arXiv:0912.5223v2 [hep-th] 24 Jan 2010
Holographic description of quantum field theory
Department of Physics & Astronomy, McMaster University,
Hamilton, Ontario L8S 4M1, Canada
(Dated: January 24, 2010)
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.
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-
We consider an action S[Φ] = SM[Φ] + SJ[Φ] with
SM[Φ] = M2Φ2,
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. First, an auxiliary field˜Φ with mass µ is introduced,
Z[J] = µ
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,
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
we choose the mass of the low energy field φ as
with dz being an infinitesimally small parameter and α being a positive constant, we have to
A = −MM
′µ, B =m
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
If we rescale the fields,
φ → e−αdzφ,
˜φ → e−αdz˜φ,
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
Sj[φ +˜φ] =
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.
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
+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.
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),
2M2(i˜ J1+ 2P1˜ J2+ 3P2˜ J3+ 4P3˜ J4)2
i(Jn− Jn+ nαdzJn)Pn
with˜ Jn= (Jn+ Jn)/2.
After repeating the steps from Eqs. (3) to (15) R times, one obtains a path integral for the
n )/2 and J(1)
n = Jn. The non-trivial solution for Eq. (17) is given by
2M2(iJ1+ 2P1J2+ 3P2J3+ 4P3J4)2?