Quantum Gravity and Closed String Field...

Quantum Gravity and Closed String Field Theory

Taejin Lee

Kangwon National UniversityKOREA

[email protected]

Hi One, Jan. 9, 2018

Taejin Lee (KNU) IBS-KIAS 2018 Hi One, Jan. 9, 2018 1 / 20

Contents

1 String Field Theory in the Proper-Time Gauge

2 Scattering Amplitudes of Open Strings on Dp-Branes

3 Quantum Gravity and String Field Theory

4 Graviton Scattering Amplitudes from the Closed String Field Theory

5 Classical Solutions with D-Brane Sources

6 Entanglement Entropy of Open Strings on Dp-Branes

7 Discussions and Conclusions8 References:

1 TL, arXiv:1708.05215 (2017), Talk given at 13th ICGAC.2 S. Lai, J. Lee, TL and Y. Yang, PLB 776, 150 (2018).3 S. Lai, J. Lee, TL and Y. Yang, JHEP 09, 130 (2017).4 TL, arXiv:1703.06402 (2017).5 TL, Phys. Lett. B 768 (2017) 248.6 TL, JKPS 71, 886 (2017).


String Field Theory in the Proper-Time Gauge

Open string field theory in the proper-time gauge

We constructed a covariant string field theory on Dp-branes, andcalculated three-string scattering amplitude and the four-string scatteringamplitude in the low energy limit.


Polyakov String Path Integral

Path integral representation of string scattering amplitudes

A =

∫D[X ]D[h] exp

[−i∫MdτdσL

],

S = − 1

4πα′

∫Mdτdσ

√−hhαβ ∂X

µ

∂σα∂X ν

∂σβηµν , µ, ν = 0, . . . , d − 1.

This is an 1.5th quantized theory.

? ? ...


Proper-Time Gauge: String Propagator

Two-dimensional world sheet metric in the proper-time gauge (n is thezero-mode of the lapse function.)

√−hhαβ =

1

n

(−1 00 n2

).

Mode expansions if we choose σ ∈ [0, |α|π],

X (σ) = x + 2∑n=1

1√nxn cos

(nσ

|α|

),

= x +∑n=1

i√n

(an − a†n

)cos

(nσ

|α|

).


String Scattering Amplitudes of Open Strings onDp-Branes

Three-String Scattering Diagram

Mapping of the three-string vertex diagram onto the upper half complexplane

(1)

(2)

(3)

(2)

Z

(1)

(3)

1ρ=τ+iσ

a

b

c

a b c


String Scattering Amplitudes of Open Strings onDp-Branes

Four-String Scattering Diagram

Mapping of the four-string vertex diagram onto the upper half complexplane

(1)

(2) (3)

(4)

ρ=τ+iσ

(1) Z

(2)(3)(4)

a

1x

b

c

d

abcdx0 1


Quantum Gravity and String Field Theory

Classical General Relativity Derived from Quantum Gravity

Boulware and Deser, Ann. Phys. 89 (1975):“A quantum particle description of local (noncosmological) gravitationalphenomena necessarily leads to a classical limit which is just a metrictheory of gravity. As long as only helicity ±2 gravitons are included, thetheory is precisely Einsteins general relativity.”

Closed String Field Theory

Closed string theory contains massless spin 2 particles in its spectrum. Thelow energy limit of the covariant interacting closed string field theory mustbe the Einstein’s general relativity. The closed string field theory mayprovide a consistent framework to describe a finite quantum theory of thespin 2 particles, the gravitons. We need to examine the graviton scatteringamplitudes of the covariant string field theory and compare them withthose of the perturbation theory of the gravity in the low energy region.


String Scattering Amplitudes of Closed String Field Theoryand Polyakov String Path Integral

Strategy of Calculation of String Scattering Amplitudes

1 Construct the covariant closed string field theory

2 Rewrite the scattering amplitudes generated by the closed string fieldtheory by using the Polyakov string path integral

3 Re-express the Polakov string path integrals in terms of the oscillatoroperators

4 Identify the Fock space (operator) representations of the string fieldtheory vertices

5 Choose appropriate external string states, corresponding to thevarious particles and evaluate the scattering amplitudes.


String Scattering Amplitudes of Closed String Field Theoryand Polyakov String Path Integral

Closed Srtring Scattering amplitude

W =

∫DX exp

(i

M∑r=1

∫Pr (σ) · X (τr , σ)dσ −

∫dτdσL

)

= [det ∆]−d/2 exp

{1

4

{∑r

ξrP20 −

∑r

∑n=1

1

nP

(r)n · P(r)

−n

+∑n,m

C rsnme

|n|ξr+|m|ξ′sP(r)−n · P

(s)−m

}

= 〈P| exp

(∑r

ξrL(r)0

)|V [N]〉.


Factorization of Three-Closed-String Scattering Amplitude

A[1, 2, 3] = g〈{k(r)}|

exp

{∑r ,s

( ∑n,m≥1

1

2N rsnm

α(r)†n

2· α

(r)†m

2+∑n≥1

N rsn0

α(r)†n

2· p

(s)

2

)}

exp

{τ0

∑r

1

αr

1

2

(p(r)

2

)2

− 1

}

exp

{∑r ,s

( ∑n,m≥1

1

2N rsnm

α(r)†n

2· α

(r)†m

2+∑n≥1

N rsn0

α(r)†n

2· p

(s)

2

)}

exp

{τ0

∑r

1

αr

1

2

(p(r)

2

)2

− 1

}|0〉.Taejin Lee (KNU) IBS-KIAS 2018 Hi One, Jan. 9, 2018 11 / 20

Three-Graviton Scattering Amplitude

Decomposition of the spin-2 field into graviton, anti-symmetric tensor, andscalar field

hµν ={1

2(hµν + hνµ)− ηµν

1

dhσσ}

+{1

2(hµν − hνµ)

}+ ηµν

{ 1

dhσσ}.

We choose the covariant gauge condition

∂µhµν = 0,

which becomes de Donder gauge condition for the graviton

∂µhµν −1

d − 2∂νh

σσ = 0.

For three-graviton scattering, we choose the external string state as

|Ψ3G 〉 =3∏

r=1

{hµν(pr )α

(r)µ−1 α

(r)ν−1

}|0〉.



We note that A[3−graviton] can be written also as

A[3−graviton] =

(2g

3

)1

28

∫ 3∏i=1

dp(i)δ

(3∑

i=1

p(i)

)

〈0|

{3∏

i=1

hµν(p(i))a(i)µ1 · a(i)ν

1

}EOpen

[3−Gauge] EOpen[3−Gauge]|0〉.

Making use of the Neumann functions of the open string

N1111 =

1

24, N22

11 =1

24, N33

11 = 22,

N1211 = N21

11 =1

24, N23

11 = N3211 =

1

2, N31

11 = N1311 =

1

2,

N11 = N2

1 =1

4, N3

1 = −1,



A[3−graviton] is precisely the three-graviton interaction term which may beobtained from the Einstein’s gravity action.

A[3−graviton] = κ

∫ 3∏i=1

dp(i)δ

(3∑

i=1

p(i)

)hµ1ν1(p(1))hµ2ν2(p(2))hµ3ν3(p(3)){ηµ1µ2p(1)µ3 + ηµ2µ3p(2)µ1 + ηµ3µ1p(3)µ2

}{ην1ν2p(1)ν3 + ην2ν3p(2)ν1 + ην3ν1p(3)ν2

}where κ = g

27·3 =√

32πG10.


Classical Solutions with D-Brane Sources: Black Hole

Closed string field theory:

S = 〈Φ|KΦ〉+g

3

(〈Φ|Φ ◦ Φ〉+ 〈Φ ◦ Φ|Φ〉

).

Classical equation of motion with a D-brane source, JD

KΦ + gΦ ◦ Φ = JD .

Perturbative solution in the weak field limit

Φ =1

KJD −

g

K

{1

KJD ◦

1

KJD

}+ · · ·

+ + +...


Entanglement Entropy of Open Strings on Dp-Branes

Bekenstein-Hawking entropy of black hole

SBH =ABH

4GN

and Entanglement entropy:

SA = −trAρA log ρA, ρA = trB |Ψ〉〈Ψ|.

On Dp-branes,

∂Xµ

∂σ

∣∣∣σ=0, π

= 0, for µ = 0, 1, . . . , p,

X i∣∣∣σ=0, π

= 0, for i = p + 1, . . . , d .

We divide the spatial dimension along x1 direction into two subregions, Aand B.



The string vacuum wave functional in the Fock space representation

Φ[Ψ] = 〈0|Ψ〉 =1

N

∫ φ{Nn}(x ,0)=ψ{Nn}(x)

φ{Nn}(x ,−∞)=0D[Φ]e−SE (Φ)

The vacuum density matrix:

ρ[Ψ,Ψ′] = 〈Ψ|0〉〈0|Ψ′〉 = Φ[Ψ]∗Φ[Ψ′].

ΑΒ

x 1



Consider the functionals Ψ = ΨA ⊕ΨB and Ψ′ = Ψ′A ⊕ΨB which coincideon the half line x1 < 0. Sum over all possible functional ΨB . Taking twocopies of the half planes, we write the reduced density matrix as

ρA(Ψ,Ψ′

)=

∫D[ΦB ]Φ[ΨA ⊕ΨB ]∗Φ[Ψ′A ⊕ΨB ]

=1

N

∫ φ{Nn}(x ,0+)=ψ{Nn}(x), x∈A

φ{Nn}(x ,0−)=ψ′{Nn}

(x), x∈AD[Φ]e−SE (Φ).

Renyi entropy for integer n

Sn =1

1− n

{logZ (n)− n logZ (1)

}, trρnA =

Z (n)

Z (1)n.

The entanglement entropy may be obtained by using the replica trick as

S = − ∂

∂n

{logZ (n)− n logZ (1)

}∣∣∣n=1

.

The n-th power of the reduced density matrix may be obtained by definingthe field theory on n-sheeted Riemann surface.


n-Sheeted Riemann Surface


Conclusions and Discussions

1 Open string field theories in the proper-time gauge on multipleDp-branes ⇒ Applications in particle physics.

2 Graviton scattering amplitudes from closed string field theory in theproper-time gauge ⇒ Kawai-Lewellen-Tye (KLT) relations for closedstring field theory, Finite quantum theory of gravity.

3 Classical solutions with Dp-brane sources⇒ Construction of black hole solutions and AdS spaces in the stringfield theory.

4 Entamglement entropy of open strings on Dp-branes⇒ Entanglement entropy for string field theories. Interpretation ofBH entropy of black hole in the context of string field theory.

5 AdS/CFT from the point of view of the covariant string field theory.


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