二维黑洞统计力学：AdS2时空与粒子动力学

186 浏览量更新于2024-07-16 收藏 1019KB PDF 举报

"这篇论文详细探讨了二维黑洞的统计力学，作者Alexei Kitaev和S. Josephine Suh来自加州理工学院。他们研究了在具有边界的近AdS2时空动力学，将其与AdS空间中两个粒子的动力学相联系。通过确定具有物理意义的波函数类别，他们为黑洞的统计力学建立了理论基础。文中展示了如何利用双面黑洞的波函数和正则化的迹线概念构建热分布函数，以及更普遍地构建任意密度矩阵。此外，他们还计算了外部操作符的相关函数，为理解和研究二维黑洞的性质提供了新的视角。该文章属于开放访问，受SCOAP3资助，发表在JHEP期刊上，关键词包括2D引力、黑洞、量子引力模型和AdS-CFT对应原理。" 这篇论文的核心在于对二维黑洞动力学的深入分析，它揭示了在特定条件下的时空结构如何简化为两个粒子在AdS空间中的相互作用。首先，作者介绍了研究背景，说明了早期AdS2时空与两个粒子动力学之间的联系。接下来，他们讨论了相关几何和经典轨迹，这是理解黑洞行为的基础。在统计力学部分，作者确定了合适的波函数类别，这些波函数能够描述黑洞的物理状态。他们引入的波函数方法不仅限于构造热分布函数，还扩展到构造任意密度矩阵，这在量子力学中是用于描述系统状态的重要工具。密度矩阵的构造对于理解黑洞的热力学性质至关重要，因为它允许我们计算系统的熵和其他热力学量。进一步，论文涵盖了外部操作符的相关函数，这对于探测黑洞外部的物理效应极其重要。这些相关函数可以帮助我们理解黑洞与周围环境的相互作用，以及可能的信息传递机制。特别是在AdS-CFT对应原理的框架下，这样的计算对于将黑洞的量子引力问题转化为更易处理的边界理论问题具有关键意义。这篇论文为二维黑洞的统计力学提供了新的见解和计算方法，不仅深化了我们对黑洞动力学的理解，也为未来在量子引力领域的研究提供了有价值的工具。其开放访问的特性使得全球的研究者都能接触到这些创新性的研究成果。

JHEP05(2019)198

a) b)

Figure 2. Typical path shapes, a) for M  1 and b) for M  1.

We will consider three more speciﬁc regularization recipes:

1. For general values of γ and L, one has to take the  → 0 limit, or at least to assume

that   min{γ

−1

, 1, L}. Then equation (3.4) holds for b

= 2/ and b

= 0. A

similar result is derived in sections 9.1–9.2 of Polyakov’s book [20], but we will give a

simpler argument. Unfortunately, the arbitrariness of  complicates the comparison

with the Schwarzian problem.

2. In the Schwarzian limit where both γ and L are large,  need not be very small. If

we assume that γ

−1

   1, then b

and b

are -independent, namely, b

≈ γ and

≈ γ/2. However, the accuracy of this approximation is not suﬃcient to match the

Schwarzian partition function.

3. The correct match is achieved if b

= γ and b

= γ/2 + 1/(8γ). This will be shown

later by calculating the density of states.

The qualitative diﬀerence between cases 1 and 2 is in the shape of a typical path

as we zoom in, see ﬁgure 2. To justify both claims, we ﬁrst separate path properties at

small distances from those at intermediate and large distances. At distances ∆x ∼ 1 in

the Schwarzian limit, one may use classical equations. We have already found from their

analysis that M ≈ γ. At short distances, the parameter M is important, but γ (as the

coeﬃcient in the area term) is not. Indeed, if ∆x  1, one may replace the hyperbolic

plane with R

. The contribution to the area from a (∆x)-size section of the path with

ﬁxed endpoints varies at most by (∆x)

. Thus, if ∆x  γ

−1/2

, the area term is negligible.

In the Schwarzian limit, the area term may actually be ignored if ∆x  1 because, as we

will see shortly, typical paths are almost straight.

Let us discuss the short-distance behavior in more detail. For this purpose, we work in

and neglect the area term. We also drop the trivial term γ(L −2π) and simultaneously

subtract γ from E

so that I

[X] vanishes but the modiﬁed action remains the same (up

to an additive constant). The number E

= −M plays the role of a chemical potential for

a small piece of a path. The simpliﬁed propagator G

(excluding the e

γ(−L+2π)

factor) is

completely characterized by the integration measure

Dµ =

j=1



(2π)

−1

δ(|X

− X

j−1

| − )



n−1

j=1

, X

= x

, X

= x

, n =



. (3.5)

A convenient analogy is a ﬂuctuating polymer chain. Suppose that one end of a chain, x

is ﬁxed at some location far away (compared to ). The probability density of the other

– 9 –

JHEP05(2019)198

end, f

, L) = G

, x

; L), satisﬁes the equation

(x, L + ) =

2π



|x − y| − 



(y) dy. (3.6)

Now imagine pulling on that end with force p

. Applying the force and passing to the

grand canonical ensemble means multiplying f

(x, L) by exp((p

)

−ML). If the chain

is long enough to attain the thermodynamic limit, the modiﬁed f

should be constant.

Thus, the original function is f

(x, L) ∝ exp(−(p

)

+ ML). Plugging this ansatz

into (3.6), we ﬁnd the dispersion relation

M =

ln I

(|p



≈











p

if |p

|  1

| if |p

|  1

(3.7)

where I

is the modiﬁed Bessel function. Typical path geometries in the two cases are

shown in ﬁgure 2. Thus in the  → 0 limit, equation (3.6) becomes ∂

= (/4)∇

. It

can be reduced to the standard diﬀusion equation that corresponds to the quadratic action

I[X] =

dτ:

∂

∇

, where β =



L. (3.8)

This proves claim 1. As for claim 2, the conditions γ  1 and M ≈ γ are consistent with

the second case of equation (3.7), namely, M ≈ |p

| for |p

| ≈ γ. To ﬁrst order in |p

|−γ,

this dispersion relation can also be written as

M ≈

2γ

. (3.9)

Hence

(x, L) = e

(γ/2)L

f(x, β), where ∂

f =

∇

f, β =

. (3.10)

From here on, we study the quadratic action (3.2). The propagator can be obtained

by solving the diﬀusion equation with a suitable initial condition:

∂

G(x

, x

; τ) =

∇

G(x

, x

; τ), lim

τ→0

G(x

, x

; τ) =

δ(x

− x

)

g(x

)

(3.11)

where the Laplacian involves the covariant derivative acting on ν-spinors,

∇

ψ = (∂

− iνω

)ψ, ν = −iγ. (3.12)

The partition function is deﬁned as the integral of e

−I[X]

over closed paths. To make the

quantity ﬁnite, we divide it by the volume of the symmetry group PSL(2, R), which is 2π

times the area of the hyperbolic plane:

Z(β) =

vol(PSL(2, R))

g(x) G(x, x; β) =

2π

G(0, 0; β). (3.13)

In the remainder of this section, we solve equation (3.11) and analyze the resulting

expression for the partition function. Without loss of generality, we may assume x

= 0;

– 10 –

JHEP05(2019)198

then the solution

G(x, 0; τ) is rotationally symmetric, i.e. independent of the polar angle

ϕ. (The ring accent indicates the disc gauge; we generally put it only where it matters.)

The Laplacian on the hyperbolic plane is related to the SL(2, R) Casimir operator Q,

− ∇

= Q + ν

. (3.14)

The representation of

SL(2, R) by spinors on H

is described in [19]. However, some results

hold only for real ν, so the corresponding arguments have to be redone. The disk gauge

expression for the Casimir operator is

Q = −(1 − u)



u∂

+ ∂



1 − u



(m − ν)

− (m + ν)



, (3.15)

where

u = r

, m = ν − i∂

. (3.16)

While

Q is not Hermitian for imaginary values of ν, it becomes Hermitian when restricted to

the m = ν subspace, which consists of rotationally symmetric functions. In this special case,

Q = −(1 − u)



u∂

+ ∂



+ γ

(1 − u). (3.17)

The functions in question depend only on u ∈ [0, 1), but we should use the correct inner

product and boundary condition at u = 0. The inner product is given by the integral over

the hyperbolic plane

i = 2π

(u)

∗

(u)

2 du

(1 − u)

. (3.18)

Therefore, normalizable functions vanish at u → 1 faster than (1 − u)

1/2

. To determine

the condition at the origin, we notice that eigenfunctions of

Q have the asymptotic form

f(u) ≈ a+b ln u for u → 0. But in two dimensions, a singularity at the origin is not allowed;

hence b = 0. A more general condition is that f(0) is ﬁnite and lim

u→0

u∂

f(u) = 0. It

guarantees the Hermicity of

Q because hf

i−h

i = 4πu



∗

(∂

)−(∂

∗





u=0

Let us ﬁnd an eigenbasis of the operator

Q acting in the Hilbert space we have just

described. The m = ν eigenfunctions are as follows [19]:

(u) =

(0) · (1 − u)

F(λ + ν, λ − ν, 1; u),

= λ(1 − λ)

, (3.19)

where F(a, b, c; x) = Γ(c)

−1

2 1

(a, b, c; x) is the scaled hypergeometric function and

(0)

is for now simply a normalization factor. The eigenvalue λ(1 −λ) must be real; hence λ is

real or has the form

+ is with a real s. Eliminating the λ ↔ 1 −λ redundancy, there are

three mutually exclusive cases: λ =

, λ >

, and λ =

+ is with s > 0. It follows from

the asymptotic expression

(u)

(0)

≈

Γ(1 − 2λ)

Γ(1−λ+ν) Γ(1−λ−ν)

(1−u)

Γ(2λ − 1)

Γ(λ + ν) Γ(λ − ν)

(1−u)

1−λ

for u → 1 (3.20)

that the ﬁrst two sets of eigenfunctions are not normalizable or δ-normalizable. Thus we

restrict to the third case. Fixing

(0) =



(2π)

−1

sinh(2πs)/(cosh(2πγ) + cosh(2πs))



1/2

we have



−iγ

1/2+is



−iγ

1/2+is



= s

−1

δ(s − s

). (3.21)

– 11 –

剩余55页未读，继续阅读

weixin_38606019

粉丝: 4
资源: 935

二维黑洞统计力学：AdS2时空与粒子动力学

二维声黑洞的热力学

二维黑洞微状态的Calogero公式

二维共形场理论对黑洞的热力学

存在指数电动力学的Lifshitz黑洞的热力学和规范/重力对偶

非线性电动力学黑洞的热力学和带电粒子吸收时弱宇宙检查的有效性

新型带电膨胀的BTZ黑洞的热力学

引力彩虹中新型带电Dilaton黑洞的热力学

二维带电膨胀的黑洞的狄拉克准正规模

带电的AdS黑洞的热力学几何形状和相变

关于二维SYM中十维局部黑洞和约束状态的注记

最新资源