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arXiv:0912.5223v2 [hep-th] 24 Jan 2010

Holographic description of quantum field theory

Sung-Sik Lee

Department of Physics & Astronomy, McMaster University,

Hamilton, Ontario L8S 4M1, Canada

(Dated: January 24, 2010)

Abstract

We propose that general D-dimensional quantum field theories are dual to (D + 1)-dimensional local

quantum theories which in general include objects with spin two or higher. Using a general prescription, we

construct a (D + 1)-dimensional theory which is holographically dual to the D-dimensional O(N) vector

model. From the holographic theory, the phase transition and critical properties of the model in dimensions

D > 2 are described.

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I.INTRODUCTION

Quantum field theory is a universal language that describes long wavelength fluctuations in

quantum systems made of many degrees of freedom. Although strongly coupled quantum field

theories commonly arise in nature, it is notoriously difficult to find a systematic way of under-

standing strongly coupled quantum field theories.

The anti-de Sitter space/conformal field theory correspondence[1–3] opened the door to under-

stand a class of strongly coupled quantum field theories. According to the duality, certain strongly

coupled quantum field theories in D dimensions can be mapped into weakly coupled gravita-

tional theories in (D + 1) dimensions in large N limits. Although the original correspondence

has been conjectured based on the superstring theory, it is possible that the underlying principle is

more general and a wider class of quantum field theories can be understood through holographic

descriptions[4–7], which may have different UV completion than the string theory.

In this paper, we provide a prescription to construct holographic theories for general quantum

field theories. As a demonstration of the method, we explicitly construct a dual theory for the

D-dimensional O(N) vector model, and reproduce the phase transition and critical behaviors of

the model using the holographic description.

The paper is organized in the following way. In Sec. II, we will convey the main idea behind

the holographic description by constructing a dual theory for a toy model. In Sec. III A, using

the general idea presented in Sec. II, we will explicitly construct a holographic theory dual to the

D-dimensional O(N) vector model. In Sec. III B, we will consider a large N limit where the

theory becomes classical for O(N) singlet fields in the bulk. In Sec. III C, the phase transition and

critical properties of the O(N) model will be discussed using the holographic theory.

II.TOY-MODEL : 0-DIMENSIONAL SCALAR THEORY

In this section, we will construct a holographic theory for one of the simplest models : 0-

dimensional scalar theory. In zero dimension, the partition function is given by an ordinary inte-

gration,

Z[J] =

?

dΦ e−S[Φ].

(1)

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We consider an action S[Φ] = SM[Φ] + SJ[Φ] with

SM[Φ] = M2Φ2,

SJ[Φ] =

∞

?

n=1

JnΦn.

(2)

Here SMis the bare action with ‘mass’ M. SJis a deformation with sources Jn. The values of

Jn’s are not necessarily small. In the following, we will consider deformations upto quartic order

: Jn= 0 for n > 4. However, the following discussion can be straightforwardly generalized to

more general cases.

For a given set of sources Jn, quantum fluctuations are controlled by the bare mass M. One

useful way of organizing quantum fluctuations is to separate high energy modes and low energy

modes, and include high energy fluctuations through an effective action for the low energy modes.

Although there is only one scalar variable in this case, this can be done through the Polchinski’s

renormalization group scheme[8]. First, an auxiliary field˜Φ with mass µ is introduced,

Z[J] = µ

?

dΦd˜Φ e−(S[Φ]+µ2˜Φ2).

(3)

At this stage,˜Φ is a pure auxiliary field without any physical significance. Then, we find a new

basis φ and˜φ

Φ = φ +˜φ,

˜Φ = Aφ + B˜φ,

(4)

in such a way that the ‘low energy field’ φ has a mass M

′which is slightly larger than the original

mass M. As a result, quantum fluctuations for φ become slightly smaller than the original field Φ.

The missing quantum fluctuations are compensated by the ‘high energy field’˜φ with mass m

′. If

we choose the mass of the low energy field φ as

M

′2= M2e2αdz

(5)

with dz being an infinitesimally small parameter and α being a positive constant, we have to

choose

A = −MM

′

m

′µ, B =m

′M

′µ,M

(6)

where

m

′2= M2

e2αdz

e2αdz− 1=

3

M2

2αdz.

(7)

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Note that m

′2is very large, proportional to 1/dz. This is because˜φ carries away only infinites-

imally small quantum fluctuations of the original field Φ. Moreover, m

arbitrary mass µ because˜φ is physical.

′is independent of the

In terms of the new variables, the partition function is written as

Z[J] =

?Mm

′

M

′ +MM

′

m

′

??

dφd˜φ e−(SJ[φ+˜φ]+M′2φ2+m′2˜φ2).

(8)

If we rescale the fields,

φ → e−αdzφ,

˜φ → e−αdz˜φ,

(9)

the quadratic action for low energy field φ can be brought into the form which is the same as the

original bare action,

Z[J] = m

?

dφd˜φ e−(Sj[φ+˜φ]+M2φ2+m2˜φ2),

(10)

where

Sj[φ +˜φ] =

4

?

n=1

jn(φ +˜φ)n,

(11)

with jn= Jne−nαdzand m = m

Jn, which is a manifestation of reduced quantum fluctuations for the low energy field φ. The new

action can be expanded in power of the low energy field,

′e−αdz. Note that jn’s become smaller than the original sources

Sj[φ +˜φ] = Sj[˜φ] + (j1+ 2j2˜φ + 3j3˜φ2+ 4j4˜φ3)φ

+(j2+ 3j3˜φ + 6j4˜φ2)φ2+ (j3+ 4j4˜φ)φ3+ j4φ4.

(12)

In the standard renormalization group (RG) procedure[8, 9], one integrates out the high energy

field to obtain an effective action for the low energy field with renormalized coupling constants.

Here we take an alternative view and interpret the high energy field˜φ as fluctuating sources for

the low energy field. This means that the sources for the low energy field can be regarded as

dynamical fields instead of fixed coupling constants. To make this more explicit, we decouple the

high energy field and the low energy field by introducing Hubbard-Stratonovich fields Jnand Pn,

Z[J] = m

?

dφd˜φΠ4

n=1(dJndPn) e−(S

′

j+M2φ2+m2˜φ2),

(13)

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where

S

′

j= Sj[˜φ]

+iP1J1− iP1(j1+ 2j2˜φ + 3j3˜φ2+ 4j4˜φ3) + J1φ

+iP2J2− iP2(j2+ 3j3˜φ + 6j4˜φ2) + J2φ2

+iP3J3− iP3(j3+ 4j4˜φ) + J3φ3

+iP4J4− iP4j4+ J4φ4.

(14)

Now we integrateout˜φ to obtain an effectiveaction for thesource fields. Themass m2for thehigh

energy field is proportional to 1/dz and only terms that are linear in dz contribute to the effective

action (for the derivation, see the Appendix A),

Z[J] =

?

dφΠ4

n=1(dJndPn) e−(SJ[φ]+M2φ2+S(1)[J,P]),

(15)

where

S(1)[J,P] =

4

?

+αdz

2M2(i˜ J1+ 2P1˜ J2+ 3P2˜ J3+ 4P3˜ J4)2

n=1

i(Jn− Jn+ nαdzJn)Pn

(16)

with˜ Jn= (Jn+ Jn)/2.

After repeating the steps from Eqs. (3) to (15) R times, one obtains a path integral for the

partition function

Z[J] =

?

ΠR

k=1Π4

n=1(DJ(k+1)

n

DP(k)

n)e−S(R)[J(k),P(k)]Z[J(R+1)],

(17)

where

S(R)[J(k),P(k)] =

R

?

+αdz

2M2(i˜J(k)

k=1

? 4

?

n=1

i(J(k+1)

n

− J(k)

n

+ nαdzJ(k)

n)P(k)

n

1

+ 2P(k)

1

˜J(k)

2

+ 3P(k)

2

˜J(k)

3

+ 4P(k)

3

˜J(k)

4)2?

(18)

with˜J(k)

n

= (J(k+1)

n

+ J(k)

n )/2 and J(1)

n = Jn. The non-trivial solution for Eq. (17) is given by

?

Z[J] =Π4

n=1(DJnDPn) e−S[J,P],

(19)

where

S[J,P] =

?∞

+

0

dz

?

i(∂zJn+ nαJn)Pn

α

2M2(iJ1+ 2P1J2+ 3P2J3+ 4P3J4)2?

5

.

(20)