On the internal motion of electrons. I.

Transcription

1 Über die innere Bewegung des Elektrons. I, Zeit. Phys. (939), 5. On the internal motion of eletrons. I. By H. Hönl and A. Papapetrou With diagrams. (Reeived 3 Marh 939) Translated by D. H. Delphenih Upon attempting to bring the fat of eletron spin into harmony with a point-like struture for the eletron from the standpoint of lassial theory, one will neessarily be led to introdue negative masses, along with positive ones. The onsideration of a mass pole m and a mass dipole p presents itself as a simple model of the eletron.. There is a lose onnetion with the work of Lubanski on the foundations of approimate solutions to the field equations of gravitation (viz., retarded potentials) as a way of obtaining the equations of motion for a pole-dipole partile.. It will be shown that the solutions to the equations of motion ehibit something that orresponds to a irular motion of the partile with a onstant (miro-) veloity and vanishing momentum (for partiles that are marosopially at rest). The equations of that irular motion admit a remarkable passage to the limit (m, p ), under whih, the miro-veloity of the partile will attain the speed of light, although the energy will remain finite. That suggests that we should bring that limiting ase into agreement with the infinite (or pratially infinite) self-energy of the eletron harge. The ase of translation will be eamined. 3. The energy funtion of the pole-dipole partile ehibits a lose analogy with the Hamiltonian operator of the Dira eletron (viz., both are bilinear forms in the omponents of the maro-momentum and miro-veloity of the partile). There are equations of motion in anonial form that also apply to the ase of an eternal eletromagneti field. 4. An epression for the angular momentum that proves to be a onstant of motion will be derived on the basis of an intuitive analysis of the energy distribution. The total angular momentum appears to split into two parts that one will refer to as the internal angular momentum (i.e., spin) and the orbital angular momentum. The enter-of-mass theorem is satisfied. That will imply a far-reahing justifiation for the model of the eletron as a rotating mass point that was reently desribed by the authors, in whih a point-like mass with the speed of light desribes an orbit with a radius ħ / µ (µ is the eletron mass). One of the authors reently proposed a model of the eletron that sought to overome the well-known problems with its dimensionality ( ). Aording to that model, the eletron is imagined to be a mass point that rotates on a irle of radius ħ / µ with a veloity of light (ħ is the Plank-Dira onstant, µ is its mass, and is the speed of light) ( ). Eletron spin (of magnitude ħ / ) will beome immediately understandable in that way, and in fat in a way that orresponds preisely to the zitterbewegung of eletrons ( ) H. Hönl, Ann. Phys. (Leipzig) 33 (938), 565; quoted as lo. it. in what follows. Cf., also Naturwiss. 6 (938), 48. ( ) In lo. it., that eletron model was introdued as Model II, as opposed to another Model I, in whih the eletron was imagined to be a rotating harge distribution that ontinuously fills up spae and also has dimension ħ / µ. From now on, Model I will be rejeted as a basis for what follows in this artile.

2 Hönl and Papapetrou On the internal motion of eletrons. I. that is assoiated with the Dira equation ( ). Furthermore, a kinemati analysis of the motion of eletrons, by analogy with the laws of motion for a symmetri top (lo. it.), indiates that the eletron model will reprodue the orrespondingly orret probability for the reombination of eletron-positron pairs (for a suffiiently general speial ase). That argument enouraged the authors to pursue the Ansatz thus-obtained further towards a orresponding desription of the properties of the eletron and to analyze it more deeply. In regard to Lorentz s etended eletron, the proposed eletron model means a return to the point eletron. In ontradition to the eletron with a harge that ontinuously fills up spae, one an raise only the generally-logial objetion that one might epet that an elementary partile should have a point-like struture, sine otherwise the introdution of etended eletrons would diretly ontradit the quantum theory of wave fields. In fat, the quantum theory of eletromagneti fields implies that the quantity of harge in an arbitrarily-bounded volume must be an integer multiple (inluding zero) of an elementary harge e, whih is ompatible with only an unetended elementary partile. However, the demand that the eletron should be point-like will lead to a fundamental problem in regard to the eistene of angular momentum for the eletron, whih has been established by eperiment. Namely, if one lets the dimensions of a partile go to zero then the angular momentum will also vanish, sine the irumferential speed of the partile annot eeed the speed of light. On the other hand, the introdution of a rotating mass point seems nearly impossible to justify dynamially. Meanwhile, the internal ontraditions for a partile of vanishing etension will ome about only when the proper rotation must take plae about an ais that goes through the enter-of-mass, and that will be assumed to lie inside of the partile, sine one taitly supposes that the partile has a positive mass distribution throughout it. Thus, in order to remove the diffiulties, it will often be neessary to admit not only positive masses, but negative ones, as well. The fat that this theoretial possibility will imply an atual, pratiable path will be shown by the following onsiderations olletively. For the time being, we would like to restrit ourselves to a simple eamination that indeed delivers a rude, but all the more harateristi, piture of the epeted onsequenes of the introdution of negative masses. We onsider a struture that is omposed of a positive point-mass, along with an equally-large negative one viz., a mass (gravitational, resp.)-dipole. We define the dipole moment by the vetor: () p = m (r + r ), in whih r + and r are the position vetors of the point-masses + m and m, respetively. If suh a partile performs a translatory motion then the resulting momentum will often be zero. On the other hand, a pure translation will orrespond to a non-vanishing angular momentum L, in general. In fat (by onsidering the relativisti variation of mass), it will be: m L = {[ r, ] [, ]} + rɺ + r r ɺ, ( ) E. Shrödinger, Berl. Ber. (93), 46.

3 Hönl and Papapetrou On the internal motion of eletrons. I. 3 or when one introdues the translational veloity v = ɺr + = ɺr : () L = [ p, v ] = v, whih is an epression that will vanish in the speial ase of p v. Observe that the angular momentum is ompletely independent of the referene point. If the dipole eeutes a rotation about the midpoint of the line that onnets both point-masses then the angular momentum relative to the midpoint will vanish. In that ase, a resultant momentum: m B = ( rɺ ) + r ɺ will remain. Upon introduing the angular veloity o of the dipole about its orresponding midpoint: ɺr + = [o, r + ], r ɺ = [o, r ], one will get: (3) B = [ o, r ]. The mass-dipole will then define a sort of ounterpart to the simple mass-pole (i.e., point-mass): During fore-free motion (i.e., translation), the dipole will possess only angular momentum, while the simple pole will have only momentum without angular momentum. By ontrast, for a dipole, a rotation of the system will be required to maintain the onstany of the momentum; i.e., in general, an eternal fore will be neessary, just as it will be for the pole to onserve its angular momentum. In what follows, we shall systematially eamine the motion of a point-like partile that arises from onsidering a mass-pole and dipole, and it will be shown that suh a partile will make it possible to give a far-reahing interpretation of the properties of an eletron. In fat, it will imply that this struture possesses not only different mehanial properties from the usual mass-points (as was indiated just now in the eample of the simple dipole), but it will also enable the reation of the gravitational field that it produes to be steered. In other words: It will allow equations of motion for the poledipole partiles to be derived from the requirement that the gravitational field must satisfy the field laws. Those equations of motion an be ompared to the harateristi equations for the Dira eletron, and a lose analogy will emerge from that whih an be pursued in fine detail (if one disregards a peuliarity that relates to the sign of the magneti moment). On that basis, it will seem legitimate for us to regard the gravitational pole-dipole partile as the lassial model for the eletron, and that sense, to speak of the internal motions of eletrons (viz., miro-motions). We emphasize that this desription of the properties of eletrons by the Dira equations is not only omplete, but simple, in priniple. For that reason, there might also be a speial attration to

4 Hönl and Papapetrou On the internal motion of eletrons. I. 4 pursuing the etent to whih the properties of Dira eletrons an also be interpreted as lassial, in the sense of the orrespondene priniple.. The derivation of the equations of motion for a pole-dipole partile from the method of retarded potentials. In what follows, we shall oupy ourselves with a given partile of vanishing etension by the introdution of a mass-pole and a mass-dipole, and indeed without onsidering any eternal fores, to begin with. In that ase, we shall base our onsiderations on the well-known approimate solutions to the strong-field equations for gravitation. It will be shown that those approimate solutions will suffie ompletely, sine they imply the equations of motion for partiles from onsiderations that pertain to large distanes from those partiles. Thus, we shall first losely follow the method of retarded potentials that was employed by Lubanski ( ), and for the onveniene of the reader, we shall desribe it briefly here one more:. Let: (4) = i, = iy, 3 = iz, 4 = t be the oordinates of the spae-time ontinuum, in whih, y, z mean the usual spatial oordinates, and t means time, and let g µ be the fundamental metri tensor of the line element: (5) ds = g µ d µ d. ds > for the element of a time-like line; s will then have the meaning of proper time along that world-line. In a spae-time region that is free from gravitational fores, it an be assumed that: (6) g µ = g µ µ = δ = for µ =, for µ. For suffiiently-weak gravitational fields (i.e., suffiiently-large distanes from the matter that produes the field), one an set: (7) g µ = δ µ + γ µ, in whih γ µ should be envisioned as a tensor field on the bakground of the pseudo- Eulidian metri that has the normal values (6) for g µ. If one introdues the gravitational potential : (8) ϕ µ = γ µ δ µ then the field equations for gravitation will demand that when one neglets all higher potentials, the ϕ µ must satisfy equations of Poisson type: ( ) Lubanski, Ata Phys. Pol. 6 (937), 356; ited as L in what follows. In addition, f. also, M. Mathisson, also in ibid. 6 (937), 67. The goal of the Lubanski paper was to re-establish the equations of motion for a material system that Mathisson had posed. ρ γ ρ

5 Hönl and Papapetrou On the internal motion of eletrons. I. 5 ϕµ = κtµ, (9) =, µ µ ( ) t (T µ is the matter tensor, and κ is the relativisti gravitational onstant), when one imposes the auiliary onditions ( ): () ϕ µ µ =. Hene, the ϕ µ will have to satisfy the system of simultaneous equations: (9 ) ϕµ =, ϕ µ µ = outside the partiles, and when one eludes eternal fores. t P L l α u α A, y, z Figure. The past-direted light-one at P: L is the world-line of the partile. A is the intersetion of L with the light-one (i.e., the redution point).. Let us onsider the time-like world-line L that the motion of our partile desribes. Let X α be the oordinates of an arbitrary point A along L. L will be determined uniquely when the X α are epressed as funtions of proper time s: X α = X α (s). From (4) and (5), the omponents of the veloity: () u α = dx dsα will then satisfy the requirement that: () u α u α =. ( ) Enzykl. d. math. Wiss., Bd. V, pp. 736, et seq.

6 Hönl and Papapetrou On the internal motion of eletrons. I. 6 Furthermore, let P ( α ) be an arbitrary. The light-one that emanates from P and is direted into the past will meet the world-line L at the point A; the omponents of the vetor PA will then be (f., Fig. ): (3) l α = X α α. A funtion n = n (P) that is single-valued in all spae will now be defined with the help of l α and u α by the relation: (4) n = l α u α. We would net like to eplain the meaning of n: From (3) and (4), we will have epliitly: (3 ) l = i (X ), l = i (Y y), l 3 = i (Z z). We will then have the veloity omponents: (5) u = i u 4, u = i y u 4, u 3 = i z u 4, u 4 = with: v =, y = v y, vz z =, = v,, if v = d / dt, v y, v z are the omponents of the veloity v that is defined in the usual way. It will follow from this and (4) that: (4 ) n = u 4 [l 4 {(X ) + (Y y) y + (Z z) z }]. Sine l α lies on the light-one, we will then have: l α l α =, (l 4 ) = (X ) + (Y y) + (Z z) = r, in whih r is the retarded distane between the point P and A, in the usual sense. Sine the vetor l α is direted into the past, moreover, suh that: l 4 = r, one an, as a onsequene of (4), and by means of: (X ) + (Y y) y + (Z z) z = r r, also write: (6) n = u 4 r ( + r ). Hene, r will be the omponent (divided by ) of the veloity in the diretion of the spae-like vetor r = PA. Disregarding the fator u 4, n will then prove to be idential with the denominator in the retarded potential for an eletrial potential harge. The use

7 Hönl and Papapetrou On the internal motion of eletrons. I. 7 of n in the formulas that follow for the potential (in plae of r in the orresponding epression for the stati potential) will then lead to one the eletrodynamial effets that are analogous to the retarded effet for gravitational phenomena. 3. In order to pursue our onsiderations further, we will require a single differential rule that we would like to derive briefly here. Let P ( α + δ α ) be a point that is infinitely lose to P ( α ). P will then orrespond to a ertain point A on L that is shifted from A along L by: s s δs = δ = δ. (Beause of the single-valued assoiation of A to P, s will beome a single-valued funtion of P.) Now, from (), we will have: suh that the hange in l α will be: δx α = u α δs, (7) δ l α = δx α δ α = u α s δ δ α. The vetor that is onstruted at P with the omponents l α + δ l α will lie on the lightone that emanates from the point P. From that, we will have: (l α + δ l α ) (l α + δ l α ) =, whih will further imply that: (8) l α δ l α = l α δ l α =. Now, upon onsidering (4), the introdution of (7) into (8) will imply that: l α α s α u δ δ = s n suh that, due to the arbitrariness in δ, one will get: l δ =, (9) s = l. n From that and (7), one an further prove that: () l α α u l α = δ +. n

8 Hönl and Papapetrou On the internal motion of eletrons. I. 8 Finally, if a funtion f(s) on L is given then when we insist upon the one-to-one orrespondene between the point P and the point A, we will find from (9) that: () We will get: () n = l f α u α = l f n ɺ, + u α l α f ɺ = df ds. l α = u + ( + l uɺ α ) n from () and (), just as we did for () and (4). 4. On the basis of the eatly-derived differential rules () and (), the following fundamental theorem, an now be stated (by a long omputation that we shall omit): If f is an arbitrary funtion of s then the differential equation: (3) f = n will be satisfied everywhere eept for the line L. One an then immediately state solutions to the first system of equations (9 ) viz., α ϕ = that behave in a pole-like fashion along the world-line L. (4) ϕ α = m n α, in whih m α, and similarly ϕ α, are symmetri in α and. The pole-like harater of the solutions omes from the fat that, from (6), the ϕ α will possess singularities like / r along L. However, the tensor m α will be severely restrited by the onditions () for the ϕ α. In the ase of a pole-like solution, we would like to ontent ourselves with the following result: It an be proved ( ) that the m α must possess the form: (5a) m α = m u α u, and furthermore: (5b) mɺ =, muɺ α =. That result will finally justify envisioning the ϕ α in (4), when ompared with (9), as the gravitational potential of a simple point mass (up to a fator of κ / π). The m α in (5a) will then be the matter tensor for the point-mass, while (5b) will epress the onservation of mass and momentum (i.e., Galilean inertial motion). ( ) L, lo. it., equation (7).

9 Hönl and Papapetrou On the internal motion of eletrons. I. 9 One will now arrive diretly at an etensive lass of solutions to the potential ϕ α = that no longer have a pole-like harater by remarking that eah solution will imply another solution when one differentiates it one or more times with respet to the oordinates λ and adds suh an epression with arbitrary oeffiients. We would like to proeed with that development up to only first order, and from (4), set: (6) ϕ α = α λ, α m m + λ, n n in general, in whih m λ, α must be symmetri in α,. We will have to refer to the seond term in the right-hand side of (6) as the dipole term, by analogy with the representation (by Mawell) of multipoles in eletrostatis; its preise physial meaning will then demand a speial analysis. First of all, with the Ansatz (6), the onditions () must be satisfied. By some lengthy, but elementary, omputations, that are based upon the rules (9) through (), one will find an epression for ϕ α / that takes the form ( ): α α α α ϕ A B C (7) = + +, 3 n n n in whih: (7a) A α = m λ, α g u u 3 u l u l λ + λ λ + λ, n (7b) B α u l u l l l λ, α λ + λ λ = mɺ gλ + 3 n n + m λ, α l uɺ uλl + ulλ l uɺ lλl l uɺ uɺ λl + uɺ lλ l α gλ m u +, n n n n n n n (7) C α = l l l l l l λ, α λ λ, α λ l uɺ λ, α λ l uɺɺ l uɺ mɺɺ 3mɺ m 3 n n n n n n l l α α l uɺ + mɺ m. n n n Sine () is satisfied identially, we must have: (8) A α = B α = C α =. In order to draw onlusions from equation (8), Lubanski deomposed the tensors m α and m λ, α aording to the omponents of the veloity u α. We shall skip over the ( ) L, lo. it., equations (3) to (3).

10 Hönl and Papapetrou On the internal motion of eletrons. I. work that Lubanski did, due to the uniqueness of that deomposition, and ontent ourselves with a summary of the end results ( ), in whih, one an set: (9a) m α = *m α + q α u + q u α + m u α u, (9b) m λ, α = n λα u + n λ u α p λ u α u, with no loss of generality. The tensors *m α, n α, p α, q α that were reently introdued into this are all orthogonal to u α : (3) *m α u =, q α u α =, n α u =, p α u α = ; furthermore, n α is antisymmetri in α,, while *m α is symmetri. 5. Of the quantities that appear in (9a) and (9b), m and p λ are the ones that are losest to having an immediate physial meaning. In order to see that, it will suffie to onsider the stati ase: u = u = u 3 =, u 4 =, and to do that, one must ompare the value of ϕ 44 that follows from the general solution to (9), namely: (3) ϕ α α κ Tret = dv π ρ with the one that is omputed from (6). The omponent T 44 of the matter tensor is equal to the mass density µ in this ase; (3) will then imply that: (3) ϕ 44 κ µ = dv π, ρ in whih ρ means the distane from the volume element dv to the referene point P, and the integration etends over the volume of the partile. The position of the volume element dv has the oordinates X + ξ, Y + η, Z + ζ. Sine the oordinates ξ, η, ζ relative to the redution point A (X, Y, Z) are small in omparison to r, the following development will be possible: ρ = (/ r) λ + ξ + = (/ r) λ ξ +, λ r λ X r ξ in whih, we have set ξ = iξ, ξ = iη, ξ 3 = iζ, in analogy with (4). With that, (3) will beome: (3 ) ϕ 44 κ (/ r) λ = µ dv µ ξ dv λ π + r. ( ) L, lo. it., the equation before (35) and equation (36).

11 Hönl and Papapetrou On the internal motion of eletrons. I. On the other hand, the orthogonality onditions (3) imply that: *m 44 =, q 4 =, n 4λ = n λ4 =, p 4 =. Thus, in the stati ase, from (9a) and (9b), we will have m 44 = m and m λ,44 = p λ, while, from (6), n = r. From that and (6), we will get: (6 ) ϕ 44 = m + p r λ (/ r). λ A omparison of (6 ) with (3 ) will now imply that: κ (33) m = µ dv π, p λ κ λ = µ ξ dv π. The vetor p λ then represents the stati moment of the mass distribution (relative to the hosen redution point), up to a fator of κ / π, or, when one passes to a point-like partile, the dipole moment, resp. That implies that m has the same meaning that it did in the ase of a simple mass point. In what follows, we will show that the quantities m α and p α depend upon m µ and p λ and their derivatives (along with u and uɺ ); hene, they do not represent any independent variables of the system. Therefore, all that remains is to give meaning to n α, whih annot be dealt with in suh a simple manner as m and p λ. It is only in the ase that was treated by Lubanski (lo. it.), in whih p λ =, that there is an argument (in regard to the ϕ 4 ) that is analogous to the one above, and n α will represent the asymmetri energy-momentum tensor of the system, up to a fator. However, one must emphasize that in the interesting ase of a point-like pole-dipole partile, the dipole moment p λ must be non-zero in any ase. One must always hoose the redution point inside of the partile, sine one demands that the elements of the world-line L must represent the instantaneous translational motion of the partiles. However, the enter-of-mass of a point-like pole-dipole system will always lie outside of the partile, so the vanishing of p λ will be impossible ( ). In other words, due to the non-vanishing of p λ, a simple meaning for n α will no longer be possible. For that reason, we will simplify our problem by the assumption that the internal situation of the partile is already haraterized suffiiently by m and p λ, and therefore we will set n α =. Naturally, that assumption (just like the trunation of the development for ϕ α after the dipole terms) orresponds to just the priniple of maimum simpliity, and it will be justified by the following result: ( ) In ontrast with the present situation, Lubanski s onsiderations relate to an etended, everywherepositive, mass distribution, so its enter-of-mass will lie in the interior of the partile. One an then let the redution point A oinide with the enter-of-mass, and therefore take p λ =. However, in return for that, n α will be an atual, well-defined part of the physial arrangement of the system, sine it will haraterize its moment of momentum. We shall not go into that in detail, sine a passage to a small partile, in the sense of Lubanski and Mathisson (lo. it.) takes plae here; f., our introdutory eplanation, as well.

12 Hönl and Papapetrou On the internal motion of eletrons. I. In the derivation of the equations of motion, we will first set n α, as well as p λ, to something non-zero, and subsequently set n α =. 6. The omputations that result from substituting the deomposition (9) into (7) are elementary, but lengthy. For that reason, we would like to restrit ourselves to the statement of the results of the omputation. First of all, the first of onditions (8) viz., A α = is satisfied identially. The seond ondition viz., B α = implies that ( ): l nɺ + m + q u pɺ u p uɺ u =. n α α α α α (34) ( * ) That equation shall be satisfied l identially. We denote the first fator by Q α : (35) Q α α α α α α = nɺ + * m + q u pɺ u p uɺ, suh that (34) will go to: (34 ) Q α = Q α u = Q α l. n We now onsider the vetor (l,,, ), in partiular. From (4) and (34 ), we will then have: Q α α α Q l Q = =. l u u One similarly proves, upon onsidering the vetors (, l,, ),, that: Q α α α α 3 α 4 Q Q Q Q = = = =, 3 4 u u u u suh that: Q α = Q α u, and from (35): (36) Q α u α α α α α = nɺ + * m + q u pɺ u p uɺ. From the latter relation, upon multiplying it by u and realling () and (3), one will net get: (37) Q α α α = nɺ u pɺ u u. Multiplying by u α and onsidering (37) and (3) will give us: (38) q α α α α = pɺ pɺ u u nɺ u. ( ) Some terms are rearranged in Lubanski [lo. it., equation (4)].

13 Hönl and Papapetrou On the internal motion of eletrons. I. 3 Upon substituting (37) and (38) in (36), it will follow that: α α α α α (39a) nɺ + nɺ u u nɺ u u + * m p uɺ =. *m α will now imply the symmetry property of (39a) diretly. Upon swithing and α in (39a), it will follow that: α α α α α (39b) nɺ + nɺ u u nɺ u u + * m p uɺ =. Upon adding (39a) and (39b), one will get: (4) *m α = ( p u α p α u ɺ ) + ɺ, whereas the omplete relation in regard to Lubanski will emerge upon subtrating: (4) n α n u u α n α u u ( p α u p u α ɺ + ɺ ) ɺ + ɺ ɺ =. All that remains is to satisfy the third of the onditions (8), namely, C α =. Upon onsidering (38) and (4), we will get, after some omputation: α α α α (4) qɺ + mu ɺ + m uɺ ɺɺ p =, whih is the epansion of a orresponding relation in Lubanski. That equation an be integrated immediately: q α + m u α pɺ α = onstant, or, if we reall the meaning of q α that orresponds to (38): (43) P α m u α α α α + nɺ u pɺ u u + pɺ = onst. Equation (43) epresses the law of onservation of energy momentum, as one will reognize upon passing to the simple mass point (p α =, n α =, P α = m u α ). P α is the generalized energy-momentum vetor: (44) P = i P, P = i P y, P 3 = i P z, P 4 = E, when P, P y, P z are the usual omponents of the momentum, and E is energy. As for m and p α in (43), one understands that they should have the values in (33), without the fator κ / π.

14 Hönl and Papapetrou On the internal motion of eletrons. I Aording to our assumption in Setion 5, we finally set n α = and get the following system of dynamial equations of motion from (4) and (43): α α (45) p uɺ p uɺ =, (46) α d P ds d ds (m uα pɺ u u α pɺ α ) =, + upon whih we will base our further onsiderations.. Solving the equations of motion for the ase of irular orbits.. We shall net eamine solutions to the equations of motion for whih the momentum of the system vanishes: (47) P = P = P 3 =. We will refer to suh a system as a partile that is marosopially at rest, and we would like to show that the equations of motion in that ase will admit an internal motion of the partile that orresponds to a irular orbit of a pole-dipole singularity that advanes with onstant veloity. The partile will then be haraterized by its proper mass m and the magnitude of its dipole moment p. We would like to omit a onsideration of the general trajetory types that are onsistent with the equations of motion here. We then set v = onst. for the irular orbit; it will follow that u 4, whih we would like to denote u, will also be onstant: (48) u 4 = u =, =. The path lies in the y-plane with the oordinate origin as its midpoint; and y, as well as u and u, will then be periodi funtions of time, while z and u 3 will vanish. From (45), the four-vetor p α and the four-aeleration uɺ α will be parallel to eah other. Hene, the spae-like vetor p will have the diretion of the radius vetor (viz., it will be the same or the opposite), and its magnitude will be onstant, on the grounds of symmetry, while p 4 will vanish. We will then have: pɺ u = ( pu ɺ ) = onstant, p 4 =. We would like to denote the onstant that arises in that way by m. It is often Lorentzinvariant, and it will be shown that it is always positive; we then define: (49) m = pɺ u.

15 Hönl and Papapetrou On the internal motion of eletrons. I. 5 As for the energy E of the partile, from (44) and (46), we will now have: E (5) = P4 = (m m ) u. On the other hand, upon multiplying (46) by u α and realling (49), one will prove that: (5) P α u α = m m + p ɺ α u α = m m, in general, so in the present ase, due to (47) and (48): E (5) u = m m. We will then get the following double equation for the energy from (5) and (5): E (53) = (m m ) u = m m. u Sine we must have u >, we then infer from this that: (54) m m >. The equality sign relates to the limiting ase of u =, =, whih we shall disuss in detail later on. One further shows from (5) that sine m >, from (54), the rest mass of the partile µ = E / that appears will always be smaller than the proper mass m that belongs to the pole term: E (55) µ = < m. The introdution of the dipole terms along with the pole term will then lead to a redution of the energy, in general. Meanwhile, the radius R and veloity v of the irular orbit still remain undetermined. We would like to show that for a given partile (i.e., for a given m and p), either of those quantities an be derived from the other one. We have: (56) = R os ωt, y = R sin ωt, (56 ) v = d = Rω sin ωt, dt v y = dy = Rω os ωt, dt and thus, from (5): (56 ) Rω u Rω u u = i sin ωt, u = i os ωt.

16 Hönl and Papapetrou On the internal motion of eletrons. I. 6 Due to the fat that m > and from (54 ), p will be direted towards the interior of the orbit: (57) p = p os ωt, p y = p sin ωt. In fat, with that hoie of sign, one an show that: (57 ) pɺ = i pɺ = dp i dt dt pω u pω u = i sin ωt, pɺ = i ds os ωt, and from that and (49) and (56 ), it will follow that: (58) m = Rpω u ; thus, we have m >. If one inserts (56 ) and (57 ) into P = or P =, aording to (46) then that will immediately yield: (59) p = (m m ) R. Now, from (53), we will have: m m m = u, and it will follow from this and (59) that: (6) and if one realls (48) then: (6 ) u = mr p, + = mr p. R and will not be determined uniquely then until yet another dynamial quantity emerges (e.g., the angular momentum); f., Setion 4, as well. By the way, equation (59) epresses the fat that a irular motion about the enterof-mass of the system will result. One oasionally larifies that fat by noting that, from (5), the moving partile will have a dynamial mass of m m (instead of the proper pole mass). (59) will then desribe the vanishing of the total moment of the mass distribution relative to the enter of the irular orbit, where one will observe that p ontradits that. However, we will obtain a more preise statement of the enter-of-mass theorem later on. (Setion 4.3).. The equation of irular motion admits a remarkable passage to the limit in whih the speed of the partile approahes the speed of light. It an be inferred from (54) that u will beome infinite (i.e., = ) when m m /. If m remains finite then from the

17 Hönl and Papapetrou On the internal motion of eletrons. I. 7 seond equation in (53), the energy E will go to zero. For a finite, non-zero energy, one must then have m! Similarly, from (59), the dipole moment p will go to zero when one adheres to a finite radius R. (A finite radius R will be required e.g. for a finite angular momentum; f., Setion 4. For a finite p, from (6), R as.) The relations shed a speial light upon the problem of the self-energy of the eletrially-harged elementary partile. From the arguments that were raised in the Introdution in favor of an inetended struture for elementary partiles, it would seem plausible to attribute an elementary point harge to the harged elementary partile (and possibly higher eletri multipoles, as well). However, from the previous disussion, the self-energy of an eletri point harge will always be regarded as being infinite. Indeed, the authors showed in a paper that appeared reently ( ) that the self-energy of an eletri point harge will prove to be finite when one onsider the gravitational fore; however, the upper limit to that energy lies so high that for atomi dimensions, it an be regarded as pratially infinite (namely, as muh as a fator of higher than the rest energy of the eletron!). It now seems appropriate to reonile the fat that m will beome infinite for a partile that moves with the speed of light (and finite rest energy, just the same) with the infinite or pratially-infinite self-energy of the harge of the partile. We now see that by taking into aount a small dipole term that vanishes in the limiting ase, the priniple of the self-energy of the harge will no longer represent an insurmountable obstale, sine a finite rest energy E and a finite orbital radius for the partile are both ompatible with an infinite self-energy. We now return to the previously-skethed piture of the struture of the eletron (assuming an infinite self-energy for the harge), in whih the mass-point (mass µ = E / ) rotates around a virtual enter (viz., enter-of-mass) at a distane that is welldefined (by the angular momentum) at the speed of light ( ). The objetions that were raised in the name of mehanis and eletrodynamis that we initially deferred an now be seen to atually weaken by the introdution of the dipole moment. By the way, it is interesting to us to disuss the minor deviations from the strit limiting ase = that are obtained when one replaes m with the upper limit e k (k is the Newtonian gravitational onstant) for the self-energy of an eletri point-harge e, as omputed in the theory of gravitation ( 3 ). For the eletron harge, e = 4.77 esu, so the mass that orresponds to that limit will be m =.85 6 g (but the eletron mass is µ =.9 7 g!). With m = m /,, we will net get from (5) that: and from that: µ = m µ, u = m µ = m µ, ( ) H. Hönl and A. Papapetrou, Zeit. Phys. (939), 65. ( ) Lo. it., pp. 569; f., the Introdution that paper, as well. ( 3 ) H. Hönl and A. Papapetrou, lo. it.

18 Hönl and Papapetrou On the internal motion of eletrons. I. 8 The deviation from the veloity of light will be immeasurably small then. The magnitude of the assoiated dipole moment p will then follow from (5) and (59): p = µ R u µr µ µr. m The dipole moment p then proves to be smaller than the moment of the mass µ relative to the enter of the orbit by a fator whose order of magnitude is. 3. The ase of vanishing momentum. The ase of P will redue to the ase of P = by a Lorentz transformation. Let the latter be thought of as defining a system of partiles that are marosopially at rest, with whih we now assoiate a moving oordinate system, y, z that moves in an arbitrary diretion relative to the internal orbit with a speed of w = *. If we hoose the diretion of translation to be the (, resp.) diretion then the Lorentz transformation will read: (6) = + wt, y = y, z = z, t = t + w /. Sine the P α onstitute a four-vetor, the transformation (6) must obviously take the energy-momentum omponents in the rest system, namely: (6) P 4 E P = P y = P z =, = = µ, to: (63) P = µ w E, P = P =, = µ. The marosopi energy-momentum omponents thus behave preisely like a simple mass point of rest mass µ, regardless of the internal motion of the partile and its ompliated struture. We would now like to seek a detailed understanding of formula (63) by onsidering a speial ase, namely: a) Translation in the orbital plane. Aording to (6), when we write u, instead of u, we will have: 4 u + wu / u =, u y = u y, u z = z u + wu / u =, u 4 =,

19 Hönl and Papapetrou On the internal motion of eletrons. I. 9 p = p, p y = p y, p z = p z =, p 4 = w p /. The invariane of pɺ u then follows from this (sine ds = ds ): pɺ u = pɺ u = m, in whih m has the meaning (58) for the orbit (in the o-moving system). Furthermore, from (46), the momentum proves to be: (64) P u + wu / pɺ = (m m ) u + pɺ = (m m ) +, while, on the other hand: (64 ) P = (m m ) u +ɺ p =. It will follow from the last two relations that: ( m m ) uw (64 ) P =, whih is idential with the first equation in (63), due to (5). Similarly, the energy will prove to be: u (65) E + wu / w pɺ / = (m m ) u 4 + pɺ 4 = (m m ) +, whih likewise agrees with (63), due to (64 ) and (5). In (64) and (65), the atually-onstant magnitudes P and E appear to be deomposed into two variable magnitudes whose flutuations will anel out. We will also find similar relations for angular momentum. b) Translation perpendiular to the orbital plane. The omputations take an espeially simple form in this ase. If we hoose the diretion of translation to be the z- ais then, from (6) (with u z =, p z = p 4 = ), we will have: wu / u = u, u y = u y, u z = u, u 4 =, p = p, p y = p y, p z =, p 4 =.

20 Hönl and Papapetrou On the internal motion of eletrons. I. u 4 will then be onstant, just as u is, while p α will remain simultaneously unhanged. It will now follow diretly from (46) that: (66a) P z ( m m ) uw/ = (m m ) u z =, (66b) E ( m m ) u = (m m ) u 4 =, whih will agree with (63), due to (5) (if one swithes z and ). 3. The Hamiltonian funtion and eternal fores. The energy funtion H of our system offers espeial interest, sine that funtion an be ompared to the Hamiltonian operator of Dira wave mehanis for the eletron, whih epresses the known of the properties of the eletron ompletely.. To that end, we eliminate pɺ α from the energy funtion that is inferred from (46): H (67) = P4 = ( m pɺ u ) u4 + pɺ 4 and introdue the momentum omponents P α, in plae of them (along with the veloity omponents u α ). In order to do that, we an appeal to the relation (5), whih is valid in general: (5) Pα u α = m m, whih will go to: (5 ) Eu = m m when one speializes to the rest system (P = ). It will follow from (5) and (5 ) that: (68) P α u α = E u. We deompose the left-hand side into spatial and temporal parts, while keeping in mind the reality onditions that orrespond to (4): (68 ) P 4 u 4 (P u) = E u. From (67) and (68 ), we will now get:

21 Hönl and Papapetrou On the internal motion of eletrons. I. (69) H u 4 = (P u) + E u. Upon dividing by u 4, in whih we onsider: u = d d dt u4 = = v ds dt ds,, and inserting E = µ, what will arise an be written omprehensively and definitively as: u (7) H = v P + v y P y + v z P z + u µ. This epression for the Hamiltonian funtion has a noteworthy analogy to the Hamiltonian operator for the fore-free motion of a Dira eletron, namely: (7) H op. = (α P + α y P y + α z P z ) + α µ, ħ ħ ħ in whih P, P y, P z must be replaed with the operators,, and α, α y, i i y i z α z, α mean matries that fulfill the well-known ommutation relations. The orrespondene will beome omplete when one observes that, from Breit ( ) and Shrödinger ( ): d (7) dt = i ħ (H op. H op.) = α, ; i.e., the α, α y, α z are the omponents of the miro-veloity of the partile. In (7), as well as in (7), we then have a bilinear form in the maro-momenta and miroveloities of the partiles. One must aordingly make the u / u 4 in the mass term orrespond to the matri α, whih an be oneived as a time-dependent operator from Shrödinger, as an α, α y, α z. The eigenvalues of the α matries are ±, so the observable values of the omponents of the miro-veloity will then be ±. As we saw in Setion., nothing stands in the way of the miro-veloity of the pole-dipole partile approahing arbitrarily lose to the speed of light ( 3 ). One must also immediately onfirm that the equations of motion eist in anonial form, if one imagines that the, P, in (7) are pairs of anonially-onjugate variables. In fat, we will then have: 4 (73) d dt = H P dp = v,, dt = H =, ( ) G. Breit, Pro. Am. Aad. 4 (98), 553. ( ) E. Shrödinger, lo. it. ( 3 ) One must ertainly observe the differene that, in the pole-dipole model, the veloity omponents an take on all values between + and, while in quantum theory, the observable values for eah veloity omponents an only be ±.

22 Hönl and Papapetrou On the internal motion of eletrons. I. The first system of equations is analogous to (7), while the seond one epresses the onservation of momentum for fore-free motion. One must emphasize that here equations (73) do not represent an equivalent substitute for the equations of motion (46), sine they do not lead to the defining equations for momentum as they would in the dynamis of point-masses.. We would like to employ the anonial equations (73) for the purpose of arriving at equations for the motion of a pole-dipole partile that arries a harge e in an eletromagneti field by formally etending them ( ). In order to do that, we introdue the vetor potential A and the salar potential Φ, from whih the eletri and magneti field strengths E and H, resp., of the eternal field will be given by ( ): (74) H = rot A, E = grad Φ A. t We now introdue the momentum of the partile in an eletromagneti field by analogy with the orresponding epression for the harged mass point, if we reall (46) and (49): (75) Π = (m m ) u + e pɺ + A and the energy funtion, by epanding (7): (76) H = e u Π + v u µ A + e Φ. 4 From that, we will now regard, Π, as pairs of anonially-onjugate variables. As before, the first system of anonial equations will yield: (73 ) d dt = H Π = v, It will follow that the seond system is: (73b ) dπ H = dt = e A A y A v z Φ + v y + v z e, ( ) The introdution of the eletromagneti field goes stritly beyond our assumption in Setion that T α = (outside the partile). The equations of motion (78) derived below then demand a firmer foundation from the standpoint of pure field theory. ( ) We shall denote vetors that relate to the eternal field e.g., A, E, H by German letters and the ones that relate to the partile by boldfae print. [Trans. note: In the original, it was an overhead arrow.]

23 Hönl and Papapetrou On the internal motion of eletrons. I. 3 If one preserves the definition (46) of the variables P, P y, P z, as before, then one an show that on the one hand, from (75): dπ dp (77) = e da + dt dt dt It will then follow from (73b ) and (77) that: dp e A A A A v v v, dt y z t = + + y + z + dp dt = Φ A e A y A A A z e + vy vz t y z, or, when one realls (74): (78) dp e = e E + [v, H]. dt That is, in fat, the epeted etension of the equations of motion (46) to the ase in whih an eletromagneti field is present, in whih the Lorentz fore appears as the resulting eternal field. We will pursue the analogy between the pole-dipole partile and the Dira eletron further in a later part of this artile, along with the appliations of the equations of motion (78). 4. Angular momentum and the enter-of-mass theorem. One an, in priniple, take two paths for the introdution of angular momentum: One an proeed by stating an epression for angular momentum using formal analogies; e.g., with the orresponding epressions for the Dira eletron (viz., orbital angular momentum plus spin), suh that the onstany of angular momentum will represent an important foundation as a onsequene of the equations of motion. One an also proeed with an intuitive analysis of the mass (energy, resp.) distribution of the system, and from there, indutively seek to arrive at a general epression for angular momentum. We would like to take the latter informative path, and from the outset, we must stik to it in order to also find the (distributed) energy of the gravitational field from the energy of the original mass distribution ( ). ( ) The distintion between the energy of matter (whih orresponds to the matter tensor T µ ) and the energy of the gravitational field is based upon the fat that a onservation law of the form: µ µ µ ( T + t ) =

24 Hönl and Papapetrou On the internal motion of eletrons. I. 4 R ω O a a + M m M Figure. Shemati arrangement of the masses m, ± M along the radius of the partile orbit.. We onsider a partile that is marosopially at rest, and whih we introdue as being represented by a pole m and a dipole p, and for the sake of illustration, we imagine that the dipole is realized by two masses ±M on both sides of m at a distane of a (f., Fig. ): (79) p = Ma, a R. Sine p is direted towards the interior of the orbit M will lie outside of the irle of radius R, and + M will lie inside of it. We must omplete that simple piture of the mass distribution by the addition of a suitable field energy, in suh a way that the behavior of the struture will be in harmony with the equations of motion (45) and (46). We first ompute the energy E I of the original mass distribution on the basis of our intuitive model: (8) + E I = m u + M ( u u ), in whih are the u ± are the orresponding veloity-dependent magnitudes for the masses ± M. We will then have: (8) u = R ω /, will be true in the general theory of relativity, in whih the t µ depend upon only the omponents of the gravitational field. In general, the gravitational energy density t µ annot be loalized uniquely, sine the t µ type: do not transform like the omponents of a tensor of rank two. By omparison, integral theorems of the P µ µ µ = ( T + t ) d d d = onst. 3 will be true, in whih P µ do behave like the omponents of a four-vetor under oordinate transformations. Cf., W. Pauli, Enzykl. d. math. Wiss., Bd. V, pp. 74, et seq. One must further presume that one an also formulate angular momentum and enter-of-mass theorems for the total system, in general (i.e., independently of the oordinate system), although evidene of that has yet to ome to light up to now.

25 Hönl and Papapetrou On the internal motion of eletrons. I. 5 (ω is the orbital frequeny), and orrespondingly: (8a) u + = ( R a) ω / = u Ra ω u +, (8b) u = ( R + a) ω / = u Ra ω + u +. From that, E I will beome: (8 ) E armω I = m u u +, and upon onsidering (79) and (58) in the limit as a : (8) E I = (m m ) u. A omparison with (5) will show that the material part E I of the energy by no means subsumes the total energy; we must further add the field energy E II to it: (83) E II = (E E I ) = m u. Remarkably, E II depends upon only m ; i.e., from (49) and (54 ), the energy of the field will be produed by the angular motion of the dipole and will always be negative. We employ a orresponding onsideration for the momentum. Analogous to (8), we have the material system: (84) P I = m u + M (u + u ). Now, we have that the magnitude of u is: (85) u = v u = R ω u, and orrespondingly, if we reall (8a) and (8b) then the magnitudes of u + and u will be: (85a) u + ( R a) ω = u + = u a Ra ω u +, R (85b) u ( R + a) ω = u = u a Ra ω + + u +. R

26 Hönl and Papapetrou On the internal motion of eletrons. I. 6 From that, we will see that in the limit as a, (84) will go to: (84 ) P I = (m m ) u p R u. The epression (84 ) obviously does not ome from (8) when one multiplies (8) by the veloity u / u. The reason for that lies in the fat that the mass distribution I is not pole-like. In fat, from (3), the etra term in (84 ) whose magnitude is pu / R = pω u / will orrespond to the momentum that originates preisely in the rotation of the dipole. From (59), we an also write: (86) P I = m u. Sine the total momentum P will vanish for the rest motion, the momentum P II for the field energy that is assoiated with P I must be: (87) P II = m u. A omparison of (87) and (83) will now show that the field energy must be regarded as being pole-like at the position of the partile. We are therefore in a position to give an immediate epression for the angular momentum. The pole-like onstituents m and E II provide no ontribution to the angular momentum J relative to the (instantaneous) position of the partile. All that will remain is the ontribution from the dipole, whih, from the onsiderations that were introdued in the ontet of (), will be represented by: (88) J = [p u]. For another referene point, relative to whih the instantaneous position of the partile will be denoted by the vetor r, we will have an epression: (89) J = [p u] + [r P], instead of (88), and it will often remain valid for the general ase of a marosopiallymoving partile, as well. Therefore, (88) will then be independent of the referene point in the ase of the rest motion. From (89), the total angular momentum is seen to deompose into two piees that one an rightfully identify as the internal angular momentum (i.e., spin) and the orbital angular momentum. We would now like to state that the quantity that is defined by (89) rigorously represents a onstant of the motion. When one observes that P = onst., and from (45), that p is parallel to uɺ, differentiating (89) with respet to proper time will show that: J ɺ = [p u] + [rɺ P].

27 Hönl and Papapetrou On the internal motion of eletrons. I. 7 Now, we have from (46) that: [ rɺ P] = [u P] = [u pɺ ], from whih: (9) J ɺ =, J = onst. The onstany of the angular momentum for the fore-free motion defines another justifiation for the Ansatz (89). By omparison, neither the intrinsi not the orbital angular momentum will be onstant in the ase of non-vanishing momentum. These relations are very similar to the ones that Shrödinger disussed for the fore-free motion of the Dira eletron ( ).. There eists a peuliarity with regard to the sign of the angular momentum. Sine p is direted towards the interior of the orbit ( ), in the ase in whih we imagine that the irular orbit of the rest motion takes plae in the y-plane in the positive sense of the partile [in the sense of equations (56)], from (88), the z-omponent of the angular momentum will beome negative: J = J z = pu. We substitute p in this from (59) and replae m m with µ / u, from (5), and then get: (9) J = µ v R, in whih v is the speed of the partile. The angular momentum of the rest motion will therefore have preisely the same magnitude as when the partile mass m rotates at a distane R with a veloity of v that has the opposite sign! From the ompleity of the internal struture of the partile, suh behavior should not need to astonish anyone. By the way, we an also assert that diretly from the form of the moment of momentum relative to the enter of the irular orbit. We first find the part of the angular momentum of the material system that is assoiated with m, p by realling (84) and (79): J I = m ur + M [u + (R a) u (R + a)] = P I R p u; in addition, we will have a ontribution from the field energy, for whih we will have: J II = P II R = P I R, due to its pole-like struture, and the assertion will be proved one more. 3. It is now easy to onfirm that the enter-of-mass theorem is satisfied for the motion of the partile. We onsider the ase of rest motion and first determine the ( ) Cf., the Introdution to this paper.

28 Hönl and Papapetrou On the internal motion of eletrons. I. 8 distane from the enter of mass of the mass distribution E I to the midpoint from the form of the momentum itself. From (8) and (79), we will have: E I = m u E R + M [ u + ( R a) u I ( R + a)] = R p u. Due to (59) and (8), we an write this as: E (9a) I = m u R, suh that, from (8) and (54 ): m (93) = R R. m m The enter-of-mass of E I will approah the position of the partile with inreasing angular veloity (for onstant R) in the interior of the orbit and will oinide with it when the partile attains the speed of light. On the other hand, due to the pole-like harater of E II, the moment of E II relative to the enter of the orbit is, when one realls (83): E (9b) P II = m u R. It will follow from (9a) and (9b) that: (9) E I + R E II = ; i.e., the total moment will vanish relative to the enter of the orbit. That enter will then be, at the same time, the enter-of-mass of the partile. With that, we have resolved the parado that a fore-free partile should rotate about a enter that lies outside of itself. 4. The foregoing onsiderations subsequently provide a far-reahing justifiation for the formerly-stated eletron model of the rotating mass point (see the Introdution). The model, whih was naïve at the time, was muh simpler than the one that we treated here, sine the ompliations of the pole-dipole struture of the partile were therefore ignored. However, only when we had introdued that struture was it possible to give the dynamial foundation for the former purely-hypothetial irular motion. The dimensions of the orbit (for the eletron at rest) prove to be in agreement with the naïve model, from the present omputations, when we also introdue the basi assumption that the miro-veloity of the partile is the veloity of light, as was suggested by the Dira equation (Setion 3.), and the momentum of momentum is assumed to be quantized with the value ħ. Equation (9) will then imply that the magnitude of the angular momentum is: (94) J = µ R = ħ,