Emergence of quantization: the spin of the electron

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1 Journal of Physics: Conference Series OPEN ACCESS Emergence of quantization: the spin of the electron To cite this article: A M Cetto et al 2014 J. Phys.: Conf. Ser View the article online for upates an enhancements. Relate content - Electron system correlate by the zeropoint fiel: physical explanation for the spin-statistics connection A M Cetto an L e la Peña - Zero-Point Fluctuations an Cosmological Constant Bernhar Haisch an Alfonso Ruea - Quantization as an emergent phenomenon ue to matter-zeropoint fiel interaction A M Cetto, L e la Peña an A Valés- Hernánez Recent citations - Is Einsteinian no-signalling violate in Bell tests? Marian Kupczynski - Electron system correlate by the zeropoint fiel: physical explanation for the spin-statistics connection A M Cetto an L e la Peña - Extene Ehrenfest theorem with raiative corrections L e la Peña et al This content was ownloae from IP aress on 31/10/2018 at 14:44

2 Emergence of quantization: the spin of the electron A M Cetto, L e la Peña an A Valés-Hernánez Instituto e Física, Universia Nacional Autónoma e México, Ap. postal , México DF, Mexico ana, luis, anreavh@fisica.unam.mx Abstract. In previous papers, the quantum behavior of matter has been shown to emerge as a result of its permanent interaction with the ranom zero-point raiation fiel. Funamental results, such as the Schröinger an the Heisenberg formalism, have been erive within this framework. Further, the theory has been shown to provie the basic qe formulas for the raiative corrections, as well as an explanation for entanglement in bipartite systems. This paper aresses the problem of spin from the same perspective. The zero-point fiel is shown to prouce a helicoial motion of the electron, through the torque exerte by the electric fiel moes of a given circular polarization, which results in an intrinsic angular momentum, of value /2. Associate with it, a magnetic moment with a g-factor of 2 is obtaine. This allows us to ientify the spin of the electron as a further emergent property, generate by the action of the ranom zero-point fiel. 1. Introuction In previous work, the quantum behavior of matter has been shown to emerge as a result of its continuous interaction with the ranom zero-point raiation fiel (zpf). This has been, in essence, the program of stochastic electroynamics (se). Funamental quantum results, such as the Schröinger an the Heisenberg formalism, have been recently obtaine within this framework [1]. Contact has been mae with (nonrelativistic) quantum electroynamics, by eriving the main formulas for the raiative corrections, notably the Lamb shift an the atomic lifetimes [1], [2]. Further, the stuy of bipartite systems has shown the zpf to lie at the origin of entanglement an the symmetrization postulate [3]. Another most funamental problem in quantum theory relates to the nature of the spin of the electron: is it something inherent, as is usually consiere, or is it a result of the ynamics, as has been postulate time an again? A theory intene to explain the genesis of quantization shoul be expecte to provie an answer to this query, instea of a priori taking the spin as one more innate property of the particle, like the mass or the electric charge. During the initial perio of se [4] the electron was in fact taken as a spinless particle, the only exception being the analysis by Braffort an Taroni [5] of some effects ue to spin. As of 1981, a number of moels for spin as an acquire property have been propose [6]-[12] (there were of course earlier moels, such as that of Huang [13]). Though base on classical moels, the various se calculations exhibit the zpf as the source of a kin of (nonrelativistic) zitterbewegung that gives rise to an intrinsic angular momentum of the electron, with a mean square value of orer 2 an projections of orer. This paper re-aresses the problem of spin from the perspective offere by present se. Before entering into the subject matter, a brief introuction to recent work on se is provie, Content from this work may be use uner the terms of the Creative Commons Attribution 3.0 licence. Any further istribution of this work must maintain attribution to the author(s) an the title of the work, journal citation an DOI. Publishe uner licence by Lt 1

3 focusing on those results that will be use for the present erivations. It then will be shown that in aition to giving rise to position, momentum an energy fluctuations, the zpf inuces a helicoial motion, as a result of the torque exerte by the ranom Lorentz (electric) force on the particle. The fiel moes of a given circular polarization are shown to give rise to an (intrinsic) spin angular momentum of the electron, of value /2. Aitionally, the corresponing magnetic moment with a g-factor of 2 is erive. The paper ens with some brief aitional remarks on the physical meaning of the results obtaine. 2. Emergence of quantum mechanics 2.1. The original stochastic problem This section contains the basic steps that take us from the original ynamical equation for the particle embee in the ranom zpf, to the Schröinger formulation of quantum mechanics [1]. The system uner stuy is a particle of mass m an charge e typically an electron, subject to an external (bining) potential V in aition to the stationary zpf. The motion of the particle is escribe by mẍ = f(x) + mτ... x + ee(t), (1) where mτ... x, with τ = 2e 2 /3mc 3, represents the reaction force ue to Larmor raiation, an f(x) is the force ue to the potential V. A nonrelativistic escription is assume to be sufficient, so that the magnetic term of the Lorentz force is neglecte. The ranom fiel E(t) is taken in the long-wavelength approximation, anticipating that those fiel moes that become eventually ominant have wavelengths much larger than the characteristic imensions of the motion. The ranom fiel has zero mean value, i.e., E(t) (i) = 0, where ( ) (i) enotes the average over all realizations (i) of the fiel, an a spectral energy ensity corresponing to a mean energy ω/2 per frequency moe, i. e., ρ 0 (ω) = ω2 1 ω3 π 2 c 3 ω = 2 2π 2 c 3. (2) This expression correspons to the autocorrelation function where E i (t)e j (t ) (i) = δ ϕ(t t ), (3) ϕ(t t ) = 4π 3 0 ρ 0 (ω) cos ω(t t )ω. (4) If there is an extra component ue to an external fiel, as coul be a thermal equilibrium raiation or any other excitation of the fiel moes, the corresponing contribution must be ae to ρ 0 (ω) in equation (2) Statistical escription from phase space to configuration space The etaile motion of a single particle epens on the specific realization (i) of the fiel, which is unknown. Therefore a statistical escription is mae, by constructing the equation of evolution for the particle phase-space probability ensity Q(x, p, t), taking the zpf as given an starting from the ynamical equations obtaine from (1), mẋ = p, ṗ = f(x) + mτ... x + ee(t). (5) Through a stanar projection proceure [14], the generalize Fokker-Planck equation (gfpe) Q t + 1 p i Q + f i Q + mτ... x i Q = e 2 ˆDi Q (6) m x i p i p i p i 2

4 is obtaine, with the iffusion operator ˆD(t) efine by means of the expression ˆD i (t)q = ˆP E i Ĝ p j E j an the projection operator ˆP an the evolution operator Ĝ given by k=0 ˆP A = A (i), ĜA(x, p, t) = [ eĝ ] 2k (1 p ˆP )E l Q, (7) l t e ˆL(t t ) A(x, p, t )t, (8) where ˆL is ˆL = 1 p i + (f i + mτ... x i ). (9) m x i p i The operator e ˆL(t t ) in equation (8) makes x(t ), p(t ) (t < t) evolve towars x(t), p(t) as final conitions, following a eterministic path. Right after particle an fiel start to interact, the system is far from equilibrium. In this initial regime the main effect of the zpf on the particle is ue to the high-frequency moes, which prouce violent accelerations an ranomize the motion. Eventually, however (after a transient perio which is estimate to be of the orer of /mc s for an electron), the interplay between the electric fiel force an raiation reaction is expecte to rive the system close to equilibrium. In this new (time-reversible) regime the Markovian approximation applies, an the gfpe (6) reuces to a true Fokker-Planck equation, Q t + 1 p i Q + (f i + mτ... x i ) Q = m x i p i D pp Q + D px Q, (10) p i p j p i x j with the iffusion coefficients given to lowest orer in e 2 by D pp = e2 t t ϕ(t t ) p t j p, D px = e2 i t ϕ(t t ) x j p. (11) i A statistical escription in configuration space is mae in terms of a hierarchy of equations, obtaine by multiplying (10) successively by p k i (k = 1, 2,..., where p i stans for the ifferent components of the momentum p ) an integrating over momentum space (assuming the integran to vanish in the limits). In the time-reversible regime, the raiative terms become small an ten to balance each other in the mean, so any aitional effect they may have on the ynamics becomes negligible. When the raiationless approximation is mae (i. e., to zero orer in e 2 ), the first two equations of the hierarchy can be recast in the form of a Schröinger-like equation for a complex function ψ(x, t), 2η2 m 2 ψ + V ψ = 2iη ψ t, (12) an the corresponing complex conjugates, with ψ ψ = 3 pq(x, p, t) = ρ(x, t), an η a free parameter. The value of this parameter is etermine by the energy-balance conition that must hol in the equilibrium regime (see below). 3

5 2.3. Some important relations for average values Equation (6) (or its Markovian version, (10)) contains a wealth of statistical information about the ynamics of the system, part of which is lost in the transition to the reuce, raiationless escription in configuration space. To recover some of this information we take the (phasespace) average of a generic function G(x, p) that has no explicit time epenence. Equation (6) multiplie from the left by G an integrate over the entire phase space gives thus t G = ẋ i G x i + f i G p i + mτ... x i G p i e 2 G p i ˆDi. (13) In particular, for any G = ξ(x, p) representing a classical ( raiationless ) integral of the motion, we have... ξ ξ ξ = mτ x i e 2 ˆDi. (14) t p i p i In the Markovian limit mentione above, this equation takes the form... ξ ξ = mτ x i + t p i D pp 2 ξ p i p j + D px 2 ξ p i x j, (15) with the iffusion coefficients given by (11). For instance, for the particle Hamiltonian (efine without the raiative terms, i. e., to zero orer in e 2 ), H = 1 2m p2 + V (x), (16) equation (15) gives H = τ... x p + 1 t m trdpp. (17) This equation gives the contributions of the issipative an iffusive terms to the energy balance. Stationarity is reache when the terms cancel each other, which shoul occur when both fiel an particle are in their groun state, i.e., τ... x p 0 = 1 m trdpp 0. (18) Notice that for the calculation of these terms (to the lowest orer of approximation) one must use the solutions of the Schröingerlike equation (12), since the system is alreay in the timereversible regime. This means, in particular, that in equations (11) the following replacements must be mae: p j p 1 x j i 2iη [ˆx i, ˆp j ], p 1 i 2iη [ˆx i, ˆx j ]. (19) Explicit calculation gives that for the energy-balance conition (18) to be satisfie, the parameter η must have precisely the value η = /2 [1]. This is, then, the point of entry of Planck s constant into the Schröinger equation, obtaine from (12), 2 2m 2 ψ + V ψ = i ψ t. (20) The time-reversible an raiationless regime, in which the mechanical system is correctly escribe by this equation, is therefore calle the quantum regime. Notice that, from this perspective, provies a irect measure of the intensity of the fluctuations impresse by the zpf upon the (quantum) particle. 4

6 Let us now go back to equation (15), an apply it to the angular momentum L, which is a classical constant of the motion for any central-force problem. By taking ξ =L =x i p j x j p i (i j), we get... L = mτ x x D A, (21) t where D A is the antisymmetric tensor with components D px Dpx ji. This equation tells us that for the angular momentum to be conserve, the raiative terms must cancel each other, mτ x... x = D A. (22) Finally, for the square of the angular momentum L 2, equation (14) gives L 2... = mτ t x i L 2 p i + so that for L 2 to be conserve, we must have... L 2 mτ x i = p i D pp D pp 2 L 2 p i p j 2 L 2 p i p j + D px D px 2 L 2 p i x j These latter results, first propose in [9], will be use in section L 2 p i x j, (23). (24) 2.4.Linear response of the particle in the stationary regime A close analysis of the stationary states attaine by the particle in the time-reversible regime escribe above, has prove particularly revealing of some general properties of quantum systems. For such analysis a more straightforwar approach, complementary to the previous one, has been evelope [1], starting again from the same equation of motion (1), an assuming that any stationary solution of it (characterize by the inex α) can be written as an expansion of the form x α (t) = x αβ a αβ e iωαβt, (25) β an similarly for any ynamical variable, A α (t) = β Ã αβ a αβ e iω αβt, (26) where a αβ stans for the ranom amplitue of the zpf moe of frequency ω αβ, E α (t) = β Ẽ αβ a αβ e iω αβt. (27) In these equations the coefficients x αβ, Ã αβ are in principle functions of the amplitues a αβ, an hence stochastic variables. By introucing these expansions into equation (1), the system is foun to respon resonantly to certain fiel moes (αβ); the rest of the zpf (to be neglecte in the expansions) represents just a backgroun noise. The values of the corresponing ω αβ (calle relevant frequencies), as well as of the coefficients x αβ, epen on the specific problem. By imposing the conition of ergoicity on the stationary solutions, the { x αβ } turn out to be inepenent of the {a αβ }, which means, accoring to (25), that the response of the system to the respective fiel moes is linear in the fiel variables. Further, those moes controlling the stationary states are foun to satisfy certain properties, which can be summarize in terms of the chain rule for the ranom fiel coefficients, a αβ a β β a β β a β (n 1) β = a αβ, whence 5

7 a αβ = e iϕ αβ with ϕ αβ = φ α φ β a ranom phase. Similarly for the corresponing relevant frequencies, the relation ω αβ + ω β β ω β (n 1) β = ω αβ applies, which means that ω αβ is of the form ω αβ = Ω α Ω β. (28) As a result of these properties, the coefficients in the above expansions satisfy a matrix algebra; thus, for instance, if ˆx is the matrix with elements x αβ, we have (x n ) α = β ( x n ) αβ a αβ e iω αβt, (29) with ( x n ) αβ given by the element αβ of the corresponing matrix prouct, ( x n ) αβ = (ˆx n ) αβ. It is clear that the time-epenence of x α (t) can be transferre to every single factor x αβ, so that the evolving matrix ˆx(t) has as elements the coefficients x αβ e iω αβt. This allows to write the equation satisfie by the stationary solutions in the form m 2ˆx(t) t 2 = ˆf(t) + mτ 3ˆx(t) t 3 + eê(t). (30) In the raiationless approximation, the last two terms are neglecte an one is left with m 2ˆx(t) t 2 = ˆf(t). (31) With these results one may construct the law of evolution for the matrix Â(t) associate with the (generic) ynamical variable A; the outcome is i Â(t) t = [Â(t), ˆΩ], (32) with the matrix elements of ˆΩ given by Ω αβ = Ω α δ αβ. Notice that, as with the previous erivation presente in section 2.2, the raiationless approximation has elete any explicit trace of the zpf. No stochastic variable is containe in equations (30)-(32); instea, they are expresse in terms of operators. In section 2.3, the missing value of the paramerter η in equation (12) was obtaine by imposing the energy-balance conition; this le unequivocally to the Schröinger equation (20). In the present case, it is again the scale of the solution what is missing in equation (32). The loss is repaire by, first, fining that the canonical commutator [ˆx, ˆp], with ˆp = m ˆx(t) t = im [ ˆx(t), ˆΩ ], (33) has the universal form [ˆx, ˆp] = CI, with C constant, an, secon, fining the value of C. This is reaily achieve by applying the above equations to the harmonic oscillator of natural frequency ω 0 in its groun state, when it is in equilibrium with the zpf moe of the same frequency. One thus obtains C = i, whence [ˆx, ˆp] = i I. (34) The value of the commutator represents therefore a irect measure of the intensity of the fluctuations impresse upon the particle by the zpf, just as was conclue in section 2.3. Finally, from the above results it can be reaily seen that the relevant frequencies are given by ω αβ = Ω α Ω β = 1 (E α E β ), (35) where E α is the eigenvalue of the Hamiltonian in state α, an the final form of equation (32) is therefore i Â(t) = [Â(t), Ĥ], (36) t i. e., the Heisenberg equation for the operator Â(t). 6

8 3. Revealing the spin of the electron To isclose the rotational effect of the zpf on the particle, let us briefly go back to the original (stochastic) equation of motion (1) an rewrite it as ṗ = f + mτ... x + ee(t), (37) where p = mẋ. We shall consier that there is no external torque; the (central) force can then be written as f = g(r)x, r = x. By taking the vector prouct of this equation with x we get... L = x ṗ = mτx x + ex E, (38) t which gives the instantaneous change of the angular-momentum ue to the torque exerte by both raiation reaction an the ranom zpf. The average of this expression over the fiel realizations is t L(i) = mτ(x... x) (i) + e(x E) (i). (39) Since in the quantum regime the ynamical variables satisfy the conition of ergoicity, accoring to our iscussion in section 2.4, this equation is equivalent to (21). Equation (39), however, has the avantage of allowing us to irectly ientify the last term as the effective torque exerte by the Lorentz force of the zpf on the particle. For the angular momentum to be conserve, we must have (compare with equation (22)) mτ(x... x) (i) = e(x E) (i). (40) Let us assume that the system is in its groun state, so that there is no orbital angular momentum. Making the usual substitution ṗ f (which is vali to zero orer in τ e 2 ) an writing f as g(r)x, we get explicitly τg(r)l (i) 0 = me (x E) (i) 0. (41) Since only the fluctuating component of x can contribute to the average (x E) (i) (because E(t) is purely ranom), it is clear that all the angular momentum thus generate is ue to the ranom motion aroun the mean trajectory followe by the particle; thus, it is inepenent of the system of coorinates, an has an internal nature, usualy taken as intrinsic The electron s intrinsic angular momentum The above iscussion suggests looking for a proceure that can bring to the surface the intrinsic angular momentum acquire by the electron through its interaction with the zpf. For this purpose we recall the experimental observation that the interaction of the electron with the raiation fiel takes place via the circular polarize moes of the fiel (or moes of a certain helicity). This is known to be the case for the photonic fiel, which, from the perspective of se, is the excite state of the raiation fiel, aitional to the zero-point component. Now, it is natural to assume that the moes of the fiel in its groun state, i. e., of the zpf, interact in a similar way with the electron. To analyze the effect of such interaction, we shoul therefore consier the zpf as compose of moes of both right- an left-hane circular polarization. When the ensemble of fiel moes is consiere in its entirety, as is usually one, any such effect is conceale; yet by focusing on one of the two subensembles of a given polarization, the effective rotation inuce on the corresponing particles shoul be isclose. Let us therefore consier a situation in which the particle is in a stationary state, uner the action of the backgroun fiel of a given circular polarization with respect to an axis k. To start 7

9 with, we shall consier the particle in its groun state, so as to ensure that there is no orbital angular momentum. We therefore take L 0 = 0 (ˆx i ˆp j ˆx j ˆp i ) 0 (42) an analyze separately the contributions arising from each of the two circular polarizations, characterize by the (circularly polarize) vectors ɛ k± = 1 2 (ɛ ki ± iɛ kj ), (43) with ɛ ki, ɛ kj unit Cartesian vectors orthogonal to the axis k. Since accoring to the results reporte in section 2.4, the response of the particle to the fiel is linear, to escribe its motion uner the action of a circular fiel moe one shoul write the variable x in cylinrical coorinates, i. e. x = x + ɛ k+ + x ɛ k + x kˆk, with x ± = 1 2 (x i ix j ), x i = 1 2 ( x + + x ), x j = i 1 2 ( x + x ). (44a) (44b) Taking into account that x ± n0 = ( x 0n), equation (42) becomes explicitly L 0 = m n ( ω n0 x + 0n x n0 ) ( x 0n x+ n0 = m ω n0 x + 2 0n x 2) 0n. (45) In the groun state, L 0 = 0; hence the two sums on the right-han sie contribute with equal magnitue an opposite sign to the k-component of the total angular momentum, as shoul be the case for a nonpolarize vacuum. Taken separately, these contributions are L + 0 = m ω n0 x + 2 0n, L 0 = m ω n0 x 2 0n. (46) n n n Since, on the other han, the mean value of the commutator (34) gives the sum rule m n ω n0 x ± 0n 2 = 2, (47) the size of each separate contribution to (45) is just /2. In orer to istinguish these contributions from the (orbital) component of the angular momentum we write S ± instea of L ± 0 ; thus L 0 = S + + S, with S ± = ± 2. (48) Direct calculation of the square of the angular momentum, using (24), is more cumbersome. As a simple expeient, let us carry out this calculation for the isotropic harmonic oscillator in its groun state. The (raiationless) approximation ṗ i = f i allows us to write in this case m... x i = ω0 2p i. Further, since L 2 = x 2 p 2 (x p) 2, we have p i L 2 p i = 2L 2, 2 L 2 p i p i = 4x 2, With these results equation (24) transforms into 2 L 2 x i p i = 0. L 2 0 = mc3 D pp e 2 ω0 2 x 2 0, (49) 8

10 with D pp given by equation (11). Now in the case of the harmonic oscillator, using equations (2), (4) an (19) we obtain a constant value for the iffusion coefficient, D pp = e2 ω 3 0 c 3, (50) whence using x 2 0 = 3ˆx 2 0 = 3 /(2mω 0 ), equation (49) becomes finally L 2 0 = m ω 0 x 2 0 = (51) This result was obtaine for the first time by Marshall in 1965 [15], an taken as an aitional contribution to the orbital angular momentum ue to the zpf. In [9] the same result was obtaine, but interprete as an intrinsic (spin) angular momentum of ouble the correct value. However, in line with the present approach, we separate again the full ensemble into two subensembles corresponing to the ifferent circular polarizations, thus obtaining L 2 0 = L L 2 0, (52) with each partial contribution to the mean square angular momentum given by L = L 2 0 = (53) Using the notation introuce above (see equation (48)), which ientifies this as an intrinsic angular momentum, we write S 2 + = S 2 = (54) The fact that this result oes not epen on the oscillator s frequency ω 0, suggests that it hols in the general case, an for the free particle in particular. Therefore, we conclue that when the transformation (44) possesses physical meaning, so that the ecompositions L 0 = L L 0 an L = L L 2 make sense, equations (48) an (54) 0 tell us that there exists an angular momentum that oes not correspon to an orbital motion of the particle an can therefore be consiere as intrinsic. For an electron, which (as state above) interacts with the raiation fiel via its circular polarize moes, the transformation (44) is inee physically meaningful, an the angular momentum thus inuce can therefore be ientifie with its spin General erivation of the spin angular momentum In the preceing section we have isclose the existence of the spin angular momentum for an electron in its groun state. Let us now exten our analysis to the general case, incluing excite states with orbital angular momentum. Accoring to the above iscussion, we shoul separate the contributions to the angular momentum arising from the two circular polarizations of the fiel. Denoting with the inex n (or k) the set of quantum numbers that characterize a state of the particle, incluing the orbital angular momentum an its projection along the z-axis, we have (for simplicity in the writing we use x i = x, x j = y, an x k = z) ˆLz n = n ˆL z n = k (x nkp ykn y nk p xkn ) = = im k ω kn (x nk y kn y nk x kn ). (55) 9

11 Uner the same proceure that le to (45), equation (55) transforms into ˆLz = m ( ω kn x + 2 n k nk x 2) nk. (56) This expression can be rewritten as with O z σ n given by (σ = ±) O ˆLz n = O z + n + O z n, (57) z σ n = σm k ω kn x σ nk 2. (58) Using again the sum rule m k ω kn x nk 2 = m k ω kn y nk 2 = 1 2, one obtains from (58) = m ( ω kn x + 2 k nk + x 2) nk = O z + n O z n, (59) which combine with (57) gives O z σ n = 1 2 ˆLz + σ 1 n 2. (60) This quantity O z σ n contains, for every polarization state σ, both the corresponing part of the orbital angular momentum, an the spin associate with that state. To construct the operator associate with the vector S introuce in section 3.1, we observe that the mean value ˆL z n oes not epen on σ, whereas the term σ /2 oes not epen on n. This inicates that the operator ˆL z an the operator to be associate with σ /2 (which we shall call ˆΣ z ) belong to ifferent Hilbert spaces. In orer to express O z σ n as the average of an operator, we must therefore exten the Hilbert space to inclue the ichotomous variable σ in aition to the quantum inex n. The result is the prouct space H = H n H 2, with H 2 a biimensional vector space spanne by an orthonormal basis having as elements the vectors { σ} = { +, }. In terms of nσ = n σ, equation (60) rewrites as with ˆΣ z an operator that has σ as eigenvector, Expressing ˆΣ z in terms of the Pauli matrices gives O z σ n = 1 2 nσ ˆL z nσ nσ ˆΣ z nσ, (61) nσ ˆΣ z nσ = σ ˆΣ z σ = σ. (62) ˆΣ z = a 0 I + a z ˆσ z + a +ˆσ + + a ˆσ, (63) where a ± = (a x ia y ) / 2, an ˆσ + = 2 +, ˆσ = 2 + are laer operators. Conition (62) impose on ˆΣ z gives a 0 = 0, a z = 1. Further, since we are here consiering the variables (x +, x, z), the polarization vectors (43) fix ẑ as the preferre axis, whence a ± = 0, ˆΣ z = ˆσ z, an equation (61) becomes ( O z σ n = nσ 1 ˆL ) 2 + Ŝ ẑ nσ, (64) with Ŝ the vector operator efine as Ŝ z = 1 2 ˆΣ z, i.e., Ŝ = 1 2 ˆσ. (65) 10

12 The ientification of the operator Ŝ with the spin of the electron is thus justifie. The inepenence of ˆL z n from σ an of ˆΣ z from n, inicates that uner the present conitions, the fluctuations associate with the spin are not correlate with those that characterize the kinematics of the particle in the configuration space: L an S constitute inepenent ynamical variables. Of course the spaces of the spin an the orbital angular momentum may become connecte by the presence of magnetic fiels. It shoul be stresse that even if ˆL an Ŝ are combine uner one expression for the angular motions, e. g. equation (64), the spin is not an orbital angular momentum. Inee, there are funamental ifferences between ˆL an Ŝ. In particular, the mean value of ˆL z, say, can be freely etermine by ajusting external parameters, an may acquire a whole spectrum of values. However, only the sign of the projection Ŝz can be subject to external ajustment; its absolute value is etermine by the funamental commutator, which in its turn is fixe by the zpf. It is because of the universal value of the commutator that the spin of the electron is the same for all electrons uner all circumstances, which reinforces its intrinsic nature. The connection of the commutator [ˆx, ˆp x ] = i with the spin of the electron eserves a further comment. As pointe out in previous sections, Planck s constant is a irect measure of the size of the fluctuations, both those of the zpf an those impresse by it on the particle. Specifically, since the commutator implies that the fluctuations of x an p x have a minimum value ajuste to the rule σxσ 2 p 2 min x = 2 /4 (σx, 2 σp 2 x are variances), one may write the numerical relation ± Ŝz ± = /2 = σ x σ px min, which emphasizes the fact that the value of the electron spin is etermine by the irreucible fluctuations of the phase-space variables x, p x aroun the instantaneous position of the particle The spin gyromagnetic factor It was iscovere experimentally that the g-factor associate with the spin magnetic moment of the electron has an approximate value g S = 2, whereas for the orbital magnetic moment the g-factor is g L = 1. This characteristic value of g S is incorporate into nonrelativistic quantum theory by han, usually without further elaboration. The issue is normally resolve by resorting to the Dirac equation, which preicts g S = 2. Given that the present theory prouces the electron spin, it seems appropriate to investigate the value preicte by it for g S. For this purpose consier the electron acte on, in aition to the external force f(x), by a static uniform magnetic fiel B = Bẑ. The contribution of the orbital angular momentum L to the Hamiltonian is given by Ĥ = ˆµ B = µ z B, (66) where ˆµ = (g L µ 0 ˆL)/ is the magnetic moment ue to ˆL, µ 0 = e /(2mc) is the Bohr magneton (with e = e ), an g L = 1. Therefore the mean energy is E = µ 0 BˆL z. (67) Consier now a situation in which the spin projection along ẑ has a well-efine value, say Ŝz = + /2. This means that one shoul consier only the subensemble that correspons to σ = +. Resorting to equation (60) to write the corresponing contribution to L z as (ˆL z + )/2, the component of E of interest is E + = µ 0B ( 1 2 ˆL z + 2 ) = µ 0B ( ) ˆL z + Ŝz. (68) An analogous result hols for the subensemble with σ =, for which Ŝz = /2, E = µ ( 0B 1 2 ˆL z ) = µ ( ) 0B ˆL z + Ŝz. (69) 11

13 The corresponing Hamiltonian escribing the total magnetic interaction of the electron follows from the sum of these contributions; it is therefore Ĥ LS = µ 0B (ˆLz + 2Ŝz). (70) This contains the correct g-factor of 2 for the spin of the electron, in the raiationless approximation. It is clear from the erivation that this value is etermine by the two egrees of freeom associate with the polarization of the zpf. Notice that the result (70) gives a precise meaning to the operator appearing in equation (64). Inee, from this latter equation we can write Ô = (ˆL + 2Ŝ)/2, whence Ĥ LS = µ 0 B (ˆL + 2Ŝ ) = ˆµ B, (71) with ˆµ = 2µ 0 Ô, which irectly relates Ô with the total magnetic-moment operator of the atomic electron. 4. Final remarks The present results give strong support to the representation of the electron spin as an acquire angular momentum. They reaffirm the image, suggeste in previous se work, of spin as a helicoial motion aroun the local mean path followe by the particle, prouce by the highfrequency (circular polarize) moes of the fluctuating vacuum. If, accoring to the results presente, a charge particle acquires spin 1/2, the question arises as to whether a similar effect shouln t be expecte to appear in the case of scalar bosons. A possible answer to this question is that the spin 1/2 is acquire by elementary particles, such as the electron, whereas composite particles may acquire it or not, epening on their internal structure. This suggests that bosons are in general composite structures, with an even number of elements (if of fermion type). Acknowlegments This work was supporte by DGAPA-UNAM through project PAPIIT IN References [1] A. M. Cetto, L. e la Peña an A. Valés-Hernánez, J. Phys. JPCS 361 (2012) an references therein; L. e la Peña, A. M. Cetto an A. Valés-Hernánez, Int. J. Moern Phys. A (to be publishe). [2] A. M. Cetto an L. e la Peña, Phys. Scripta T151 (2002) ; A. M. Cetto, L. e la Peña an A. Valés-Hernánez, Rev. Mex. Fis. 59 (2013) 433; arxiv: (quant-ph), 26 Jan [3] A. Valés-Hernánez, Investigación el origen el enreamiento cuántico ese la perspectiva e la electroinámica estocástica lineal, Ph. D. thesis, UNAM (Mexico, 2010); A. Valés-Hernánez, L. e la Peña an A. M. Cetto, Foun. Physics 41 (2011) 843. [4] L. e la Peña an A. M. Cetto, The Quantum Dice (Kluwer, Dorrecht, 1996) an references therein. [5] P. Braffort an A. Taroni, C. R. Aca. Sc. Paris 264 (1967) [6] S. M. Moore an J. A. Ramírez, Lett. Nuovo Cim. 33 (1982) 87; S. M. Moore, Lett. Nuovo Cim. 40 (1984) 385. [7] G. Cavalleri, Lett. Nuovo Cim. 43 (1985) 285. [8] A. Ruea, Foun. Phys. Lett. 6 (1993) 139. [9] A. Jáuregui an L. e la Peña, Phys. Lett. A, 86 (1981) 280; L. e la Peña an A. Jáuregui, Foun. Phys. 12 (1982) 441. [10] A. Barranco, S. A. Brunini an H. M. Franca, Phys. Rev. A 39 (1989), [11] D. Hestenes, Foun. Phys. 15 (1985) 63. [12] K. Muralihar, Apeyron 18 (2011) 146. [13] K. Huang, Am. J. Phys. 20 (1952) 479. [14] U. Frisch, in Probabilistic Methos in Applie Mathematics, vol. I, A. T. Barucha-Rei, e. (Acaemic Press, NY, 1968); see also L. e la Peña an A. M. Cetto, J. Math. Phys. 18 (1977) [15] T. W. Marshall, Nuovo Cim. 38 (1965)