Holographic description of quantum field theory
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. Comment: 23 pages, 1 figure; v2) references added; appendix expanded
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ABSTRACT: We propose a local renormalization group procedure where length scale is changed in spacetime dependent way. Combining this scheme with an earlier observation that high energy modes in renormalization group play the role of dynamical sources for low energy modes at each scale, we provide a prescription to derive background independent holographic duals for field theories. From a first principle construction, it is shown that the holographic theory dual to a D-dimensional matrix field theory is a (D+1)-dimensional quantum theory of gravity coupled with matter fields of various spins. The gravitational theory has (D+1) first-class constraints which generate local spacetime transformations in the bulk. The (D+1)-dimensional diffeomorphism invariance is a consequence of the freedom to choose different local RG schemes.Journal of High Energy Physics 04/2012; 2012(10). · 5.62 Impact Factor
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ABSTRACT: We propose a method for determining the exact correspondence between the Wilsonian cut-off scale on the boundary and its holographically dual bulk theory. We systematically construct the multi-trace Wilsonian effective action from holographic renormalisation and evolve it by integrating out the asymptotically Anti-de Sitter bulk geometry with scalar probes. The Wilsonian nature of the effective action is shown by proving that it must be either double-trace, closing in on itself under successive integrations, or have an infinite series of multi-trace terms. Focusing on composite scalar operator renormalisation, we relate the Callan-Symanzik equation, the flow of the scalar anomalous dimension and the multi-trace beta functions to their dual RG flows in the bulk. Establishing physical renormalisation conditions on the behaviour of the large-$N$ anomalous dimension then enables us to extract the energy scales. Examples of pure AdS, GPPZ flow, black brane in AdS, M2 and M5 branes are studied before we generalise our results to arbitrary numbers of mass and thermal deformations of an ultra-violet CFT. Relations between the undeformed Wilsonian cut-off, deformation scales and the deformed Wilsonian cut-off are discussed, as is phenomenology of each considered background. We see how a mass gap, the emergent infra-red CFT scaling, etc. arise in different effective theories. We also argue that these results can have alternative interpretations through the flow of the conformal anomaly or the Ricci scalar curvature of boundary branes. They show consistency with the c-theorem.Journal of High Energy Physics 12/2011; 2012(6). · 5.62 Impact Factor
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ABSTRACT: We study a conjectured connection between the AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric so computed has the properties expected from AdS/CFT. We also discuss the cMERA in a time-dependent background induced by quantum quench and estimate its corresponding metric.Journal of High Energy Physics 08/2012; 2012(10). · 5.62 Impact Factor
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?