零多边形Wilson环的量子力学解析

145 浏览量更新于2024-07-16 收藏 870KB PDF 举报

"这篇论文探讨了零多边形Wilson环在量子力学中的应用，特别是与最大超对称轨距理论中的散射振幅的关系。文章指出，这些散射振幅可以视为零多边形轮廓上超级Wilson回路的对偶表示。通过分析超级Wilson回路的操作员产品展开，研究发现其动力学由多颗粒GKP激发的演化控制，这些激发源自潜在的开放自旋链的光谱问题。作者采用Baxter Q-Operator和Sklyanin的变量分离方法，对GKP激发的特征函数和特征值进行了明确的构造，并展示了如何在超级Wilson循环中定义多粒子六边形变换。他们还证明了在't Hooft耦合的先锋阶中，这些变换具有保质量的形式，这对于保留粒子数的转换尤其重要，这一结果是在更一般的情况中提出的。" 这篇发表在《核物理B》上的研究由A.V.Belitsky、S.E.Derkachov和A.N.Manashov等人合作完成，他们在不同的学术机构任职。文章在2014年3月15日在线发布，经过了多次修订和接受，由Stephan Stieberger编辑审阅。量子力学中的零多边形Wilson环是一个关键的概念，它在最大超对称轨距理论中扮演着重要角色。这里的“零多边形”指的是具有零面积的多边形，通常在二维空间中表示，但在四维时空中则有更复杂的几何含义。超级Wilson回路是量子场论中的一种操作员，它可以用来描述强相互作用粒子的路径积分。通过对这些回路的操作员产品展开，研究者能够揭示出其背后的动力学特性。多颗粒GKP激发是Wilson回路动力学的核心元素，它们来源于一个潜在的开放自旋链的光谱问题。开放自旋链是量子链系统的一个实例，其中自旋站点之间的相互作用导致了复杂的动力学行为。Baxter Q-Operator是一种强大的数学工具，常用于解决量子可积系统，而Sklyanin的变量分离方法则是一种解决一维量子系统的有效手段。在这项工作中，作者不仅解决了模型的解析形式，还给出了GKP激发的特征函数和特征值的具体构造。他们展示的多粒子六边形变换定义了在Wilson循环内的动态过程，同时证明了在't Hooft耦合的先锋阶下，这些变换保持了粒子数守恒，这是对之前一般性结论的重要验证和扩展。这篇研究深入探讨了零多边形Wilson环在量子力学中的动力学性质，以及它们与最大超对称轨距理论散射振幅的对偶关系，对于理解和计算高能物理中的复杂现象具有重要意义。通过利用数学工具如Baxter Q-Operator和Sklyanin方法，作者为理解这些物理过程提供了新的理论框架。

A.V. Belitsky et al. / Nuclear Physics B 882 (2014) 303–351 311

eigenfunctions and ﬁnding corresponding eigenvalues. For the case at hand, we are dealing with

an open spin chain that does not possess SL(2, R) symmetry due to boundary interactions. In

addition, as we can conclude from the one-particle eigenfunction (1.21), the system is not en-

dowed with a vacuum states. As a consequence of this last fact, traditional methods based on the

Algebraic Bethe Ansatz [24,25] are not applicable and instead one has to rely on the method of

the Baxter Q-operator [26] and the Separation of Variables (SoV) [27].

Our subsequent consideration is organized as follows. After introducing in the next section

natural Hilbert space for the

Hamiltonian and a scalar product that makes it self-adjoint, we

ﬁnd a complete set of integrals motion commuting with H

. As a next step on the way to solve

the spectral problem, we recall the factorization of R-matrices [24,25] into intertwiners that

play a crucial role in our analysis. In Section 3, we use them to construct the Hamiltonians and

demonstrate their equivalence with the ones arising in the problem of renormalization of the

light-cone operators (1.23). Then in Section 4, we construct the Baxter Q-operator as well as

its conjugate making use of available techniques developed for SL(2) invariant magnets. Both

operators are then used to generate commuting Hamiltonians. In Section 5, we devise an iterative

procedure to obtain the eigenfunctions of the Hamiltonian. We then cast them in a form of a

contour integral representation as well as an integral representation in the upper half-plane. The

latter is instrumental in demonstration of their orthogonality with respect to the scalar product

introduced earlier. To adopt our consideration for multiparticle contributions to the null polygonal

Wilson loops, we deﬁne a new scalar product on the real half-line in Section 6 and establish its

relation to the one for the analytically continued functions in the upper half-plane. Then we

explicitly compute the square and hexagon leading order transitions in Section 7, where we

also prove the factorized ansatz for the latter suggested in Ref. [16]. Finally, we conclude by

pointing applications of the current construction to the operator product expansion of polygonal

Wilson loops and outline the generalization to include GKP excitations of different spins. Several

appendices contain conventions and computational details that we found inappropriate to include

in the main body of the paper. We also provided there a complementary construction the wave

function in the representation of Separated Variables and its relation to the eigenvalues of the

Baxter Q-operator.

2. Open spin chain

The Hamiltonian (1.24) deﬁnes a non-periodic one-dimensional lattice model of interacting

spins S

= (S

−

) acting at each site deﬁned by the coordinate of the GKP excitation.

They form an inﬁnite-dimensional representation of the sl(2, R) algebra,



−



= 2S





=±S

. (2.1)

Our choice of the representation S

±,0

for the spin generators S

±,0

on the ﬁelds X is driven

by the form of the pairwise Hamiltonians (1.28) acting on the elementary ﬁelds X(x),i.e.

±,0

,X(x

)]=iS

±,0

X(x

) [21,22]. They are taken in the form of differential operators act-

ing on lattice sites

= x

∂

+ 2sx

−

=−∂

= x

∂

+ s, (2.2)

where all labels of the representation s are the same for any n. The latter condition deﬁnes a

homogeneous open spin chain. The spin variable s is assumed to be real and bounded from

below by 1/2.

312 A.V. Belitsky et al. / Nuclear Physics B 882 (2014) 303–351

2.1. Scalar product

It is convenient to deﬁne a scalar product on the space of functions Ψ

,...,x

). Its choice

is a matter of convenience but it has to be adapted to the physical problem under consideration.

As was explained in Section 1, we are interested in the τ -dependence of correlation functions of

the light-cone operators O

which read schematically

0|O







,...,x





aτH

,...,x

)|0



aτE









,...,x





†

,...,x

). (2.3)

Here the functions Ψ

are determined by the operator matrix elements between the eigenstate

 of Hamiltonian and the vacuum,

,...,x

) =E

,...,x

)|0, and the sum

runs over all eigenstates. Although the variables x

are the light-cone coordinates of the ﬁelds X

entering the Lagrangian of the theory and are, therefore, real, nevertheless distribution amplitudes

,...,z

) can be regarded as functions of z

in complex plane. It is known that they are

analytic functions in upper half-plane and vanish at inﬁnity. As a consequence of this change, the

sl(2, R) generators in Eq. (2.2) depend and act on the z

variables accordingly.

In what follows we will consider the spectral problem (1.29) on the space of functions of

N comple

x variables z

analytic in the upper half-plane and endowed with a standard SL(2, R)

invariant scalar product [28]. Such a formulation appeared to be very useful for constructing the

SoV representation for the closed and open SL(2, R) spin chains [29,30]. The scalar product on

the space of functions holomorphic in the upper half-plane

is deﬁned as follows [28]

Φ|Ψ =





Φ(z

)



∗

Ψ(z

), (2.4)

where z

= x

+ iy

. The integration measure reads

2s − 1

(2y

)

2s−2

θ(y

) (2.5)

and the integration runs over the upper half-plane due to the presence of the step-function θ(y

Under this scalar product the generators (2.2) are anti-self-adjoint



0,±



†

=−S

0,±

, (2.6)

with hermitian conjugation deﬁned conventionally as

Φ|GΨ =



†





. (2.7)

The Hilbert space of the model is given by the tensor product of Hilbert spaces at each site



n=1

, such that the generalization of the scalar product (2.4) to multivariable functions is

straightforward,

Φ|Ψ =





k=1



Φ(z

,...,z

)



∗

Ψ(z

,...,z

). (2.8)

In QCD terminology Ψ

,...,x

) is known as the distribution amplitude.

The Hilbert spaces of holomorphic functions are well studied mathematical topic, for a review, see Ref. [31].

剩余48页未读，继续阅读

weixin_38587130

粉丝: 4
资源: 937

零多边形Wilson环的量子力学解析

圆锥形扭曲场和空多边形Wilson环

C++ 多边形相交、多边形合并算法，支持凹凸多边形

rhino grasshoper 多边形莫比乌斯环.gh

关于轻型Wilson多边形的几何缩放：αs中的高阶

关于具有圆形边缘的多边形轮廓上的Wilson圈的异常共形Ward身份

多边形裁剪 多边形裁剪 多边形裁剪

OpenGL绘制花环.rar_glfw多边形_layfv3_opengl_opengl 正多边形_opengl 画花

不同方式的多边形填充.rar_填充多边形_多边形_多边形填充

多边形：允许使用多边形

最新资源

多边形裁剪多边形裁剪多边形裁剪