Elsevier

Annals of Physics

Volume 432, September 2021, 168566
Annals of Physics

The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere

https://doi.org/10.1016/j.aop.2021.168566Get rights and content

Highlights

  • •

    Curvature can induce a gauge structure.

  • •

    Gauge field possesses an angular momentum.

  • •

    Geometric momentum and angular momentum are mutually related simply on hypersphere.

Abstract

A particle that is constrained to freely move on a hyperspherical surface in an N≥2 dimensional flat space experiences a curvature-induced gauge potential, whose form was given long ago (Ohnuki and Kitakado, 1993). We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential and its components obey the commutation relations pi,pj=−iħJij/r2, in which ħ is the Planck’s constant, and pi (i,j=1,2,3,…N) denotes the i-th component of the geometric momentum, and Jij specifies the ij-th component of the generalized angular momentum containing both the orbital part and the coupling of the generators of continuous rotational symmetry group SO(N−1) and curvature, and r denotes the radius of the N−1 dimensional hypersphere.

Introduction

In quantum mechanics, a constrained dynamical system is usually associated with a gauge structure, and it is quite well-understood in, for instance, gravitational field [1], [2], condensed matter physics [3], [4], quantum fields [5] and particle physics [6]. We are recently interested in the constrained motion, i.e., a particle remains and freely moves on a hypersurface [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. There are also a lot of papers paying attention to the curvature-induced gauge structure for the system, and the form of the gauge potential is well-known [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] . The form of the gauge potential is clearly a coupling of the curvature and all generators of a rotational symmetry group, which in quantum mechanics is not necessarily understood to represent the spin though it is the important situation. Even so, the gauge potential is far beyond fully understood. For instance, once a classical bracket (cf. Eq. (13)) involves both the momentum and the orbital angular momentum, the corresponding quantum commutator (cf. Eq. (19)) based on the quantization of the classical bracket contains both the momentum operator and the angular momentum operator, and it appears transparent. However, the quantum commutator is highly non-trivial because the momentum explicitly contains the extrinsic curvature, and the previous studies unintentionally missed this geometric nature of the momentum till 2011 when the geometric momentum came into sight [8]. The main aim of the present study is to show that the relation (19) holds true with reasonable inclusion of the gauge potential into the geometric momentum.

This paper is organized in the following manner. In Section 2, how Ohnuki and Kitakado obtained the gauge potential is outlined and commented. In Section 3, the Dirac formalism of quantization for the particle on the sphere is invoked, where the Dirac brackets between the momentum components and the orbital angular momentum components play central role. To show that the fundamental quantum conditions in the Dirac scheme of quantization are completely compatible with the gauge potential, we must utilize the proper form of the momentum. In Sections 4 A proof of, 5 A proof of, we explicitly prove two sets of fundamental quantum conditions in the Dirac scheme, respectively. Section 6 is a brief conclusion and discussion.

Section snippets

Ohnuki and Kitakado gauge potential on hyperspherical surface

By a hyperspherical surface SN−1 (N⩾2), we mean a spherical surface in N dimensional flat space EN. The surface equation is given by, xixi=r2,(i=1,2,3,…,N), and ni=xir,where r denotes the radius of the sphere, and xi is the ith coordinate and ni is the ith component of the normal vector. Throughout the paper, the Einstein’s summation convention is adopted, which implies the indices repeated twice in a term are summed over the range of the index unless specified, and the Roman indices i,j,k,l,m,

Dirac formalism of quantization on the sphere SN−1 and an SO(N,1) algebra

Let us now consider following equation of the hyperspherical surface equation, fx=12rxi2−r2=0.From it we know n=∇f. Dirac formalism for constrained motion gives following Dirac brackets [9], xi,xjD=0,xi,pjD=δij−ninj,pi,pjD=−Lijr2=−xipj−xjpir2, where Lij=xipj−xjpi is the usual ij-component of the orbital angular momentum satisfying, xk,LijD=xiδkj−xjδki,pk,LijD=piδkj−pjδki,Ljk,LlmD=δjlLkm−δjmLkl+δkmLjl−δklLjm.

Now we utilize the full Dirac formalism of quantization procedure to define a quantum

A proof of pi,pj=−iħJij/r2

We start from p=Π−A, from which the commutators pi,pl are, pi,pl=Πi−Ai,Πl−Al.We split N components pi of the momentum p into two categories of pα α=1,2,…,N−1 and pN, pα=Πα+1r2xifiα=Πα−Aα,pN=ΠN−AN=ΠN, where we used AN=0. The calculations of pi,pl will be done separately, and we first study pα,pβ and secondly deal with pα,pN.

In order to compute pα,pβ, we split the commutator pα,pβ into four parts, pα,pβ=Πα−Aα,Πβ−Aβ=Πα,Πβ−Πα,Aβ−Aα,Πβ+Aα,Aβ.On the right-hand-side of above expression, the first

A proof of pk,Jij=iħpiδkj−pjδki

The proof of fundamental quantum conditions pk,Jij=iħpiδkj−pjδki (20) is also straightforward. To do it, we split N components pi of the momentum p into two categories of pα α=1,2,…,N−1 and pN. In consequence, the commutators pk,Jij are divided into two categories as pλ,Jij and pN,Jij. To process pλ,Jij, we compute [pλ,Jαβ] and pλ,JiN one by one. The detailed steps of calculation of commutators [pλ,Jαβ] is given in the following.

We split commutators [pλ,Jαβ] into four parts, [pλ,Jαβ]=[Πλ−Aλ,Lα

Conclusions and discussions

For a particle that is constrained on an N−1-dimensional hypersphere, the classical bracket is pi,pjD=−Lij/r2. In contrast to the usual understanding of the corresponding quantum condition remains pi,pj=−iħLij/r2 in which Lij represents orbital angular momentum, we argue that it must be replaced by pi,pj=−iħJij/r2 in which Jij includes two parts, and one of them is the usual orbital angular momentum Lij and another comes from the gauge potential fij. In other words, we adapt the definition of

CRediT authorship contribution statement

Z. Li: Calculations, Software, Writing – original draft. L.Q. Lai: Calculations, Software, Writing – original draft. Y. Zhong: Methodology, Check. Q.H. Liu: Conceptualization, Writing – review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work is financially supported by National Natural Science Foundation of China under Grant No. 11675051.

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