二维球面上的几何动量与狄拉克费米子：规范势的新视角

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"这篇论文详细探讨了在二维球体上协变的几何动量、规范势与狄拉克费米子之间的关系。作者们提出了一个关于超表面ΣN-1上自旋粒子量子条件的通用框架，其中广义角动量在超球面SN-1上的定义是几何动量的一般协变结果。此外，他们专注于二维球面上狄拉克费米子的研究，揭示了存在一种具有几何起源的广义角动量，其三个笛卡尔分量构成了su(2)代数。他们还指出，自旋半粒子不会受到由曲率引起的几何势的影响。这项工作发表于Eur.Phys.J.C，并且是开放访问的。" 在本文中，作者首先介绍了几何动量的概念，这是在N维平坦空间中粒子在(N-1)维曲面ΣN-1上的动量表现形式，其依赖于曲面的平均曲率。当动量变得一般协变时，适应于曲面上的自旋粒子，其中的自旋连接部分可以被视为规范势。这是物理学中的一个重要概念，因为规范势在量子场论中扮演着核心角色，特别是在描述粒子间的相互作用。论文的主要发现有两个。第一个发现是建立了一个自旋粒子在超表面ΣN-1上的量子条件的通用框架。在这个框架下，广义角动量在超球面SN-1上的定义是由几何动量的一般协变性带来的结果。广义角动量在研究自旋粒子的动力学时起着关键作用，因为它不仅涉及粒子的位置，还涉及其自旋状态。第二个发现集中在二维球面上的狄拉克费米子。狄拉克费米子是具有半个整数自旋的基本粒子，如电子。作者展示了在这个特定背景下，存在一种广义角动量，其三个笛卡尔分量满足su(2)代数，这意味着它们遵循类似于旋转对称性的规则。这种几何起源的广义角动量为理解和描述狄拉克费米子的行为提供了新的视角。此外，作者还指出，对于自旋半粒子，它们在二维球面上的运动不会受到由曲率产生的几何势的影响。这一发现可能对理解曲率空间中狄拉克费米子的行为有深远的影响，尤其是在量子霍尔效应、拓扑绝缘体和量子计算等领域的应用。这篇论文深入探讨了自旋粒子在曲面几何中的动力学特性，特别是对于狄拉克费米子，它揭示了新的物理现象，并为量子场论和固体物理学的研究提供了重要的理论工具。

Eur. Phys. J. C (2019) 79:712

https://doi.org/10.1140/epjc/s10052-019-7231-4

Regular Article - Theoretical Physics

Generally covariant geometric momentum, gauge potential and a

Dirac fermion on a two-dimensional sphere

Q. H. Liu

1,2,a

,Z.Li

,X.Y.Zhou

,Z.Q.Yang

,W.K.Du

School for Theoretical Physics, College of Physics and Electronics, Hunan University, Changsha 410082, China

Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China

Received: 2 July 2019 / Accepted: 17 August 2019 / Published online: 24 August 2019

Abstract For a particle that is constrained on an (N − 1)-

dimensional (N ≥ 2) curved surface 

N −1

, the Cartesian

components of its momentum in N -dimensional ﬂat space

are believed to offer a proper form of momentum for the par-

ticle on the surface, which is called the geometric momentum

as it depends on the mean curvature. Once the momentum

becomes generally covariant as to be applicable to spin par-

ticles on the surface, the spin connection part in it can be

interpreted as a gauge potential. The principal ﬁndings are

twofold. The ﬁrst is a general framework of quantum condi-

tions for a spin particle on the hypersurface 

N −1

, and the

generalized angular momentum is deﬁned on hypersphere

N −1

as one consequence of the generally covariant geo-

metric momentum. The second is devoted to a study of a

Dirac fermion on a two-dimensional sphere and we show

that there is the generalized angular momentum whose three

cartesian components form the su(2) algebra, demonstrated

to be of geometric origin but obtained before by considera-

tion of dynamics of the particle. Moreover, we show that there

is no curvature-induced geometric potential for the spin half

particle.

1 Introduction

In quantum mechanics, there are fundamental quantum

conditions (FQCs) [x

, x

]=0, [x

, p

]=i

hδ

, and

[ p

, p

]=0, which are deﬁned by the commutation rela-

tions between positions x

and momenta p

(i, j, k, l =

1, 2, 3,...,N ) where N denotes the number of dimensions

of the ﬂat space in which the particle moves [1]. In position

representation, the momentum operator takes simple form

as p =−i∇ where ∇≡e

∂/∂x

is the ordinary gradient

operator, and N mutually orthogonal unit vectors e

span the

N dimensional Euclidean space E

. Hereafter the Einstein

e-mail: quanhuiliu@gmail.com

summation convention over repeated indices is used. When

the particle is constrained to remain on a hypersurface 

N −1

embedded in E

, in classical mechanics, we can deﬁne Dirac

bracket {A, B}

instead of the usual Poisson one {A, B} for

the two functions A and B of both x

and p

[2–6],

{A, B}

≡{A, B}−{A,ϕ

−1

αβ

{ϕ

, B}, (1)

where ϕ

= 0(α = 1, 2) are two constraint conditions,

and ϕ

≡ f (x)

(

= 0

)

is the equation of surface, and ϕ

≡

p ·∇f (x)

(

= 0

)

shows the motion to be tangential to the

surface, and the matrix elements C

αβ

is deﬁned by C

αβ

{ϕ

,ϕ

}. The equation of surface f (x) = 0 can be so chosen

that

∇ f (x)

= 1 thus n ≡∇f (x) = e

is the normal vec-

tor at a local point u

(μ, ν, α, β = 1, 2, 3,...,N −1) on the

surface. The relation between positions x in E

and the local

coordinates u

on 

N −1

is x = x(u

), and x

= ∂x/∂u

is the μth contravariant component of the natural frame on

the point. The usual canonical commutation relations are

hypothesized to be given by [ A, B]=i

{

[A, B]

}

Hermitian

[2] where O

Hermitian

stands for a Hermitian operator of an

observable O. Explicitly, Dirac brackets for variables x

and

are given by [x

, x

]

= 0, [x

, p

]

= (δ

− n

and [p

, p

]

=−(n

, j

− n

) p

. The FQCs [2]are

thus [3–6],

, x

]=0, [x

, p

]=i

h(δ

− n

), and

[ p

, p

]=−i



, j

− n

) p



Hermitian

, (2)

This set of the FQCs (2) is highly non-trivial, from which it

is in general impossible to uniquely construct the momenta

. Our propose of the proper form of the momentum for a

spinless particle was [7–11],

p =−i

h(∇



+ Mn/2), (3)

where ∇



≡e

(δ

−n

)∂

=∇−n∂

= x

∂

is the gradi-

ent operator, and the mean curvature M ≡−∇



·n is deﬁned

123

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