Proposed physical explanation for the electron spin and related antisymmetry
aa r X i v : . [ qu a n t - ph ] J a n Proposed physical explanation for the electronspin and related antisymmetry
A. M. Cetto, L. de la Pe˜na and A. Vald´es-Hern´andezInstituto de F´ısica, Universidad Nacional Aut´onoma de M´exicoJanuary 3, 2018
Abstract
We offer a possible physical explanation for the origin of the electronspin and the related antisymmetry of the wave function for a two-electronsystem, in the framework of nonrelativistic quantum mechanics as pro-vided by linear stochastic electrodynamics. A consideration of the sepa-rate coupling of the electron to circularly polarized modes of the randomelectromagnetic vacuum field, allows to disclose the spin angular momen-tum and the associated magnetic moment with a g -factor 2, and to estab-lish the connection with the usual operator formalism. The spin operatorturns out to be the generator of internal rotations, in the correspondingcoordinate representation. In a bipartite system, the distinction betweenexchange of particle coordinates (which include the internal rotation an-gle) and exchange of states becomes crucial. Following the analysis ofthe respective symmetry properties, the electrons are shown to couple inantiphase to the same vacuum field modes. This finding, encoded in theantisymmetry of the wave function, provides a physical rationale for thePauli principle. The extension of our results to a multipartite system isbriefly discussed.Keywords: Pauli principle; electron spin; spin-symmetry connection;stochastic electrodynamics; bipartite entanglement; zero-point radiationfield. Can one speak of the origin of the electron spin? Does it make sense to lookfor a physical agent behind the exclusion principle? These two most promi-nent features of the electron are customarily taken in nonrelativistic quantummechanics as empirical facts. Nevertheless, the concept of spin as a derivedquality rather than an innate one has given rise over the years to a diversity ofsuggestive proposals, as is illustrated by our short but multifarious selection ofrelevant work [1]-[12]. On the other hand, despite the various proofs existing inthe literature, the physical gears behind the spin-statistics connection are stillunclear ([10]-[11] and references therein). Here we address this state of affairs in1n integrated approach using the tools of stochastic electrodynamics ( sed ), thetheory developed to explain the quantum behavior of matter as a result of itsinteraction with the fluctuating radiation vacuum or zero-point field ( zpf ) [13],taken as a real field. In other words, the purpose of this paper is to reveal the physics behind the spin and its symmetry properties, in contrast to relativisticquantum mechanics, where the electron spin appears as a natural element ofthe formalism.It should be noted that the electron spin has received some, though ratherlimited attention in sed . In particular, by using a harmonic-oscillator modelfor the particle and separating the zpf into components of circular polarization,de la Pe˜na and J´auregui [5] obtained the spin angular momentum and the asso-ciated magnetic moment as acquired properties, the former within a numericalfactor of order 1. Sachidanandam [6] arrived at similar results for an electron ina uniform magnetic field. More recently, Muralidhar [7] derived the electron spinby taking the zero-point energy of the (free) electron as an energy of rotationwithin the region of space surrounding the particle. These various sed -inspiredcalculations are quite suggestive, in that they all lead to a result of order ~ for the mean square value of the spin and of order ~ for the spin projections,exhibiting the zpf as the source of a kind of (nonrelativistic) zitterbewegung.This confirms the recurrent proposal of a close relationship between spin andzitterbewegung, which started with Schr¨odinger and extends to our days, as isexemplified in the cited literature[1]-[13].Here we use the tools provided by linear stochastic electrodynamics ( lsed )to tackle the issue of the spin-statistics connection for a two-electron system,which leads us to propose a specific physical mechanism for the Pauli exclusionprinciple, not examined in previous work. We start by recalling that accordingto lsed , the spin appears as an intrinsic angular momentum of size ~ /
2, with anassociated magnetic moment with g -factor g S = 2 [9]. This paves the way for theintroduction of the corresponding spin-operator formalism and for establishingcontact with its usual (nonrelativistic) quantum treatment. We then recallthat two identical particles forming part of the same system couple to thesame mode of the vacuum field; this is the mechanism behind the entanglementof their state vector, as indicated previously ([13]-[14]). The present work goesfurther with respect to Refs. [9], [14], [15], in that here we take advantage of thefact that the lsed expressions still contain explicitly the relevant field variablesdescribing the spin orientation, and show that a proper consideration of therotation angle associated with the spin angular momentum, combined with acareful analysis of the exchange properties of the (entangled) state vector, leadsto the antisymmetry of the latter. This result reveals an antiphase coupling ofthe two electrons to the common field modes. Since no more than two particlescan couple in antiphase to the same mode, we disclose here the origin of thePauli principle. The analysis can be extended to a multi-electron system, as isbriefly discussed in the final part of the paper.For a systematic and comprehensive account of sed in its present form andin particular for a detailed explanation of the emergence of entangled states, werefer the reader to Refs. [13] and [14], respectively. Ref. [9] contains previous2ork on the emergence of spin in lsed . Additionally, some results presented inthis paper draw from previous work on the subject, specifically Ref. [15]. Let us start by recalling from the theory of electric dipole transitions in atomsthat in a single transition the absorbed or emitted radiation carries one unitof angular momentum, involving an interaction of the electron with (photonicor external) radiation field modes of circular polarization [16]. Since in sed wework normally in the electric dipole approximation, we shall assume that thecoupling of electrons to the zpf also involves circularly polarized field modes.Accordingly, we expand the zpf vector potential in terms of plane modes offrequency ω k = c | k | , wave vector k , and circular polarization γ (see e. g. Ref.[17], section 10.6), A ( r , t ) = 12 r ℏ V X k ,γ √ ω h − i ˆ ǫ γ k ( ϕ ) a ( k , γ ) e i ( k · r − ωt ) + c.c. i , (1)with V the volume of integration. The permittivity ǫ of free space is takenequal to 1, and ω = ω k . The intensity of the field is proportional to Planck’sconstant, which fixes the energy per normal mode at ℏ ω/
2, as corresponds tothe zero-point term (see below).Let ˆ e i , i = 1 , , , be a right-handed triadic of Cartesian unit vectors, with ˆ e pointing in the direction of k ; the right and left circular polarization vectorsare then given by ˆ ǫ + k ( ϕ ) = √ ( ˆ e + i ˆ e ) e iϕ , ˆ ǫ − k ( ϕ ) = i √ ( ˆ e − i ˆ e ) e − iϕ , where ϕ is the angle of rotation around the ˆ e -axis. Further, the mode amplitudes a ( k , γ ) = e iζ ( k ,γ ) vary at random from realization to realization of the field; thisis where stochasticity comes in. For a maximally incoherent field such as the free zpf , the phases pertaining to different modes ( k , γ ) are statistically independent.It is convenient to absorb the ϕ -factors appearing in the polarization vectors ˆ ǫ γ k ( ϕ ) into the mode amplitudes, so that the latter become a ( k , γ, ϕ ) = e iζ ( k ,γ ) e iγϕ ( k ,γ ) , γ = ± , (2)and the polarization vectors reduce to ˆ ǫ + k = √ ( ˆ e + i ˆ e ) , ˆ ǫ − k = i √ ( ˆ e − i ˆ e ).Equation (1) takes then the form A ( r , t ) = 12 r ℏ V X k ,γ √ ω h − i ˆ ǫ γ k a ( k , γ, ϕ ) e i ( k · r − ωt ) + c.c. i . (3)Integrating over the entire volume, with the help of Eq. (2), one readilyobtains for every mode contained in (3) a fixed (nonrandom) energy H γ k =3 ω/ ℏ ω k / , a nonrandom linear momentum P γ k = ℏ ω ˆk / c, and a fixedintrinsic angular momentum along the direction of propagation J γk of value J γ k = Z V ( E × A ) γ k d r = γ ℏ ˆk , γ = ± . (4)These values coincide with the results reported in the literature [17]. Since thereare as many modes in the k direction as in the − k direction, the total linearmomentum vanishes for every ω. Further, for the free field the contributionsof the two polarizations compensate each other for every k , so that the totalintrinsic angular momentum vanishes as well. This may explain why, in sed as well as in qed , these terms are normally omitted. Yet every individual fieldmode ( k , γ ) does have an intrinsic angular momentum of value ± ℏ / For an analysis of the effect of the vacuum field on the angular momentum of theelectron, the approach provided by lsed is particularly convenient. As shownelsewhere (Ref. [13] and references therein) the theory furnishes a description ofthe stationary states of the mechanical system once it has reached the quantumregime —i.e., when the system has acquired ergodic properties and detailedenergy balance has been attained between particle and vacuum field. Addi-tional effects of the radiation terms are then negligible and may be omitted inthe radiationless approximation. The ensuing description is, formally, entirelyequivalent to the Heisenberg quantum description; yet it still contains relevantinformation about those field modes that play a central role in sustaining thestationary states. This information also plays a key role in the analysis thatfollows.Let us consider a charged, pointlike particle subject to an external conserva-tive force, typically an atomic electron. According to lsed , a generic dynamicalvariable G ( t ) pertaining to the particle in a stationary state α has the form G α ( t ) = ˜ G αα + X β = α ˜ G αβ a αβ e iω αβ t , (5)where the index β = α represents any other stationary state; the set { αβ } depends on the specific problem. The ˜ G αβ turn out to be the matrix elements(in the energy representation) of the respective quantum operator ˆ G , so that˜ G αα is the expectation value of G in state α . The field amplitudes a αβ pertainto those modes of the field (of frequency ω αβ ) to which the particle respondsresonantly, and are given by (cf. Eq. (2)) a αβ ( ϕ ) = e iζ αβ e iγ αβ ϕ , γ αβ = ± . (6)4s manifested by the dependence of G α on the a αβ , the particle variables aredriven linearly, so to say, by such field modes (under stationarity neither ˜ G αβ nor ω αβ depend on the stochastic amplitudes a αβ ). A notable feature of thequantum regime is that when ergodicity is imposed, the phases of the a αβ be-come correlated in such a way that the chain rule a αβ ′ a β ′ β = a αβ holds (nosummation over repeated indices) [13], with β ′ any stationary state, which im-plies ζ αβ = ζ α − ζ β , ω αβ = ω α − ω β , and γ αβ = γ α − γ β . Physically this is a resultof the effect of the radiating particle on the field under stationary conditions;mathematically, this guarantees that the product of two (or more) dynamicalvariables can also be written as a linear expansion of the form (5). Further, therelation ω α = E α / ~ is shown to hold, with E α the energy associated with state α , meaning that the resonance frequencies are just the transition frequencies asgiven by Bohr’s rule [13].Equation (5) applies in particular to the components of x ( t ) and p ( t ) = m ˙ x ( t ) in state α . The average value of the angular momentum component L z = xp y − yp x becomes thus h L z i α = h α | ˆ L z | α i = im X β ω βα (˜ x αβ ˜ y βα − ˜ y αβ ˜ x βα ) , (7)with ˜ x αβ = h α | ˆ x | β i , ˜ y αβ = h α | ˆ y | β i , and | α i a vector in H , the correspondingHilbert space of states.Since the electron is driven by the (electric component of the) circularlypolarized zpf modes, we shall consider separately the two circular polarizationsof the field modes contributing to h L z i α , which are those that propagate alongthe z axis. It is therefore convenient to use cylindrical variables for the particle: x + = x · ˆ ǫ + = √ ( x + iy ) , x − = x · ˆ ǫ − = √ ( ix + y ), and x k = z . In terms ofthese, (7) rewrites as h L z i α = m X k ω βα (cid:18)(cid:12)(cid:12)(cid:12) ˜ x + αβ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˜ x − αβ (cid:12)(cid:12)(cid:12) (cid:19) . (8)To calculate the separate terms in (8) we resort to the commutator [ˆ x, ˆ p ] = i ~ , which in lsed is shown to be a further consequence of the particle-fieldinteraction in the quantum regime.[15] In cylindrical variables it takes the form ~ = m X β ω βα (cid:18)(cid:12)(cid:12)(cid:12) ˜ x + αβ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ x − αβ (cid:12)(cid:12)(cid:12) (cid:19) . (9)By combining with Eq. (8) we get h L z i α = h M z i + α + h M z i − α , (10)with h M z i + α = h L z i α + ~ , h M z i − α = h L z i α − ~ . (11)Notice that even if (and when) h L z i α is zero, the separate contributions to h M z i + α and h M z i − α do not vanish. Each one contains one-half of the orbital5ngular momentum along the z direction plus an intrinsic angular momentumcomponent ± ~ / S z . By summing over thepolarizations, the spin terms cancel each other and we are left with the orbitalterm only. So even though its contributions are concealed by the summation,the spin emerges as manifestation of the coupling of the particle to the separatepolarized field modes of the zpf .To express h M z i ± α as the average of an operator ˆ M z , we note that h L z i α does not depend on the spin state and the latter does not depend on the atomicstate. We therefore decompose H as H = H ⊗H S , with H the non-spin Hilbertspace, spanned by the orbital state vectors, denoted henceforth by | α i , and H S a bidimensional space spanned by the orthonormal vectors {| σ i} = ( | + i , |−i )representing the eigenstates of a spin operator which we call ˆ S . In terms of | α i = | α , σ i = | α i ⊗ | σ i , Eqs. (11) become h α | ˆ M z | α i = h α | ˆ L z | α i + h σ | ˆ S z | σ i , (12)with σ = ± . In terms of the Pauli matrix ˆ σ z we have ˆ S z = ~ ˆ σ z /
2, and (12)becomes h α | ˆ M · ˆ z | α i = h α σ | (cid:16) ˆ L + ˆ S (cid:17) · ˆ z | α σ i , (13)where ˆ S = ~ ˆ σ . (14)The independence of h ˆ L z i from σ and of h ˆ S z i from α , indicates that under thepresent conditions, the fluctuations associated with the spin (a non-relativisticzitterbewegung, taken as internal ) are not correlated with those that character-ize the mean instantaneous kinematics of the particle. This is a characteristicnonrelativistic independence. (The spaces of the two angular momenta may ofcourse become connected by the presence of magnetic forces.) By internal weare referring to the jiggling around the local mean position of the particle; whena translational motion is superimposed, this corresponds to a kind of helicoidalmotion.Along with energy and momentum, a given mode of the zpf is thus seen toalso transfer a minimum angular momentum to the particle, equal to its meanspin value ~ /
2, independent of the binding force. It should be stressed thatthis ineluctable angular momentum does not refer to a spinning motion of the(pointlike!) particle, but rather to a rotation around its instantaneous positionalong its (comparatively smooth) trajectory. This additional motion endorsesthe notion frequently encountered (already in Schr¨odinger, Ref. [1]) that thespin has its origin just in the zitterbewegung. At the same time it explains whythe spin cannot be associated with the instantaneous mean local coordinates ofthe particle, since it represents a motion around the latter. Further, it elucidatesthe reason for the presence of spin in Dirac’s equation, since this (relativistic) In Ref. [9], this rotational motion is shown to be generated by the torque due to theelectric component of the zpf ; see Eq. (41) and the related discussion in that paper. g S -factor associated withthe electron spin —which in the nonrelativistic case is also normally introducedby hand. For this purpose consider an atomic electron acted on by a staticuniform magnetic field B = B ˆ z in addition to the binding Coulomb force.The additional contribution to the Hamiltonian is ˆ H = − ˆ µ · B , with ˆ µ = − ( g L µ ˆ L ) / ~ , µ = | e | ~ / (2 mc ) the Bohr magneton ( e = − | e | ), and g L = 1.The corresponding mean energy E = µ B h ˆ L z i / ~ can be separated using Eqs.(10) and (11) into E ± = µ B (cid:16) h ˆ L z i + 2 h ˆ S z i ± (cid:17) / ~ . The partial Hamiltoniansdescribing the magnetic interaction of the electron with right and left modesare therefore ˆ H + LS = ˆ H − LS = µ B (cid:16) ˆ L z + 2 ˆ S z (cid:17) / ~ , and the full Hamiltonianreads ˆ H LS = ˆ H + LS + ˆ H − LS = µ ~ B (cid:16) ˆ L z + 2 ˆ S z (cid:17) , (15)with the correct g S -factor of 2 for the spin magnetic moment. It is clear fromthis derivation that the value of g S is linked with the two polarizations of the zpf .[5] By writing Eq. (15) as ˆ H LS = − ˆ µ · B , the operator ˆ M turns out to beproportional to the total magnetic moment operator ˆ µ of the atomic electron, ˆ µ = − µ ~ ( ˆ L + 2 ˆ S ) = − µ ~ ˆ M . (16)Now we reformulate the state vectors | α i = | α , σ i = | α i ⊗ | σ i consideringthe internal rotation angle ϕ introduced in section 2. As Eq. (5) indicates, G α depends on ϕ through the amplitudes (6), with γ αβ = γ α − γ β = ± , asexplained after Eq. (6). We profit from the structure of Eq. (5) to transfer thedependence on ϕ to the state vectors | α i , and define | α ( ϕ ) i = e iγ α ϕ | α , σ i , (17)meaning that the angle ϕ is now associated with the particle.To determine the set of values { γ α } , let us assume that there exist (atleast) three different possible values, say γ α , γ β , and γ δ . Then γ α − γ δ = ± γ β − γ δ = ± γ δ = 1+ γ β ∓
1, which gives for γ δ the values γ β or γ β +2 , contrary to γ δ = γ β ∓ γ α is a dichotomous parameter, like σ . One of its values can be madeto refer to polarization +1, the other to − , the case of equal values beingexcluded by γ α − γ β = ±
1. They differ then in sign, so that γ β = − γ α = γ α ± , or γ α = ± / . This leads us to identify the values of the parameter γ α with theeigenvalues ± / ~ . Notice further that thevectors (17) become then eigenfunctions of the operator − i∂ ϕ with eigenvalues ± /
2, whence in this ϕ -representation — ϕ being the internal rotation angle—the spin operator ˆ S z becomes ˆ S z = − i ~ ∂ ϕ , (18)in analogy with the orbital angular momentum operator ˆ L z = − i ~ ∂ φ . Symmetry properties of the bipartite state vec-tor
We recall that in nonrelativistic quantum mechanics the antisymmetry of thestate vector for fermions is normally postulated, or borrowed from relativisticquantum field theory. As mentioned above, although there exist several pro-posed quantum-mechanical derivations of the spin-symmetry connection, thephysical reasons for this connection remain unascertained (see Refs. [10] and[11] for a discussion).In the following we will analyse the spin-symmetry connection by applyingthe tools of lsed to a stationary state of a two-electron system. This will proveto have the advantage of resorting to a more complete description of the state ofthe bipartite system, which involves the relevant field variables and the internalrotation angles in addition to the variables that represent the (quantum) statesof the two particles according to the usual description. The symmetry propertiesof the bipartite state vector will therefore be determined by considering theexchange of the full set of variables.A major outcome of lsed is that any stationary state of a system of twoidentical particles becomes described in terms of an entangled state vector ofthe form ([13], [14]) | ψ i AB = √ (cid:16) | A i + λ AB | B i (cid:17) , (19)with the entanglement parameter λ AB given by the product of the random fieldamplitudes, the subindices referring to particles 1 and 2, λ AB = (cid:0) e iζ αα ′ (cid:1) (cid:0) e iζ α ′ α (cid:1) . (20)In (19), state | A i is given by | A i = | α ( ϕ ) i | α ′ ( ϕ ) i = e − iσϕ | α , σ i e − iσ ′ ϕ | α ′ , σ ′ i , (21)and similarly for | B i , with α, α ′ and σ, σ ′ interchanged; A ( α, α ′ ), B ( α ′ , α ),with α ′ = α , are different composite states having the same total energy E A = E α + E α ′ = E B . In expressions such as (21), the first state vector refersalways to particle 1 and the second one to particle 2. As is clear from (20),the entanglement results from the fact that identical particles couple throughcommon relevant zpf modes, which allows for the emergence of correlationsbetween them.Let us analyse the symmetry properties of the state vector | ψ i AB underdifferent exchange operations. The expression (19) lends itself to two such oper-ations: one can either exchange states A and B , or exchange particles 1 and 2.In contrast to quantum mechanics, these operations are not equivalent becausethey act distinctly either on the particles (which are identical) or on the states(which are different, by construction). In particular, the exchange of particlesinvolves the internal angular coordinate ϕ (see below), a variable foreign to theusual quantum description. 8he first operation ( A, B ) → ( B, A ) leads to | ψ i BA = √ (cid:16) | B i + λ BA | A i (cid:17) . (22)According to (20), λ BA = (cid:0) e iζ α ′ α (cid:1) (cid:0) e iζ αα ′ (cid:1) = λ ∗ AB = λ − AB , whence from (19)and (22), | ψ i BA = λ − AB | ψ i AB . (23)Since | A i and | B i represent two different states, in principle | ψ i BA may bedifferent from | ψ i AB .Now we perform an exchange of particles (1 , → (2 , | ψ i AB = | ψ i AB . (24)For this operation we need to consider explicitly the dependence of | ψ i AB onthe intrinsic (internal) angular coordinate ϕ . As discussed in Ref. [12], onemust take care that the rotations that take particle 1 to the azimuthal angleof particle 2 and vice versa, are both made in the same sense —say clockwise.When ϕ > ϕ , ϕ transforms into ϕ and ϕ transforms into 2 π + ϕ , so thatone gets | ψ i AB = √ (cid:16) e − πiσ ′ | B i + λ BA e − πiσ | A i (cid:17) = − λ BA √ (cid:16) λ AB | B i + | A i (cid:17) , (25)since e − i πσ = e − i πσ ′ = − λ AB λ BA = 1. When ϕ < ϕ , ϕ transformsinto ϕ , ϕ transforms into 2 π + ϕ , and the exchange applied to | ψ i givesagain Eq. (25). Therefore we have in both cases | ψ i AB = − λ BA | ψ i AB . (26)By comparing this result with Eq. (24) we obtain λ AB = − , (27)which introduced into (19) or (23) leads to the well-known antisymmetric formof the state vector | ψ i AB = √ (cid:16) | A i − | B i (cid:17) = − | ψ i BA , (28)indicating that the permutation of fermion states A ↔ B produces an overallchange of sign, just as in quantum theory.The above outcome has a further important physical implication: from (20)we note that λ AB = − (cid:0) e iζ αα ′ (cid:1) = e πi (cid:0) e iζ αα ′ (cid:1) , indicating that thecoupling of particles 1 and 2 to the mode of frequency ω αα ′ occurs out of phase ,with a phase difference of π . For particles with symmetric wave functions (i. e.9ith λ AB = 1), by contrast, the coupling occurs in phase , as seen from (20). While an arbitrary number of (identical) particles can couple in phase to a singlemode, no more than two particles can couple with a phase difference of π to thesame mode. This throws a new light on the Pauli exclusion principle. The above analysis can be extended to a multielectron system, subject again to acommon zpf , thanks to the fact that the chain rule discussed in Sect. 3 remainsin force for an arbitrary product of spin phases. To determine the resulting stateof the system one must consider the various possible configurations
A, B, C, ... of stationary states corresponding to the same total energy E A = E B = E C = ... Yet direct application of this procedure becomes rather cumbersome, as the de-generacy increases rapidly with the number of particles. A convenient approachis to consider first any pair of electrons of the system, say those in states α, α ′ .By taking successively every possible pair, all relevant frequencies will be ac-counted for, and all the respective symmetries will thus be included. Since as aresult no pair of electrons can be in the same (single-particle) state, the state ofthe entire system will be described by a totally antisymmetric, multiply entan-gled state vector built of different bipartite single-particle states that carry thefactor ( − pσ = ( − p in front of each term, where p stands for the number oftranspositions in the permutation needed to reach the corresponding exchangedstate, starting from the initial state.The calculations presented here confirm the physical picture of the spin ofthe electron as a helicoidal motion (a zitterbewegung) around the local meantrajectory, adding an effective structure to the originally pointlike particle. Theoperator − i ~ ∂ ϕ turns out to be the generator of internal rotations, accordingto Eq. (18). Further, for a bi- or multielectron system, consideration of thepermanent presence of the background field resonantly connecting the particlesserves to unveil the persistent mystery of the physics behind the spin-statisticsassociation. The mediation of the zpf turns out to have a definite role in fixingthe statistics. Nevertheless, the above arguments appear so far to be insufficientto deal with the universe of bosonic particles, which in general are hadrons andsubject also to nuclear interactions; other mechanisms most likely intervenein the definition of the total exchange symmetry properties of such compoundsystems. Acknowledgment s.
The authors acknowledge helpful comments and sugges-tions from an anonymous referee. This work was supported in part by DGAPA-UNAM, through project IN104816. Note that also in this latter case the assumption that the two-particle state vector doesnot change sign upon particle exchange, holds true. eferences [1] E. Schr¨odinger 1930. ¨Uber die kr¨aftefreie Bewegung in der relativistischenQuantenmechanik, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math.
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