Holography of Information in AdSCFT Robert de Mello Kocha1and Garreth Kempb2 aSchool of Science Huzhou University Huzhou 313000 China

2025-05-06 0 0 621.6KB 30 页 10玖币

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Holography of Information in AdS/CFT

Robert de Mello Kocha,1and Garreth Kempb,2

aSchool of Science, Huzhou University, Huzhou 313000, China,

aSchool of Physics and Mandelstam Institute for Theoretical Physics,

University of the Witwatersrand, Wits, 2050,

South Africa

bDepartment of Mathematics and Applied Mathematics,

University of Johannesburg, Auckland Park, 2006, South Africa.

ABSTRACT

The principle of the holography of information states that in a theory of quantum

gravity a copy of all the information available on a Cauchy slice is also available near the

boundary of the Cauchy slice. This redundancy in the theory is already present at low

energy. In the context of the AdS/CFT correspondence, this principle can be translated

into a statement about the dual conformal ﬁeld theory. We carry out this translation

and demonstrate that the principle of the holography of information holds in bilocal

holography.

1robert@zjhu.edu.cn

2garry@kemp.za.org

arXiv:2210.11066v1 [hep-th] 20 Oct 2022

Contents

1 Introduction 1

2 Bilocal Holography 4

3 Convergence of the Operator Product Expansion 8

4 OPE and Holography of Information 12

4.1 Twobulkﬁeldsacting ............................. 13

4.2 Three bulk ﬁelds acting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Generic bulk observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Discussion and Conclusions 18

A OPE Observations 20

A.1 Instructive examples of OPE convergence . . . . . . . . . . . . . . . . . . . 22

A.2 Converting between ηand σ.......................... 24

1 Introduction

Locality is a cherished principle in physics. Relativistic causality - the fact that no physical

information carrying signal can propagate faster than the speed of light - is implemented

in the statement that spacelike separated ﬁelds commute. This ensures that laboratories

in spacelike separated spacetime regions function independently. A deep result express-

ing this independence in the algebraic formulation of quantum ﬁeld theory is the split

property. As a consequence of the split property, we can specify the state of quantum

ﬁelds independently on diﬀerent parts of a Cauchy slice. This lore is being challenged

[1, 2, 3, 4, 5] in the setting of quantum gravity, by the principle of the holography of

information, which claims that

In a theory of quantum gravity, a copy of all the information available on a Cauchy slice

is also available near the boundary of the Cauchy slice. This redundancy in description is

already visible in the low-energy theory.

This principle demands a dramatic revision of intuition built on locality. For example,

the principle of holography of information implies that given the state near the boundary

of the Cauchy slice, the rest of the state is determined: the split property fails and we

are not guaranteed that laboratories in spacelike separated regions of spacetime function

independently! The principle of the holography of information is a source of a dramatic

new non-locality1.

The argument of [1, 2, 3, 4, 5] is compelling in its simplicity. The principle of the

holography of information has two basic ingredients. The ﬁrst is the Reeh-Schlieder

Theorem [15], which is a Theorem about relativistic quantum ﬁeld theory. We simply

state the theorem and refer the reader to [16] for a readable account with details. Denote

the vacuum of the quantum ﬁeld theory as |Ωiand use H0to denote the vacuum sector2

of the full Hilbert space H. The vacuum sector consists of all states that can be created

from the vacuum by applying local ﬁeld operators. Assuming3that the algebra of local

ﬁelds is generated by a hermitian scalar ﬁeld φ(xµ), we introduce a smeared ﬁeld φf≡

Rd~xf(xµ)φ(xµ) and a set of states (both nand the functions fiare varied to get the full

set)

|Ψ{f1,··· ,fn}i=φf1φf2···φfn|Ωi(1.1)

Let Σ be a Cauchy hypersurface. Consider an arbitrarily small open set V ⊂ Σ and let

UVbe a small neighbourhood of Vin spacetime. The Reeh-Schleider theorem states that

even after restricting the functions fito support in UV, the states |Ψ{f1,··· ,fn}igenerate

H0. This remarkable result reﬂects the enormous amount of entanglement in the quantum

ﬁeld theory vacuum. The second ingredient that goes into the principle of the holography

of information is that, as a consequence of the Gauss law, the energy of a state in gravity

can be measured from near the boundary. This implies that the projector onto the state

of lowest energy, PΩ=|ΩihΩ|, is an element of the boundary algebra of operators. The

principle now follows [1, 2, 3, 4, 5]: First, note that any observable in H0can be written

as a linear combination of operators of the form |aihb|where |aiand |biare allowed to be

any states in H0. Using the Reeh-Schleider theorem we know the complete set of these

operators can be written in the form

|aihb|=φf(a)

1φf(a)

2···φf(a)

n(a)|ΩihΩ|φf(b)

1φf(b)

2···φf(b)

n(b)

(1.2)

Since the Gauss law implies that PΩis an element of the boundary algebra of operators,

and since a product of operators in the boundary algebra is again an element of the

boundary algebra we conclude that the complete set of operators |aihb|belong to the

boundary algebra. Consequently, any observable4in H0is an element of the boundary

algebra of operators and the principle is proved.

1For related studies we refer the interested reader to [6, 7, 8, 9, 10, 11, 12, 13, 14].

2The vacuum sector is not necessarily the full Hilbert space as there may be superselection sectors.

This happens, for example, when there are conserved charges that are not carried by any local operator.

In a non-trivial superselection sector an analogue of the Reeh-Schlieder theorem holds, so the existence

of non-trivial superselection sectors should not distract us.

3This assumption is to simplify the discussion and is easily relaxed [16].

4This would include operators that one naively thought were localized deep in the bulk of spacetime.

This unusual localization of quantum information in quantum gravity is the focus

of this paper. More concretely, the AdS/CFT correspondence gives a non-perturbative

deﬁnition of quantum gravity on negatively curved spacetimes in the form of a conformal

ﬁeld theory. Our goal in this article is to use the AdS/CFT correspondence to search for

signatures of the principle of the holography of information directly in the conformal ﬁeld

theory. Concretely, we use the bilocal holography of the free O(N) model to study these

questions in higher spin gravity. The construction and key results of bilocal holography

are reviewed in Section 2. The conformal ﬁeld theory is described using a bilocal collective

ﬁeld. A key formula from Section 2 is the mapping (2.4) which locates the bulk operator,

corresponding to a given bilocal operator, in the bulk AdS4spacetime. We also review

the important fact that there is some freedom in the reconstruction of the bulk ﬁelds in

the conformal ﬁeld theory. Results establishing the convergence of the operator product

expansion in unitary conformal ﬁeld theories, in Minkowski spacetime, are reviewed in

Section 3. In Section 4 we present our central result: the principle of the holography of

information, in bilocal holography, can be veriﬁed using the operator product expansion.

We speculate on how the principle is realized in AdS/CFT, in more general situations, in

Section 5.

A potential point of confusion can be clariﬁed immediately: the reader might wonder if,

in the setting of the AdS/CFT correspondence, the holography of information is trivially

true. After all, doesn’t the statement of the AdS/CFT correspondence, that the dynamics

of the bulk is coded into the dynamics of a conformal ﬁeld theory living on the boundary,

imply the holography of information? This is a misunderstanding of the principle. The

principle of the holography of information is a statement about the quantum gravity

theory itself. The proof of the principle [1, 2, 3, 4, 5], as reviewed above, does not invoke

AdS/CFT in any way at all, and consequently it also holds (for example) for a theory

of quantum gravity in ﬂat spacetime where a holographic dual is not even established.

Our goal is to use AdS/CFT to map the principle of the holography of information into a

statement about the conformal ﬁeld theory. This statement should be proved using only

conformal ﬁeld theory methods i.e. without appealing to AdS/CFT or to the holographic

gravity dual. If this succeeds, it provides non-trivial support for the principle.

Finally, the setting of our study is higher spin gravity which diﬀers in some important

ways from usual Einsteinian gravity. The spectrum of higher spin gravity includes not

just a massless spin two graviton, but rather there are massless gauge ﬁelds for every

even integer spin. It is clear that higher spin gravity will not share all the features of

Einsteinian gravity and there may be important diﬀerences between the two. Nonetheless,

we believe that this is a reasonable arena in which to test the holography of information.

Higher spin gravity is a quantum theory - so the Reeh-Schlieder theorem applies, and

it does enjoy the gauge invariance that is responsible for the Gauss law. Thus the key

ingredients needed to prove the principle are present.

2 Bilocal Holography

The AdS/CFT correspondence [17, 18, 19] relates a conformal ﬁeld theory (with loop

expansion parameter ~) to a theory of quantum gravity (with loop expansion parameter

N). Changing the loop expansion parameter requires a non-trivial rearrangement of the

conformal ﬁeld theory degrees of freedom. It can be achieved by collective ﬁeld theory

[20, 21] which expresses the theory in terms of invariant variables. The key insight is that

the collective ﬁeld variables have no explicit Ndependence, so that the 1

Nexpansion is

manifestly generated as the loop expansion of the collective ﬁeld theory. Since collective

ﬁeld theory provides a constructive approach to holography, it is the ideal framework for

this study. In what follows we work at the leading order in the large Nexpansion.

The conformal ﬁeld theory we study, the free O(N) model, has the Lagrangian

L=1

2∂µφa∂µφa(2.1)

and is deﬁned in 2 + 1 dimensions. There is compelling evidence [22] that this theory is

AdS/CFT dual [23, 24] to higher spin gravity [25, 26, 27] in AdS4spacetime. Hologra-

phy for vector models, using a collective ﬁeld description, was ﬁrst proposed in [28] and

then developed in a series of papers5[29, 30, 31, 32, 33, 44, 35], to which the reader is

referred for more details. The discussion is most transparently carried out using a light-

front quantization, since it is then possible to choose light cone gauge and to reduce to

physical degrees of freedom. Denote the conformal ﬁeld theory coordinates with little

letters as x+, x−, x and the coordinates of the dual AdS4spacetime with capital letters

as X+, X−, X, Z, with Zthe extra holographic coordinate. For the O(N) model, at each

time x+we change from the original ﬁeld φa(x+, x−, x) to a new set of gauge invariant

variables, given by the bilocal ﬁelds

σ(x+, x−

1, x1, x−

2, x2) = φa(x+, x−

1, x1)φa(x+, x−

2, x2) (2.2)

where the index ais summed. The bilocal packages the complete set of independent single

trace equal x+gauge invariant ﬁelds. This collective ﬁeld is a function of 5 coordinates. In

what follows, it is convenient to perform a Fourier transform in the x−coordinate, which

trades x−

1and x−

2for the conjugate momenta p+

1and p+

2. We also perform a Fourier

transform in the AdS spacetime, trading coordinate X−for coordinate P+. The change

of ﬁeld variable from φato σis associated with a Jacobian which is highly non-linear

and leads to an inﬁnite sequence of interaction vertices [37]. The single trace spectrum of

primary operators includes a scalar of dimension ∆ = 1 and higher spin currents Jµ1···µ2s

of every even integer spin 2sand dimension ∆ = 2s+ 1. As usual, every single trace

primary corresponds to a ﬁeld of the dual higher spin gravity: there is a massless gauge

ﬁeld AM1···M2sof every even integer spin, as well as a scalar ﬁeld. The bilocal develops a

5Related but distinct ideas were recently put forward in [36].

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HolographyofInformationinAdS/CFTRobertdeMelloKocha;1andGarrethKempb;2aSchoolofScience,HuzhouUniversity,Huzhou313000,China,aSchoolofPhysicsandMandelstamInstituteforTheoreticalPhysics,UniversityoftheWitwatersrand,Wits,2050,SouthAfricabDepartmentofMathematicsandAppliedMathematics,UniversityofJohannesbu...

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Holography of Information in AdSCFT Robert de Mello Kocha1and Garreth Kempb2 aSchool of Science Huzhou University Huzhou 313000 China

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