General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere

TL;DR

This paper develops a covariant geometric momentum framework for particles on curved surfaces, interprets the spin connection as a gauge potential, and analyzes a Dirac fermion on a sphere, revealing algebraic structures and absence of geometric potential.

Contribution

It introduces a general covariant geometric momentum formalism and applies it to Dirac fermions on a sphere, uncovering algebraic properties and the lack of curvature-induced potential.

Findings

General covariant geometric momentum framework established.

Generalized total angular momentum forms an su(2) algebra.

No curvature-induced geometric potential for the Dirac fermion.

Abstract

For a particle that is constrained on an ($N-1$)-dimensional ($N\geq2$) curved surface, the Cartesian components of its momentum in $N$-dimensional flat space is believed to offer a proper form of momentum for the particle on the surface, which is called the geometric momentum as it depends on the mean curvature. Once the momentum is made general covariance, the spin connection part can be interpreted as a gauge potential. The present study consists in two parts, the first is a discussion of the general framework for the general covariant geometric 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 total angular momentum whose three cartesian components form the $su(2)$ algebra, obtained before by consideration of dynamics of the particle, and we demonstrate that there is no curvature-induced geometric potential for the fermion.

Equations

$$[x_i,x_j]=0,[x_i,p_j]=i\hslash ({\delta }_{ij}-n_i{n}_j),\text{ and }[p_i,p_j]=-i\hslash {\left\{(n_i{n_k}_{,j}-n_j{n_k}_{,i})p_k\right\}}_{Hermitian},$$

$$\mathbf{p}=-i\hslash ({\nabla }_{\Sigma }+Mn/2),$$

$$[p_i,p_j]=-i\hslash \left\{(n_i{n_k}_{,j}-n_j{n_k}_{,i})p_k+p_k(n_i{n_k}_{,j}-n_j{n_k}_{,i})\right\}/2$$

$${\partial }_{\mu }⟶D_{\mu }\equiv {\partial }_{\mu }+i{\Omega }_{\mu }$$

$$\mathbf{p}=-i\hslash ({\nabla }_{\Sigma }+\frac{M\mathbf{n}}{2}+i{\mathbf{r}}^{\mu }{\Omega }_{\mu })=-i\hslash ({\nabla }_{\Sigma }+\frac{M\mathbf{n}}{2})+\hslash {\mathbf{r}}^{\mu }{\Omega }_{\mu },$$

$${\partial }_{\mu }⟶D_{\mu }\equiv {\partial }_{\mu }+i{W}_{\mu }$$

$$\mathbf{p}=-i\hslash ({\nabla }_{\Sigma }+\frac{M\mathbf{n}}{2}+i{\mathbf{r}}^{\mu }{W}_{\mu })=-i\hslash ({\nabla }_{\Sigma }+\frac{M\mathbf{n}}{2})+\hslash {\mathbf{r}}^{\mu }{W}_{\mu },$$

$$F_{\mu \nu }\equiv {\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }-i[A_{\mu },A_{\nu }]=i[D_{\mu },D_{\nu }].$$

$$\mathbf{A}\equiv {\mathbf{r}}^{\mu }{A}_{\mu }=-\hslash {\mathbf{r}}^{\mu }{\Omega }_{\mu },\text{ or }-\hslash {\mathbf{r}}^{\mu }{W}_{\mu }.$$

$$f(\mathbf{x})\equiv \frac{1}{2R}\left(\sum _{i=1}^N{x}_i^2-R^2\right)=0.$$

$${\left[x_i,x_j\right]}_D=0,{\left[x_i,p_j\right]}_D=\left({\delta }_{ij}-n_i{n}_j\right),\text{ and }{\left[p_i,p_j\right]}_D=-\frac{L_{ij}}{R^2},$$

$$[L_{ij},L_{k\ell }{]}_D=-{\delta }_{i\ell }{L}_{kj}+{\delta }_{ik}{L}_{\ell j}+{\delta }_{jk}{L}_{i\ell }-{\delta }_{j\ell }{L}_{ik},\text{ and }{\left[L_{ij},p{}_{\ell }\right]}_D=\left({\delta }_{i\ell }p{}_j-{\delta }_{j\ell }p{}_i\right).$$

$$\left[p_i,p_j\right]=-i\hslash \frac{J_{ij}}{R^2},\left[J{}_{ij},p{}_{\ell }\right]=i\hslash \left({\delta }_{i\ell }p{}_j-{\delta }_{j\ell }p{}_i\right),\text{ and }[J{}_{ij},J{}_{k\ell }]=i\hslash O\left([J{}_{ij},J{}_{k\ell }{]}_D\right).$$

$${\left[x_i,H\right]}_D=\frac{p_i}{H}{c}^2,\text{ and }{\left[p_i,H\right]}_D=-n_i\kappa \frac{{\left(cp\right)}^2}{H},$$

$$p_i=\frac{1}{c^2}H{\left[x_i,H\right]}_D,\text{ and }\mathbf{n}\wedge {\left[\mathbf{p},H\right]}_D=0.$$

$$p_i=\frac{1}{i\hslash }\frac{H\left[x_i,H\right]+\left[x_i,H\right]H}{2c^2},\text{ and }\mathbf{n}\wedge \left[\mathbf{p},H\right]-\left[\mathbf{p},H\right]\wedge \mathbf{n}=0.$$

$$[\mathbf{x},H]\equiv i\hslash \frac{\mathbf{p}}{\mu };\mathbf{n}\wedge [\mathbf{p},H]-[\mathbf{p},H]\wedge \mathbf{n}=0.$$

$$\mathbf{p}=-i\hslash ({\nabla }_{\Sigma }+\frac{M\mathbf{n}}{2});H=-\frac{{\hslash }^2}{2\mu }{\nabla }_{LB}^2+V_G,$$

$$x=\sin \theta \cos \phi ;y=\sin \theta \sin \phi ;z=\cos \theta ,$$

$$p_x={\Pi }_x+{\sigma }_z\frac{\hslash }{2}\frac{\cos \theta }{\sin \theta }\sin \phi ,p_y={\Pi }_y-{\sigma }_z\frac{\hslash }{2}\frac{\cos \theta }{\sin \theta }\cos \phi ,\text{ and }{p}_z={\Pi }_z,$$

$${\Pi }_x$$

$${\Pi }_y$$

$${\Pi }_z$$

$$\mathbf{A}={\sigma }_z\frac{\hslash }{2r}(-\frac{\cos \theta }{\sin \theta }\sin \phi ,\frac{\cos \theta }{\sin \theta }\cos \phi ,0)={\sigma }_z\frac{\hslash }{2}\frac{1}{\sqrt{x^2+y^2+z^2}}(-\frac{zy}{x^2+y^2},\frac{zx}{x^2+y^2},0),$$

$$\mathbf{B}\equiv \nabla \times \mathbf{A}=-{\sigma }_z\frac{\hslash }{2}\frac{{\mathbf{e}}_r}{r^2},$$

$$\left[p_i,p_j\right]=-i\hslash {\epsilon }_{ijk}\frac{J_k}{R^2},\left[J{{}_i,p}_j\right]=i\hslash {\epsilon }_{ijk}{p}_k,\text{ and }[J{}_i,J{}_j]=i\hslash {\epsilon }_{ijk}{J}_k.$$

$$J_x=L_x+{\sigma }_z\frac{\hslash }{2}\frac{\cos \phi }{\sin \theta },J_y=L_y+{\sigma }_z\frac{\hslash }{2}\frac{\sin \phi }{\sin \theta },\text{ and }{J}_z=L_z,$$

$$-i\hslash {\gamma }^{\mu }\left({\partial }_{\mu }+{\Omega }_{\mu }\right)\Psi -{\gamma }^{0}m\Psi =0.$$

$$H=-i\hslash \left({\sigma }_x\left({\partial }_{\theta }+\frac{1}{2}\frac{\cos \theta }{\sin \theta }\right)+{\sigma }_y\frac{1}{\sin \theta }{\partial }_{\phi }\right)+{\sigma }_{z}m.$$

$$V_G=a_{0}I+a_x{\sigma }_x+a_y{\sigma }_y+a_z{\sigma }_z$$

$$\left[p_x,H^{\prime }\right]$$

Full text

Relativistic wave equations; Curved surface; Canonical quantization; Momentum

I Introduction

In quantum mechanics, there are fundamental quantum conditions (FQCs) $[x_{i},x_{j}]=0,$ $[x_{i},p_{j}]=i\hbar\delta_{ij},$ and $[p_{i},p_{j}]=0$, which are defined by the commutation relations between positions $x_{i}$ and momenta $p_{i}$ ($i,j,k,l=1,2,3,...,N$) where $N$ denotes the number of dimensions of the flat space in which the particle moves  Dirac . In position representation, the momentum operator takes simple form as $\mathbf{p}=-i\hslash\nabla$ where $\nabla\equiv\mathbf{e}_{i}\partial/\partial x_{i}$ is the ordinary gradient operator, and $N$ mutually orthogonal unit vectors $\mathbf{e}_{i}$ span the $N$ dimensional Euclidean space $E^{N}$. Hereafter the Einstein summation convention over repeated indices is used. When the particle is constrained to remain on a hypersurface $\Sigma^{N-1}$ embedded in $E^{N}$, the FQCs become  weinberg ,

[TABLE]

where $O_{{}_{Hermitian}}$ stands for a Hermitian operator of an observable $O$, and the equation of surface $f(\mathbf{x})=0$ can be so chosen that $\left|\nabla f(\mathbf{x})\right|=1$ so $\mathbf{n\equiv}\nabla f(\mathbf{x})=$ $\mathbf{e}_{i}n_{i}$ being the normal at a local point on the surface. This set of the FQCs* *(1) is highly non-trivial, from which it is in general impossible to uniquely construct the momenta $p_{i}$. Our propose of the proper form of the momentum for a spinless particle was liu07 ; liu11 ; liu13 ; liu132 ; liu18 ,

[TABLE]

where ${\nabla_{\Sigma}\equiv}\mathbf{e}_{i}(\delta_{ij}-n_{i}n_{j})\partial_{j}$ $=\nabla-\mathbf{n}\partial_{n}$ $=\mathbf{r}^{\mu}\partial_{\mu}$ is the the gradient operator, and $\mathbf{r}^{\mu}$ is $\mu-$th contravariant component of the natural frame on the point ($u^{1},u^{1},...u^{\mu},...u^{N-1}$) on the surface $\Sigma^{N-1}$, and $u^{\mu}$ ($\mu,\nu,\alpha,\beta=1,2,3,...,N-1$) denote the local coordinates, and the mean curvature ${M\equiv-\nabla_{\Sigma}}\cdot\mathbf{n}$ is defined by the sum of the all principal curvatures. Since the mean curvature ${M}$ is an extrinsic curvature, this form of momentum (2) is fundamentally different from the canonical ones in curvilinear coordinates for it depends on the geometric invariants. Thus it can be conveniently called as geometric momentum liu07 ; liu11 ; liu13 ; liu132 ; liu18 ; exp ; wang17 . This momentum can be obtained by many different ways including: the hermiticity requirement on derivative part $-i\hbar{\nabla_{\Sigma}}$ liu07 , and compatibility of constraint condition $\mathbf{n\cdot p+p\cdot n}=0$ which means that the motion is perpendicular to the surface normal vector $\mathbf{n}$ liu11 ; liu13 , and thin-layer quantization or confining potential formalism which instead considers that particle is confined onto the surface $\Sigma^{N-1}$ by means of introduction of a confinement potential along the normal direction of the the surface liu132 , and dynamical quantum conditions (DQCs) liu18 , etc. wang17 ; ikegami It was demonstrated that this momentum (2) satisfies last one of the FQCs (1), when it explicitly takes following simplest form liu18 ,

[TABLE]

Experimental justification was performed by comparison of the interference spots formed by the surface plasmon polariton propagating on a cylindrical surface, predicted by the introduction of the geometric momentum or not exp , respectively. Some of previous discussions deal with quite general case liu13 ; liu18 ; ikegami , some of them liu07 ; liu11 ; liu132 ; wang17 are mainly for a particle on $\Sigma^{2}$.

The geometric momentum (2) suffices to act on state function that has a single component. However, state functions on the surfaces are usually multi-component such as spinors RMP1957 ; 2010PR ; LeeDH ; Iorio , requiring that the momentum be made general covariance. In fact, the general covariant geometric momentum (GCGM) is at hand, though not yet explicitly written before. The present paper shows that the GCGM is a useful and convenient physical quantity.

This paper is mainly divided into two parts. Sections II-IV are devoted to build up a general formulation between the GCGM and quantization conditions. Section V and VI study the Dirac fermion on $S^{2}$. In section II, the introduction of the GCGM is made and its dependence on the gauge potential is transparent. In section III, though we do not know in general whether the GCGM satisfies the quantization condition (3), or satisfies other forms of the last one of the FQCs (1) $[p_{i},p_{j}]=-i\hbar\left\{({n_{i}}{n_{k}}_{,j}-{n_{j}}{n_{k}}_{,i})p_{k}\right\}_{Hermitian}$, the FQCs $\left[p_{i},p_{j}\right]$ for a Dirac fermion on $S^{N-1}$ have a well-defined consequence to define a generalized total angular momentum. In section IV, we show how the self-consistent consideration of the quantization conditions leads us to the DQCs for a relativistic particle on $\Sigma^{N-1}$. In section V, we deal with the Dirac fermion on $S^{2}$, and use FQCs and GCGM to reproduce the same generalized total angular momentum obtained before by means of a purely dynamical consideration. In section VI, we use DQCs and GCGM to check whether the curvature-induced geometric potential presents for a Dirac fermion on $S^{2}$, and results show that no such a potential. Final section VII is a brief conclusion.

II General covariant geometric momentum and gauge potential

This section is to show that the GCGM is at the ready, and its dependence on the gauge potential is transparent.

To note ${\nabla_{\Sigma}\equiv}\mathbf{e}_{i}(\delta_{ij}-n_{i}n_{j})\partial_{j}=\mathbf{r}^{\mu}\partial_{\mu}$, and the usual derivative $\partial_{\mu}(\equiv\partial/\partial u^{\mu})$ in (2) can be made general covariant by a simple replacement RMP1957 ; 2010PR ; LeeDH ; Iorio ,

[TABLE]

and we immediately have,

[TABLE]

where $\Omega_{\mu}=\left(-i/8\right)\omega_{\mu}^{ab}\left[\gamma_{a},\gamma_{b}\right]$ in which $\omega_{\mu}^{ab}$ are the spin connections 2010PR ; LeeDH ; Iorio ; ogawa and $\gamma_{a}$ ($a,b=0,1,2,...N$) are Dirac spin matrices. In comparison of (5) with the usual kinematical momentum ${\mathbf{p}}=-i\hbar{\nabla-q}\mathbf{A}$ in presence of magnetic potential $\mathbf{A}$, we see that an equivalent magnetic potential $\mathbf{A}$ can be defined by ${q}\mathbf{A\equiv}{-}\hbar\mathbf{r}^{\mu}\Omega_{\mu}$, in which the charge ${q}$ can be understood as an effective interaction strength between the charge with the field. Once writing $\Omega_{\mu}$ as a product of $\omega_{\mu}^{ab}/4$ and $Q_{ab}\equiv\left[\gamma_{a},\gamma_{b}\right]/(2i)$, we can take the eigenvalues of the matrices $Q_{ab}$ as an effective interaction strength ogawa . This form of GCGM (5) is applicable to particles, relativistically or not, massively or not.

Two observations concerning the GCGM are in following.

1. Once the surface $\Sigma^{N-1}$ is embedded into higher flat space in $E^{N+p}$ ($N\succeq 2$) with a positive integer $p\succeq 1$, we have another way of making the derivative $\partial_{\mu}$ in (2) covariant by replacement ogawa ; Nconnection ; maraner ,

[TABLE]

and we have as well,

[TABLE]

where $\hbar W_{\mu}=N_{\mu}^{AB}L_{AB}/2$ in which $N_{\mu}^{AB}$ ($A,B=N,N+1,...N+p$) stand for the normal connections determined by the so-called normal fundamental form, and $L_{AB}$ are angular momentum in the normal space. It was realized that the normal connections $N_{\mu}^{AB}/2$ and spin connections $\omega_{\mu}^{ab}/4$ can take identical form for $S^{N-1}$, which was used to explore an origin of spin other than that is generally accepted to be connected with relativity OK . Thus, the relationship between spin and space embedding is far from fully understood.

2. Starting from replacement (4) or (6), we can define the gauge potential $A_{\mu}\equiv-\hbar\Omega_{\mu}$, or $A_{\mu}\equiv-\hbar W_{\mu}$. Therefore the field strength $F_{\mu\nu}$ can be defined by  Nconnection ; maraner ; ogawa ; ohnuki ,

[TABLE]

Whether this gauge field $F_{\mu\nu}$ is abelian or non-abelian depends on whether the commutators $[A_{\mu},A_{\nu}]$ vanish or not. In terms of the GCGM, we have the gauge potential in Cartesian coordinates,

[TABLE]

Now the introduction of the GCGM is complete. However whether it satisfies the quantization condition (3), or satisfies other forms of the last one of the FQCs (1) $[p_{i},p_{j}]=-i\hbar\left\{({n_{i}}{n_{k}}_{,j}-{n_{j}}{n_{k}}_{,i})p_{k}\right\}_{Hermitian}$, is not so easily resolved in general. We leave it as an open problem though we believe it is true. For the special case of a Dirac fermion on $S^{N-1}$, this problem turns out to be another one defining instead the generalized total angular momentum, which will be discussed in next section.

III Fundamental quantum conditions $\left[p_{i},p_{j}\right]$ for a

Dirac fermion on $S^{N-1}$

The hypersphere $S^{N-1}$ of radius $R$ in $N$-dimensional flat space $E^{N}$ can be,

[TABLE]

The fundamental set of Dirac brackets is simply weinberg ; liu11 ; liu13 ,

[TABLE]

where $L_{ij}\equiv x_{i}p_{j}-x_{j}p_{i}$ is the $ij$-component of the orbital angular momentum. In addition, we have an $SO(N,1)$ group with generators $p_{i}$ and ${L_{ij}}$ because we have also  liu13 ; ohnuki ,

[TABLE]

These relations (11) and (12) hold irrespective of particle being massive or not, relativistic or not. However, in classical mechanics, there is no spin; and these relations (12) are obtained by considering the purely orbital motion. In our approach, we require that these relations (11) and (12) hold true in sense of $\left[u,v\right]=i\hbar O\left(\left[u,v\right]_{D}\right)$. Explicitly, we have,

[TABLE]

Here we re-denote $L_{ij}$ by the symbol $J{{}_{ij}}$, a symbol denoting generalized total angular momentum in quantum mechanics. Our discussion needs a flat space with $N$ cartesian coordinates $x_{i}$ ($i=1,2,3,...,N$) as the prerequisite. So, for a Dirac fermion on $S^{N-1}$, the FQCs are set up and given by (13), which lead us to defining the generalized total angular momentum $J$ in quantum mechanics.

When quantizing a classical system, we put symmetries on the top priority: liu11 ; liu13 ; liu18 ; liu133 ; liu15 Our philosophy is: *The symmetry expressed by the Poisson or Dirac brackets in classical mechanics preserves in quantum mechanics; and so the Hamiltonian is determined by the symmetry. *It can be considered a specific demonstration of the fundamental philosophical idea stating that symmetry dictates interactions in quantum mechanics yang . The philosophy leads us to set out FQCs and DQCs for the non-relativistic and spinless particle, and the most profound consequence is to successfully reproduce of the geometric potential in Hamiltonian and the geometric momentum liu18 , respectively. In next section, the DQCs affecting the form of Hamiltonian for a relativistic particle on $\Sigma^{N-1}$ will be formulated.

IV Dynamical quantum conditions for a relativistic particle on

$\Sigma^{N-1}$

For a relativistic particle whose classical Hamiltonian is $H=\sqrt{\left(pc\right)^{2}+\left(\mu c^{2}\right)^{2}}$ with $c$ being the velocity of light and $\mu$ being the mass of the particle, we can obtain two Dirac brackets,

[TABLE]

where $\kappa$ is the first curvature of the geodesic on the hypersurface $\Sigma^{N-1}$  liu16 . Notice that Eqs. (14) have two important consequences,

[TABLE]

These two relations indicate that in quantum mechanics momentum $\mathbf{p}$ and Hamiltonian $H$ must be compatible with following two quantum conditions,

[TABLE]

These two sets of quantum conditions constitute the so-called DQCs for the relativistic particle on $\Sigma^{N-1}$, which put requirement on the form of Hamiltonian operator.

Three remarks concerning the DQCs are in following.

1. In classical mechanics for a particle, constrained or not, the relativistic velocity $\mathbf{v}\equiv\mathbf{p}c^{2}/H$ (14) can be rewritten as the familiar form, $p=\mu v/\sqrt{1-v^{2}/c^{2}}$. In quantum mechanics, the DQCs imply a definition of the velocity operator $\mathbf{v}\equiv c\mathbf{\alpha}=\left(H^{-1}\mathbf{p+p}H^{-1}\right)c^{2}/2$  position  with $\mathbf{\alpha}$ are $4\times 4$ Pauli matrices and in Pauli-Dirac representation we have \mathbf{\alpha=}\left(\begin{array}[c]{lc}0&\mathbf{\sigma}\\ \mathbf{\sigma}&0\end{array}\right), while the momentum $\mathbf{p}$ is defined as $\mathbf{p}\equiv\left(H\left[\mathbf{x},H\right]+\left[\mathbf{x},H\right]H\right)/(2i\hbar c^{2})$ which is identical to $-i\hslash\nabla$ for motion in flat space. However, it is not the case in quantum mechanics once the motion is constrained. In the quantum mechanics, the relativistic Hamiltonian operator $H$ for a particle of any spin in flat space can be easily constructed and it acts on the multi-component wave functions. However, the construction of such a Hamiltonian for a spin particle on a curved space or curved space is not an easy task at all. Fortunately, such a Hamiltonian for a Dirac fermion on $S^{2}$ is easily found 2010PR ; LeeDH ; Iorio ; Abrikosov . For a Dirac fermion on $S^{2}$, $\mathbf{n}\wedge\left[\mathbf{p},H\right]-\left[\mathbf{p},H\right]\wedge\mathbf{n}=0$ (16) clearly leads to no presence of geometric potential, as discussed in section VI.

2. For a spinless particle that moves non-relativistically, two Dirac brackets (15) become $[\mathbf{x},H]_{D}\equiv\mathbf{p/}\mu$ and $\mathbf{n}\wedge\left[\mathbf{p},H\right]_{D}=0$. DQCs take following forms liu11 ; liu13 ; liu15 ; liu18 ,

[TABLE]

Quantum conditions (1) and (17) constitute the so-called *enlarged canonical quantization scheme *which gives the unambiguous forms of both the momentum and Hamiltonian $H$ for a non-relativistic free particle liu18 ; liu15 ,

[TABLE]

where $\nabla_{LB}^{2}={\nabla_{\Sigma}}\cdot{\nabla_{\Sigma}}$ is the usual Laplace-Beltrami operator on the surface $\Sigma^{N-1}$, and $V_{G}\equiv-\frac{\hbar^{2}}{4\mu}K+\frac{\hbar^{2}}{8\mu}{M}^{2}$ is the celebrated geometric potential  jk ; dacosta ; fc ; packet1 ; exp1 ; exp2 in which $K$ is in fact the trace of square of the extrinsic curvature tensor ikegami , and ${\mathbf{p}}$ is very geometric momentum without spin connection liu11 ; liu13 ; exp . Physical consequences resulting from *geometric potential and geometric momentum *are experimentally confirmed exp ; exp1 ; exp2 , and more experimentally testable results are under explorations packet1 .

3. In comparison with the overall successes of the DQCs (18) for a non-relativistic and spinless particle on $\Sigma^{N-1}$, we can only say that for a relativistic and spin particle on $\Sigma^{N-1}$ the GCGM and the Hamiltonian must be simultaneously compatible with the DQCs (16). Since the GCGM is already given, we must look for the proper form of the Hamiltonian. It is well-accepted that Hamiltonian for spinless and non-relativistic particle contains the geometric potential, so it is taken for granted that there must be some form of the geometric potential in Hamiltonian for a spin and relativistic particle. Unfortunately, due to the fact that a full understanding of spin connection is still lacking, we can not help but deal with a system case by case. In the rest part of the paper, we mainly deal with the curvature-induced geometric potential for a Dirac fermion on $S^{2}$. Before it, we give specific FQCs and DQCs.

V Generalized total angular momentum for a Dirac fermion on $S^{2}$

The surface $S^{2}$ of unit radius can be parameterized by,

[TABLE]

where $\theta$ is the polar angle from the positive $z$-axis with $0\leq\theta\leq\pi$, and $\varphi$ is the azimuthal angle in the $xy$-plane from the $x$-axis with $0\leq\varphi<2\pi$. After some lengthy but straightforward calculations, we can reach a very simple expression for the general covariant geometric momentum whose three components are given by,

[TABLE]

where \sigma_{z}=\left(\begin{array}[c]{lc}1&0\\ 0&-1\end{array}\right) is the $z$-component Pauli matrix, and $\Pi_{i}$ are geometric momentum (18) for the particle on $S^{2}$  liu11 ; liu13 ; liu15 ; liu18 ,

[TABLE]

It has been recognized that spin connection can be interpreted in terms of gauge potential. In GCGM (20), the gauge potential $\mathbf{A}$ is evidently,

[TABLE]

in which the radius $r=\sqrt{x^{2}+y^{2}+z^{2}}$ is recovered. The magnetic strength $\mathbf{B}$ is,

[TABLE]

where $\mathbf{e}_{r}\equiv(x,y,z)/r$. Evidently, the magnetic field is produced by a monopole of unit charge at the center of the sphere $-\delta(r)$, and the eigenvalues $\pm\hbar/2$ of the Pauli matrix $\sigma_{z}\hbar/2$ are the effective interaction strength.

The FQCs (13) for a Dirac fermion on $S^{2}$ are explicitly,

[TABLE]

These six operators $p_{i}$ (20) and $j_{i}$ (27) constitute all generators of an $SO(3,1)$ group. In consequence, we have following generalized total angular momentum,

[TABLE]

where ${L}_{{x}}=i\hbar(\sin\varphi\partial_{\theta}+\cot\theta\cos\varphi\partial_{\varphi})$, ${L}_{{y}}=-i\hbar(\cos\varphi\partial_{\theta}-\cot\theta\sin\varphi\partial_{\varphi})$ and ${L}_{{z}}=-i\hbar\partial_{\varphi}$ are usually $x$, $y$ and $z$ -component of the orbital angular momentum, respectively. This generalized total angular momentum was first constructed explicitly by Abrikosov in 2002, Abrikosov who observed the Hamiltonian for massless Dirac fermion to be invariant under a $SU(2)$ group transformation and identified (27) as a consequence. Then, Abrikosov demonstrated it is really generalized total angular momentum Abrikosov for the eigenvalues of $J^{2}\equiv J_{i}J_{i}$ are $j(j+1)\hbar^{2}$ with $j=1/2,3/2,5/2,...$. In other words, Abrikosov obtained the generalized total angular momentum (27) on the base of dynamics. In contrast, we obtain the same result (27) from both the FQCs (13) and GCGM (5). Moreover, in this section, our result (27) applies for particle, massive or massless, relativistic or non-relativistic, irrespective the form of Hamiltonian. In the history, Ohnuki and Kitakado ohnuki  in 1993 created the so-called fundamental algebra for quantum mechanics on $S^{N-1}$ and obtained generators of $SO(N)$ which when $N=3$ reduces to be $J_{x}=L_{x}+Sy/(r^{2}+rz),$ $J_{y}=L_{y}-Sx/(r^{2}+rz),$ and $J_{z}=L_{z}$, in which $S$ is a real number rather than operator in our situation. However, the monopole is the same. Especially, Ohnuki and Kitakado also considered the momentum operators on $S^{N-1}$ but they obtained the geometric one (2) rather than GCGM (5).

VI No geometric potential for a Dirac fermion on $S^{2}$

The so-called geometric potential is the additional term in Hamiltonian resulting from quantization. Recently, whether such a curvature-induced geometric potential presents is a topic of considerable controversy maraner ; packet9 ; jose , and all use the confining potential formalism but have opposite results. Our approach based on the DQCs (16) is totally different from the confining potential formalism, which are transparent and convincing.

The general covariant Dirac equation for a fermion on a two-dimensional sphere is 2010PR ; LeeDH ; Iorio ,

[TABLE]

where $m\equiv\mu c$ is the reduced mass. The Hamiltonian can be shown to be given by 2010PR ; Abrikosov ,

[TABLE]

where \sigma_{x}=\left(\begin{array}[c]{lc}0&1\\ 1&0\end{array}\right), \sigma_{y}=\left(\begin{array}[c]{lc}0&-i\\ i&0\end{array}\right) are, respectively, the $x,y$ -component of Pauli matrices. Now, whether a geometric potential exists in the relativistic Hamiltonian (29) is going to be resolved.

First, let us assume that the most general form of the geometric potential is given by,

[TABLE]

where \left(a_{0},a_{x},a_{y},a_{z}\right)\are function of $\theta$ and $\varphi$. The trial Hamiltonian is now $H^{\prime}=H+V_{G}$.

Secondly, we compute three commutators $\left[p_{i},H^{\prime}\right]$ and the results are, respectively,

[TABLE]

[TABLE]

[TABLE]

VII Conclusions

For a particle that is constrained on an ($N-1$)-dimensional curved surface, the geometric momentum (2) has stood both theoretical examinations and experimental testifications. To make it general covariance so as to be applied to the spin particles, a simple replacement suffices of ordinary derivative $\partial_{\mu}$ in gradient operator ${\nabla_{\Sigma}\equiv}\mathbf{e}^{\mu}\partial_{\mu}$ by its general covariant derivative $\partial_{\mu}\longrightarrow\partial_{\mu}+i\Omega_{\mu}$. A general formalism when quantizing a classical system is established, and we have FQCs and DQCs. The FQCs for a Dirac fermion on $S^{N-1}$ lead us to a generalized total angular momentum, while the DQCs for a Dirac fermion on $\Sigma^{N-1}$ offer us a way to check whether the geometric potential presents for relativistic spin particle on any hypersurface $\Sigma^{N-1}$. In present paper, we obtain the generalized total angular momentum for $S^{2}$, which was reported before on the dynamical consideration, and show that for a Dirac fermion on $S^{2}$, no geometric potential is permissible.

Acknowledgements.

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