BMN矩阵模型中的准正态模式与黑洞物理

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"这篇研究论文探讨了非交换矩阵动力学中的准正态模式，并与AdS/CFT对应中的黑洞物理现象进行了对比分析。作者通过数值模拟BMN矩阵模型的戴森流体解，研究了系统的实时动力学行为，特别是在均衡分布被淬灭后，观察到了标量单轨迹可观测物的准正态振荡。他们识别出了最低的准正态模，并确定了其频率与能量的关系。此外，将BMN矩阵模型视为N=4超对称Yang-Mills（SYM）理论的一个截断版本，他们还计算了在AdS 5-Schwarzschild背景下的双标量场的准正态模频率，并发现了与BMN模型中相似的结果。该论文发表于JHEP09(2017)048，由Springer出版，2017年9月12日发布。" 详细知识点: 1. **非交换矩阵动力学**: 非交换矩阵动力学是一种描述量子系统动力学的方法，其中矩阵元素不遵循经典的交换律。在这种理论中，矩阵元素之间的相互作用可能导致非平凡的动力学行为。 2. **BMN矩阵模型**: BMN矩阵模型是弦理论的一种简化模型，它在特定极限下可以作为N=4超对称Yang-Mills（SYM）理论的近似。该模型允许研究高能行为和AdS/CFT对应。 3. **AdS/CFT对应** (Anti-de Sitter/Conformal Field Theory correspondence): 这是一种强弱对偶理论，将量子场论（如N=4 SYM）与引力理论（如AdS时空中的弦理论）联系起来，使得在一种理论中难以解决的问题可能在另一种理论中得到解答。 4. **戴森流体解决方案**: 在本研究中，戴森流体指的是矩阵模型中的特殊解，它描述了系统的集体动力学行为，类似于物理中的流体动力学。 5. **准正态模式** (Quasi-normal modes): 准正态模式是在封闭系统中振荡解的一种形式，通常出现在黑洞物理学中，它们代表了黑洞辐射的频率。这些模式随着时间衰减，且频率和衰减率与系统参数相关。 6. **数值模拟**: 由于解析解的困难，研究人员通过数值方法模拟了BMN矩阵模型的实时动力学，以观察和理解系统的动态行为。 7. **标量单轨迹可观测物**：在量子场论中，单轨迹可观测物是指只涉及一个粒子或场的算符，它们是理解和研究系统性质的重要工具。 8. **能量与频率的关系**：研究中确定了最低准正态模的频率与能量的函数关系，这有助于揭示系统动力学的基本特征。 9. **N=4超对称Yang-Mills理论** (N=4 SYM): 这是一个具有四个超对称性的规范理论，它在理论物理中扮演着重要角色，尤其是在AdS/CFT对应中的应用。 10. **AdS 5-Schwarzschild背景**: AdS 5-Schwarzschild是五维反德西特时空中的一个黑洞解决方案，它是研究黑洞物理和AdS/CFT对应的重要背景。通过以上分析，我们可以看出这篇论文深入探讨了非交换矩阵模型与AdS/CFT对应该领域的前沿问题，提供了对黑洞物理、量子场论和弦理论之间关联的更深入理解。

JHEP09(2017)048

where the matrices {A

}

c=1,2,3

, and {Y

}

q=1,...,15

generate rotations in R

, and

, respectively.

The canonical form of A

is that of an anti-symmetric matrix whose upper triangular

part has 0 everywhere, except for +1 in the position (¯a, ¯c), corresponding to the plane

¯a

, X

¯c

) which is being rotated. Similarly for Y

. For example, the combination Tr



−



is the charge associated to rotations in the plane (X

, X

). From the equations of

motion (2.15)–(2.14), it can be explicitly checked that

= 0 , ∀ k, q . (2.17)

3 Statistics and dynamics

In the previous section we have introduced the BMN matrix model as a dynamical system

focusing on various aspects of the time evolution of a single conﬁguration. In this section

we describe a more general framework in which the degrees of freedom are interpreted as

the microscopic elements of a statistical ensemble. The statistical framework that we are

advocating is an example of a“Dyson ﬂuid”, and constitutes our starting point for the

study of non-equilibrium dynamics in the BMN matrix model.

The deﬁnition of equilibrium that we shall use throughout the paper is the following.

Given: a set of initial conditions, and an algebra of observables, a dynamical system is said

to be at the equilibrium w.r.t. the given initial conditions, if the expectation value of any

observable in the algebra is time independent. Let us point out that the when a statistical

equilibrium is reached, the microscopic degrees of freedom are not necessarily steady, and

in fact they can have an highly non trivial dynamics. As we are going to show, the process

of equilibration in the BMN matrix model has precisely this feature.

Before presenting our numerical results we describe in great detail our choice of initial

conditions, and we emphasize some important properties of the observables that we shall

study. Several aspects of our analysis will be generic, therefore we expect our results

to provide the key elements towards the understanding of the process of equilibration in

non-commutative dynamical systems.

3.1 Big-Bang initial conditions

We consider the following class of out-of-equilibrium initial conditions:

= 0 , X

= 0 , P

∈ TGU , (3.1)

where TGU stands for “Traceless Gaussian Unitary ensemble”. In practice, the momenta

are parametrized as:

−1

∀I , (3.2)

and each coeﬃcient p

is extracted randomly from a single gaussian distribution, which

we take to have standard deviation σ and zero mean. Because of this property, we expect

, P

] 6= 0 for all of the randomly generated initial conditions, apart from a set of zero

– 6 –

JHEP09(2017)048

measure. The constraint (2.11) is satisﬁed at the initial time, and therefore A

will remain

zero during the time evolution.

Intuitively, we designed the initial conditions (3.1) having in mind a big-bang in which

the variables {X

} start at the origin with random momenta. As soon as the {X

} do

not commute, interactions are turned on. This can be understood by looking at the force

terms in (2.8)–(2.9), namely

(3)

∝ iε

abc

(4)

∝

, [X

, X

]] . (3.3)

For a given initial condition we shall ﬁnd [P

, P

] 6= 0 at the initial time, and after an

inﬁnitesimal time step of δt,

(δt) = X

(0) + δt P

(0) = δt P

(0) , (3.4)

thus F

(3)

6= F

(4)

6= 0. With these forces turned on, the degrees of freedom are coupled,

and will evolve non trivially under the Hamiltonian ﬂow ϕ.

We generate a ﬁnite set of initial conditions of the form (3.1)–(3.2), which will be

denoted by E(0), where E stands for ensemble. By deﬁnition E(t) = ϕ

(E(0)). Notice that

E(0) belongs to a Lagrangian submanifold of the phase space, centered at X

= P

= 0, and

extended only along the directions of the momenta. The numerical integration is achieved

through an improved second order leap-frog algorithm [15]. The time step used in the

simulation is δt = 0.05. The stability of the integration has been checked by increasing

and decreasing δt of a factor of two. Once the ensemble E(t) is obtained, we compute the

expectation value of any observable O through the estimator,

hOi(t) ≡

vol(E(t))

X ∈ E

O(X) , (3.5)

where vol(E(t)) equals the number of conﬁgurations, and we determine the error on hOi

by means of a jackknife analysis.

On E(t), the SO(3) ×SO(6) charges vanish because the {X

= 0}, whereas the energy

is non zero. Since the energy is purely kinematical at the initial time, we can calculate its

expectation value and its standard deviation, analytically,

hEi = n

tot

− 1) σ

, SD[E] =

2 n

tot

− 1) σ

. (3.6)

It is convenient to deﬁne the parameter h and rescale σ in such a way that hE i does not

depend on N,

σ =

− 1

. (3.7)

Then, hE i = h n

tot

. For concreteness we will always specify the dependence on the expec-

tation value of the energy by referring to E = E

The statistical framework developed so far has a general validity. For the BFSS matrix

model, the study of the ensemble greatly simpliﬁes because when m = 0 the equations of

– 7 –

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