Noncommutative Phase Spaces by Coadjoint Orbits Method

We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the anisotropic Newton-Hooke groups in two- and three-dimensional spaces. Through these constructions the positions and the momenta of the phase s...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2011
Hauptverfasser:	Ngendakumana, A., Nzotungicimpaye, J., Todjihounde, L.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2011
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/148081
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Noncommutative Phase Spaces by Coadjoint Orbits Method / A. Ngendakumana, J. Nzotungicimpaye, L. Todjihounde // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-148081
record_format	dspace
spelling	irk-123456789-1480812019-02-17T01:25:44Z Noncommutative Phase Spaces by Coadjoint Orbits Method Ngendakumana, A. Nzotungicimpaye, J. Todjihounde, L. We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the anisotropic Newton-Hooke groups in two- and three-dimensional spaces. Through these constructions the positions and the momenta of the phase spaces do not commute due to the presence of a magnetic field and a dual magnetic field. 2011 Article Noncommutative Phase Spaces by Coadjoint Orbits Method / A. Ngendakumana, J. Nzotungicimpaye, L. Todjihounde // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E60; 22E70; 37J15; 53D05; 53D17 DOI: http://dx.doi.org/10.3842/SIGMA.2011.116 http://dspace.nbuv.gov.ua/handle/123456789/148081 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the anisotropic Newton-Hooke groups in two- and three-dimensional spaces. Through these constructions the positions and the momenta of the phase spaces do not commute due to the presence of a magnetic field and a dual magnetic field.
format	Article
author	Ngendakumana, A. Nzotungicimpaye, J. Todjihounde, L.
spellingShingle	Ngendakumana, A. Nzotungicimpaye, J. Todjihounde, L. Noncommutative Phase Spaces by Coadjoint Orbits Method Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ngendakumana, A. Nzotungicimpaye, J. Todjihounde, L.
author_sort	Ngendakumana, A.
title	Noncommutative Phase Spaces by Coadjoint Orbits Method
title_short	Noncommutative Phase Spaces by Coadjoint Orbits Method
title_full	Noncommutative Phase Spaces by Coadjoint Orbits Method
title_fullStr	Noncommutative Phase Spaces by Coadjoint Orbits Method
title_full_unstemmed	Noncommutative Phase Spaces by Coadjoint Orbits Method
title_sort	noncommutative phase spaces by coadjoint orbits method
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/148081
citation_txt	Noncommutative Phase Spaces by Coadjoint Orbits Method / A. Ngendakumana, J. Nzotungicimpaye, L. Todjihounde // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT ngendakumanaa noncommutativephasespacesbycoadjointorbitsmethod AT nzotungicimpayej noncommutativephasespacesbycoadjointorbitsmethod AT todjihoundel noncommutativephasespacesbycoadjointorbitsmethod
first_indexed	2025-07-12T18:11:06Z
last_indexed	2025-07-12T18:11:06Z
_version_	1837465725429612544
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 116, 12 pages Noncommutative Phase Spaces by Coadjoint Orbits Method Ancille NGENDAKUMANA †, Joachim NZOTUNGICIMPAYE ‡ and Leonard TODJIHOUNDE † † Institut de Mathématiques et des Sciences Physiques, Porto-Novo, Benin E-mail: nancille@yahoo.fr, leonardt@imsp-uac.org ‡ Kigali Institute of Education, Kigali, Rwanda E-mail: kimpaye@kie.ac.rw Received May 24, 2011, in final form December 13, 2011; Published online December 18, 2011 http://dx.doi.org/10.3842/SIGMA.2011.116 Abstract. We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the aniso- tropic Newton–Hooke groups in two- and three-dimensional spaces. Through these con- structions the positions and the momenta of the phase spaces do not commute due to the presence of a magnetic field and a dual magnetic field. Key words: classical mechanics; noncommutative phase space; coadjoint orbit; symplectic realizations; magnetic and dual magnetic fields 2010 Mathematics Subject Classification: 22E60; 22E70; 37J15; 53D05; 53D17 1 Introduction Noncommutative phase spaces provide mathematical backgrounds for the study of magnetic fields in physics. Noncommutativity appeared in nonrelativistic mechanics first in the work of Peierls [14] on the diamagnetism of conduction electrons. In relativistic quantum mechanics, noncommutativity was first examined in 1947 by Snyder [16]. During the last 15 years, non- commutative mechanics has been an important subject which attracted quite a lot of attention (see, e.g., [3, 4, 5, 6, 9, 19]). Noncommutative phase space is defined as a space on which coordinates satisfy the commu- tation relations: {qi, qj} = Gij , {qi, pj} = δij , {pi, pj} = Fij , where δij is a unit matrix, whereas Gij and Fij are functions of positions and momenta. The physical dimensions of Gij and Fij are respectively M−1T and MT−1, where M represents a mass while T represents a time. The aim of this paper is to introduce the construction of noncommutative spaces by using different minimal couplings and the realization of some of them as coadjoint orbits [10, 11, 17]. The maximal coadjoint orbits of the anisotropic Newton–Hooke groups in two dimensions and in three dimensions are shown to be models of noncommutative spaces. The paper is organized as follows. In Section 2 noncommutative phase spaces are introduced by generalizing the usual Hamiltonian equations to the cases where a magnetic field and a dual magnetic field are present. Section 3 is devoted to the study of planar mechanics in the following three situations: • when a massive charged particle is in an electromagnetic field, mailto:nancille@yahoo.fr mailto:leonardt@imsp-uac.org mailto:kimpaye@kie.ac.rw http://dx.doi.org/10.3842/SIGMA.2011.116 2 A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde • when a massless spring is in a dual magnetic field, • when a pendulum is in an electromagnetic field and in a dual one. It is shown in this paper that under the presence of these fields • the massive charged particle acquires an oscillatory motion with a certain frequency, • the massless spring acquires a mass, • the pendulum appears like two synchronized oscillators. The second and third results mentioned above are quite new. In Section 4 we construct, for the first time, the coadjoint orbits of the anisotropic Newton–Hooke groups in dimension two and dimension three. Then, we obtain noncommutative phase spaces in presence of a magnetic field and a dual magnetic field. 2 Noncommutative phase spaces In this paragraph we recall Hamiltonian mechanics in both Darboux’s coordinates (Section 2.1) and noncommutative coordinates (Section 2.2), the noncommutativity coming from the presen- ce of two fields Fij and Gij . We will distinguish three cases of noncommutative coordinates corresponding to the presence of the magnetic field only, of the dual magnetic field only and of the both fields simultaneously. 2.1 Commutative coordinates It is known that a symplectic manifold is a 2n-dimensional manifold equipped with a closed nondegenerate 2-form σ. If σab are the matrix elements of the matrix representing the symplectic form σ and if σab are solutions of σbcσ ca = δab , then a Poisson bracket of two functions f and g belonging to C∞(V,R) is given by {f, g} = σab ∂f ∂za ∂g ∂zb . (1) The space C∞(V,R) endowed with the Poisson bracket given by (1) is an infinite Lie algebra [1]. If za = (pi, q i) are the canonical coordinates (Darboux’s coordinates) on V , the symplectic form on V is σ = dpi ∧ dqi, that means there is no coupling to a gauge field, and the Poisson bracket (1) becomes {f, g} = ∂f ∂pi ∂g ∂qi − ∂f ∂qi ∂g ∂pi . It follows that {pk, pi} = 0, {pk, qi} = δik, {qk, qi} = 0. That means the momenta pi as well as the positions qi are commutative. It is also known that if Xf is the Hamiltonian vector field associated to f , then Xf (g) = {f, g} and the evolution equations under the flow Φexp(sXf ) on V generated by Xf are dza ds = Xf (za), which are exactly the usual Hamiltonian equations when f is the energy. Let us now introduce noncommutative coordinates by coupling the momentum pi with a mag- netic potential Ai and the position qi with a potential A∗i. Noncommutative Phase Spaces by Coadjoint Orbits Method 3 2.2 Noncommutative coordinates Let us consider the change of coordinates πi = pi − 1 2 Fikq k, xi = qi − 1 2 pkG ki. (2) The matrix form of (2) is( πi xi ) = ( δki −1 2Fik −1 2G ik δik )( pk qk ) . As ( δki −1 2Fik −1 2G ik δik ) = ( δji −1 2Fij 0ij δij )( δsj − 1 4FjmG ms 0js 0js δjs )( δks 0sk −1 2G sk δsk ) , the transformation (2) is a change of coordinates if det ( δsj − 1 4FjmG ms ) 6= 0. It follows that [20, equation (5)] {πi, πk} = Fik, {πi, xk} = δki , {xi, xk} = Gik, (3) i.e. the new momenta as well as the new configuration coordinates are noncommutative. The Jacobi identity implies that the Fij ’s depend only on positions, that the Gij ’s depend only on momenta and that the two 2-forms σ1 = Fij(x)dxi ∧ dxj and σ2 = Gij(π)dπi ∧ dπj are closed. Let the Poisson brackets of two functions f , g in the new coordinates be given by {f, g}new = ∂f ∂πi ∂g ∂xi − ∂f ∂xi ∂g ∂πi = Yf (g). It follows that YH = XH +Gij ∂H ∂qi ∂ ∂qj + Fij ∂H ∂pi ∂ ∂pj . The derivative of any function f with respect to time t, in terms of F and G, is then given by df dt = XH(f) +Gij ∂H ∂qi ∂f ∂qj + Fij ∂H ∂pi ∂f ∂pj , (4) and the equations of motion are then given by dqk dt = ∂H ∂pk +Gki ∂H ∂qi , dpk dt = −∂H ∂qk + Fik ∂H ∂pi . If for example H = δijpipj 2m + V is the Hamiltonian with the potential energy V depending only on the configuration coordina- tes qi, the equations of motion are then dqk dt = pk m +Gki ∂V ∂qi , dpk dt = − ∂V ∂qk + Fik pi m . They are equivalent to the modified Newton’s second law [5, 15, 20] m d2qk dt2 = − ∂V ∂qk + Fik pi m +mGki d dt ( ∂V ∂qi ) . This means that the noncommutativity of the momenta implies that the particle is accelerated and is not free even if the potential V vanishes identically. 4 A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde 3 Couplings in planar mechanics In this section we construct explicitly these noncommutative phase spaces by introducing cou- plings. We start with the usual coupling of momentum with a magnetic potential. Then, we introduce the coupling of position with a dual potential and finish with a mixing model. 3.1 Coupling of momentum with a magnetic field 3.1.1 Commutative coordinates Consider a four-dimensional phase space (a cotangent space to a plane) equipped with the Dar- boux’s coordinates (pi, q i). This means that the momenta as well as the positions commute. Consider also an electron with mass m and an electric charge e, moving on a plane with the electromagnetic potential Aµ = ( Ai = −1 2Bεikq k, φ = Eiq i ) where a symmetric gauge has been chosen, ~E being an electric field while ~B is a magnetic field perpendicular to the plane. It is known that the dynamics of the particle is governed by the Hamiltonian H = ~p 2 2m − eφ (5) and that the equation of motion is m d2~q dt2 = e ~E, where the right hand side is the electric force. 3.1.2 Noncommutative coordinates From the classical electromagnetism, it is known that the coupling of the momentum with the magnetic potential is given by the relations πi = pi + eB 2 εikq k, xi = qi. (6) The coordinates πi and xi are such that {xi, xk} = 0, {πi, xk} = δki , {πi, πk} = −eBεik. In the presence of an electromagnetic field, the momenta are noncommutative while the positions are commutative. Using the equation (3), we have Fij = −eBεij , Gij = 0ij . (7) Use of (5) and (7) into (4) gives rise the equations of motion m d2~q dt2 = e ( ~E + ~p m × ~B ) , (8) where the right hand side represents the Lorentz force. Moreover, in noncommutative coordi- nates, the Hamiltonian (5) becomes H = ~π 2 2m − e ~E · ~x+ mω2~x 2 2 + ~ω · ~L, Noncommutative Phase Spaces by Coadjoint Orbits Method 5 where ω is the cyclotron frequency, ~L = ~x× ~p is the orbital angular momentum and ~ω = eB 2m ~n with ~n the unit vector in the direction perpendicular to the plane. In the presence of a magnetic field, the massive particle has become an oscillator with frequency ω given above and then the equation of motion is m d2~x dt2 = e ( ~E + ~π m × ~B ) , (9) where we recognize again the Lorentz force ~fLorentz = e ~E + e ~πm × ~B. Note that the relations (8) and (9) have the same form. The Newton’s equations are then covariant under the coupling (6). In the next two subsections, we present quite new theories associated with an unusual coupling of position with a dual magnetic field. 3.2 Coupling of position with a dual field 3.2.1 Commutative coordinates Consider a massless spring with k as a Hooke’s constant and a dual charge e∗ in a dual magnetic field B∗. Suppose that the dynamics of the spring is governed by the Hamiltonian H = k ~q 2 2 − e∗~p · ~E∗, (10) where we have used the symmetric gauge. Moreover the analogue of the Newton’s second equation is 1 k d2~p dt2 = e∗ ~E∗, where d2~p dt2 is a yank, i.e. the second derivatives of momentum with respect to the time variable t, e∗ ~E∗ is a velocity while e ~E is a force as in the previous subsection. 3.2.2 Noncommutative coordinates Let us consider the coupling of the position with the dual potential A∗i depending on the momenta pi, πi = pi, xi = qi + e∗B∗ 2 pkε ki. (11) In this case, the Poisson brackets become {xi, xj} = −e∗B∗εij , {pk, xi} = δik, {pk, pi} = 0. Therefore, in the presence of the dual field, positions do not commute while the momenta commute. Then Fij = oij , Gij = −e∗B∗εij . (12) Use (10) and (12) into (4) gives rise to the Newton’s analogue equations are then 1 k d2~p dt2 = e∗ ~E∗ + e∗k~q × ~B∗, (13) 6 A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde the right hand side being a velocity. In noncommutative coordinates the Hamiltonian is H = k~x 2 2 − e∗~π · ~E∗ + ~π 2 2ms − ~ω · ~L, where the spring mass ms is defined by 1 ms = k e∗2B∗2 4 , while the vector ~ω is given by ~ω = k e∗B∗ 2 ~n. The Hooke’s constant k can be written as k = msω 2. (14) In the presence of the dual field, the spring then acquires a mass ms and the equations of motion are given by 1 k d2~π dt2 = e∗ ~E∗ + e∗k~x× ~B∗. (15) The vector ~f∗ = e∗( ~E∗+k~x× ~B∗) can be considered as a dual Lorentz force with the dimension of velocity. It represents for the spring what the Lorentz force represents for a charged particle. Here also the coupling (11) preserves the covariance of the Newton’s analogue equations. Com- paring (13) and (15) and using (14) into (15) we can conclude that e∗ω2( ~E∗+k~x× ~B∗) is a kind of jerk [13]. 3.2.3 Coupling with a magnetic field and with a dual f ield Now consider the case of a massive pendulum with mass m and Hooke’s constant k under the action of an electromagnetic potential Aµ = (Ai, φ) and a dual electromagnetic potential A∗µ = (A∗i , φ ∗) with Ai = −1 2Bεikq k, φ = Eiq i, A∗i = −1 2B ∗pkεki and φ∗ = piE ∗ i , where ~E is an electric field and ~E∗ its dual field while ~B is a magnetic field and ~B∗ its dual field. The corresponding motion is governed by the Hamiltonian H = ~p 2 2m + k~q 2 2 − eφ− e∗φ∗. Let xi = qi + e∗B∗ 2 pkε ki, πi = pi + eB 2 εikq k be the minimal coupling in the symmetric gauge; that is Gij = −e∗B∗εij , Fij = −eBεij . We assume that the cyclotron frequency acquired by the massive charged particle is equal to the frequency of the massless spring: eB 2 = mω, e∗B∗ 2 = 1 msω , Noncommutative Phase Spaces by Coadjoint Orbits Method 7 where ms is the acquired mass by the spring while µ = m ·ms m+ms is the reduced mass of the two synchronized massive oscillators. It follows that {xi, xj} = −e∗B∗εij , {πk, xi} = γδik, {πk, πi} = −eBεki (16) with γ = 1 + m ms and m = µγ. In the presence of the two kind of field, the positions as well as the momenta do not commute. The Hamiltonian in noncommutative coordinates is written as H = ~π 2 2µ + Mω2~x 2 2 − eφ− e∗φ∗, where M = m+ms is the total mass, φ = ~E · ~x+ ~n · ~E × ~π msω and φ∗ = ~π · ~E∗ + ~n ·mω~x× ~E∗. Note that M = msγ. The motion’s equations in noncommutative coordinates are then d~x dt = ~π µ + e∗ [ γ ~E∗ + k~x× ~B∗ − e ~B∗ × ~E ] , d~π dt = −kγ~x+ e [ γ ~E + ~π m × ~B − e∗ ~B × ~E∗ ] , where the Hooke’s constant k is given by (14). If the mass m of the particle is very smaller than the mass ms acquired by the spring, i.e. m� ms, then γ becomes 1 and µ = m�M = ms. In that limit, the brackets (16) become {xi, xj} = −e∗B∗εij , {πk, xi} = δik, {πk, πi} = −eBεki, the Hamiltonian becomes H = ~π 2 2m + msω 2~x 2 2 − e [ ~E · ~x+ ~n · ~E × ~π msω ] − e∗ [ ~π · ~E∗ + ~n ·mω~x× ~E∗ ] and the equations of motion are given by d~x dt = ~π m + e∗ [ ~E∗ + k~x× ~B∗ − e ~B∗ × ~E ] , d~π dt = −k~x+ e [ ~E + ~π m × ~B − e∗ ~B × ~E∗ ] . The velocity ee∗ ~B∗ × ~E and the force ee∗ ~E∗ × ~B result from the coexistence of the two fields. 4 Noncommutative phase spaces as coadjoint orbits of anisotropic Newton–Hooke groups It is well known that the dual of a Lie algebra has a natural Poisson structure whose symplectic leaves are the coadjoint orbits. These orbits will provide naturally noncommutative phase spaces. In this section, we use orbit construction to realize noncommutative phase spaces on the ani- sotropic Newton–Hooke groups in two- and three-dimensional spaces, the anisotropic Newton– Hooke groups ANH± being Newton–Hooke groups NH± without the rotation parameters [2]. Their Lie algebras have the structures [Ki, E] = Pi, [Pi, E] = ±ω2Ki, i = 1, 2, . . . , n, 8 A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde where ~K = ∂ ∂~v , ~P = ∂ ∂~x , E = ∂ ∂t + ~v · ∂ ∂~x ± ω2~x · ∂ ∂~v . Standard methods [8, 10, 11, 12] show that the structure of the central extensions of the Lie algebras ANH± is • in one-dimensional space [K,E] = P, [P,E] = ±ω2K, [K,P ] = M, • in two-dimensional spaces [Ki,Kj ] = 1 c2 J3εij , [Ki, E] = Pi, [Ki, Pj ] = Mδij , [Pi, Pj ] = ± 1 r2 J3εij , [Pi, E] = ±ω2Ki, • in three-dimensional spaces [Ki,Kj ] = 1 c2 Jkε k ij , [Ki, E] = Pi, [Ki, Pj ] = Mδij , [Pi, Pj ] = ± 1 r2 Jkε k ij , [Pi, E] = ±ω2Ki, where r is a constant with the dimension of length, c is a constant with the dimension of speed while Jk is a rotation parameter around the kth axis. 4.1 One-dimensional space case In this case m is a trivial invariant. The other invariant, the solution of the Kirillov’s system, is U = e− p2 2m ± mω2q2 2 , where q = k m . We denote the two-dimensional orbit by O(m,U). It is not interesting for our study because there are one momentum and one position. Note that the symplectic realizations of ANH− and ANH+ are respectively given by L(v,x,t)(p, q) = ( p cos(ωt)−mωq sin(ωt)−mv cos(ωt), p mω sin(ωt) + (q + x) cos(ωt)− v ω sin(ωt) ) and L(v,x,t)(p, q) = ( p cosh(ωt) +mωq sinh(ωt)−m(v cosh(ωt)− ωx sinh(ωt)), p mω sinh(ωt) + (q + x) cosh(ωt)− v ω sinh(ωt) ) . Let (p(t), q(t)) = L(0,0,t)(p, q), it follows that p(t) = p cos(ωt)−mωq sin(ωt), q(t) = p mω sin(ωt) + q cos(ωt) Noncommutative Phase Spaces by Coadjoint Orbits Method 9 for ANH− and p(t) = p cosh(ωt) +mωq sinh(ωt), q(t) = p mω sinh(ωt) + q cosh(ωt) for ANH+. The equations of motion are then given by dp dt = ±mω2q, dq dt = p m for ANH± or equivalently d2q dt2 = ±ω2q; which is a second order differential equation whose solutions are trigonometric functions for ANH− case and hyperbolic ones in ANH+ case. It is for this reason that ANH− describes a universe in oscillation while ANH+ describes a universe in expansion. 4.2 Two-dimensional spaces case Let mM∗ + hJ∗3 + kiK ∗i + piP ∗i + eE∗ be the general element of the dual of the central extended Lie algebra. Then m and h are trivial invariants under the coadjoint action of ANH± in two-dimensional spaces. The other invariant, the solution of the Kirillov’s system, is explicitly given by U = e− ~p 2 2µe ± µeω 2~q 2 2 with µe = m± h ωr2 , ~q = ~k µe , (17) where hω0 = mc2 denotes the wave-particle duality, µe is an effective mass. The restriction of the Kirillov’s matrix on the orbit is given by Ω =  0 h c2 m 0 − h c2 0 0 m −m 0 0 ± h r2 0 −m ∓ h r2 0  . By using relations (17), the duality wave-particle and the equality c = ωr, we obtain that the Poisson brackets of two functions defined on the orbit are given by {H, f} = ∂H ∂pi ∂f ∂qi − ∂H ∂qi ∂f ∂pi +Gij ∂H ∂qi ∂f ∂qj + Fij ∂H ∂pi ∂f ∂pj , i, j = 1, 2 with Gij = − εij mω0 , Fij = −(m− µe)ωεij . It follows that the magnetic field B and its dual field B∗ are such that e∗B∗ = 1 mω0 , eB = (m− µe)ω. The effective mass is then given in terms of the magnetic field as µe = m− eB ω . 10 A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde The Hamilton’s equations are then dπi dt = −∂H ∂qi ± (m− µe)ωεik ∂H ∂pk , dxi dt = ∂H ∂pi + εik 2mω0 ∂H ∂qk . The inverse of Ω is Ω−1 =  0 ± ω µe − 1 µe 0 ∓ ω µe 0 0 − 1 µe 1 µe 0 0 1 µeω0 0 1 µe − 1 µeω0 0  , where we have used the wave-particle duality and (17). Finally the orbit is equipped with the symplectic form σ = dpi ∧ dqi + 1 µeω0 εijdpi ∧ dpj ± µeωεijdqi ∧ dqj . We observe that with the anisotropic Newton–Hooke group in two-dimensional spaces, the ob- tained phase spaces are completely noncommutative while the phase space obtained with the Galilei group in [3] is only partially noncommutative. This is due to the difference in the struc- ture of their extended Lie algebras. In the Newton–Hooke case space translations as well pure Newton–Hooke transformations do not commute while only pure Galilei transformations do not commute in the Galilei case. Note that the nontrivial Lie brackets of the extended Newton– Hooke Lie algebra in two-dimensional space are given by [J,Ki] = Kjε j i , [J, Pi] = Pjε j i , [Ki, Pj ] = Mδij , [Ki, E] = Pi, [Pi, E] = ω2Ki, which means that the generators of space translations as well as pure Newton–Hooke transfor- mations commute. One can not then associate a noncommutative phase space to the Newton– Hooke group. It is then the absence of the symmetry rotations (anisotropy of the plane) which guaranties the noncommutative phase space for the anisotropic Newton–Hooke group. 4.3 Three-dimensional spaces case Let mM∗ + hiJ ∗i + kiK ∗i + piP ∗i + eE∗, i = 1, 2, 3 be the general element of the dual of the central extended Lie algebra. Then m and hi are trivial invariants under the coadjoint action of ANH± in three-dimensional spaces. We need another invariant. As we can verify, the Kirillov’s form, in the basis (Ki, Pi, E), is given by Bαβ =  hkε k ij c2 mδij pi −mδij ± hkε k ij r2 ±ω2ki pj ∓ω2kj 0  . The other invariant which is a solution of the Kirillov’s system is U = e− pipj ( Φ−1± )ij 2m − mω2qiqj ( Φ−1± ) ij 2 + ω2piq j ( Φ−1± A )i j , where Aij = hkε k ij mc2 , Φ± = I ± ω2A and qi = ki m . Noncommutative Phase Spaces by Coadjoint Orbits Method 11 We see that Φ± is a metric for R3. Let us denote the maximal coadjoint orbit by O (m,~h,U) . The restriction of the Kirillov’s form on the orbit is then Ω = m ( Aij δji −δij ±ω2Aij ) and its inverse is Ω−1 = 1 m ±ω2 ( AΦ−1± ) ij ( Φ−1± )j i − ( Φ−1± )i j ( AΦ−1± )ij  . The maximal orbit is then equipped with the symplectic structure σ = ( Φ−1± )i j dpi ∧ dqj + 1 m ( AΦ−1± )ij dpi ∧ dpj ±mω2 ( AΦ−1± ) ij dqi ∧ dqj and it follows that the Poisson brackets of two functions defined on the orbit is then {f, g} = (Φ−1± )ji ( ∂f ∂pi ∂g ∂qj − ∂f ∂qi ∂g ∂pj ) + Fij ∂f ∂pi ∂g ∂pj +Gij ∂f ∂qi ∂g ∂qj . This implies that {pi, pj} = Fij , {pi, qj} = ( Φ−1± )j i , {qi, qj} = Gij , where the magnetic field Fij and the dual magnetic field Gij are given by Fij = ±mω2 ( AΦ−1± ) ij and Gij = 1 m ( AΦ−1± )ij . Moreover the Hamilton’s equations are dpk dt = − ( Φ−1± )i k ∂H ∂qi ±mω2 ( AΦ−1± ) ik ∂H ∂pi , dqk dt = ( Φ−1± )k i ∂H ∂pi + 1 m ( AΦ−1± )ik ∂H ∂qi . With the anisotropic Newton–Hooke groups ANH± in three-dimensional spaces, we also have realized phase spaces where the momenta as well as the positions do not commute. 5 Conclusion We know that we can introduce the classical electromagnetic interaction through the modified symplectic form σ = dpi∧dqi+ 1 2Fijdq i∧dqj [1, 7, 17]. This has been initiated by J.M. Souriau [17] in the seventies. Recently many authors (see, e.g., [3, 4, 18, 19]) generalized this modification of the symplectic form by introducing the so-called dual magnetic field such that σ = dpi ∧ dqi + 1 2Fijdq i ∧ dqj + 1 2G ijdpi ∧ dpj . The fields F and G are responsible of the noncommutativity respectively of momenta and positions. In our paper we have introduced these fields firstly by minimal coupling momenta with magnetic potentials (the usual one), secondly by minimal coupling of positions with dual potentials and lastly by mixing the two couplings. We have also realized phase spaces endowed with modified symplectic structures as coadjoint orbits of the anisotropic Newton–Hooke groups in two and three-dimensional spaces. In all these cases, the fields are constant because they are coming from central extensions of Lie algebras. 12 A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde References [1] Abraham R., Marsden J.E., Foundations of mechanics, 2nd ed., Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] Derome J.R., Dubois J.G., Hooke’s symmetries and nonrelativistic cosmological kinematics. I, Nuovo Ci- mento B 9 (1972), 351–376. [3] Duval C., Horváthy P.A., The exotic Galilei group and the “Peierls substitution”, Phys. Lett. B 479 (2000), 284–290, hep-th/0002233. [4] Duval C., Horváthy P.A., Exotic Galilean symmetry in the non-commutative plane and the Hall effect, J. Phys. A: Math. Gen. 34 (2001), 10097–10107, hep-th/0106089. [5] Duval C., Horváth Z., Horváthy P.A., Exotic plasma as classical Hall liquid, Internat. J. Modern Phys. B 15 (2001), 3397–3408, cond-mat/0101449. [6] Grigore D.R., Transitive symplectic manifolds in 1 + 2 dimensions, J. Math. Phys. 37 (1996), 240–253. [7] Guillemin V., Sternberg S., Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. [8] Hamermesh M., Group theory and its applications to physical problems, Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Inc., Reading, Mass. – London, 1962. [9] Horváthy P.A., The non-commutative Landau problem, Ann. Physics 299 (2002), 128–140, hep-th/0201007. [10] Kirillov A.A., Elements of theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin – New York, 1976. [11] Kostant B., Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 87–208. [12] Nzotungicimpaye J., Galilei–Newton law by group theoretical methods, Lett. Math. Phys. 15 (1988), 101– 110. [13] Nzotungicimpaye J., Jerk by group theoretical methods, J. Phys. A: Math. Gen. 27 (1994), 4519–4526. [14] Peierls R., On the theory of diamagnetism of conduction electrons, Z. Phys. 80 (1933), 763–791. [15] Romero J.M., Santiago J.A., Vergara J.D., Newton’s second law in a non-commutative space, Phys. Lett. A 310 (2003), 9–12, hep-th/0211165. [16] Snyder H.S., Quantized space-time, Phys. Rev. 71 (1947), 38–41. [17] Souriau J.-M., Structure des systèmes dynamiques, Mâıtrises de Mathématiques Dunod, Paris, 1970. [18] Vanhecke F.J., Sigaud C., da Silva A.R., Noncommutative configuration space. Classical and quantum mechanical aspects, Braz. J. Phys. 36 (2006), 194–207, math-ph/0502003. [19] Vanhecke F.J., Sigaud C., da Silva A.R., Modified symplectic structures in cotangent bundles of Lie groups aspects, Braz. J. Phys. 39 (2009), 18–24, arXiv:0804.1251. [20] Wei G.-F., Long C.-Y., Long Z.-W., Qin S.-J., Fu Q., Classical mechanics in non-commutative phase space, Chinese Phys. C 32 (2008), 338–341. http://dx.doi.org/10.1007/BF02734453 http://dx.doi.org/10.1007/BF02734453 http://dx.doi.org/10.1016/S0370-2693(00)00341-5 http://arxiv.org/abs/hep-th/0002233 http://dx.doi.org/10.1088/0305-4470/34/47/314 http://arxiv.org/abs/hep-th/0106089 http://dx.doi.org/10.1142/S0217979201007361 http://arxiv.org/abs/cond-mat/0101449 http://dx.doi.org/10.1063/1.531388 http://dx.doi.org/10.1006/aphy.2002.6271 http://arxiv.org/abs/hep-th/0201007 http://dx.doi.org/10.1007/BF00397830 http://dx.doi.org/10.1088/0305-4470/27/13/025 http://dx.doi.org/10.1016/S0375-9601(03)00191-9 http://arxiv.org/abs/hep-th/0211165 http://dx.doi.org/10.1103/PhysRev.71.38 http://arxiv.org/abs/math-ph/0502003 http://arxiv.org/abs/0804.1251 http://dx.doi.org/10.1088/1674-1137/32/5/002 Introduction Noncommutative phase spaces Commutative coordinates Noncommutative coordinates Couplings in planar mechanics Coupling of momentum with a magnetic field Commutative coordinates Noncommutative coordinates Coupling of position with a dual field Commutative coordinates Noncommutative coordinates Coupling with a magnetic field and with a dual field Noncommutative phase spaces as coadjoint orbits of anisotropic Newton-Hooke groups One-dimensional space case Two-dimensional spaces case Three-dimensional spaces case Conclusion References

Noncommutative Phase Spaces by Coadjoint Orbits Method

Institution

Ähnliche Einträge