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A local analysis of noncommutativity of two different components of momentum for a particle on hypersurface

Q. H. Liu, Xing Lin Yang, Z. Li

Published Sep 18, 2017• 0 citations
OpenAlex

Abstract

As a nonrelativistic particle constrained to remain on an $N-1$ ($N\geq 2$) dimensional hypersurface embedded in an $N$ dimensional Euclidean space, two different components $p_{i}$ and $p_{j}$ ($i,j=1,2,3,...N$) of the Cartesian momentum of the particle are not commuting with each other. In fact, the so-called \textit{fundamental quantum conditions} $[p_{i},p_{j}]\left( \neq 0\right) $ depend on products of positions and momenta in uncontrollable ways, thus their physical consequences are little known. The local analysis technique is utilized to crack this long-standing problem. The first finding is that this noncommutativity on a \textit{local point }of the hypersurface turns out to be transparent and the quantum conditions are then operator-ordering free and dependent on the local curvature of the surface. The second finding is that, for a small circle lying on each of the \emph{% tangential planes} covering the \textit{local point}, the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy; and along each of the \emph{normal sectional curves} centering the given point the noncommutativity leads to a translation plus an additional rotation and the amount of the rotation is one half of the tangential angle change of the arc.

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As a nonrelativistic particle constrained to remain on an N−1N-1N−1 (N≥2N\geq 2N≥2)
dimensional hypersurface embedded in an NNN dimensional Euclidean space, two
different components pip_{i}pi​ and pjp_{j}pj​ (i,j=1,2,3,...Ni,j=1,2,3,...Ni,j=1,2,3,...N) of the Cartesian
momentum of the particle are not commuting with each other. In fact, the
so-called \textit{fundamental quantum conditions} [pi,pj](≠0)[p_{i},p_{j}]\left( \neq 0\right) [pi​,pj​](=0) depend on products of positions and momenta in uncontrollable ways,
thus their physical consequences are little known. The local analysis technique
is utilized to crack this long-standing problem. The first finding is that this
noncommutativity on a \textit{local point }of the hypersurface turns out to be
transparent and the quantum conditions are then operator-ordering free and
dependent on the local curvature of the surface. The second finding is that,
for a small circle lying on each of the \emph{% tangential planes} covering the
\textit{local point}, the noncommutativity leads to a rotation operator and the
amount of the rotation is an angle anholonomy; and along each of the
\emph{normal sectional curves} centering the given point the noncommutativity
leads to a translation plus an additional rotation and the amount of the
rotation is one half of the tangential angle change of the arc.
As a nonrelativistic particle constrained to remain on an N−1N-1N−1 (N≥2N\geq 2N≥2) dimensional hypersurface (embedded in an NNN dimensional Euclidean space), two different components pip_{i}pi​ and pjp_{j}pj​ (i,j=1,2,3,...Ni,j=1,2,3,...Ni,j=1,2,3,...N) of the Cartesian momentum are not commuting with each other in the sense of the Dirac bracket [pi,pj]D≠0[p_{i},p_{j}]_{D}\neq 0[pi​,pj​]D​=0. The noncommutativity on a local point of the hypersurface is examined and results show that it is arisen from the local curvature of the surface. In quantum mechanics, with the infinitesimal displacement operator constructed via the momentum, and letting the infinitesimal displacement operator acting on a quantum state along a small circle on the hypersurface, we obtain a rotation operator, and the amount of the rotation is an angle anholonomy.
The geometric momentum was originally introduced for defining the momentum of particle constrained on a hypersurface, but it is in fact not necessarily defined on a curved surface only. If a coordinate system contains a family of hypersurfaces and a normal vector on hypersurface used as a unit vector, the geometric momentum can be defined on the family of hypersurfaces and can be used to determine a complete set of commuting observables. For instance, the spherical polar coordinate system is such a kind of coordinate, in which for a given value of radial position, the spherical surface is a hypersurface. It is well-known that any vector in the space can be decomposed into components along each axis of the spherical polar coordinates, but the geometric momentum has a different decomposition, for it requires a projection of the momentum on the hypersurface, and then needs to decompose the projection into the Cartesian coordinates of the original space where the whole spherical coordinates are defined. Explicitly, with a relation-iħ▽= p Σ + p n where-iħ▽ can be usual momentum operator in Cartesian coordinates, and p Σ is the momentum component on the hypersurface which turns out to be the geometric momentum, and p n is the momentum component along the radial direction, we have a nontrivial definition of radial momentum as p n ≡-iħ▽- p Σ. Once-iħ▽ and p Σ are measurable, p n is then indirectly measurable. The three-dimensional isotropic harmonic oscillator can be described in both the Cartesian and the spherical polar coordinates, whose quantum states thus can be examined in terms of both momentum and geometric momentum distributions. The distributions of the radial momentum are explicitly given for some states. The radial momentum operator that was introduced by Dirac has clear physical significance, in contrast to widely spreading belief that it is not measurable due to its non-self-adjoint.
In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum ${p}_{\ensuremath{\theta}}$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as pi{p}_{i}pi​ (i=1,2,3)(i=1,2,3)(i=1,2,3) are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between pi{p}_{i}pi​ generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.
For describing particles moving on the two dimensional curved surfaces, we can use either the intrinsic local coordinates or the Cartesian coordinates. The representation of the momentum operators differs from each other in these two kinds of coordinates, the former ones depend on the intrinsic geometrical quantities, but the latter case depend on a geometrical invariant, namely the mean curvature. Taking the operator-ordering problem into consideration, the kinetic operator for the former case can be expressed in a possibly unique way, while that for latter case can be expressed in two different ways.
For a non-relativistic particle that freely moves on a curved surface, the fundamental commutation relations between positions and momenta are insufficient to uniquely determine the operator form of the momenta. With introduction of more commutation relations between positions and Hamiltonian and those between momenta and Hamiltonian, our recent sequential studies imply that the Cartesian system of coordinates is physically preferable, consistent with Dirac' observation. In present paper, we study quantization problem of the motion constrained on the two-dimensional sphere and develop a discriminant that can be used to show how the quantization within the intrinsic geometry is improper. Two kinds of parameterization of the spherical surface are explicitly invoked to investigate the quantization problem within the intrinsic geometry.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.