共形场理论中纠缠引起的非线性重力研究

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共形场理论中纠缠引起的非线性重力在共形场理论中，纠缠熵的概念扮演着非常重要的角色。在本文中，我们将证明非线性引力方程的出现是由于纠缠熵的存在。我们通过将标量主或应力张量算符的源添加到定义真空状态的欧氏路径积分中来定义CFT激发态。这些状态的纠缠熵可以通过渐近AdS几何图形（通过Ryu-Takayanagi公式）来表示。首先，我们需要了解什么是共形场理论（Conformal Field Theory，CFT）。CFT是一种 Quantum Field Theory，它具有共形对称性，也就是说，它对旋转、平移、缩放和特殊相对论变换具有不变性。CFT广泛应用于描述二维和高维系统中的critical phenomena，如phase transitions和quantum Hall effects。在CFT中，我们可以定义激发态通过将标量主或应力张量算符的源添加到定义真空状态的欧氏路径积分中。这些激发态的纠缠熵可以通过渐近AdS几何图形（通过Ryu-Takayanagi公式）来表示。纠缠熵是量子系统中的一种度量，它可以描述系统中粒子的相互作用和相关性。在我们的研究中，我们证明了这些激发态的几何结构必定能满足爱因斯坦方程直至第二阶的摄动，并且应力能张量由与原始主算子相关的物质场产生。这些结果表明，纠缠熵在共形场理论中扮演着非常重要的角色，它可以导致非线性引力的出现。我们的工作没有对AdS / CFT对偶性进行任何假设，因此我们的结果既可以作为对AdS / CFT对应关系的一致性检查，又可以直接证明时空和引力物理可以从保形场论中的纠缠描述中出现。我们的研究结果为理解共形场理论和引力物理之间的关系提供了新的见解，并且可能对我们的理解宇宙的本质产生深远的影响。我们的研究结果表明，纠缠熵在共形场理论中扮演着非常重要的角色，它可以导致非线性引力的出现。我们的工作为理解共形场理论和引力物理之间的关系提供了新的见解，并且可能对我们的理解宇宙的本质产生深远的影响。

JHEP08(2017)057

Basic idea of the proof. Suppose that a family of geometries M (ε) described by the

metric

g(ε) = g

(0)

+ ε δg

(1)

+ ε

δg

(2)

+ . . . (1.15)

correctly calculates via HRRT the entanglement entropy for some family |Ψ(ε)i of states

up to some order in ε. Then S

in (1.14) must agree with S

grav

in (1.13).

We can also show that the remaining terms on the left sides of (1.13) and (1.14) must

agree with each other: applying (1.14) to an inﬁnitesimal ball of radius R, we ﬁnd that the

terms on the right hand side are subleading in powers of the radius, so we have that

dε

grav

) =

dε

) =

dε

(hH

i) . (1.16)

As shown in [9], this implies that in Feﬀerman-Graham gauge where ∆g

= ∆g

zµ

= 0,

the leading order asymptotic metric must be related to the CFT stress tensor as

∆g

µν

16π

d−3

µν

d−2

+ . . . , (1.17)

where the dots indicate terms at lower orders in z. Using this relation and the explicit

forms for E

grav

and H

it then follows that E

in (1.14) must agree with hH

i in (1.13) [9].

Since the left sides of these equation have now been shown to agree, we have that for the

geometries M(ε) and the states Ψ(ε),

dε

S(ρ

||ρ

(0)

) =

ω(g,

dε

, L

g) +

G . (1.18)

This central result is valid for any ε; in order to proceed, we now consider terms at speciﬁc

orders in ε expanded about ε = 0.

First order. Setting ε = 0 in the expression (1.18), the left side vanishes since the

relative entropy between a reference state and a perturbation to that state vanishes at ﬁrst

order in the perturbation. The ﬁrst term on the right side also vanishes, since the L

g = 0

for the unperturbed AdS metric. Further, the matter ﬁelds do not enter the expressions at

this order. Thus, only the last term G is nonzero, and the the explicit form of the equation

becomes

(1)



= 0 , (1.19)

where E

(1)

is the Einstein tensor linearized about AdS and 

is a volume form. A simple

argument [9] shows that this is possible for all balls A if and only if E

(1)

= 0, so we see

that the ﬁrst order Einstein equations must be satisﬁed for any geometry that reproduces

the entanglement entropy of a perturbed CFT state.

We can further argue that a unique

such geometry exists for every state of every CFT: given the CFT stress-energy tensor

In practice, one ﬁrst shows that the equations of motion in the boundary directions are satisﬁed,

(1)

µν

= 0. The vanishing of the remaining components E

(1)

zµ

and E

(1)

can be thought of as constraint

equations and follow from standard general relativity arguments. We refer to [9] and references therein for

details.

– 8 –

JHEP08(2017)057

expectation value, we can associate a unique metric by solving the linearized Einstein

equations with boundary conditions determined by (1.17). Reversing the steps that lead

to (1.19) then implies that the HRRT formula for this geometry correctly calculates the

entanglement entropies.

Second order. At second order, our master equation (1.18) becomes explicitly

(2)

S(ρ

||ρ

(0)

) ≡

dε

S(ρ

||ρ

(0)

)



ε=0

ω(g

(0)

, δg

(1)

, L

δg

(1)

) −

2 ξ

(2)



(1.20)

On the left side, δ

(2)

S(ρ

||ρ

(0)

) is quadratic in the ﬁrst order perturbation to the state and

deﬁnes the quantum Fisher information metric at the point ρ

(0)

in the space of density

matrices. The ﬁrst term on the right deﬁnes the canonical energy; in a gravitational theory,

it is interpreted as the perturbative energy for graviton and matter ﬂuctuations with energy

deﬁned with respect to the timelike Killing vector ξ

The results of the previous section show that the metric perturbation δg

(1)

satisﬁes

Einstein’s equations and is therefore determined in terms of the CFT stress tensor via a

boundary-to-bulk propagator. The tensor E

(2)

is the second order Einstein tensor minus

the matter stress-tensor sources, which includes terms quadratic in the ﬁrst order metric

and matter perturbations and terms linear in the second order metric perturbation.

In order to proceed, we now restrict to the class of states (1.2) that we expect to be

well-described by some classical spacetime in the case of a holographic theory. In this case,

we can compute the left side of (1.20) perturbatively in the sources λ

. At quadratic order

in the source λ

for some operator O

, the result for δ

(2)

S(ρ

||ρ

(0)

) in terms of λ

depends

only on the two-point function of O

in the CFT (and its analytic continuation). With

some work, it is possible to show that the CFT result for the second order relative entropy

can be reexpressed in gravitational language,

making use of the same auxiliary anti-de

Sitter space as computes the leading order entanglement entropies. We deﬁne a ﬁeld φ

(1)

(0)

for each sourced operator O

in (1.2), such that φ

(1)

satisﬁes the linearized equations of

motion on AdS appropriate to the dimension and spin of O

, and has boundary behaviour

determined by hO

i as we would have in genuine examples of holography. Then we can

show that

(2)

S(ρ

||ρ

(0)

) =

∗

ω(g

(0)

, δg

(1)

, L

δg

(1)

) . (1.21)

Deriving this equation will be the main task of the paper and will be the content of section 3.

Finally, assuming that

= a

∗

, we can combine (1.21) and (1.20) to obtain

(2)



= 0 . (1.22)

This should not be too surprising: the two-point function of an operator O

in any CFT is the same as

the two-point function of an operator of the same dimension in a holographic CFT. We know that the latter

can be alternatively obtained via a gravity calculation. Thus, we can re-express the two-point function for

any CFT in gravitational language.

– 9 –

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