Gauge-invariant formulation of the electromagnetic interaction in Hamiltonian mechanics

Transcription

1 INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 5 (1) FEBRERO 24 Gauge-invariant forulation of the eletroagneti interation in Hailtonian ehanis G.F. Torres del Castillo Departaento de Físia Mateátia, Instituto de Cienias, UAP, 7257 Puebla, Pue., Méxio Reibido el 22 de agosto de 23; aeptado el 17 de septiebre de 23 Making use of the fat that for an arbitrary autonoous ehanial syste any onstant of otion an be used as Hailtonian, the equations of otion of a harged partile in an eletroagneti field are written in Hailtonian for without introduing potentials for the eletroagneti field. It is shown that the Hailtonian and the Poisson braket obtained here oinide with those appearing in the standard Hailtonian forulation. Keywords: Hailton equations; eletroagneti interation. Haiendo uso del heho de que para un sistea eánio autónoo arbitrario ualquier onstante de oviiento puede usarse oo hailtoniana, las euaiones de oviiento de una partíula argada en un apo eletroagnétio se esriben en fora hailtoniana sin introduir poteniales para el apo eletroagnétio. Se uestra que la hailtoniana y el paréntesis de Poisson obtenidos aquí oiniden on los que apareen en la forulaión hailtoniana estándar. Desriptores: Euaiones de Hailton; interaión eletroagnétia. PACS: 45.2.Jj 1. Introdution As it is well-known, the equations of otion of a harged partile in a given eletroagneti field an be forulated aking use of the Lagrangian foralis, introduing a veloitydependent potential that ontains the potentials of the field, and also using the Hailtonian foralis (see, e.g., Ref. 1). The fat that the eletroagneti potentials appear in the Lagrangian, and not the eletroagneti fields theselves, is a drawbak beause the potentials are not uniquely deterined by the eletroagneti fields; if the potentials possess soe syetry, the eletroagneti fields also have that syetry, but if the eletroagneti fields possess soe syetry, the potentials ay not share it. Hene, in the standard forulation, the Lagrangian or the Hailtonian ight not exhibit all the syetries of the syste. For instane, the unifor agneti field B = (,, B 3 ), with B 3 = onst., is invariant under all translations in spae, but the vetor potential of a nonvanishing agneti field annot be invariant under all translations. As shown in Ref. 2, the equations of otion of any autonoous ehanial syste an be written in Hailtonian for (i.e., df/dt = {f, H}, for any differentiable funtion, f, defined on the phase spae) with the Hailtonian being any onstant of otion (not neessarily the total energy) provided that the Poisson braket is suitably defined. Furtherore, if the nuber of degrees of freedo of the syste is greater than 1, for eah hoie of the Hailtonian funtion, there are infinitely any suitable Poisson brakets (see also Refs. 3 and 4). Using the fat that, in the fraework of Newtonian ehanis, the kineti energy of a harged partile is not odified by a agnetostati field, we show that the equations of otion of a harged partile in a agnetostati field an be written in Hailtonian for, with the usual Hailtonian funtion of a free partile and the agneti field ebodied in the Poisson braket, without having to introdue auxiliary quantities suh as the vetor potential. We also show that the Hailtonian and the Poisson braket eployed here oinide with those appearing in the standard Hailtonian forulation for a partile in a agneti field. The general ase of a harged partile in an arbitrary eletroagneti field is dealt with in a siilar way, onsidering the orresponding relativisti equations of otion. In Se. 2 we give a suary of the Hailtonian foralis. In Se. 3 we onsider the equations of otion, aording to the Newtonian ehanis, of a harged partile in a agnetostati field and in Se. 4 we onsider the equations of otion of a harged partile in an arbitrary eletroagneti field in the fraework of relativisti ehanis. 2. Hailton s equations Hailton s equations are usually written in the for dq i dt = H, dp i dt = H q i (1) (i = 1, 2,..., n), where H is the Hailtonian funtion, (q 1,..., q n, p 1,..., p n ) is a set of anonial oordinates on the phase spae and n is the nuber of degrees of freedo. Hailton s equations (1) are equivalent to the forula df = {f, H}, (2) dt for any differentiable funtion, f, defined on the phase spae that does not depend expliitly on the tie, where {, } de-

2 GAUGE-INVARIANT FORMULATION OF THE ELECTROMAGNETIC INTERACTION IN HAMILTONIAN MECHANICS 89 notes the Poisson braket, defined by {f, g} = f q i g f g q i. (3) (Here, and in what follows, there is suation over repeated indies.) In the ase of an autonoous syste, we an assue that H does not depend expliitly on the tie and Eqs. (1) iply that H is a onstant of otion. The for of Eqs. (1) and (3) is invariant under the replaeent of q i, p i by new oordinates q i, p i, if the latter are obtained fro q i, p i by eans of a anonial transforation; however, Eq. (2) is appliable in any oordinate syste. In ters of an arbitrary oordinate syste in the phase spae, (x 1, x 2,..., x 2n ), the Poisson braket (3) is expressed as {f, g} = ( x µ x ν q i xµ x ν q i ) f x µ g x ν µν f g = σ x µ x ν (4) (µ, ν,... = 1, 2,..., 2n), where we have introdued the definition σ µν {x µ, x ν }. (5) The Poisson braket (3) satisfies the Jaobi identity, therefore, {{f, g}, h} + {{g, h}, f} + {{h, f}, g} = ; {{x µ, x ν }, x λ } + {{x ν, x λ }, x µ } + {{x λ, x µ }, x ν } = whih, aording to Eqs. (4) and (5) aounts to ρλ σµν σ x ρ + σρµ σνλ x ρ + σρν σλµ =. (6) xρ Conversely, if a set of funtions σ µν = σ νµ satisfies Eqs. (6), then the braket ( ) ( ) f g {f, g} = σ µν x µ x ν satisfies the Jaobi identity. Fro Eqs. (2) and (4) it follows that the Hailton equations in an arbitrary oordinate syste are dt µν H = σ x ν. (7) The equations of otion of a given autonoous syste expressed in ters of an arbitrary oordinate syste an be written in the Hailtonian for (7) in infinitely any ways; one just has to hoose soe onstant of otion and use it as Hailtonian in Eqs. (7), whih, together with Eqs. (6), deterine funtions σ µν. If n > 1, there are infinitely any ways of hoosing the the funtions σ µν satisfying Eqs. (6), (7), and the onditions σ µν = σ νµ [2,4]. 3. Partile in a agnetostati field The equations of otion for a harged partile of ass in a agnetostati field with indution B, and possibly in a fore field derivable fro a potential, are given by dp dt = e v B U, (8) where e is the eletri harge of the partile, v denotes its veloity, is the speed of light in vauu and U is soe potential that depends only on the position of the partile. The vetor p is the usual linear oentu p = v. (9) Fro Eqs. (8) it follows that (p 2 /2) + U is a onstant of otion. Hene, we an hoose H = p2 + U. (1) 2 Then, letting (x 1,..., x 6 ) = (q 1, q 2, q 3, p 1, p 2, p 3 ), where (q 1, q 2, q 3 ) are the Cartesian oordinates of the partile and (p 1, p 2, p 3 ) are the Cartesian oponents of p, and aking use of Eqs. (8) and (9) one finds that Eqs. (7) an be written expliitly in atrix for as ( ) e ( ) e ( ) e p 1 p 2 p 3 (B 3 p 2 B 2 p 3 ) q 1 (B 1 p 3 B 3 p 1 ) q 2 (B 2 p 1 B 1 p 2 ) q 3 =(σ µν ) q 1 q 2 q 3 p 1 p 2 p 3. (11) By inspetion, one finds that an antisyetri atrix (σ µν ) that satisfies this last equation is (σ µν eb 3 )= 1 eb 2 (12) 1 eb 3 eb 1 eb 2 1 eb 1 i.e., {q i, q j } =, {q i, p j } = δ i j, {p i, p j } = e ε ijkb k, (13) Rev. Mex. Fís. 5 (1) (24) 88 92

3 9 G.F. TORRES DEL CASTILLO and it an be readily verified that this braket satisfies the Jaobi identity [substituting Eqs. (12) into Eqs. (6) or diretly using Eqs. (13)] as a onsequene of the fat that the divergene of B vanishes. For an arbitrary ehanial syste, any differentiable funtion, G(x µ ), is the infinitesial generator of a loal oneparaeter group of anonial transforations whose orbits are the solutions of the syste of differential equations ds [f. Eq. (7)] or, equivalently, = σµν G x ν (14) ds = {xµ, G}. (15) The funtion G(x µ ) is a onstant of otion if and only if the Hailtonian is invariant under the anonial transforations defined by G [i.e., if and only if H is onstant along the solutions of (14)]. Conversely, given a loal one-paraeter group of anonial transforations, x µ = x µ (s), Eq. (14) defines up to an additive onstant its infinitesial generator G and, if the Hailtonian is invariant under these transforations, G is a onstant of otion. (The integrability onditions of Eqs. (14) for G are the onditions for the transforations x µ = x µ (s) to be anonial.) Owing to the presene of the oponents of B in the funtions σ µν [or, equivalently, in the Poisson brakets (13)], it turns out that a translation or rotation is a anonial transforation if and only if the agneti field is invariant under that transforation. Taking, for sipliity, U =, any translation or rotation leaves the Hailtonian H = p 2 /2 invariant; however, not all of the lead to the existene of onstants of otion. Only the translations or rotations that leave the agneti field invariant orrespond to anonial transforations that leave the Hailtonian invariant, thus iplying the existene of onstants of otion. (Note that these onditions involve only the agneti field itself, without aking referene to the vetor potential whih is defined up to gauge transforations.) For exaple, if the agneti field is invariant under translations along the q 1 -axis, then B i q 1 =, for i = 1, 2, 3, and fro B = it follows that B 2 q 2 + B 3 q 3 =, whih eans that B 3 dq 2 B 2 dq 3 is an exat differential, at least loally; hene, there exists a funtion Φ(q 2, q 3 ) suh that B 3 dq 2 B 2 dq 3 = dφ. (16) Then, aking use of Eqs. (13) and (16), one finds that {q i, p 1 + e Φ} = δi 1, {p i, p 1 + e Φ} = (17) and by oparing with Eq. (15) one onludes that p 1 + (e/)φ is a generating funtion of translations along the q 1 -axis, whih is a onstant of otion as a onsequene of the invariane of H under these translations. It should be rearked that Φ is defined up to an additive onstant only. Aording to Darboux s theore, Eqs. (6) iply the loal existene of anonial oordinates (i.e., oordinates Q i, P i, suh that {Q i, Q j } = = {P i, P j }, {Q i, P j } = δj i ). If the vetor field A is a vetor potential for B, that is, B = A, then fro Eqs. (13) one finds that {q i, p j + e A j} = δ i j, {p i + e A i, p j + e A j} =, (18) whih eans that Q i q i, P i p i + e A i are anonial oordinates. If the Hailtonian (1) is written in ters of these oordinates one obtains the standard expression H = 1 2 (P e A)2 + U. (19) It should be rearked that fro Eqs. (18) it follows that P k is the infinitesial generator of translations along the q k -axis if and only if A i / q k =, for i = 1, 2, 3. In spite of this fat, it is ustoary in quantu ehanis to replae the anonial oenta P k by the differential operators i / q k, whih is the infinitesial generator of translations along the q k -axis in the oordinate representation. Another exaple, that perhaps illustrates this point ore learly, is provided by the agneti field B = g r r 3, (2) whih would be produed by a agneti onopole plaed at the origin (though this field does not satisfy the ondition B = at the origin). The agneti field (2) is invariant under rotations about any axis passing through the origin, but there is no spherially syetri vetor potential for this field. The infinitesial generator of a loal one-paraeter group of anonial transforations an be derived fro Eq. (14). Denoting by (ω µν ) the inverse of the atrix (σ µν ), fro Eq. (14) we have (ω µν )= dg = ω µν dx ν ds dxµ. (21) In the present ase, fro Eq. (12) one finds that eb 3 eb 2 1 eb 3 eb 1 1 eb 2 eb 1 1, (22) Rev. Mex. Fís. 5 (1) (24) 88 92

4 GAUGE-INVARIANT FORMULATION OF THE ELECTROMAGNETIC INTERACTION IN HAMILTONIAN MECHANICS 91 and for rotations about the q i -axis, and dq j ds = ε jiq dp j ds = ε jip, where s is the angle of the rotation. Thus, if L i denotes the infinitesial generator of rotations about the q i -axis, fro Eq. (21) we obtain ( ) eb l dl i = ε jkl ε kiq ε ji p dq j + ε ji q dp j = d(ε ijk q j p k ) + e (B iq j dq j B j q j dq i ) (23) and substituting Eq. (2) it follows that L i = ε ijk q j p k eg q i r, (24) whih does not oinide with ε ijk q j P k, the infinitesial generator of rotations of the anonial variables. With the aid of Eqs. (13) and (2), one verifies that {L i, L j } = ε ijk L k. 4. Partile in an eletroagneti field In order to express the equations of otion of a harged partile in an arbitrary eletroagneti field in a for siilar to that presented in the foregoing setion, it is onvenient to onsider the fro the point of view of relativisti ehanis. Sine the derivations are alost idential to those given in the previous ase, we shall oit soe details in what follows. In this setion the lower ase Greek indies take the values, 1, 2, 3. Let q µ be the Cartesian oordinates of a harged partile (with respet to soe inertial frae) with rest ass and eletri harge e. If τ denotes the partile proper tie, p µ = dq µ dτ (25) is the usual four-oentu (see, e.g., Ref. 5). The relativisti equations of otion for the partile in a given eletroagneti field are given by dp µ dτ = e F µ dq ν ν dτ = e F µ νp ν, (26) where the F µν are the oponents of the eletroagneti field tensor. (The tensor indies are lowered or raised with the aid of (η µν ) = diag(1, 1, 1, 1) and its inverse in the usual way, e.g., p µ = η µν p ν, p µ = η µν p ν.) Fro Eqs. (26) and the antisyetry of F µν it follows that H p µp µ 2 (27) is a onstant of otion. It an be readily verified that Eqs. (25) and (26) an be written as if dq µ dτ = {qµ, H}, dp µ dτ = {pµ, H} {q µ, q ν } =, {q µ, p ν } = δ µ ν, {p µ, p ν } = e F µν (28) [f. Eqs. (13)]. The Jaobi identity is satisfied by virtue of the Maxwell equations F µν q λ + F νλ q µ + F λµ q ν =. Thus, a translation, rotation or boost is a anonial transforations if and only if it leaves the eletroagneti field invariant. If A µ denotes a four-potential for the eletroagneti field (i.e., F µν = A ν / q µ A µ / q ν ) then fro Eqs. (28) one finds that Q µ = q µ and P µ = p µ + ea µ / are anonial oordinates but, again, this does not ean that P µ is the infinitesial generator of translations along the q µ -axis, unless A ν / q µ = for ν =, 1, 2, Conluding rearks The anonial oordinates are distinguished by the fat that in ters of the the Hailton equations (7) and the Poisson braket (4) redue to the sipler fors (1) and (3), respetively, but in the ase of partiles interating with the eletroagneti field it is preferable to eploy nonanonial oordinates, whih ay have learer eaning, and without having to introdue gauge-dependent quantities. The Poisson braket obtained here as the siplest solution of Eqs. (6) and (11) is preisely the one found in the standard Hailtonian forulation [2]. As pointed out above, the usual anonial oentu P i or P µ need not be the infinitesial generator of translations along the q i - or q µ -axis. Aknowledgeent The author aknowledges Dr. M. Montesinos for useful disussions. Rev. Mex. Fís. 5 (1) (24) 88 92

5 92 G.F. TORRES DEL CASTILLO 1. M.G. Calkin, Lagrangian and Hailtonian Mehanis, (World Sientifi, Singapore, 1996). 2. G.F. Torres del Castillo and G. Mendoza Torres, Rev. Mex. Fís. 49 (23) M. Montesinos, Phys. Rev. A 68 (23) G.F. Torres del Castillo, The Lagrangians of a onediensional ehanial syste, subitted to Rev. Mex. Fís. (23). 5. J.D. Jakson, Classial Eletrodynais, 2nd ed. (Wiley, New York, 1975). Rev. Mex. Fís. 5 (1) (24) 88 92