Found Phys (2014) 44:266–295DOI 10.1007/s10701-014-9784-2

Complex Vector Formalism of Harmonic Oscillatorin Geometric Algebra: Particle Mass, Spin andDynamics in Complex Vector Space

K. Muralidhar

Received: 21 December 2012 / Accepted: 21 February 2014 / Published online: 11 March 2014© Springer Science+Business Media New York 2014

Abstract Elementary particles are considered as local oscillators under the influenceof zeropoint fields. Such oscillatory behavior of the particles leads to the deviations intheir path of motion. The oscillations of the particle in general may be considered ascomplex rotations in complex vector space. The local particle harmonic oscillator isanalyzed in the complex vector formalism considering the algebra of complex vectors.The particle spin is viewed as zeropoint angular momentum represented by a bivector. Ithas been shown that the particle spin plays an important role in the kinematical intrinsicor local motion of the particle. From the complex vector formalism of harmonicoscillator, for the first time, a relation between mass m and bivector spin S has beenderived in the form σ 3mc2J± = λ�s · SJ±. Where, �s is the angular velocitybivector of complex rotations, c is the velocity of light. The unit vector σ 3 acts asan operator on the idempotents J+ and J− to give the eigen values λ = ±1. Theconstant λ represents two fold nature of the equation corresponding to particle andantiparticle states. Further the above relation shows that the mass of the particle maybe interpreted as a local spatial complex rotation in the rest frame. This gives an insightinto the nature of fundamental particles. When a particle is observed from an arbitraryframe of reference, it has been shown that the spatial complex rotation dictates therelativistic particle motion. The mathematical structure of complex vectors in spaceand spacetime is developed.

Keywords Zeropoint energy · Spin · Particle mass · Geometric algebra

K. Muralidhar (B)Physics Department, National Defence Academy, Khadakwasla,Pune 411023, Maharastra, Indiae-mail: kundetimuralidhar@gmail.com

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1 Introduction

The wave mechanical theory of an oscillator in quantum mechanics provides thebasis for understanding wide variety of systems. The ground state energy of quantumoscillator has been found to be finite and corresponds to the zeropoint energy of theoscillator. The origin of zeropoint energy is presumed to be due to quantum mechanicaleffect and considered to be uniformly present throughout space in the form of randomlyfluctuating electromagnetic zeropoint fields [1]. The oscillatory nature of the particleis mainly attributed to the random fluctuations defined by the zeropoint fields. ThePlanck’s idea of zeropoint energy was studied by Marshal [2] in terms of classicalstochastic electrodynamics and found the relation between classical and quantumoscillators. Boyer showed that for a harmonic oscillator, the fluctuations producedby zeropoint fields on the particle are exactly in agreement with quantum theory[3,4]. The zeropoint energy associated with a particle can be obtained in the classicalstochastic electrodynamics and this energy may be assumed to be confined within awave packet of the particle. This leads to the explanation of uncertainty principle andmany other important quantum mechanical phenomena [5]. In particular Haisch et al.[6,7] argued the possibility of particle mass due to zeropoint energy by consideringaccelerated motion of the particle in zeropoint field. Considering the charged particleas a Brownian particle and expressing Fokker–Planck terms of action angle variablesand with an approximation to a stationary solution at a constant frequency of theoscillator, the average component of adiabatic invariant was obtained as spin angularmomentum [8]. In the classical theory of electron magnetic moment, the momentumand velocity are considered to deviate from their linearity and this condition yieldsmagnetic moment and spin equation [9]. One can express the momentum as a sum ofparallel and normal components to the velocity vector of the electron and the scalarproduct of normal component of momentum and velocity gives the spin energy andhence the oscillatory motion of electron. In quantum mechanics, the zeropoint energyof a harmonic oscillator endows the particle confinement to a finite region of spaceand leads to uncertainty principle and the particle possesses a minimum value ofmomentum which is normally treated as a deviation from the translational momentumof electron. This minimum value of momentum gives rise to the kinetic energy aszeropoint energy of the particle. This leads to an internal circulatory motion of chargedparticle and further the internal angular momentum of this circulatory motion is thespin of the particle [10].

The origin of spin was mysterious in the early years of its inception in quantummechanics and it was hypothesized with some intuitive model that electron rotatesabout its own symmetry axis and ascribed to account for the doublet fine structure[11,12]. Over the years the half integral spin has been defined as an inherent and fixedproperty of electron and the existence of spin can be derived from the fundamentalpostulates of quantum mechanics and the property of symmetry transformations [13].The spin angular momentum of electron may be considered as a circulatory current ofthe charged matter and it is a kinematic property of massive elementary particles andcorresponds to a rotation group symmetry SU(2). Understanding spin posed severalquestions in the early years of its invention and to certain extent even today. Belinfantetreated spin as a circulating flow of energy, or momentum density in the electron

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wave field [14,15]. The Dirac theory of electron automatically includes the effect ofspin which leads to the conclusion that the spin is a quantum relativistic effect [16].The Dirac electron executes rapid oscillations superimposed on its normal averagetranslational motion. This oscillatory motion is known as zitterbewegung as was firstshown by Schrödinger that the electron appears vibrating rapidly with a very highfrequency equal to 2mc2h−1 with internal velocity equal to the velocity of light.These oscillations are confined to a region of the order of Compton wavelength. Inconventional Dirac theory the electron position coordinate is given by [17]

x(t) =[x(0)+ (c2 pH−1t)

]+ i

hc

2

(α − cpH−1

)H−1exp (−2i Ht/h) (1)

The first term on right of the above equation represents normal translational motion ofthe electron and the second term, the imaginary part which is of the order of Comptonwavelength, represents the zitterbewegung or oscillatory motion of the particle. Thedimensional coordinate in Eq. (1) can be considered as complex or equivalently non-hermitian character of corresponding operator. Then the position coordinate can beexpressed in a complex form as X (t) = x(t) + iξ(t) . Where, x(t) represents theaverage position of the electron and ξ(t) to the oscillating part. Similarly from theDirac theory the operator cα acts like electron momentum and can be expressed in theform P(t) = p(t)+ iπ(t). Here, p(t) represents the average motion and iπ representsthe oscillatory part. As a consequence, one can discern the internal structure of anelementary particle like electron or quark, arises from the oscillatory nature of theparticle. These oscillations may be considered as complex rotations and the internalmotion is then defined by spin angular momentum.

The idea that the electron spin generated from the circular motion of zitterbewegungwas advocated by several researchers. Some important approaches of zitterbewegungand internal structure of the particle are as follows. In the Huang’s [18] approach ofzitterbvewegung, the electron executes an internal circular motion about the directionof electron spin with radius equal to an order of Compton wavelength of electron. Theintrinsic spin of an electron is attributed to the angular momentum of circular motionwhich gives the intrinsic magnetic moment. Barut and Bracken [19] considered therelative momentum instead of microscopic momentum of Schrödinger, the oscillatorycoordinate was referred in the rest frame of the electron and the zitterbewegung wasdescribed in terms of momentum P , position Q and the Hamiltonian H = mc2β as afinite three dimensional isotropic quantum mechanical oscillator within the compactphase space. It has been shown that the Lie algebra generated by position and momen-tum satisfy the commutation relation

[Qi , Pj

] = i hδi jβ. The system required higherdimensional group structure SO(5) and harmonic oscillator dynamics. Holten [20] dis-cussed the classical and quantum electrodynamics of spinning particles. The distinctconsequence of the theory was that the spinning particle emerges as a modification ofrelativistic time dilation formula by a spin dependent term and in the investigation ofDirac electron, he analyzed the zitterbewegung as a circular motion about the direc-tion of spin. In the Hestenes model of Dirac electron [21–23], the spin was consideredas a dynamical property of electron motion and described as circulation of electronmass and charge. The zitterbewegung is a local circulatory motion of electron spin

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and leads to the magnetic moment. In the analysis of harmonic oscillator wave packetthe angular momentum has been decomposed into angular momentum of centroid andintrinsic angular momentum. The electron spin is considered as the zeropoint angularmomentum. The correlation of variance p in momentum with variance x in posi-tion is a consequence of the correlation between position and momentum determinedby the ground state angular momentum. In a recent exposition of Dirac electron, Sid-harth [24] pointed out that the motion of electron consists of usual Newtonian motionover which a very high frequency zitterbewegung motion is superimposed. The highfrequency motion represents the circulatory motion with a radius of Compton wave-length and internal velocity equal to velocity of light. The circulation is given by theintegral

∫pds and the angular momentum of the circulatory motion then becomes

equal to the half integral spin of electron.The geometrical origin of spin was suggested by Newman [25] by applying com-

plex translation to the coordinate in Minkowski space. The spin angular momentumhas been identified as an imaginary part of complex displacement. Barducci et al.[26] investigated path integrals for fermions in Grassmann algebra and consideredthose paths as spin. In the approach of geometric algebra using multivector valuedLagrangian, Barut and Zanghi [27] arrived at the bivector form of classical internalspin of Dirac electron. In the extensions of semi-classical theories, the spin was iden-tified with a bivector and the point particle executes circular motion by absorbingenergy from zeropoint field [28,29].

A quite natural consequence of spin considerations above univocally suggests theelectron structure connection with spin angular momentum. A comprehensive descrip-tion of symmetric rigid body models are extensively discussed by Rohrlich [30].Recently the spinning rigid body model of electron was considered by Kiessling [31]and proved the conservation laws of energy and momentum. A ring like extendedelectron structure was visualized as early as in 1919 by Compton [32]. The classicalmodels of electron as point particle with spin was first formulated by Frenkel [33] andThomas [34], the spin was assumed as a proper mechanical moment due to hiddenfuzzy mass and charge. This model was further studied by Mathisson [35] and elabo-rated by Wyssenhoff and Raabe [36]. These treatments of spinning electron structurehave reemerged several times over the years [37]. In the Wyssenhoff’s improved modelthe internal motion is along the circle perpendicular to the spin with uniform angularvelocity given by ω = mc2(γ s)−1 where, m is the electron mass, γ Lorentz factorand s spin of the electron. The electron structure cannot be a point or charged spherebut mostly considered as an extended structure with an internal angular momentumsuch that the position of point like charge is distinct from a centre of mass point andthis particular feature that motion of charge does not coincide with centre of masswas found to be an actual characteristic of zitterbewegung motion [38]. In the theorydeveloped by Bhabha and Corben [39], the elementary particle with charge and dipoledistribution with an anomalous coupling to electromagnetic field was considered. Theelementary particle structure has been given as the rotation of the particle around thecentre of mass position such that the classical equations developed explain the zit-terbewegung of electron and reveal the physical origin of magnetic moment. In thekinematical treatment of spinning particles, Rivas [40] defines a classical elementaryparticle as a Lagrangian system whose kinematical space is a homogeneous space of

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the kinematical group. In this theory an important feature is the separation between thecentre of mass and centre of charge. The spin of the particle is expressed in terms ofkinematical variables and it is related to the rotation and internal motion of the chargearound the centre of mass of the particle.

In the theories of Madelung fluid [41], considering the existence of zitterbewegungand within the stochastic mechanics framework, the spinning particle appears as anextended object while the quantum potential is tentatively related to internal motion[42]. To describe the internal motion, the coordinate is split into two parts; coordinateg describing the centre of mass motion with chosen reference frame and an internalcoordinate X with reference to centre of mass rest frame, x = g + X . Differentiatingwith respect to time gives the velocity x = g + X or v = w + V [43]. From thedynamical point of view the conserved electric current is associated with the helicaltrajectories of electron with x and v. The quantum potential term in the Hamiltonianis simply related to the non-classical energy term to zitterbewegung and spin. In theabsence of spin the quantum potential term vanishes and the motion becomes classicalor Newtonian. The quantum potential enters into the Schrödinger equation owing tothe presence of spin [44]. It has been shown that in the Lorentz factor one must considervelocity asw instead of v as normally chosen by several authors. In this sense the propertime is connected with the centre of mass and not with the centre of charge becausethe orbital motion of charge is expected to be light-like. Since the charge moves withvelocity of light in the rest frame of centre of mass the proper time does not exists.The theory gives direct evidence that the quantum behavior of micro-systems appearsto be due to the existence of spin and it seems appropriate to write h = 2 |s| instead of|s| = h

2 . Then the celebrated de Broglie equation can be expressed as E = 2 |s|ω. Allthis carries further evidence that quantum mechanics of micro-systems may be a directconsequence of spin [45,46]. The world line of electron is a helical path on a cylinderand the electromagnetic field of such motion has been shown to be a Coulomb-likeoscillating field with frequency equal to the zitterbewegung frequency and the averageof such field over zitterbewegung time period turns out to be the normal Coulomb fieldof electron [47]. As already pointed out the proper time is only connected with centreof mass but not with the internal rotation, in the local rest frame the relativity is notpresent at all. The internal circular motion around the centre of mass when observedfrom an arbitrary frame appears as helical motion of the particle. The relativistic effectsobserved are simply a consequence of this helical motion of the particle [48,49]. Thetime period of rotation as observed from an arbitrary frame appears dilated by a Lorentzfactor when compared with the time period of rotation in the rest frame of centre ofmass and interestingly the derivation of which was obtained from Galilean kinematicsby using Pythagoras theorem. In the classical stochastic electrodynamics approachequating power absorbed by circular motion and power radiated, it has been foundthat the radius of rotation is equal to the Compton wavelength.

The matter as a local disturbance of space and particle motion through space con-tinuum in the form of a wave; the proposal goes back to Clifford in his space theoryof matter [50]. In accordance with classical charged particle theories, if the chargedparticle is assumed to be completely electromagnetic then its rest mass is purely elec-tromagnetic in its origin [30]. In the celebrated theory of unification of weak andelectromagnetic interactions by Weinberg and Salam, initially the particles are con-

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sidered as mass less and their apparent mass arises from self interaction [51]. However,the concept of mass less particles is based on the mass energy conservation and theparticle mass is energy [52]. Thus it is often identified that the particle mass can beattributed entirely to its internal energy associated with the internal angular momentumthat is spin [53]. The connection between spin rotation and mass in the Wigner’s groupapproach is that in the rest frame of the particle the first Casimir constant representsmass and the corresponding group is the little group containing all rotations in spaceand hence the rotation group symmetry SU(2) [54]. It has been shown by Batty-Prattand Racey [55] in their theory of geometric model of particles, that a moving vortexspinning in the spherical mode satisfies the Dirac equation of electron and hence themass and energy can be explained as being manifestations of rotation of space. Withthis they conclude that mass and energy are cyclical disturbances of the continuumand their measures are proportional to angular frequency of rotation.

The ideas expressed by several researchers about the structure of an elementaryparticle may be consolidated in the following manner. An elementary particle is notdefinitely a point particle with charge and mass or a spherical rigid body with chargedistribution. In accordance with Dirac theory, the sub-structure of electron is a pointcharge rotating in a circular motion with spin angular momentum. The frequency ofrotation is equal to the zitterbewegung frequency and the radius of rotation is equalto half the Compton wavelength. This circular motion is responsible for the observedmagnetic moment of electron and an instantaneous electric dipole oscillating withzetterbewegung frequency. The mass of the particle is related to the frequency ofzitterbewegung. The circular motion is observed from the rest frame positioned at thecentre of rotation which is also the centre of mass. Thus the centre of mass position andcentre of charge position are separated by the radius of rotation. The internal motion ofthe particle is related to the so called quantum potential and in turn responsible for thequantum behavior of the particles in general. In the stochastic approach, a particle inthe presence of zeropoint fields oscillates from its mean position and such oscillationsmay be considered as complex rotations. In the geometric algebraic approach the spinof a particle is the orientation of rotation plane which is represented by a bivector. Theproper time is related to the motion of centre of mass as observed from an arbitraryframe and the particle with internal structure moves in a helical path. Such motion leadsto the basic understanding of relativistic effects and relativistic quantum mechanics.Finally the mass and spin of a particle may be simply a local rotation of space.

The purpose of this work is twofold; first to investigate the role of spin in theinternal structure of a particle and its motion in complex vector formalism and secondto develop the mathematical structure of complex vector space. A resonant couplingof charged particle to the zeropoint field makes the particle to oscillate and suchoscillations may be considered as complex rotations. The imaginary part of suchcomplex rotations gives the classical origin of particle spin [56] and the bivector natureof spin angular momentum is then identified with the zeropoint angular momentum[57]. In the present article, considering an elementary particle in free space as anisotropic oscillator, the energy of the harmonic oscillator is derived in the framework ofcomplex vectors. From the analysis of harmonic oscillator, the mass of the elementaryparticle in terms of spin bivector is derived. Throughout this article the centre of massis taken as the centre of oscillations of the particle in the zeropoint field and in this

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sense the particle charge centre might well be differed from the centre of mass leadingto finite size of the particle. The relativistic particle motion is analyzed in the complexvector formalism. The paper is organized as follows. In Sects. 2 and 3 we presentthe derivation for the energy of harmonic oscillator and particle mass relation to thespin bivector respectively. The mathematical structure of complex vectors in spaceand spacetime is discussed in Sect. 4. Conclusions are given in Sect. 5. Throughoutthis article the word particle is used in the sense of a particle like electron.

2 Complex Vector Formalism of Harmonic Oscillator

Consider the mean path of a particle is defined by x = x(t) and the velocity of theparticle v = x(t). In the presence of zeropoint field, the induced random fluctuations ofzeropoint field make the particle to oscillate from its mean position. Here we considerthe resonance coupling of the fluctuating random zeropoint field with the particle suchthat the particle oscillates at a single characteristic frequency. Then the particle ingeneral can be considered as an isotropic and homogeneous harmonic oscillator andthus the path of the particle is blurred. Let the normal deviation in the position of theparticle be denoted by ξ and the corresponding canonical deviation in the momentumby π . Then the Hamiltonian of the harmonic oscillator in the rest frame of the particlecan be expressed as

H = mω20ξ

2

2+ π2

2m= ω0

2

(mω0ξ

2 + π2

mω0

). (2)

where, ω0 is the characteristic angular frequency of oscillations around the centre ofmass point of the particle and m is the mass of the particle. Let us consider a parameterz defined by

z2 = H

hω0= 1

2h

(mω0ξ

2 + π2

mω0

)= k2

(a2 + b2

). (3)

where, k2 = 12h , a2 = mω0ξ

2 and b2 = π2

mω0.

In the quantum mechanical treatment of harmonic oscillator, it is a common procedureto factorize the sum of squares into a product of two complex numbers. Now, the para-meter z can be written as a complex number z = k(a+ib). The product zz∗ = |z|2 , z∗being the complex conjugate, the complex numbers correspond to even multivectors inthe geometric algebra of Euclidean space and in general the unit imaginary is replacedby a bivector iσ 3 = σ 1σ 2.

z = k(a + iσ 3b) (4)

Where, σ 3 is the unit vector along 3-axis, i = σ 1σ 2σ 3 is the pseudoscalar and the set{σk; k = 1, 2, 3} represents a right handed orthogonal unit vectors in Euclidean space.When a and b are considered as pure scalars, the even multivector z can be written ina parametric from of a rotor representing rotation in iσ 3 plane.

z = k(cos θ + iσ 3sin θ) = k exp(iσ 3θ) (5)

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Found Phys (2014) 44:266–295 273

Where, the scalars a and b are expressed as a = cos θ and b = sin θ . The aboveequation shows that the oscillations of the particle are equivalent to local complexrotations. Since the deviations in position and momentum of an oscillating particle arebasically vectors one must treat the even multivector z as a complex vector rather thana simple complex number. This can be done by utilizing the property of pseudoscalari2 = −1 and expressing a and b as vectors.

a = (mω0)1/2ξ and b = (mω0)

−1/2π

Then a complex vector can be defined as a sum of vector a and bivector i b.

Z = k(a + i b) (6)

A reversion operation changes the order of vectors and denoted by over bar. Thereversion operation on the complex vector Z gives

Z = k(a − i b). (7)

As the pseudoscalar i commutes with all vectors in three dimensional space, thegeometric product Z Z can be expressed as

Z Z = k2(a2 + b2)+ 2k2 i(a ∧ b). (8)

The particle spin is the zeropoint angular momentum associated with the zeropointenergy. The bivector product, a ∧ b = ξ ∧π is the internal zeropoint angular momen-tum of the particle and represents the spin bivector S = ξ ∧ π [57]. In a classicalapproach the magnitude of spin can be obtained using stochastic electrodynamics. Instochastic electrodynamics approach by Boyer [8], considering a stationary solutionat constant frequency of the charged particle oscillator in random zeropoint field, anaverage component of adiabatic action invariant 〈J1〉 was found to be equal to h

2 whichis in accordance with the adiabatic hypothesis of quantum theory. Further, consider-ing the extended structure of the particle which is discussed in the introduction, themagnitude of spin angular momentum of the particle may be assumed as h

2 . In viewof these considerations, the magnitude of spin angular momentum of the particle canbe assumed as h

2 and this assumption transforms a classical oscillator into a quantumoscillator. The spin bivector is the dual of spin vector s, S = i s. Then Eq. (8) becomes

ZZ = H

hω0+ 1

hiS = H

hω0− 1

hs. (9)

The product ZZ is seen to be a multivector containing scalar and vector parts. Similarlyit is easy to show

ZZ = H

hω0− 1

hiS = H

hω0+ 1

hs. (10)

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From Eqs. (9) and (10), the inner and outer products of the complex vectors Z and Zcan be obtained as

Z.Z = 1

2(ZZ + ZZ) = H

hω0(11)

Z ∧ Z = 1

2(ZZ − ZZ) = − 1

hs. (12)

The magnitudes of the vectors a and b are proportional to h1/2 and both the productska and kb are dimensionless. Hence the product Z .Z is a scalar and it does not dependon the frequency ω0. Then in the Eq. (11), in terms of complex vectors Z and Z theHamiltonian H is a linear function of hω0. If the particle has constant spin, the 3-axiscan be chosen along the spin direction. By choosing the spin vector along σ 3 directionthe particle spin vector can be expressed as s = h

2 σ 3. Now replacing spin vector inEq. (12) gives

ZZ − ZZ = σ 3. (13)

Now we consider the correspondence between the set of orthogonal basis vectors {σ k}to the Pauli spin matrices. The basis vectors σ k satisfy the commutation relationssimilar to those of Pauli spin matrices and hence both sets are said to be isomorphic.With this correspondence the spin vector can be expressed as

s = h

2σ 3 → h

2

[1 00 −1

]. (14)

The meaning of the above equations becomes explicit when we introduce the basisspinor space for spin-up and spin-down states of the particle.

J+ =(

10

)and J− =

(01

)(15)

The corresponding quantities in geometric algebra are the idempotents defined as

J+ = 1

2(1 + σ 3) and J− = 1

2(1 − σ 3). (16)

These idempotents when multiplied with the vector σ 3 give

σ 3J+ = +J+ and σ 3J− = −J−.

These equations can be expressed in an elegant form by introducing a constant λ.

σ 3J± = λJ± (17)

Thus the vector σ 3 acts like an operator to give the eigen values λ = ±1 whichrepresent spin-up and spin-down states. Now multiplying the Eq. (13) from right byJ+, gives

(ZZ − ZZ)J+ = +1J+. (18)

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Substituting s = h2 σ 3 in Eqs. (9) and (10) and multiplying by J+ from right we find

ZZJ+ =(

H

hω0− 1

2

)J+ (19)

ZZJ+ =(

H

hω0+ 1

2

)J+. (20)

When the particle oscillations and the fluctuations of the zeropoint field are inequilibrium, the oscillations are at resonant simple harmonic oscillations. Since allthese oscillations are at random because of the random fluctuations of the zeropointfield, we consider stochastic average of all such oscillations and such average motionmay be considered as a circular motion corresponding to the average ground stateenergy of the harmonic oscillator. In the present approach we are considering theparticle structure with centre of charge and centre of mass points separated by theradius of rotation. One must consider this radius of rotation emerges from the averageof number of random oscillations of the particle oscillator in the random fluctuatingzeropoint field. This radius of rotation is considered as a vector which is perpendicularto the momentum vector π . In the ground state energy of the particle, the motion maybe considered as a circular motion with constant magnitude |ξ |. In this special case ofminimum steady state energy or ground state of the harmonic oscillator, the kineticand potential energies are equal or the total energy is twice the kinetic energy and fromthe definition of constants a and b in Eq. (3) we have a2 = b2. Now it can be shown

that Z2 = Z2 = 0. Then the complex vectors Z and Z are complex null vectors. The

complex vector Z is now expressed as Z = ka(σa + iσb). Where, σ a and σ b are unitvectors along a and b. Further it can be shown that

Z ZJ+ = 1J+ and Z ZJ+ = 0. (21)

These relations combined with Eqs. (19) and (20) directly yield the ground state energyE0 of the oscillator

H0 = E0 = hω0

2(22)

In this case the system is considered as an elementary free particle oscillator whichresponds to resonant oscillations and in this article the analysis is mainly concernedwith this ground state energy. As discussed in the introduction, the ground state energycorresponds to the zeropoint energy of the oscillator per mode [3,4] and an averagevalue is obtained by taking a stochastic average of all such modes.

However, in the more general considerations the frequency of oscillations of theparticle may not necessarily be at resonance with the fluctuations of the spectrum ofrandom zeropoint fields and in such cases one may take into account the higher energystates of the particle oscillator. The random zeropoint field in interaction with the par-ticle oscillator would be expected to produce random displacements producing a shiftin the average energy minimum without any change in the particle spin. The constancyof the spin will be shown below. Hence there is always a possibility of frequenciesωn = nω0 with which the particle may oscillate. The particle oscillator may acquire

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additional energy from the spectrum of zeropoint fields and the Hamiltonian containsthe additional energy terms. Now, generalizing the Hamiltonian with the condition

Hhω0

≥ 12 , suppose that H = H ′ + hω0. Then the Eqs. (19) and (20) can be expressed

as

ZZJ+ =(

H ′

hω0+ 1

2

)J+; ZZJ+ =

(H ′

hω0+ 3

2

)J+. (23)

These equations satisfy the relation (ZZ − ZZ)J+ = 1J+. Similarly this relation issatisfied in all the cases when H = H ′′ + 2hω0, H = H ′′′ + 3hω0 etc. Continuingthe same procedure until the energy reaches its minimum value H = H0 + nhω0one can find that in each case the relation (ZZ − ZZ)J+ = 1J+ is satisfied. In thecase of higher energy states of the oscillator, the magnitudes of vectors a and b arenot equal and the average harmonic oscillations are in general elliptical. Thus theargument yields that the terms in brackets of Eqs. (19) and (20) must be integers. Withthis correspondence the geometric products ZZ and ZZ are expressed as

ZZJ+ = nJ+ and ZZJ+ = (n + 1)J+. (24)

Comparing Eq. (24) with Eqs. (19) and (20) and replacing H by the energy En ofthe oscillator, the required result for the energy of the harmonic oscillator is obtained.

En = (n + 1

2)hω0 (25)

In the harmonic oscillator problem treated by Dirac contains the complex formwith position and momentum treated as operators and using the commutator rela-tion [x, p] = i h and successive procedure of finding higher order operators lead tothe energy of the harmonic oscillator [16]. An extension of this idea in the Swinger’soscillator model shows a connection between the algebra of angular momentum andtwo independent harmonic oscillators denoted by annihilation and creation operators(a and a†) and provides an inherent connection between field oscillators and spin[13]. A mathematical equivalence of electromagnetic field to the harmonic oscillatorof same frequency with the commutator relation

[a, a†

] = 1 yields zeropoint energyof the oscillator and also to a particle term in Einstein fluctuation formula whichobviously relates the particle to the zeropoint energy [4].

In the general case the complex vectors Z and Z can be expressed as

Z = kσ a(a − bσ 3) and Z = kσ a(a + bσ 3). (26)

Since the unit vector σ 3 is absorbed by the idempotent, multiplying Z and Z fromright by J+ gives

ZJ+ = kσ a(a − b)J+ and ZJ+ = kσ a(a + b)J+. (27)

Now, one can easily verify the following relations.

Z ZJ+ = k2(a + b)2J+; Z ZJ+ = k2(a − b)2J+ (28)

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Found Phys (2014) 44:266–295 277

If we define ZZ as an operator N , then NJ+ = nJ+,

ZJ+ = √nσ aJ+, ZJ+ = √

n + 1σ aJ+ (29)

Comparing these equations with Eq. (27) gives

2ka = √n + 1 + √

n, 2kb = √n + 1 − √

n and 4k2ab = 1. (30)

Thus for higher energies of the oscillator the magnitude of a increases with corre-sponding decrease in the magnitude of b. Since 4k2ab = 1 the spin remains constantwith its magnitude ab = h

2 . Now, using Eq. (18) we have

N ZJ+ = Z(n + 1)J+; N ZJ+ = (Z Z − 1)ZJ+ = Z(n − 1)J+ (31)

In the case of ground state of the oscillator the above equations reduce to

N ZJ+ = (+1)ZJ+ and N ZJ+ = (−1)ZJ+. (32)

These relations and Eq. (31) reveal that the complex vectors Z and Z are analogous(not equal) to the creation and annihilation operators and satisfy similar commuta-tion relations. If we work with the idempotent J− the complex vectors reverse theiractions. However, it is very clear that the algebra of complex vectors Z and Z is notisomorphic to the algebra of creation and annihilation operators in quantum mechan-ics. The complex vector formalism presented here elucidates how spin is related to thezeropoint energy of the particle. Further, we have some correspondence between thecomplex vector formalism of harmonic oscillator to the quantum oscillator throughequivalence between commutator product and bivector product.

3 Particle Mass and Relativity

From the Eq. (9) we discern that spin plays an important role in the kinematic behaviorof the particle in its internal motion and it may be further explored to find someimportant results by writing it in the following form

ZZhω0 = En − ω0s. (33)

Multiplying this equation by J+ from right and using the relation ZZJ+ = 0 for theground state of the oscillator the above Eq. (33) is expressed as

(E0 − ω0s)J+ = 0. (34)

Let us denote ωs = 2ω0 and write the angular velocity bivector as �s = −iωs =−iσ 3ωs . The average value of zeropoint energy associated with the particle is 〈E0〉 =hω0π

. The average value of spin is obtained by multiplying it by a factor 2π

. Using

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278 Found Phys (2014) 44:266–295

both these average values in the above Eq. (34) gives particle mass with the use ofequivalence mc2 = hω0.

σ 3mc2J+ = +�s.SJ+ (35)

From Eq. (10) with similar arguments we find

σ 3mc2J− = −�s.SJ−. (36)

Using Eq. (17) and writing the above equations into a single form we have

σ 3mc2J± = λ�s.SJ±. (37)

The constant λ = ±1 represents the eigen values of σ 3. Thus the constant λ representsthe twofold nature of the Eq. (37). The values +1 and −1 correspond to clockwiseand counterclockwise rotations respectively. For +1 value of λ, Eq. (37) representsa particle with mass m, spin half and with negative charge, a particle like electron.On the other hand for −1 value of λ, Eq. (37) represents a particle with mass m,spin half and positive charge. Thus particle and antiparticle conjugation is achievedwith positive and negative values of λ. Since �s.S is positive for either clockwise orcounterclockwise rotations the relation between spin and particle or anti particle massis expressed as

mc2 = �s.S. (38)

This equation elucidates the relation between the two important parameters of anelementary particle, mass and spin. Thus the mass of the particle turns out to be thelocal internal rotational energy given by the term �s.S. The relation expressed in theEq. (38) is not new in the literature but the way the relation derived is different fromall the previous works till now to the knowledge of the author. In the reference [15],considering circulating energy flow in the fields as the spin angular momentum, theratio of energy to spin has been shown to be equal to the angular frequency of rotation.Thus the elementary particle may be visualized as the local rotation in the spin planeand such rotations emerge as a consequence of zeropoint field and its ubiquitouspresence. The rest mass of the particle to internal energy and the rotational motion isconnected to the origin of charge [46] in a sense that the square of charge is proportionalto the spin angular momentum [58]. In a general view we designate inertia to the massof a body which was identified by Galileo, quantified by Newton, stated by Mackas the local property connected to the cosmic distribution and according to Einstein,mass is energy [59]. Thus the mass is seen as the energy of oscillations of the particleand the zeropoint energy confined to a region of space of dimensions of the orderof Compton wavelength of the particle [24]. In the standard model, particles acquiremass by Higgs field and one has to choose a different coupling for each particle. In therenormalization procedure of quantum field theory with finite cut-off for the radiativelyinduced mass, it has been shown that mass depends on particle spin in the limit whenthe bare mass tends to zero [60]. However, in the quantum electrodynamics it is wellknown that the sum of bare mass and renormalized mass equals the electron mass [4].Recently Pollock interpreted particle mass (fermionic and bosonic) arising from thezeropoint vacuum oscillations by introducing a matrix mass term in the Dirac equation

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[61,62]. The spin frequency of the particle complex rotation is written as ωs = 2mc2

h .Employing the same procedure as above, Eq. (10) can also be expressed as

(σ 3mc2 +�s.S)J+ = σ 3hω0J+. (39)

For particles of mass zero, like a photon, substituting m = 0 in the Eq. (39) yields itsspin sp = hσ 3. This can be expressed in the following form for helicity states of massless particles.

spJ± = hσ 3J± = λh J± (40)

Here the vector σ 3 represents the polarization direction and as a consequence itsoperation on J± gives the values λ = ±1. For spin half particles the polarizationdirection coincides with the spin quantization axis.

3.1 Particle Dynamics

The local spin rotation produced by the zeropoint fields makes it to propose the localcomplex rotation as the mass of the particle. This local complex rotation will be treatedin a rest frame with its origin at the centre of rotation or equivalently the centre of massand may be called as the particle rest frame. When the particle moves with velocityv = dx

dt , the centre of local complex rotation is denoted with the position vector x. Thecomplex vector connected with both the motion of the particle along its mean positionand local complex rotation can be expressed as

X (t) = x(t)+ iξ. (41)

This form is similar to the coordinate considered by Barut and Zanghi [27] with theexception that iξ is a bivector. Differentiating Eq. (41) with respect to time gives

U = v + iu. (42)

Where, U = d Xdt and u = dξ

dt . Since the deviation in position ξ is considered asorthogonal to x , the velocity u of the particle internal motion is orthogonal to theparticle translational velocity v. A reversion operation on U gives U = v − iu. Andthe scalar product

U .U = v2 + u2. (43)

In the particle rest frame v = 0 and U.U = u2. Since the particle internal velocityin the particle rest frame u = c the velocity of light, |U | = u = c. However, whenthe particle is observed from an arbitrary frame different from the rest frame of theparticle centre of mass, as the centre of mass moves with velocity v, the particlemotion contains both translational and internal rotational motion of the particle. Thenthe particle internal velocity can be seen as

u2 = c2 − v2. (44)

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280 Found Phys (2014) 44:266–295

Or u = c(1 − β2)1/2 = cγ−1 (45)

Where, β = vc and the factor γ is the usual Lorentz factor. The angular frequency of

rotation of the particle internal motion is equal to the ratio between the velocity c andradius of rotation ξ , ωs = c

ξ. However, when observed from an arbitrary frame the

angular frequency ω would be equal to the ratio between u and ξ .

ω = u

ξ= ωsγ

−1 (46)

Expressing u and ξ as vectors the angular velocity is expressed as a bivector. Thus wehave −iω = u

ξor −iω ∧ ξ = u and the Eq. (46) is then expressed as

� = �sγ−1. (47)

Where, � = −iω. Thus the angular frequency of rotation decreases when observedfrom an arbitrary frame and the decrease depends on the velocity of the centre ofmass. Considering the helical motion of the particle, this method of calculation fortime dilation was first shown in a simple manner by Cavelleri [48]. The above analysisshows that the basic reason for the relativistic effects that we observe is due to theinternal rotation which is a consequence of fluctuating zeropoint fields. We have seenthat the angular velocity bivector is the ratio between u and ξ and it represents anoriented plane along −iσ 3. Then the proper time is connected with this rotation plane.The ratio ξ

u gives the total time period of rotation and represents a bivector in the spinplane iσ 3. As the particle moves this time plane moves along the path of centre ofmass and generates proper time. Then the proper time interval is identified as iσ 3dτor the metric iσ 3cdτ and from which a complex vector can be expressed in the form

TV = dx + iσ 3cdτ. (48)

Now, the product TV .T V represents the interval c2dt2 in the rest frame of the particle.

TV .T V = dx2 + c2dτ 2 (49)

Or c2dτ 2 = c2dt2 − dx2 (50)

The geometrical concept of proper time as a plane defined by the angular momentumhas been considered recently by Machicote [63]. The idea of proper time as a rotationalentity or a bivector contradicts the steady flow of time. However recent experiments usethe rotating electric field in circularly polarized light as an ottosecond clock to probeatomic processes [64–66] such that proper time may be treated as plane of rotation.In the rest frame of the particle dτ = dt and the particle path is light-like. The timeinterval in the local frame or rest frame of the particle can be defined as the ratio ξ

cand represents vectorial time. Thus the concept of Newtonian time can still be definedin the rest frame of the particle [63]. If the particle moves over a distance along therotation path in dt time, the proper time interval will be dτ = γ−1dt . Similarly theparticle centre of mass point moves over a proper distance dx ′, the distance observed

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in the arbitrary frame would be dx ′ = γ dx . These equations are simply time dilationand length contraction relations. The relative velocity is then equal to γ v and therelativistic momentum is then expressed as

p = γmv. (51)

This momentum in the rest frame of the particle is zero. However, as considered inthe complex formalism of harmonic oscillator in the previous section, the deviation ofparticle momentum π which is the internal momentum associated with the complexrotation of the particle in the local rest frame allows defining the momentum complexvector as

P = p + iπ . (52)

The scalar product P.P = p2 +π2 and noting P.Pc2 = E2 gives the relativistic energyrelation in terms of momentum.

E2 = p2c2 + m2c4 (53)

The above analysis confirms the fact that the special theory of relativity emerges fromthe internal complex rotations in the local space. By using the expression pc = Eβ

we get as usual E = γmc2. The particle velocity in the rest frame of the particle iszero and in this case the time dilation factor γ = 1 and the energy of the particle inits rest frame is mc2.

3.2 The Role of Spin in Particle Motion

The orbital angular momentum of the particle in a curved path of the centre of masspoint is a bivector quantity given by L = x ∧ p. Expanding the bivector product X ∧Pgives

X ∧ P = x ∧ p + ξ ∧ π + i(ξ ∧ p)− i(x ∧ π). (54)

In the Eq. (54), the vector term i(x ∧π) represents a phase factor and the term i(ξ∧ p)represents an additional vector term to spin bivector ξ ∧ π . Then the total spin maybe expressed in the form of complex vector.

St = S + i SL (55)

Where, SL = ξ ∧ p representing an oriented plane perpendicular to spin plane. This islike the spin normally obtained in classical theories [9] (the assumption of nonlinearityof p andv leads to a deviationx in the mean path of the particle and with identificationx as ξ). The bivector SL can be expressed using Eq. (51) and ξ = h

2mc .

SL = iσ 2γβh

2(56)

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282 Found Phys (2014) 44:266–295

The scalar product St .St = S2 + S2L = −γ 2S2. Considering St .St = −S2

0, we have|S0| = γ |S| and

S0 = iσ 3 |S0| = γ S. (57)

This shows that as the particle moves with velocity v, the magnitude of spin appearsto be increased by a Lorentz factor. From Eqs. (47) and (57) one can arrive at

�.S0 = �s.S. (58)

It means that the particle mass remains unchanged even if the particle centre of massis in motion or in other words the Eq. (58) leads to the nonexistence of the so calledrelativistic mass. Thus in general, either in Newtonian or relativistic theories the par-ticle mass m is independent of a reference frame [67]. Thus the property of materialparticle mass is a conserved quantity. It is quite interesting to note that even in thelimit v → c, Eq. (58) is valid. The bivector spin rate can be obtained by differentiatingS = ξ ∧ π with respect to time.

d S

dt= S = u ∧ π + ξ ∧ π = (−�) ∧ S (59)

Where, π is replaced by −iω ∧ π = � ∧ π . Since S.S = S.(−�) ∧ S = 0, thedifferential of square of spin bivector is zero and S2 = S.S = − ∣∣S2

∣∣ is a constantof motion and S.u = ξ ∧ mu.u = 0. This condition is an important property of anelementary particle, which is responsible for the intrinsic magnetic moment. For anelectron with charge e and mass m, in the present complex vector treatment and inthe centre of mass frame, the charge rotation produces a current equal to eωs and theoriented area of rotation is ξ∧u

ωs. Then the intrinsic bivector magnetic moment M0 lies

in the same orientation of spin S.

M0 = e

mcS (60)

And there appears to be an instantaneous dipole moment d = eξ associated with theparticle. The total of electric and magnetic moments can be written as a complexvector.

MV = d + M0 = e

mc(mcξ + S) . (61)

The electromagnetic field in the rest frame of the particle centre of mass in geometricalgebra can be expressed as a complex vector F = E + i B. The interaction of MV

with the electromagnetic field gives the interaction energy and can be expressed as

Em = MV .F = eξ.E + e

mcS.i B. (62)

The average of electric dipole interaction energy term over one complete internalrotation goes to zero. The last term in Eq. (62) is a constant and represents magneticdipole interaction energy term and gives the gyromagnetic ratio of electron g = 2.

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3.3 The Quantum Nature of Particles

Inspection of Eq. (37) shows that the zeropoint field modes of frequencyωs are respon-sible for local complex rotations. However from Eq. (46) the frequency as observedfrom an arbitrary frame is ω = ωsγ

−1. The difference between ω and ωs corre-sponds to the particle velocity. In other words when the particle moves with velocityv an important consequence is that the particle itself induces certain modification inthe field to take place at a lower frequency ωB . To look into this in more detail, theEq. (42) can be expressed using Eqs. (46) and (47) as

�s = �B + i�. (63)

Where,�B = β�s and�B = iωB . Thus the motion of a free particle is convenientlyvisualized as a superposition of frequencies ωs andωB such that the particle motion asobserved from an orbitrary frame appears to be a modulated wave containing internalhigh frequency ωs and an envelope frequency ωB . Then the wavelength associatedwith this frequency is

λB = 4πc

γωB= h

p. (64)

This gives the genesis of de Broglie relation and the quantum nature of elementaryparticles in general. This result is simply the consequence of superposition of internalcomplex rotations on translational motion of the particle.

3.4 Particle Motion in Spacetime

In view of the Eqs. (43), (51) and (53) we may arrive at an opinion that the propertime is generated only with particle complex rotational motion. When the particle is inmotion, the direction of internal velocity u is no more in the same direction as that inthe rest frame of the particle and it takes a new direction. Let us consider this directionby a unit vector σ 0 as observed from an arbitrary frame of reference. In general theLorentz boost can be expressed in terms of rapidity factor ϕ and a unit vector v alongparticle velocity and is given by

L = exp

(−vϕ2

)and L = exp

(vϕ

2

). (65)

Here, L is the inverse of L. Then the active Lorentz rotation of vector σ 0 gives a unitcomplex vector along the direction of proper velocity complex vector [68].

u0 = exp

(−vϕ2

)σ 0 exp

(vϕ

2

)= σ 0 cosh ϕ + (σ 0v) sinh ϕ (66)

This is simply the active boost of σ 0 through the rapidity factor in the direction ofparticle velocity and generates a new complex unit vector u0. The bivector σ 0 ∧ v

represents spacetime plane of rotation. Multiplying this unit vector u0 with velocity of

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light c gives the proper velocity complex vector u0 of the particle. The unit vector σ 0represents the fourth dimension in four dimensional spacetime. The proper velocityvector is then expressed as

u0 = cγ (σ 0 + σ 0 ∧ β). (67)

Multiplying Eq. (67) from right by the unit vector σ 0 gives the velocity four vector.

uμ = u0σ 0 = cγ (1 − β) (68)

This type of split is called spacetime split [69]. The square of proper velocity vectoris obtained as

u20 = u0σ 0σ 0u0 = uμuμ = cγ (1 − β)cγ (1 + β) = c2. (69)

Then the square of proper velocity u20 represents the time-like case. Multiplying

Eq. (67) throughout by dt gives

dτu0 = cdtσ 0 − σ k ∧ σ 0dxk . (70)

This equation allows the definition of spacetime basis vectors.

γ0 = γ 0 = σ 0; γk = σ kσ 0 = σ kγ0 (71)

With this notation Eq. (70) can be expressed as

dτu0 = γμdxμ, (μ = 0, 1, 2, 3). (72)

Then the square of proper length is

c2dτ 2 = γμdxμγνdxν = ημνdxμdxν . (73)

Where, γμγν = ημν is the spacetime metric. In a similar manner the proper momentumvector can be expressed as

p = γ (mσ 0 + σ 0mv). (74)

Where, the velocity of light is taken as c = 1. Multiplying Eq. (74) from right by theunit vector σ 0 gives four-vector momentum.

pμ = pσ 0 = p0 − p (75)

pμ pμ = pσ 0σ 0 p = p2 = (p0 − p)(p0 + p) = m2 (76)

Thus p2 is a constant of motion. In the quantum notation, this is stated as the eigenstate of matter p2 has an eigen value m2. A Lorentz rotation of spin bivector S givesthe proper spin bivector S0.

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S0 = exp

(−vϕ2

)S exp

(vϕ

2

)= γ S (77)

S0.p = S.p = 0 (78)

Thus the spin bivector in spacetime is purely a spatial bivector. Then the tensor com-ponents of spin bivector can be expressed as S00 = 0 and Si j = S. The spin vectorcomponents are obviously written as s0 = 0 and sk = s. With the identification ofcomponents W0 = p.s and Wk = p0sk , a proper complex vector which correspondsto Pauli–Lubanski proper vector can be defined as

W = σ 0W0 + σ 0 ∧ Wk . (79)

Multiplying from right by σ 0 gives the Pauli–Lubanski four-vector Wμ.

Wμ = Wσ 0 = W0 − Wk (80)

WμWμ = Wσ 0σ 0W = W 2 = (W0 − Wk)(W0 + Wk) = −m2s2 (81)

From Eq. (78) we have W.p = 0 and hence p2 and W 2 are two independent constantsof motion and in general known as first and second Casimir invariants.

In the rest frame of the particle the rotation in spin plane can be expressed by arotor.

R = exp

(�s

2t

)(82)

In the present contest Eq. (82) can also be expressed in the form as

R = exp

(�s.S

2St

). (83)

Then the rotor equation in the rest frame of the particle can be written as

R = R1

2S�s .S. (84)

The over dot on R represents differentiation with respect to time. Since rotor satisfiesthe condition R R = 1. The rotor equation in the rest frame of the particle is nowexpressed as

R2S = λ�s .S R = σ 3mc2 R. (85)

The presence of λ represents the two fold nature of the Eq. (85) and we arrive at therotor equation for particle and antiparticle respectively.

R2S = +mc2 R. (86)

R2S = −mc2 R (87)

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Using the notation ∂t = ∂∂ct , the Eq. (86) can be rewritten as

∂t R2S − mcR = 0. (88)

This is the form of Dirac equation in the rest frame of the particle. Multiplying withσ 0 from left gives

σ 0∂t R2S − σ 0mcR = 0. (89)

When the particle is in motion as observed by an arbitrary observer, the equation ofmotion of the particle is obtained by applying Lorenz boost to the Eq. (89).

Lσ 0∂t L L R2S − Lmcσ 0 R = 0 (90)

Using Eq. (72), we express Lσ 0∂t L = γ μ∂μ = ∂ . Since γ0 is invariant under spatialrotation, now Eq. (90) can be written as

∂L R2S − mcL R γ0 = 0. (91)

A homogeneous Lorentz transformation is obtained by Lorentz boost followed byrotation. Then the Lorentz rotation is the product � = L R and satisfies the condition�� = 1. Now the equation of motion of the particle can be expressed in the followingform.

∂�2S − mc�γ0 = 0. (92)

Introducing a general spinor of the form [23]

ψ = �12 ei ε2�. (93)

Where, ρ is the probability density and can be defined asψψ = ρ and the phase factoreiε is equal to +1 for ε = 0 and −1 for ε = π . In terms of this spinor the Eq. (92)becomes

∂ψ2S − mcψγ0 = 0. (94)

This is the form of Dirac equation for spin half particles. For antiparticles from Eq.(87), we arrive at

∂ψ2S + mcψγ0 = 0. (95)

However, in the presence of external electromagnetic fields we need to add a potentialterm q

c A = qc γ

μAμ in the Eq. (94).

c∂ψ2S − q Aψ − mc2ψγ0 = 0 (96)

Where, A is the electromagnetic vector potential and q is the particle charge. For anelectron q = −e. Eq. (96) is the well known Dirac–Hestenes equation in spacetimealgebra. The geometrical interpretation of Eq. (96) has been extensively discussed byHestens in several of his articles [23,53,70] and also by Boudet [71].

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4 Complex Vector Space and Spacetime

In Sect. 2 we have considered the resonant oscillations of the particle oscillator in thefluctuating zeropoint fields. For the ground state energy of the particle oscillator thecomplex vectors Z and Z are found to be complex null vectors. The complex vectorsdefined in Eqs. (6) and (7) actually form a physical space when the spin direction ofthe particle is chosen along σ 3 direction and considering local complex rotation. Nowthe vectors a, b and s can be expressed in terms of these complex vectors.

a = Z + Z

2k, b = Z − Z

2ikand 2s = h(ZZ − ZZ) (97)

Since the spin direction is chosen along σ 3, one can express the right handed ortho-normal unit vectors {σ k} in terms of Z and Z .

σ 1 = Z + Z

2ka, σ 2 = Z − Z

2ikband σ 3 = (ZZ − ZZ) (98)

And from the scalar product Z .Z the unit scalar is expressed as

1 = (ZZ + ZZ)

2k2(a2 + b2). (99)

The algebra of complex vectors Z and its conjugate Z generates complex vector space.The basis elements {1, σ k; k = 1, 2, 3} form a closed complex four dimensional linearspace [72]. Since the complex vectors are analogous to the creation and annihilationoperators in quantum mechanics, the physical space formed by the complex vectors isalso analogous to the physical space formed by the creation and annihilation operators.Such physical space from fermion creation and annihilation operators was introducedby Baylis [73]. The association of complex vector Z with its conjugate Z generatesa space in one dimension and a unit vector σ 1 along a line can be defined as in Eq.(98). The set of elements {1, σ 1} form geometric algebra G(1) of one dimension. Thealgebra G(1) does not contain any pseudoscalar. Complexifying G(1) generates thegeometric algebra of two dimensions G(2).

G(2) = G(1)+ iG(1) (100)

The unit vector σ 2 is defined in terms of complex vectors as in Eq. (98) and theproduct σ 1σ 2 = i represents the pseudoscalar or unit bivector. The pseudoscalar is anoriented unit plane. The vector σ 2 is now defined as σ 2 = σ 1i . The set of elements{1, σ 1, σ 2, i} form geometric algebra G(2) of two dimensions. A complex vector inG(2) can be defined as a combination of a vector and a bivector.

CV = d + iδ (101)

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The square of complex vector C2V = d2 − δ2. When, d = δ the complex vector is a

null complex vector. The product CV CV = d2 + δ2 + 2δi d. Thus the product CV CV

is an odd multivector in G(2). The geometric algebra G(3) can be generated from thecomplexification of the two dimensional geometric algebra G(2) [74].

G(3) = G(2)+ iG(2) (102)

The unit vector σ 3 can also be defined as σ 3 = iσ 2σ 1. Multiplying vectors σk bypseudoscalar forms unit bivectors Bk = iσ k = σ iσ j (i, j, k = 1, 2, 3). Each unitbivector represents an oriented plane and B1, B2 and B3 form basis unit right handedorthonormal bivector oriented planes in the Euclidean space. The square of basisbivector B2

k = −1. Then the unit imaginary or root minus one can be defined in twodifferent ways either with pseudoscalar or basis bivector. Since the unit imaginary isassociated with some physical plane, the correspondence with bivector is of particularimportance. In the previous sections, the spin direction has been chosen along σ 3vector and B3 = iσ 3 represents a unit bivector spin plane normal to σ 3. The productof all the basis bivectors B1 B2 B3 = +1. The set of elements {1, σ k, Bk, i} formgeometric algebra G(3) of three dimensions.

4.1 The Complex Scalar, Complex Vector and Idempotents

In the geometric algebra G(3), a multivector can be expressed as a sum of complexscalar and a complex vector parts M = CS +CV and the physical interpretation of thestructure is completely related to the pseudoscalar. The combination of a scalar andtrivector is called a complex scalar CS and the combination of a vector and bivectoris called a complex vector CV .

CS = α + iδ and CV = d + i g (103)

In a complex scalar the pseudoscalar replaces the function of unit imaginary such thatCS has correspondence with normal complex numbers. The product of two complexscalars form another complex scalar, the algebra is closed under multiplication andforms a sub-algebra of G(3). The geometric product of two complex vectors is amultivector. The inner product of two complex vectors is a complex scalar and theouter product is a vector. A reversion operation on CV gives CV = d − i g and theproduct

CV CV = d2 + g2 + 2i d ∧ g. (104)

Thus we have the outer product CV ∧ CV = 2i d ∧ g and the inner product CV .CV =d2+g2. It should be noted that the square of a complex vector is a complex scalar. Thusthe complex vector algebra is not closed and does not form sub-algebra of geometricalgebra G(3). However, the algebra we have considered in this article is the algebraof complex vectors. In the case the vectors d and g are orthogonal and equal inmagnitude then the complex vector is a complex null vector. This gives the definition

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of unit complex null vectors in the form

Dk = 1

2(σ i + iσ j ) and Dk = 1

2(σ i − iσ j ). (105)

These unit complex null vectors satisfy the property D2k = D

2k = 0 and the set

{Dk} may be called as basis complex null vectors in the geometric algebra G(3). Theproducts

Dk Dk = 1

2(1 + σ k) and Dk Dk = 1

2(1 − σ k). (106)

Thus these products are represented by self annihilating idempotents of the algebra.In particular the products D3 D3 and D3 D3 correspond to the spinors J+ and J−considered in Sect. 2.

J+ = D3 D3 = 1

2(1 + σ 3) and J− = D3 D3 = 1

2(1 − σ 3) (107)

These idempotents satisfy the following relations J±J± = J± and J+J− =J−J+ = 0. In conventional quantum mechanics these idempotents correspond tothe helicity states. Geometrically one can visualize null complex vectors as points inspace with zero magnitude, scalars represent the magnitude of geometric elements.Normally the points are considered as scalars in geometric algebra [75]. However,it is more appropriate to treat scalars as the objects of magnitude as in conventionalvector algebra. The vectors as usual are oriented line segments, the bivectors representoriented planes and pseudoscalar represents an oriented unit volume in space.

4.2 The Bivector Spin S and Complex Rotations

The rotor R represents counterclockwise rotations in the plane B3 and R representsclockwise rotations. For infinitesimal rotations in the plane σ 1σ 2, R → −B3θ

2 andwriting θ by ωs t we get

V = R − R = σ 1σ 2ωs t. (108)

Differentiating Eq. (108) with respect to time gives the angular velocity bivector.However, the origin of these complex rotations associated with an elementary particleis mainly due to the presence of zeropoint fields and the zeropoint angular momentumis identified with bivector spin [57]. From Eq. (108) the spin bivector can be expressedas

S = h

2ωs

dV

dt= h

2σ 1σ 2. (109)

And the bivector product in the Eq. (97) is now written in the form

2S = i h(ZZ − ZZ). (110)

This gives the basic mathematical expression for spin bivector in terms of complexnull vectors and it is defined by the bivector product of Z and Z .

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4.3 Extension to Higher Dimensional Space

The method followed in the generation of G(3) can be extended in the same mannerto generate higher space dimensions. The four dimentional space can be generatedfrom defining i4 = σ 1σ 2σ 3σ 4 and the vector σ 4 = i4 i . The geometric algebra offour dimensional space can be expressed as

G(4) = G(3)+ i4G(3). (111)

The set of elements{1, σ k, σ iσ j , σ iσ jσ k, i4; i, j, k = 1, 2, 3, 4

}form geometric

algebra G(4) of four space dimensions. The geometric algebra of higher dimensionalspace can be generated in a similar manner as above by defining a pseudoscalar.

G(n) = G(n − 1)+ inG(n − 1) (112)

The geometric algebra G(n) of n-dimensions contains the total number of elements2n .

4.4 Spacetime

In the complexification process of complex space, the geometric algebra G(3) is gen-erated from G(2) and G(2) from G(1) by considering unit pseudoscalar identificationwith square root of minus one. The algebra of four dimensional spacetime can begenerated from the geometric algebra by considering square root of plus one withthe identification of vector σ 0 ≡ √+1 or σ 2

0 = +1. An active Lorentz rotation ofthis vector gives a unit vector γ0 along the direction of proper velocity and this unitvector is identified with the unit vector along the future light cone. The set of ortho-normal right handed unit vectors {γμ;μ = 0, 1, 2, 3} form Minkowski spacetime.The set of basis vectors {σ k} form an algebra for rest frame relative to the time likevector γ0 and the vectors σ k are interpreted as relative bivectors, σ k = γkγ0. Theproducts of these relative vectors represent spatial bivectors γiγ j in spacetime andBk = iσ k = σ iσ j = γ jγi . Thus σ k in spacetime represent relative bivector planesand iσ k are purely spatial bivector planes. The product B1 B2 B3 = +1 representsthree dimensional hyperplane in spacetime. A unit complex bivector Dk in spacetimeis a combination of relative bivector and a spatial bivector.

Dk = 1

2(σ i + iσ j ) (113)

The vector γ0 anticommutes with relative bivectors and on the other hand it commuteswith spatial bivectors. That is γ0σ kγ0 = −σ k and γ0 iσ kγ0 = iσ k . The reversionoperation on unit complex bivector gives

Dk = 1

2(−σ i − iσ j ). (114)

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And multiplying Dk on both sides by γ0 leads to an operation known as relativereversion, which is similar to the hermitian operation in Dirac algebra.

D†k = γ0 Dkγ0 = 1

2(σ i − iσ j ) (115)

Now the products

Dk D†k = 1

2(1 + σ k) and D†

k Dk = 1

2(1 − σ k). (116)

These products are similar to the products in Eq. (106) except for the relative bivectornature of σ k . The unit pseudoscalar I in spacetime is the product of vectors γ0γ1γ2γ3and it is related to i by the equation

i = σ 1σ 2σ 3 = γ0γ1γ2γ3 = I. (117)

The pseudoscalar I is a 4-vector and represents relative oriented volume in spacetime.The spacetime algebra is generated from G(3) by considering a space relative to theunit vector γ0 in complex vector space but not by complexification.

G(1, 3) = G(3)+ γ0G(3) (118)

Here we introduced the square root of +1 to achieve the four dimensional spacetimegenerated from G(3). The set of elements {1, γμ, γμγν, γμ I, I } form the geometricalgebra of spacetime known as spacetime algebra developed by Hestens [69]. A generalelement in the algebra G(1, 3) is represented by a multivector.

M = α + d + F + gI + I δ (119)

Where, α and δ are scalars, d and g are spacetime vectors and F = c+ i d is a complexbivector and the last term I δ is a four-vector. Multivectors can be decomposed intoeven part and odd part.

M+ = α + F + I δ and M− = d + gI (120)

Multivectors of even grade is generated by the set of relative bivectors {σ k} and forma subalgebra of spacetime algebra. The product M

+M+ is a complex scalar. Then the

even multivector can be expressed in form of a spinor.

M+ = ρ12 e

iε2 R (121)

Where, R is rotor and satisfies the relation R R = 1. This is the form of a spinorrepresenting Lorentz rotation and along with Lorentz boost we express the spinor inthe form [53]

ψ = L M+ = ρ12 e

iε2 L R = ρ

12 e

iε2 �. (122)

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Where, � = LR represents Lorentz boost followed by rotation and the spinor ψ isthe Hestens spinor. The Lorentz rotor satisfies the relation �� = 1 and the set of allrotors form spin group of spacetime. The angle ε is known as the Takabayasi–Hestensangle and gives a better interpretation for the passage of electron to positron.

4.5 Generation of Higher Dimensional Complex Spacetime

Higher dimensional spacetime can be generated from spacetime by definingpseudoscalar I4 with additional spatial vector γ4 and I4 = γ0γ1γ2γ3γ4. Then we haveγ4 = I I4 and γ 2

4 = −1. The set of unit orthonormal vectors {γμ;μ = 0, 1, 2, 3, 4}form a five dimensional spacetime. The geometric algebra G(1, 4) contains 25 = 32number of elements and can be represented by the equation

G(1, 4) = G(1, 3)+ I4G(1, 3). (123)

And in general this is expressed as

G(1, n + 1) = G(1, n)+ In+1G(1, n). (124)

Thus in general the geometric algebra of higher dimensional space is generated fromthe identification of the corresponding pseudoscalar to the square root of minus onein the case of odd spatial dimensions and to the square root of plus one in the case ofeven spatial dimensions (n > 2) which gives the underlying complex nature of thealgebra.

5 Conclusions

A particle may be considered to oscillate from its mean position due to the presenceof random fluctuations of zeropoint field and such oscillations represent complexrotations. The particle spin is found to be the characteristic of such complex rotations.Considering complex null vectors in the particle rest frame, the ground state energyof the free particle oscillator is derived. From the complex vector description of theharmonic oscillator a relation between spin and mass is found to be σ 3mc2J± =λ�s.SJ±. The constant λ = ±1 represents the two fold nature and hence givesthe particle and antiparticle mass relations to spin. Since �s.S is positive for eitherclockwise or counterclockwise rotations we have mc2 = �s.S. Thus the mass of theparticle may be interpreted as a local spatial complex rotation in the rest frame. Whena particle is observed from an arbitrary frame of reference, it has been shown that thespatial complex rotation dictates the relativistic particle motion. In the current complexvector formalism, the centre of mass and centre of charge are treated as separate andhence the position and momentum of the particle are expressed as complex vectors. Byexpressing the physical quantities in terms of complex vectors the relativistic particlemotion is achieved. It has been shown that the relativity as we observe is because of theexistence of zeropoint fields. However, it may require an extensive study to prove thesame at a larger macroscopic level. Introducing complex vector time, the proper time

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is found to be an advancing local time bivector plane of rotation. The proper bivectorspin is found to depend on the time dilation factor and hence the product of bivectorspin and bivector angular velocity is a conserved quantity and thus the particle massis independent of any reference frame. The quantum nature of particles is found tobe due to the superposition of internal complex rotations on translational motion ofthe particle. The relativistic dynamics of the particle in complex spacetime has beendiscussed and the Hestenes–Dirac equation in spacetime algebra is derived.

There may be defined set of modes of zeropoint field to which the particle mayrespond. The selection of particular resonant frequency may be a matter of chance.For the ground state of the particle the chance with which the particular resonantfrequency is selected surely depends on the particle mass itself. In the classical modelof fundamental particle discussed by Batty-Pratt and Racey [55] the frequency ofrotation corresponds to the mass of the particle and more recently Grössing et al. [76],in a different approach, showed that the free particle oscillator turns out to be stationaryat resonant frequency and the energy of the oscillator is the zeropoint energy. However,the reason for the ground state of the free particle oscillator to be singled out for therepresentation of a fundamental particle is obscure. The present theory is one of thepossible extended particle theories and gives a better understanding of the nature offundamental particles. In the more general considerations the particle may acquireadditional energy from the spectrum of zeropoint field and in this case the particleoscillator energy including higher energy terms is derived. It has been shown that thecomplex vectors Z and Z are analogous to the creation and annihilation operators inquantum mechanics. Such unit complex vectors acting on lepton wave function wereshown to be raising and lowering operators in lepton isospace [77]. Interestingly thestructure of complex vectors is bosonic and the product of these vectors contributes tothe particle mass. It seems a complete understanding of the complex vectors would playan important role in exploring the particle theories in geometric algebra. Finally, it hasbeen shown that the complex vectors Z and Z form the complex space and spacetimeand the mathematical structure of the complex vector formalism is discussed.

References

1. Boyer, T.H.: The classical vacuum. Sci. Am. 253, 70 (1985)2. Marshal, T.W.: Random electrodynamics. Proc. R. Soc. Lond. A 276, 475–491 (1963)3. Boyer, T.H.: Random electrodynamics: the theory of classical electrodynamics with classical electro-

magnetic zeropoint radiation. Phys. Rev. D 11, 790–808 (1975)4. Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Academic Press,

Boston (1994)5. de La Pena, L., Cetto, A.M.: The Quantum Dice: An Introduction to Stochastic Electrodynamics.

Kluwer Academic, Dordrecht (1996)6. Haisch, B., Rueda, A., Puthoff, H.E.: Inertia as zeropoint field Lorentz force. Phys. Rev. A49, 678

(1994)7. Rueda, A., Haisch, B.: Contribution to inertial mass by reaction of vacuum to accelerated motion.

Found. Phys. 28, 1057 (1998)8. Boyer, T.H.: Connection between the adiabatic hypothesis of old quantum theory and classical elec-

trodynamics with classical electromagnetic zero-point radiation. Phys. Rev. A 18, 1238 (1978)9. Bohm, A.: Quantum Mechanics Foundations and Applications. Springer, New York (2003)

10. Hestenes, D.: Spin and uncertainty in the interpretation of quantum mechanics. Am. J. Phys. 47,399–415 (1979)

123

294 Found Phys (2014) 44:266–295

11. Uhlenbeck, G.E., Goudsmit, S.: Spinning electrons and the structure of spectra. Nature 117, 264 (1926)12. Bichowsky, F.R., Urey, H.C.: Possible explanation of the relativity doublets and anomalous Zeeman

effect by means of a magnetic electron. Proc. Nat. Acad. Sci. 12, 80 (1926)13. Sakurai, J.J.: Modern Quantum Mechanics. Pearson Education, New Jersey (164)14. Belinfante, F.J.: On the spin angular momentum of mesons. Physica 6, 887–898 (1939)15. Ohanion, H.C.: What is spin? Am. J. Phys. 54, 500 (1986)16. Dirac, P.A.M.: Physical Principles of Quantum Mechanics. Clarendon Press, Oxford (1947)17. Sakurai, J.J.: Advanced Quantum Mechanics, p. 128. Pearson Education, New Jersey (1967)18. Huang, K.: On the zitterbewegung of Dirac electron. Am. J. Phys. 20, 479 (1952)19. Barut, A.O., Bracken, A.J.: Zitterbewegung and the internal geometry of electron. Phys. Rev. D 23,

2454 (1981)20. Van Holten, J.W.: On the electrodynamics of spinning particles. Nucl. Phys. B 356, 3–26 (1991)21. Hestenes, D.: Mysteries and insights of Dirac theory. Annales de la Foundation Louis de Broglie 28,

390–408 (2003)22. Hestenes, D.: Zitterbewegung in radiative processes. In: Hestenes, D., Weingartshofer, A. (eds.) The

Electron, pp. 21–36. Kluwer Academic, Dordrecht (1991)23. Hestenes, D.: Zitterbewegung in quantum mechanics. Found. Phys. 40, 1–54 (2010)24. Sidharth, B.G.: Revisiting zitterbewegung. Int. J. Theor. Phys. 48, 497–806 (2009)25. Newman, E.T.: Classical geometric origin of magnetic moments, spin-angular momentum and the

Dirac gyromagnetic ratio. Phys. Rev. D 65, 104005 (2002)26. Barducci, A., Casalbuoni, R., Lusanna, L.: A Possible Interpretation of theories involving Grassmann

variables. Lett. Nuovo Cim. 19, 581 (1977)27. Barut, A.O., Zanghi, A.J.: Classical model of the Dirac electron. Phys. Rev. Lett. 52, 2009–2012 (1984)28. Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge

(2003)29. Doran, C.J.L., Lasenby, A.N., Gull, S.F., Somaroo, S., Challinor, A.D.: Spacetime algebra and electron

physics. Adv. Imag. Electron Phys. 95, 271–386 (1996)30. Rohrlich, F.: Classical Charged Particles. World Scientific, Singapore (2007)31. Kiessling, M.K.H.: Classical electron theory and conservation laws. Phys. Lett. A 258, 197 (1999)32. Compton, A.H.: The size and shape of electron. Phys. Rev. 14, 247 (1919)33. Frenkel, J.: Die Elektrodynamik des rotierenden elektrons. Z. Phys. 37, 243 (1926)34. Thomas, L.H.: Kinematics of an electron with an axis. Philos. Mag. 3, 1–22 (1927)35. Mathisson, M.: Neue mechanik materietter systeme. Acta. Phys. Pol. 6, 163–200 (1937)36. Weyssenhoff, J., Raabbe, A.: Relativistic dynamics of spin fluids and spin particles. Acta. Phys. Pol.

9, 7 (1947)37. Hovarthy, P.A.: Mathisson’s spinning electron: Noncommutative mechanics and exotic Galilean sym-

metry, 66 years ago. Acta. Phys. Pol. B 34, 2611–2622 (2003)38. Bunge, M.: The picture of the electron. Nuovo Cim. B 1, 977 (1955)39. Bhabha, J.H., Corben, H.: General classical theory of spinning particles in a Maxwell field. Proc. R.

Soc. Lond. A 178(974), 273–314 (1941)40. Rivas, M.: Kinematical Theory of Spinning Particles: Classical and Quantum Mechanical Formalism

of Elementary Particles. Kluwer Academic, Dordrecht (2002)41. Salesi, G.: The spin and Madelung fluid. Mod. Phys. Lett. A 11, 1815–1853 (1996)42. Ghirardi, G.C., Omero, C., Recami, A., Weber, T.: The stochastic interpretation of quantum mechanics:

a critical review. Riv. Nuovo Cim. 1, 1–34 (1978)43. Recami, E., Salesi, G.: Kinematics and hydrodynamics of spinning particles. Phys. Rev. A 57, 98–105

(1998)44. Salesi, G., Recami, E.: A veleocity field and operator for spinning particles in (nonrelativistic) quantum

mechanics. Found. Phys. 28, 763–773 (1998)45. Salesi, G., Recami, E.: Hydrodynamical reformulation and quantum limit of the Barut–Zanghi theory.

Found. Phys. Lett. 10, 533–546 (1997)46. Pavsic, M., Recami, E., Rodrigues, W.A., Maccarrone, G.D., Raciti, F., Saleci, G.: Spin and electron

structure. Phys. Lett. B 318, 481 (1993)47. Vaz, J., Rodriguez, W.A.: Zitterbewegung and the electromagnetic field of the electron. Phys. Lett. B

319, 243 (1993)48. Cavalleri, G.: h derived from cosmology and origin of special relativity and QM. Nuovo Cim. B 112,

1193–1205 (1997)

123

Found Phys (2014) 44:266–295 295

49. Bosi, L., Cavalleri, G., Barbero, F., Bertazzi, G., Tonni E., Spavieri, G.: Review of stochastic electro-dynamics with and without spin. In: Proceedings of Physical Interpretation of Relativity Theory (PIRTXI), London, UK, pp. 12–15 Sep (2008)

50. Clifford, W.K.: On the space theory of matter (1864–1876: Printed 1876), 2, 157–158. In: Newman,J.R. (ed.) In: Proceedings of the Cambridge Philosophical Society. The World of Mathematics, vol. 1.George and Unwin, London (1960)

51. Ryder, L.: Quantum Field Theory. Cambridge University Press, Cambridge (2008)52. Einstein, A.: The Meaning of Relativity. Princeton University Press, Princeton (2006)53. Hestenes, D.: Geometry of Dirac theory. In: Symposium on Mathematics of Physical Spacetime.

Facultad de Quimica, Universdad National Autonoma de Mexico city, pp. 67–96 (1981)54. Wigner, E.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149

(1939)55. Batty-Pratt, E.P., Racey, T.J.: Geometric model of the fundamental particles. Int. J. Theor. Phys. 19,

437–475 (1980)56. Muralidhar, K.: Classical origin of quantum spin. Apeiron 18, 146 (2011)57. Muralidhar, K.: The spin bivector and zeropoint energy in geometric algebra. Adv. Stud. Theor. Phys.

6, 675–686 (2012)58. Tiwari, S.C.: The nature of electronic charge. Found. Phys. Lett. 19, 51–62 (2006)59. Jammer, M.: Concepts of Mass in Contemporary Physics and Philosophy. Princeton University Press,

Princeton (2000)60. Modanese, G.: Inertial mass and vacuum fluctuations in quantum field theory. Found. Phys. Lett. 16,

135–141 (2003)61. Pollock, M.D.: On the Weyl gravitational conjuncture and massive spinor theory. Acta Phys. Pol. B

41, 779–794 (2010)62. Pollock, M.D.: On vacuum fluctuations and particle masses. Found. Phys. 42, 1300–1308 (2012)63. Machicote, J.E.R.: Time as a geometrical concept involving angular relations in classical mechanics

and quantum mechanics. Found. Phys. 40, 1744–1778 (2010)64. Ueda, K., Ishikawa, K.L.: Auttoclocks play devil’s advocate. Nat. Phys. 7, 371–372 (2011)65. Krausz, F., Ivanov, M.: Attosecond physics. Rev. Mod. Phys. 81, 163–234 (2009)66. Chappell, J.M., Iqbal, A., Lannella, N., Abbott, D.: Revisiting special relativity: a natural algebraic

alternative to Minkowski spacetime. PLoS One 45(9), 1480–1490 (2012)67. Okun, L.B.: The concept of mass (mass, energy, relativity). Sov. Phys. Usp. 32, 629 (1989)68. Sobczyk, G.: Special relativity in complex vector algebra. (2007)69. Hestenes, D.: Spacetime physics with geometric algebra. Am. J. Phys. 71, 691–704 (2003)70. Hestenes, D.: Local observables in the Dirac theory. J. Math. Phys. 14, 893 (1973)71. Boudet, R., et al.: Quantum mechanics in the geometry of space-time elementary theory. In: Babaev,

E. (ed.) Springer Briefs in Physics. Springer, Berlin (2011)72. Sobczyk, G.: Unitary geometric algebra. Adv. Appl. Clifford Algebras 22, 827–836 (2012)73. Baylis, W.E., Cabrera, R., Keselica, J.D.: Quantum/classical interface: fermion spin. Adv. Appl. Clif-

ford Algebras 20, 517–545 (2010)74. Sobczyk, G.: Geometric matrix algebra. Linear Algebra Appl. 429, 1163–1173 (2008)75. Vold, T.G.: An introduction to geometric algebra with an application in rigid body mechanics. Am. J.

Phys. 61, 491–504 (1993)76. Grössing, G., Pascasio, J.M., Schwabl, H.: A classical explanation of quantization. Found. Phys. 41,

1437–1453 (2011)77. Hestenes, D.: Spacetime structure of weak and electromagnetic interactions. Found. Phys. 12, 153–168

(1982)

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