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

For a particle that is constrained on an (N-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N-1$$\end{document})-dimensional (N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}) curved surface ΣN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^{N-1}$$\end{document}, the Cartesian components of its momentum in N-dimensional flat space are 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 becomes generally covariant as to be applicable to spin particles 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 conditions for a spin particle on the hypersurface ΣN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^{N-1}$$\end{document}, and the generalized angular momentum is defined on hypersphere SN-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{N-1}$$\end{document} as one consequence of the generally 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 angular momentum whose three cartesian components form the su(2) algebra, demonstrated to be of geometric origin but obtained before by consideration of dynamics of the particle. Moreover, we show that there is no curvature-induced geometric potential for the spin half particle.


I. INTRODUCTION
In quantum mechanics, there are fundamental quantum conditions (FQCs) [x i , x j ] = 0, [x i , p j ] = i δ 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 1 .In position representation, the momentum operator takes simple form as p = −iℏ∇ where ∇ ≡ e i ∂/∂x i is the ordinary gradient operator, and N mutually orthogonal unit vectors 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 Σ N −1 embedded in E N , the FQCs become 2 , [x i , x j ] = 0, [x i , p j ] = i (δ ij − n i n j ), and [p i , p j ] = −i (n i n k,j − n j n k,i )p k Hermitian , (1) where O Hermitian stands for a Hermitian operator of an observable O, and the equation of surface f (x) = 0 can be so chosen that |∇f (x)| = 1 so n ≡∇f (x) = 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 the first two terms [3][4][5][6][7] of the following expression, where Ω µ = (1/8) ω ab µ [γ a , γ b ] in which ω ab µ are the spin connections 8,9 and γ a (a, b = 0, 1, 2, ...N) are Dirac spin matrices, defined by the sum of the all principal curvatures.When the particle is spinless, the spin connection term is absent, and the curvature term Mn/2 can be determined by many different ways including the hermiticity requirement on derivative part −i ∇ Σ , 3 compatibility of constraint condition n • p + p • n = 0 which means that the motion is perpendicular to the surface normal vector n, 4,5 thin-layer quantization or confining potential formalism which instead considers that particle is confined onto the surface Σ N −1 by means of introduction of a confinement potential along the normal direction of the the surface, 6 etc. 12,13 Some of previous discussions [3][4][5][6]12,13 are mainly for a particle on Σ 2 , but it is not difficult for them to be extended to the situation of a particle on a hypersurface Σ N −1 in E N (N 4), although some of them directly deal with the general case 7,12 . Exprimental justification was performed by comparison of the interference fringes formed by the surface plasmon polariton on a cylindrical surface, predicted by the introduction of the geometric momentum or not 14 .With the geometric momentum p = −i (∇ Σ + Mn/2) for spinless particle at hand, the general covariant form is in fact a straightforward generalization but no such an attempt was made before.We note  8,9 , and we reach (2).Then, a comparison of ( 2) with the so-called mechanical momentum p with the magnetic field A as p = −i ∇ − qA in which A is the gauge field, results in that the general covariant geometric momentum ( 2) is associated with a gauge potential A with charge q be absorbed in A, This gauge potential exists (3) in general, dependent on the curvature of the surface and spin properties only, irrelevant to relativistic or not, massive or not.An overall understanding of this gauge potential is an open problem.One difficulty appears that there are multiple definitions of the gauge potential [8][9][10][11] , e.g., one can use the spin connection part ω ab µ to construct the gauge potential, and one use Ω µ instead, etc.In present paper, we mainly limit ourself on a Dirac fermion on two-dimensional sphere S 2 , though our formalism is established on S N −1 .
In section II, we set out the FQCs for a Dirac fermion on S N −1 .Because FQCs are not enough to to give the satisfactory form of Hamiltonian, we also establish the dynamical quantum conditions (DQCs) which puts requirement on the Hamiltonian.In section III, for a Dirac fermion on S 2 , the explicit form of the gauge potential is revealed, and is demonstrated what is produced by the line distribution of monopole.Utilizing FQCs for the fermion on S 2 , we reproduce the same total angular momentum operator obtained by means of a purely dynamical consideration.In section IV, we apply the DQCs to check whether geometric potential presents for a Dirac fermion on S 2 , and results show that no such a potential.
Final section V is a brief conclusion.

II. FUNDAMENTAL AND DYNAMICAL QUANTUM CONDITIONS FOR
The equations of motion of a quasi-relativistic spin-1/2 massless fermion in two-dimensional materials attracts much attention [15][16][17][18][19][20][21] .However, in present section, we try to set out the fundamental quantum conditions for a relativistic massive Dirac fermion on The hypersphere The fundamental set of Dirac brackets is simply 2,4,5 , where L ij ≡ 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 5,10 , These relations ( 5) and ( 6) hold irrespective of particle being massive or not, relativistic or not.However, in classical mechanics, there is no spin; and these relations ( 6) are obtained by considering the purely orbital motion.However, we postulate that the symmetry dictated by the brackets in classical mechanics preserve in quantum mechanics, i.e., these relations ( 5) and ( 6) hold true in sense of [u, v] Here we re-denote L ij by the symbol J ij , a symbol denoting possible 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 (7).
It is easily to verify that the general covariant geometric momentum (2) satisfies the quantum conditions (7), in a manner similar to verify that usual geometric momentum (2)   satisfies the usual quantum conditions (7) without spin terms, and the details will be given elsewhere.In other words, the ij-component J ij of the total angular momentum J is, where is the ij-component of the magnetic field.Three immediate comments follow.The spin enters the total angular momentum through both spin connection ω αβ µ and natural frame r µ j , so it is not the usual one J ≡ L + S. The gauge field exhibits in angular momentum in the form of strength, and appears in general covariant geometric momentum in the form of potential, instead.The gauge field is not far beyond well understood, hoping to be investigated in future.
With the general covariant geometric momentum ( 2 with c being the velocity of light and µ being the mass of the particle, we can after some calculations obtain two Dirac brackets, where κ is the first curvature of the geodesic on the hypersurface Σ N −1 22 .Notice that Eqs. ( 9) have two important consequences, These two relations indicate that in quantum mechanics momentum p and Hamiltonian H must be compatible with following two quantum conditions, These two sets of quantum conditions can be called as DQCs which put requirement on the form of Hamiltonian operator.
In classical mechanics for motion, constrained or not, the relativistic velocity v ≡ pc 2 /H 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 multicomponent spinor wave functions.However, the construction of such a Hamiltonian for 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 19,24 .For a Dirac fermion on S 2 , n∧[p, H]−[p, H]∧n = 0 (11) clearly leads to no presence of geometric potential, which will be discussed in section IV.
The DQCs for a spinless particle that moves non-relativistically produce profound consequences.Two Dirac brackets ( 10) become [x, H] D ≡ p/µ and n ∧ [p, H] D = 0. DQCs take following forms 4,5,7,25 , Quantum conditions ( 1) and ( 12) constitute the so-called enlarged quantization scheme which gives the unambiguous forms of both the momentum and Hamiltonian H for a nonrelativistic free particle 7,25 , where ∇ 2 LB = ∇ Σ • ∇ Σ is the usual Laplace-Beltrami operator on the surface Σ N −1 , and M 2 is the celebrated geometric potential [26][27][28][29][30][31] in which K is in fact the trace of square of the extrinsic curvature tensor 12 , and p is very geometric momentum without spin connection 4,5,14 .Physical consequences resulting from geometric potential and geometric momentum are experimentally confirmed 14,30,31 , and more experimentally testable results are under explorations 29 .

III. FUNDAMENTAL QUANTUM CONDITIONS AND TOTAL ANGULAR MO-MENTUM FOR A DIRAC FERMION ON S 2
The surface S 2 of unit radius can be parameterized by, x = sin θ cos ϕ; y = sin θ sin ϕ; z = cos θ, (14)   where θ is the polar angle from the positive z-axis with 0 ≤ θ ≤ π, and ϕ is the azimuthal angle in the xy-plane from the x-axis with 0 ≤ ϕ < 2π.After some lengthy calculations, we can reach a very simple expression for the general covariant geometric momentum whose three components are given by, where  is the z-component Pauli matrix, and Π i are geometric momentum (13) for the particle on S 2 4,5,7,25 , It is well-known that spin connection can be interpreted in terms of an effective gauge potential 8,11 .In momentum (15), the gauge potential A is evidently, It shows a vortex in xy-plane, rotating clockwise around the z-axis.The effective magnetic strength B is, where ρ ≡ x 2 + y 2 and e ρ ≡ (x, y)/ x 2 + y 2 .The effective magnetic field is divergencefree except at z-axis, and is produced by a monopole-like charge line distribution, Putting aside the Pauli matrix σ z , we see that the total flux penetrating the whole sphere The FQCs (7) for a Dirac fermion on S 2 are explicitly, These six operators p i (15) and j i (22) constitute all generators of an SO(3, 1) group.In consequence, we have following total angular momentum, where L x = i (sin ϕ∂ θ + cot θ cos ϕ∂ ϕ ), L y = −i (cos ϕ∂ θ − cot θ sin ϕ∂ ϕ ) and L z = −i ∂ ϕ are usually x, y and z -component of the orbital angular momentum, respectively.This total angular momentum was first constructed by Abrikosov in 2002, 24 who observed the Hamiltonian for massless Dirac fermion to be invariant under a SU(2) group transformation and identified (22) as a consequence.Then, Abrikosov demonstrated it is really total angular momentum 24 for the eigenvalues of J 2 ≡ J i J i are j(j + 1) 2 with j = 1/2, 3/2, 5/2, ....In other words, Abrikosov obtained the total angular momentum ( 22) on the base of dynamics.
In contrast, we obtain the same result ( 22) from both the FQCs ( 7) and general covariant geometric momentum (2).Moreover, in this section, our result (22) applies for particle, massive or massless, relativistic or non-relativistic, irrespective the form of Hamiltonian.
IV. DYNAMICAL QUANTUM CONDITIONS AND NO GEOMETRIC POTEN- 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 32,33 .The main difficulty lies on the different understandings of dimensional reduction associated with spin connections 33 .Our approach is totally different and our results are based on the DQCs (11).In contrast to the confining potential formalism, our treatment is convincing and transparent.
The general covariant Dirac equation for a fermion on a two-dimensional sphere is 8,9 , where m ≡ µc is the reduced mass.The Hamiltonian can be shown to be given by 8,24 , where  are, respectively, the x, y -component of Pauli matrices.When the mass is zero, the Hamiltonian reduces to that given by Abrikosov, 24 Now, whether a geometric potential exists in the relativistic Hamiltonian ( 24) is going to be resolved.
First, let us assume that the most general form of the geometric potential is given by, 26)   where (a 0 , a x , a y , a z ) are function of θ and ϕ.The trial Hamiltonian is now Secondly, we compute three commutators [p i , H ′ ] and the results are, respectively, [p x , H ′ ] = − 2 σ x cos ϕ (sin θ∂ θ + cos θ) + σ y cos ϕ∂ ϕ − sin ϕ [p y , H ′ ] = − 2 σ x sin ϕ (sin θ∂ θ + cos θ) + σ y sin ϕ∂ ϕ + cos ϕ [p z , H ′ ] = − 2 σ x cos θ∂ θ + a simple replacement suffices of ordinary derivative ∂ µ in gradient operator ∇ Σ ≡e µ ∂ µ by its general covariant derivative ∂ µ −→ ∂ µ + Ω µ .We immediately identify that there is gauge potential A ≡i r µ Ω µ associated with both the curvature of the surface and the spin of the particle.As a consequence, the total angular momentum for the spin particle on S N −1 is different from the usual one which is a simple addition of the orbital and spin angular momentum, in which the strength of the gauge field presents.This way of obtaining the total angular momentum is different from the one based on the dynamical consideration.For a Dirac fermion on S 2 , our total angular momentum is the same reported by Abrikosov.The general covariant geometric momentum together with DQCs offer a convincing way to check whether geometric potential presents for relativistic spin particle on any hypersurface Σ N −1 , which for general case remains an open problem.However, explicit calculations show that for a Dirac fermion on S 2 , no geometric potential is permissible.
) for the fermion on S N −1 , we are not much sure what proper form of the Hamiltonian is.It is worthy of repeating our philosophy again: the symmetry dictated by the brackets in classical mechanics preserve in quantum mechanics.For a relativistic particle whose classical Hamiltonian is H = (pc) 2 + (µc 2 ) 2