1

X1!X1!# ! 2$%! ! &2s (108)


relation

!X1!T ! &2s (109)



!X1 !&2s

!T" #








remote distances.






!X1!P1 ! 1 (110)


!X1 ! 1

!P1! X2

1

!P!V1(111)

17


!X1!X1 ! !"


relation !"!# ! 1

X1!X1!# ! 2$%! ! &2s (108)


relation

!X1!T ! &2s (109)



!X1 !&2s

!T" #








remote distances.






!X1!P1 ! 1 (110)


!X1 ! 1

!P1! X2

1

!P!V1(111)

17

Fluctuations of and also

contribute to uncertainties, but they are either of the same order or of non-leading order.


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

: average energy densityalong the world sheets, and hence for the uncertainty

!|X1|!X1 ! !"


relation !"!# ! 1

|X1|!X1!# ! 2$%! ! &2s (108)


relation

!X1!T ! &2s (109)



!X1 !&2s

!T" #








remote distances.






!X1!P1 ! 1 (110)


!X1 ! 1

!P1! X2

1

!P!V1(111)

17


!|X1|!X1 ! !"


relation !"!# ! 1

|X1|!X1!# ! 2$%! ! &2s (108)


relation

!X1!T ! &2s (109)



!X1 !&2s

!T" #








remote distances.






!X1!P1 ! 1 (110)


!X1 ! 1

!P1! X2

1

!P!V1(111)

17

No thermalization !

This is as it should be: There is no allowed invariants (observables) on the world-sheets which can be defined in restricted localized regions of We cannot localize the string fields in finite space-like regions if we assume the validity of the space-time uncertainty relation

with respect to target space-time. Again no legitimate observables restricted space-like regions, other than S-matrices,


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

and seems to support ‘black hole complementarity’.

No thermalization !

This is as it should be: There is no allowed invariants (observables) on the world-sheets which can be defined in restricted localized regions of We cannot localize the string fields in finite space-like regions if we assume the validity of the space-time uncertainty relation

with respect to target space-time. Again no legitimate observables restricted space-like regions, other than S-matrices,


approximately by

c(!, ") =#

eX2

1 (104)


X1(!0, ") = 0, !0 = F (") (105)


$X1

$"! X1(!, ") ! " F (") (106)










% ! c/& ! nc =#n

eX2

1







e2 ! n2



#X21 ! % (107)

16

entangled pure states thermal mixed state

cannot happen, at least in association with the existence of space-time horizons, in string theory

and seems to support ‘black hole complementarity’.

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 remarkNo 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 remarkNo 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 !

unsolved questions in string - 東京大学hep1.c.u-tokyo.ac.jp/~tam/kek2014.pdfunsolved questions...

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