Page 1
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.
1
Page 2
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)
2
Page 3
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)
Page 4
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)
4
Page 21
[5] S. R. Das and A. Jevicki, Phys. Rev. D 68, 044011 (2003).
[6] R. Gopakumar, Phys. Rev. D 70, 025009 (2004); ibid. 70, 025010 (2004).
[7] I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, J. High Energy Phys. 10, 079 (2009).
[8] J. Polchinski, Nucl. Phys. B 231, 269 (1984).
[9] For a review, see J. Polonyi, arXiv:hep-th/0110026v2.
[10] J. de Boer, E. Verlinde and H. Verlinde, J. High Energy Phys. 08, 003 (2000).
[11] R. Arnowitt, S. Deser, and C. Misner, Phys. Rev. 116, 1322 (1959).
[12] M. A. Vasiliev, arXiv:hep-th/9910096.
[13] A. C. Petkou, J. High Energy Phys. 03, 049 (2003).
[14] I. R. Klebanov and E. Witten, Nucl. Phys. B 556, 89 (1999).
[15] I thank Sean Hartnoll for pointing this out to me.
[16] G. ’t Hooft, Nucl. Phys. B 72, 461 (1974).
[17] D. J. Gross and I. R. Klebanov, Nucl. Phys. B 344, 475 (1990).
[18] D. E. Berenstein, M. Hanada and S. A. Hartnoll, J. High Energy Phys. 02, 010 (2009).
[19] S.-S. Lee, Phys. Rev. B 80, 165102 (2009).
[20] I thank Guido Festuccia for pointing this out to me.
VI.APPENDIX A
The action in Eq. (14) can be expanded in power of the high energy field as
S
′
j= SJ[φ] +
4
?
n=1
i(Jn− jn)Pn
+(j1− 2iP1j2− 3iP2j3− 4iP3j4)˜φ
+(j2− 3iP1j3− 6iP2j4)˜φ2
+(j3− 4iP1j4)˜φ3
+j4˜φ4.
(70)
Integrating over˜φ to the order of 1/m2, one obtains
Z[J] =
?
dφΠ4
n=1(dJndPn)
m
√m2+ Be−
?
M2φ2+SJ[φ]+i?4
n=1(Jn−jn)Pn+
A2
4(m2+B)
?
,
(71)
21
Page 22
where
A = (ij1+ 2P1j2+ 3P2j3+ 4P3j4),
B = (j2− 3iP1j3− 6iP2j4).
(72)
The cubic and higher order terms in˜φ do not contribute to the linear order in 1/m2∼ dz. If we
keep only those terms that are linear in dz in Eq. (71), we obtain the action,
S = M2φ2+ SJ[φ] + i
4
?
n=1
(Jn− jn)Pn+
A2
4m2+
B
2m2.
(73)
However, it is not easy to take the continuum limit (dz → 0) in this expression for the following
reason. We can regard jnand Jnas being defined at coordinates z and z + dz respectively, where
z is the logarithmic energy scale in the renormalization group flow. Then Pnis defined in the
interval (or at the middle point of the interval), [z,z + dz]. Usually, A2dz can be interpreted as
the integration of A2between z and z + dz in the continuum limit. This would be the correct if
A were a fixed constant of the order of 1. In the present case, however, A contains the dynamical
field Pnwhose typical amplitude is order of m ∼
the integrand gives a non-trivial contribution to the integration, leading to a discrepancy between
1
√dz. Therefore, an error of order of 1/m in jnin
the result in Eq. (73) and the one obtained in the naive continuum limit.
To fix this problem, we consider the following trick. First we absorb the factor
1
(m2+B)in front
of A2in the action into the measure of P3; we change the variable P3to P
√m2+ B
m
which leads to
′
3in Eq. (71),
P3 =
P
′
3+
√m2+ B − m
4mj4
(ij1+ 2P1j2+ 3P2j3),
(74)
Z[J] =
?
dφΠn?=3(dJndPn)dJ3dP
′
3e−
?
M2φ2+SJ[φ]+S′?
,
(75)
where
S
′= i
?
n?=3
(Jn− jn)Pn+A
?√m2+ B
′2
4m2
+i(J3− j3)
m
P
′
3+
√m2+ B − m
4mj4
(ij1+ 2P1j2+ 3P2j3)
?
(76)
with A
′= (ij1+2P1j2+3P2j3+4P
′
3j4). If we take the leading order terms, the above expression
becomes
S
′= i
?
n?=3
(Jn− jn)Pn+A
′2
4m2+ i(J3− j3)P
′
3.
(77)
22
Page 23
Dropping the prime sign in P3, the partition function becomes
Z[J] =
?
dφΠ4
n=1(dJndPn) e−S′′,
(78)
where
S
′′= M2φ2+ SJ[φ] +
4
?
n=1
i(Jn− jn)Pn+
1
4m2(ij1+ 2P1j2+ 3P2j3+ 4P3j4)2.
(79)
However, this expression is not completely satisfactory either. If one integrates out Jnand Pnin
this expression, one obtains an action for φ which is different from the result one obtains after
integrating out˜φ directly from Eq. (12) to the order of dz[20]. The difference is the contribution
from the tadpole diagram. The tadpole diagram simply shifts the local couplings, and it turns out
that its contribution can be accounted for by replacing jnwith (jn+ Jn)/2 in the last term of Eq.
(79) as
S
′′′= M2φ2+ SJ[φ] +
4
?
n=1
i(Jn− jn)Pn+
1
4m2(i˜j1+ 2P1˜j2+ 3P2˜j3+ 4P3˜j4)2,
(80)
where˜jn = (jn+ Jn)/2. Although this is an infinitesimal change, it still gives a non-trivial
contribution because (Jn− jn) ∼ O(1/m) and Pn(Jn− jn) ∼ O(1). It is straightforward to
show that with this action one reproduces the same action as the one obtained by integrating out˜φ
directly from Eq. (12) to the order of dz. In Eq. (80), the mean value of jnand Jnis multiplied to
Pn. Just as theerror ofthe trapezoidal methodin theusual discreteintegrationis suppressed to dz3,
the difference between Eq. (80) and the integration in the continuum limit becomes sub-leading in
dz even though Pn∼ 1/√dz. Therefore, Eq. (80) can be readily extended to the continuum limit.
If we use jn= Jne−nαdzand keep the leading order term, we obtain Eqs. (15) and (16). It is noted
that if one uses Eq. (73) and take the naive continuum limit, some couplings are spuriously shifted
due to the amplified error introduced in the continuum limit.
23