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1 correspondence with Dirac’s γ-matrices. In accordance F ∧ a = 2 (Fa + aF ), (12) with (2), the components gµν of the metric tensor are given by so that · 1 · gµν = γµ γν = 2 (γµγν + γν γµ) . (5) Fa = F a + F ∧ a (13) This will be recognized as isomorphic to a famous formula expresses a decomposition of Fa into vector and pseu- of Dirac’s. Of course, the difference here is that the γµ dovector parts. For F = b ∧ c it follows that are vectors rather than matrices. The unit pseudoscalar i for spacetime is related to the (b ∧ c) · a = b(c · a) − c(b · a) . (14) frame {γν } by the equation Many other useful identities can be derived to facilitate i = γ0γ1γ2γ3 = γ0 ∧ γ1 ∧ γ2 ∧ γ3 . (6) coordinate-free computations. They will be introduced as needed throughout the paper. It is readily verified from (6) that i2 = −1, and the geo- Any fixed timelike vector such as {γ0} defines an iner- metric product of i with any vector is anticommutative. tial frame that determines a unique separation between By multiplication the γµ generate a complete basis of 4 space and time directions. Algebraically, this can be ex- k-vectors for STA, consisting of the 2 = 16 linearly in- pressed as the “spacetime split” of each vector x designat- dependent elements ing a spacetime point (or event) into a time component x · γ0 = ct and a spatial position vector x ≡ x ∧ γ0 as 1, γµ, γµ ∧ γν , γµi, i . (7) specified by the geometric product Obviously, this set corresponds to 16 base matrices for the Dirac algebra, with the pseudoscalar i corresponding xγ0 = ct + x . (15) to the Dirac matrix γ5. The entire spacetime algebra is obtained from linear We call this a γ0-split when it is important to specify the x combinations of basis k-vectors in (7). A generic element generating vector. The resulting quantity ct + is called M of the STA, called a multivector, can thus be written a paravector in the expanded form This “split” maps a spacetime vector into the STA subalgebra of even multivectors where, by “regrading,” 4 the bivector part can be identified as a spatial vector. M = α + a + F + bi + βi = hMik , (8) Accordingly, the even subalgebra is generated by a frame k=0 of “spatial vectors” {σ ≡ γ γ ; k =1, 2, 3}, so that X k k 0 where α and β are scalars, a and b are vectors, and F σ1σ2σ3 = γ0γ1γ2γ3 = i. (16) is a bivector. This is a decomposition of M into its k- vector parts, with k =0, 1, 2, 3, 4, where h...ik means “k- Obviously, this rendition of the STA even subalgebra is vector part.” Of course, hMi0 = α, hMi1 = a, hMi2 = F , isomorphic to the Pauli algebra, though the Pauli alge- hMi3 = bi, hMi4 = βi. It is often convenient to drop the bra is not a subalgebra of the Dirac algebra because the subscript on the scalar part, writing hMi = hMi0. matrix dimensions are different. We say that a k-vector is even (odd) if the integer k We use boldface letters exclusively to denote spatial is even (odd). Accordingly, any multivector can be ex- vectors determined by a spacetime split. Spatial vectors pressed as the sum of even and odd parts. A multivector generate a coordinate-free spatial geometric algebra with is said to be “even” if its parts are even k-vectors. The the geometric product even multivectors compose a subalgebra of the STA. We will be using the fact that spinors can be represented as ab = a · b + a ∧ b = a · b + ia × b, (17) even multivectors. Computations are facilitated by the operation of re- where a×b = −i(a∧b) is the usual vector cross product. version. For M in the expanded form (8) the reverse M For the even part hMi+ = Q of the multivector M, a can be defined by spacetime split gives us f M = α + a − F − bi + βi . (9) Q = z + F, (18) where scalar and pseudoscalar parts combine in the form For arbitrary multivectorsf M and N of a complex number ^ (MN)= NM. (10) z = α + iβ, (19) It is useful to extend the definitionse f of inner and outer products to multivectors of higher grade. Thus, for bivec- and the bivector part splits into the form of a “complex tor F and vector a we can define inner and outer products vector” · 1 ˜ F a = 2 (Fa − aF ), (11) F = E + iB = −F. (20) 3

Thus, the even subalgebra in STA has the formal struc- Note that the cross product on the right is distinguished ture of complex quaternions. However, the geometric from the commutator product on the left of this equation interpretation of the elements is decidedly different from by a boldface of the product symbol. Also, it should the usual one assigned to quaternions. Specifically, the be understood that the equivalence of commutator and bivector iB corresponds to a “real vector” in the quater- outer products in this equation does not generally obtain nion literature. This difference stems from a failure to for arbitrary multivectors. distinguish between vectors and bivectors dating back to Concerning the spacetime split of products between Hamilton. For complex quaternions, it reduces to failure even and odd multivectors, for a bivector F = E + iB to identify the imaginary unit i as a pseudoscalar. Geo- and spacteime vector a with the split aγ0 = a0 + a, we metric interpretation is crucial for application of quater- have nions in physics. Reversion in the subalgebra is defined by (F · a)γ0 = E · a + a0E + a × B. (32)

† This may be recognized as the form for a spacetime split Q ≡ γ0Qγ0. (21) of the classical Lorentz force. We will use it as a template This is equivalent to “complexe conjugation” of quater- for other spacetime splits later on. nions. In particular, Concerning differentiation, the derivative with respect to any multivector variable M is denoted by ∂M , so the † F ≡ γ0Fγ0 = E − iB, (22) derivative with respect to a vector variable n is denoted by ∂n. As the derivative with respect to a position vec- so that e tor x is especially important, we distinguish it with the 1 † 1 † special symbol E = 2 (F + F ), iB = 2 (F − F ). (23) ∇ ≡ ∂x = σk∂k, (33) Moreover, in agreement with standard vector calculus. Thus, for a † 2 2 × F F = E + B +2E B, (24) relative vector field A = A(x) The identity (17) gives us

∇A = ∇· A + ∇ ∧ A = ∇· A + i∇ × A, (34) F 2 = F · F + F ∧ F = E2 − B2 + iE · B, (25) which relates the curl to the standard vector cross prod- which are familiar expressions from electrodynamics. the bivector F is said to be simple if uct. For field theory, the derivative with respect to a space- F ∧ F =0 ⇔ E · B =0, (26) time point must be defined. Though that can be done in a completely coordinate-free way [2], for a rapid sur- and that is said to be timelike, spacelike or null, respec- vey it is more expedient here to exploit the reader’s prior tively, when F 2 = E2 − B2 is positive, negative or zero. knowledge about partial derivatives. Sometimes it is convenient to decompose the geometric For each spacetime point x the reciprocal of a standard product F G into symmetric and antisymmetric parts frame {γµ} determines a set of “rectangular coordinates” {xµ} given by F G = F ◦ G + F × G, (27) µ µ µ x = γ · x and x = x γµ . (35) where the symmetric product is defined by In terms of these coordinates the derivative with respect 1  F ◦ G ≡ 2 (F G + GF ), (28) to a spacetime point x is an operator defined by and the commutator product is defined by  µ 1 ≡ ∂x = γ ∂µ, (36) F × G ≡ 2 (F G − GF ). (29) where ∂µ is given by In particular, for quaternions the symmetric product serves as a “complex inner product,” while the commu- ∂ ∂ = = γ ·  . (37) tator product serves as an ”outer product for complex µ ∂xµ µ vectors.” Comparison with (17) shows that for “real vec- tors” The square of  is the usual d’Alembertian 2 µν µν µ · ν a ◦ b = a · b, (30) = g ∂µ∂ν where g = γ γ . (38) and The matrix representation of the vector derivative  will be recognized as the so-called “Dirac operator,” origi- a × b = a ∧ b = i(a × b), (31) nally discovered by Dirac when seeking a “square root” of 4 the d’Alembertian (38) in order to find a first order “rel- in explaining the hydrogen spectrum and the magical ap- ativistically invariant” wave equation for the electron. In pearance of spin, it was discovered that the electron had STA however, where the γµ are vectors rather than ma- an antiparticle twin, the positron, conjoined with it in trices, it is clear that  is a vector operator, and we the Dirac equation. Dirac introduced a surgical proce- see that it is as significant in Maxwell’s equations as in dure called “Hole theory” that suppressed the positron Dirac’s. to keep it from interfering with the electron. Eventu- The symbol ∇ ≡ ∂x is often used elsewhere [2, 3] in- ally, electron and positron were identified with positive stead of  ≡ ∂x, but it has the disadvantage of con- and negative energy states and separated by a procedure fusability with ∇ ≡ ∂x in some contexts. Besides, the called “second quantization.” That has become the sur- triangle is suggestive of three dimensions, while the  is gical procedure of choice in QED. Here we take a new suggestive of four. That is why the  was adopted in the look at the anatomy of the Dirac equation to see what first book on STA [4], and earlier by Sommerfeld [5] and makes the electron tick. That leads to an alternative Morse and Feshbach [6]. surgical procedure for separating the electron twins with Note that the symbol ∂t for the derivative with respect new physical implications. to a scalar variable t denotes the standard partial deriva- In terms of STA the Dirac equation can be written in tive, though the coordinate index is used as the subscript the form in (37). e γµ(∂ Ψi~ − A Ψ) = m cΨγ , (44) In STA an electromagnetic field is represented by a µ c µ e 0 bivector-valued function F = F (x) on spacetime. Since  is a vector operator the expansion (13) applies, so we where me is electron mass and now we use e = ±|e| for can write the charge coupling constant, while the Aµ = A · γµ are components of the electromagnetic vector potential. The F =  · F +  ∧ F, (39) symbol i denotes a unit bivector, which can be written in the following equivalent forms: where  · F is the divergence of F and  ∧ F is the curl. Corresponding to the split of a spacetime point (15), i ≡ γ2γ1 = iγ3γ0 = iσ3 = σ1σ2 (45) the spacetime split of the vector derivative  = ∂x gives us a paravector derivative The notation i emphasizes that it plays the role of the unit imaginary that appears explicitly in matrix versions  γ0 = ∂0 + ∇, (40) of the Dirac equation. Let us refer to (44) as the real Dirac equation to dis- ·  −1 where ∂0 = γ0 = c ∂t. Hence, for example, the tinguish it from the standard matrix version. It is well d’Alembertian takes the familiar form established that the two versions are mathematically iso- morphic [2, 3, 7]. However, the real version reveals ge- 2 = ∂2 − ∇2, (41) 0 ometric structure in the Dirac theory that is so deeply hidden in the matrix version that it remains unrecog- and the divergence of the vector field A = (cϕ + A)γ0 splits to nized by QED experts to this day. That fact is already evident in the identification of the imaginary unit i as

 · A = ∂0 ϕ + ∇· A. (42) a bivector. As we see below, this identification couples complex numbers in quantum mechanics inextricably to Finally, it is worth mentioning that to evaluate vector spin, with profound implications for physical interpre- derivatives without resorting to coordinates, a few ba- tation. It is the first of several insights into geometric sic formulas are needed. For vector n and bivector F , structure of Dirac theory that will guide us to a reformu- we shall have use for the following derivatives of linear lation and new interpretation. functions: Employing the vector derivative puts the real Dirac equation in the coordinate-free form ∂ n =4, ∂ Fn =0, ∂ (n · F )=2F. (43) n n n e Ψi~ − AΨ= m cΨγ , (46) c e 0 III. ANATOMY OF THE DIRAC WAVE where A = A γµ is the electromagnetic vector potential. FUNCTION µ The spinor “wave function” Ψ = Ψ(x) admits to the Lorentz invariant decomposition Considering the central role of Dirac’s equation in the 1 spectacular successes of quantum mechanics and QED, it Ψ= ψeiβ/2 with ψ(x)= ρ 2 R(x), (47) seems indubitable that this compact equation embodies some deep truth about the nature of the electron, and where ρ = ρ(x) and β = β(x) are scalar-valued functions, perhaps elementary particles in general. However, suc- and “rotor” R = R(x) is normalized to cess came with problems that called for action by the doc- tors of Quantum Mechanics. Soon after the initial success RR = RR =1. (48)

e e 5

The Lorentz invariant “β-factor” in the general form (47) To clarify the matter, we decompose the rotor R into the for a “Real Dirac spinor” has been singled out for spe- product cial consideration. As this factor is so deeply buried in −iϕ matrix representations for spinors, its existence has not R = Ve . (55) been generally recognized and its physical interpretation Then has remained problematic to this day. We shall see it as a candidate for corrective surgery on the Dirac wave −Iϕ Iϕ 2Iϕ e1 = Rγ1R = e a1e = a1e , (56) function. We shall also be considering singular solutions Ψ± of where e the Dirac equation (46) called Majorana states and de- fined by I ≡ R iR = V iV and a1 = Vγ1V, (57)

with an analogous equation for e2. This exhibits the wave Ψ± = Ψ(1 ± σ2)=Ψγ±γ0, (49) e e e function phase ϕ as an angle of rotation in a spacelike where plane with tangent bivector I = I(x) at each spacetime point x. Moreover, the direction of that plane is deter- γ± = γ0 ± γ2. (50) mined by the spin bivector defined by ~ ~ We shall see that STA reveals properties of these states S ≡ isv = R iR = I. (58) that make them attractive candidates for distinct elec- 2 2 tron and positron states. Thus, we have a connection betweene spin and phase with the phase as an angle of rotation in the “spin plane.” In general, the Lorentz rotation (51) has a unique de- A. Local Observables composition into a spatial rotation followed by boost, which is generated by the rotor product [2] We begin physical interpretation of the Dirac wave function with identification of “local observables.” At R = VU (59) each spacetime point x, the rotor R = R(x) determines 1/2 with Uγ0U = γ0 and V = (vγ0) . a Lorentz rotation of a given fixed frame of vectors {γµ} into a frame {e = e (x)} given by µ µ For simplicity, we oftene refer to rotors V and U by the same names “boost” and “spatial rotation” used for the eµ = RγµR. (51) Lorentz transformations they generate. We can further decompose the rotor product into In other words, R determines ae unique frame field on spacetime. Whence, the wave function determines four R = U1V0U1U = U1V0U2 (60) vector fields where e ΨγµΨ= ψγµψ = ρeµ. (52) V0 = exp {α1σ2} = cosh α1 + σ2 sinh α1 (61) Note that the β-factore has cancellede out of these expres- σ sions because the pseudoscalar i anticommutes with the is a boost in a fixed direction 2 = γ2γ0, while U1 and U2 are spatial rotations. vectors γµ. It can be shown [2, 3, 7] that two of the vector fields (51) correspond to well known quantities in matrix Dirac B. Electron clock and chirality theory. The quantity As the notion of an electron clock was central to de ψγ0ψ = ρv with v = Rγ0R = e0. (53) Broglie’s seminal contribution to quantum mechanics [8], is the Dirac currente . The Born interpretatione identifies its relevance to interpretation of the Dirac equation de- this as a “probability current;” whence, ρ is a probability serves thorough investigation. The clock mechanism can density. (We shall consider an alternative interpretation be defined by considering a Dirac plane wave solution of for ρ later on.) The quantity the form (55) with momentum p, wherein the phase has the specific form ϕ = k · x. Then ϕ = k, and the Dirac ~ ~ equation (46) gives us s = Rγ R = e (54) 2 3 2 3 iβ/2 iβ/2 ~kRe = mecRe γ0, (62) can be identified as the electrone “spin vector,” though it looks rather different than its matrix counterpart. which we solve for Physical interpretation of e and e is more subtle, as 1 2 mec −iβ these vectors are not recognized in standard Dirac theory. k = ~ ve . (63) 6

This has two solutions with opposite signs given by Finally, we note that the Born probability density has cos β = ±1 and momentum p = mecv = ±~k, been set to ρ = 1 on the propagating hyperplane, thus Equation v·x = cτ describes a propagating hyperplane implying that all points on the hyperplane are equally with unit normal v, so (63) gives probable positions x0 for the electron at initial time τ0. However, for any initial position x0, the velocity v =x ˙ 2 p · x = mec τ. (64) integrates to a unique position

Accordingly, the vector e1 in (56) rotates in (or on) the x(τ)= vτ + x0. (67) hyperplane with frequency Thus, the plane wave solution consists of an ensemble 2m c2 dϕ of equally probable particle paths composing a congru- ω ≡ e = ±2 . (65) e ~ dτ ence (or fibration) of non-intersecting, timelike paths that sweep out (fibrate) a region of spacetime. The handedness is opposite for the two solutions This will be recognized as the zitterbewegung frequency of Schr¨odinger. It is precisely twice the de Broglie frequency C. Electron clock with zitter because the wave function phase angle is precisely half the rotation angle of the observables in (56). The sign of There is another plane wave solution that has been the phase specifies the sense of rotation, which is opposite largely overlooked in the literature. We simply switch for electron and positron. (55) into the form (with ρ = 1) We can now give the vector e1 a picturesque physical −iϕ interpretation as the hand on de Broglie’s electron clock, ψ = e V0, (68) with its rotation given by (56). The face of the clock is the bivector I in (57), and the reference point for an ini- which is of type (60) with constant V0 given by (61). This solves the Dirac equation with ϕ = p·x/~ and p = mecγ0. tial time on the clock face is given by the vector a1. This description of the electron clock is completely general, as To verify that: the equations hold for an arbitrary electron wave func- ψi~ = −m cγ iσ ψiσ = m cψγ . (69) tion. Indeed, equation (57) shows that the electron clock e 0 3 3 e 0 can be described as an “inertial clock,” because it retains Note that γ0iσ3 = iγ3 commutes with ψ, whereas γ0 and the mark of initial time even as interactions change the iσ3 do not. Generalization to a solution for arbitrary rotor R and hence spin direction and the attitude of the constant p = mecV γ0V is obviously given by a boost to clock in spacetime. ψ′ = V ψ. Of course, interactions can change the clock frequency Now, using (61) wee can express the wave function (68) by changing the phase ϕ. Nevertheless, the free electron as the sum of positive and negative energy solutions: frequency remains as a reference standard for the electron −ik·x +ik·x clock. This suggests that we define the free electron clock ψ = cosh α1e + σ2 sinh α1e ≡ ψ+ + ψ−. (70) period τe as the fundamental unit of electron time. Its empirical value, which I propose to call the zit, is The analog of hermitian conjugate in standard matrix † Dirac algebra is defined by ψ = γ0ψγ0. Whence, the 2π h × −21 velocity is given by 1 zit = τe = = 2 =4.0466 10 sec (66) ωe 2mec e † v = ψγ0ψ = ψψ γ0 Approximately: 1 zit ≈ 4 zepto-sec; 1 sec ≈ 1/4 zetta-zit. 2 2 † † = {|ψ+| + |ψ−| + ψ+ψ + ψ−ψ }γ0 Remarkably, direct measurement of the “zit” may be pos- e e − + sible with electron channeling experiments [9]. 2 † i2φ = |ψ| γ0 +2 < ψ−ψ+ >γ2e . (71) The two signs in (65) indicate clocks with opposite “handedness” or chirality, as we shall say. We identify the In agreement with [10], this exhibits zitterbewegung as negative sign with an electron clock and the the positive arising from interference between positive and negative sign with a positron clock. Indeed, in standard theory energy states, as originally formulated by Schr¨odinger. the two signs are interpretated as states with opposite However, it also exhibits zitterbewegung as circulation of energy and the negative energy state is identified with electron velocity in the spin plane. I have coined the term the positron. However, we have seen that the sign is zitter to distinguish this interpretation of zitterbewegung actually determined by cos β = ±1 without reference to a from alternatives. concept of energy. This suggests that we interpret β as a This result settles a long-standing controversy about “chirality parameter.” Be that as it may, we can see that the interpretation of zitterbewegung. To this day, stud- the vector e2 specifies the clock-face direction of motion ies of Dirac wave packets (e.g. [11]) fail to recognize for the clock hand e1. Hence “antiparticle conjugation” the connection of zitterbewegung to spin. Instead, it is should be defined to reverse the direction of e2 while identified as a high frequency interference effect, often at- keeping the direction of e1 unchanged. tributed to interaction with the vacuum with a negative 7 energy component ψ− presumed to express presence of which may be recognized as a Foldy-Wouthuysen (FW) positrons. On the contrary, in the zitter model here the transformation [12]. “negative energy” term has nothing to do with positrons. The FW transformation is commonly used to eliminate Instead, it is a structural feature of electron motion in- negative energy components in electron wave functions, volving electron spin and phase. often because they are regarded as “unphysical.” With- We can associate our zitter plane wave with particle out going into arguments supporting this practice, the motion in the same way we did it for the plane wave in point here is that it suppresses the role of zitter in de- the preceding subsection. Without loss of generality, we scribing electron motion. can write p = mecγ0, so ϕ = p · x/~ = ωeτ/2 defines a To ascertain what the Dirac equation can tell us about plane propagating in the direction of p with proper time the physical significance of zitter, β and electron clock, τ. Then (68) and (71) gives us a parametric equation for we study the properties of local observables thoroughly in the particle velocity: the next section. This will help us address such questions as: Is zitter an objectively real physical property of the 1 1 electron? Should electron phase (de Broglie’s clock) be − iωeτ iωeτ iωeτ v(τ)= e 2 v0 e 2 = aγ0 + bγ2 e , (72) regarded as a feature of electron zitter? What is the role of zitter in quantization? Of course, the answers will lead where a and b are constants, while ωe is the free particle to more questions and speculation. zitter frequency. For v =x ˙, this integrates to

x(τ)= γ acτ + bλ e + x , (73) 0 e 1 0 D. Flow of Local Observables where λe = c/ωe and We turn now to a general analysis of conservation laws iωeτ e1(τ)= γ1 e , (74) implied by the Dirac equation as a foundation for physi- cal interpretation. To facilitate comparison with conven- is the electron clock vector. tional Dirac theory, we first express the conservation laws The particle path x(τ) specified by (73) is a timelike in terms of the wave function. Then we peel them apart helix with pitch bλe/a. Thus, the zitter plane wave solu- to reveal their structure in terms of local observables. tion consists of an ensemble of equally probable particle A conservation law for the Dirac current Ψγ0Ψ˜ = ρv paths that fibrate a region of spacetime with a congru- is easily derived from the Dirac equation (46) and takes ence of non-intersecting, timelike helices. the form Though the circular frequency ωe is constant, the cir- cular speed increases with radius bλe without reaching  · (ρv)=0. (78) the limiting case λeωe = c at the speed of light. In that σ limit, V0 → 1+ 2 in (60), and we get the Majorana wave This can be interpreted as flow of a fluid with proper function Ψ+ defined in (49), so the velocity vector (72) density ρ. Precisely what kind of fluid depends on the becomes a null vector interpretation of other local observables, in particular, observables describing the flow of energy, momentum, u(τ)=Ψ γ Ψ + 0 + charge and electromagnetic potential. Following a sys- 1 i 1 i i − ωeτ ωeτ ωeτ tematic approach in defining these observables within the = e 2 γ+e 2 = γ0 + γ2 e . (75) e Dirac theory, we shall discover hidden structure that has In this case, zitter with electron clock is intrinsic to elec- been generally overlooked. tron motion, whereas in the previous case described by The original formulation of the Dirac equation was (72) the zitter can vanish with b = 0. based on interpreting Thus, we have three distinct kinds of free particle e (plane wave) states: Kind A, given by (55), with no zit- p µ = i~ ∂µ − Aµ (79) ter; Kind B, given by (68) and (72), with zitter velocity c ranging between zero and the speed of light; and Kind as a gauge invariant energymomentum operator. The un- C, given by (75), with zitter velocity λeωe = c. derbar notation here designates a linear operator. Specif- Kind B is related to Kind A by a unitary transforma- ically, the operator i designates multiplication by the unit tion. For example, (68) is related to (55) by imaginary in the matrix version of Dirac theory, and right −iϕ † −i†ϕ multiplication by the unit bivector i = iγ3γ0 in the STA γ1(e V0)γ1 = V0e , (76) version, as specified in where γ† = γ∗ = −γ and the right side is interpreted as e 1 1 1 p Ψ= ~ ∂ Ψiγ γ − A Ψ. (80) a positive energy factor with i replaced by i†. It can be µ µ 3 0 c µ generated by the continuous unitary transformation Equivalence of operators in the matrix version to expres- ψ → W ψW †, where W = eγ1 α0 , (77) sions in the present STA version is discussed in [13]. 8

The energymomentum operator also led to the defini- We need the Dirac equation (46) to evaluate this. Since tion of an energy momentum tensor T (n) with compo- µ nents ∂µΨiγ3 ∂ Ψ =0, (88) 1 µν µ · ν µ ν D E T = T γ = hγ0Ψγ p Ψi, (81) we have e ~ where 2 2 2 e ~( Ψ)iγ3Ψ = [ Ψiγ3Ψ − Ψiγ3 Ψ] 1 2 T µ = T (γµ). (82) D E e e = ρ ( ∧ A) · v + ∂ (ρ vAµ). (89) e c e c µ e The stress tensor T (n) is defined physically as a vector- valued tensor field specifying, at each spacetime point, Whence the energymomentum flux through a hypersurface with e T`(` )= ∂ T µ = F · (ρv) ≡ ρf, (90) unit normal n. Its adjoint T (n) can be defined by µ c γµ · T (γν )= T (γµ) · γν = T µν. (83) where F =  ∧ A is an external electromagnetic field. This has precisely the form for the Lorentz force on a Note the overbar notation T (n) to indicate the adjoint classical charged fluid, and it supports the interpretation of a linear function T (n) specified by an underbar. In of the Dirac current eρv as a charge current. this case the linear functions are vector-valued, but the A conservation law for angular momentum can be same notation is used for bivector-valued linear functions derived from invariance of the Lagrangian (84) under below. Lorentz rotations [14], but we derive it directly from The Dirac equation (46) can be derived from the La- properties of the stress tensor, as it makes structure more grangian explicit. From (90) we get

e µ µ L = ~ Ψiγ Ψ − AΨγ Ψ − m cΨΨ . (84) ∂µ(T ∧ x)= T ∧ γµ + ρf ∧ x. (91) 3 c 0 e D E To see how this equation gives us angular momentum As is well known, ae major advantagee of thise approach conservation, we need to analyze the first term on the is that conservation laws consistent with the equations right. In doing so we find other interesting results as of motion can be derived from symmetries of the La- byproducts. grangian. The most elegant and efficient way to do this First, note that is the method of multivector differentiation introduced by Lasenby, Doran and Gull in [14]. In particular, from µ γ ~(∂µΨ)iγ3Ψ = ~(Ψ)iγ3Ψ+ (iρs). (92) translation invariance of the Lagrangian they derived the 1 stress tensor D E Then, combine thise with the Dirac equatione (46) in the µ form T (n)= γµ (p Ψ)γ0Ψn e D E e ~(Ψ)iγ Ψ= m cρeiβ + Aρv (93) = γµ (~∂ Ψiγ γ )γ Ψn − Aρ (v · n). (85) 3 e c µ e3 0 0 c D E to get e This is equivalent to the stresse tensor most commonly µ iβ employed in Dirac theory. ∂nT (n)= γµT (γ )= (iρs)+ mecρe . (94) However, when Lasenby, Doran and Gull generalized their method in a ground breaking paper on Gauge The- The scalar part of this equation gives us the trace of the ory Gravity [15], translation invariance gave instead the stress tensor: adjoint stress tensor T r(T )= ∂n · T (n)= mecρ cos β, (95) e T (n)= ~(n · Ψ)iγ3Ψ − (A · n)ρ v. (86) 1 c and the pseudoscalar part gives us D E This raises a question as to whiche stress tensor is correct  · (ρs)= mecρ sin β. (96) for the electron: T (n) or T (n)? We leave the question open for the time being while we examine and compare This displays a peculiar relation of β to mass and spin of their properties. questionable physical significance. However, β plays no The dynamics of flow is determined by the divergence role in the bivector part of (94), which gives us of the stress tensor: µ µ µ µ γµ ∧ T (γ )= T (γ ) ∧ γ =  · (iρs)= ∂µS , (97) T`(` )= ∂ T (γµ)= ∂ T µ µ µ where ~ 2 e µ = ( Ψ)iγ3Ψ − ∂µ(ρ vA ). (87) µ µ µ µ 1 c S = S(γ )= γ · (iρs)= ρi(s ∧ γ ). (98) D E e 9 is identified as a bivector-valued spin flux tensor. This sign difference can be interpreted geometrically as Equation (97) gives us an explicit relation between the an opposite orientation of spin S to velocity v or mo- stress tensor and its adjoint: menta p and pc. To probe the difference between the momenta p and pc more deeply, we express them as ex- · µ  · T (n) − T (n)= n (γµ ∧ T ) = (n ∧ ) (iρs). (99) plicit functions of local observables. The dynamics of the local observables e = Rγ R is And inserting this into (90), we find that the divergence µ µ determined by the linear bivector-valued function of the stress tensor is equal to the divergence of its ad- joint: e Ωµ =Ω(γµ) ≡ 2(∂µR)R.˜ (109) e ∂ T (γµ)= ∂ T (γµ)= F · (ρv)= ρf. (100) Thus, µ µ c · This equivalent divergence of the stress tensor and its ad- ∂ν eµ =Ων eµ. (110) joint has been overlooked in the literature. Let’s consider In particular, the derivatives of the velocity and spin vec- the difference more closely. tors are The flux of momentum in the direction of the Dirac current is especially significant, because that is the flow ∂ν v =Ων · v and ∂ν s =Ων · s, (111) direction of electric charge. Accordingly, we define a mo- mentum density ρp along this flow by while the derivative of the spin bivector S = isv is given by the commutator product: ρp ≡ T (v). (101) ∂µS =Ωµ × S. (112) The adjoint determines a “conjugate momentum” density ρpc defined by Now, with the wave function in the form Ψ= ψeiβ/2 = Re(α+iβ)/2, (113) ρpc ≡ T (v). (102) We will be looking to ascertain the physical difference its derivatives can be related to observables by between these two kinds of momenta. First we note a ~(∂ Ψ)iγ Ψ= ρ[(i∂ α + ∂ β)s +Ω Sv], (114) small difference in angular momentum. µ 3 µ µ µ Returning now to the question of angular momentum with the product expansion conservation, inserting (97) into (91), we get the desired e conservation law: ΩµS = Pµ + ∂µS + iqµ (115)

µ J`(` )= ∂µJ = ρf ∧ x. (103) where we have identified components Pµ = γµ · P of the “canonical momentum vector” P defined by where the total angular momentum tensor flux is a ~ ~ bivector-valued tensor with orbital and spin parts defined P =Ω · S = e · ∂ e = − e · ∂ e , (116) by µ µ 2 1 µ 2 2 2 µ 1

J µ = J(γµ)= T µ ∧ x + Sµ. (104) and the pseudoscalar part is given by · · Accordingly, the angular momentum flux along the Dirac iqµ =Ωµ ∧ S = iΩµ (sv)= i(∂µs) v. (117) current is given by Finally, by inserting (114) into (86) we get the compo- nents of the stress tensor in the transparent form J(v)= ρ(p ∧ x + S), (105) e T = ρ[v (P − A ) + (v ∧ γ ) · ∂ S − s ∂ β]. (118) where S(v) = ρS confirms our earlier identification of µν µ ν c ν µ ν µ ν S = isv as a spin bivector. Alternatively, we can define a “conjugate” angular mo- This gives us an informative expression for the conjugate mentum tensor momentum: e µ µ µ T (v)= ρ{[(P − A) · v]v + S˙ · v − sβ˙} = ρpc. (119) Jc = T (γ ) ∧ x − S , (106) c which by the same argument yields the conservation law The first two terms in this expression for momentum flow along a streamline of the Dirac current make per- µ e ∂µJ = ρf ∧ x, (107) · c fect physical sense. Note that the factor (P − c A) v serves as a gauge invariant variable mass determined by But the conjugate angular momentum flow has a spin of the frequency of the electron clock, which is specified by opposite sign: ~ ~ P · v =Ω · S = e · e˙ = − e · e˙ , (120) J c(v)= ρ(pc ∧ x − S). (108) v 2 1 2 2 2 1 10 where where again the role of β is problematic. The Gordon current can be regarded as a reformulation  e˙µ = v · eµ =Ωv · eµ and Ωv =Ω(v). (121) of the Dirac equation in terms of local observables, as our derivation of (127) shows. For this reason, it plays The second term S˙ · v =v ˙ · S in (119) specifies a contri- a fundamental role in our analysis of alternative physical bution of spin to linear momentum due to acceleration. interpretations in subsequent sections. But first we try However, a physical interpretation for the last term in- to make some sense of β. volving the directional derivative β˙ = v · β remains problematic. From the stress energy components (118), we also get a E. Problems with β remarkably simple expression for the momentum density: e We begin our study of β by reformulating the Dirac T (v)= ρ(P − A)= ρp. (122) iβ/2 c Lagrangian (84) with Ψ = ψe to make the role of β explicit and then to relate it to the explicit role of other And for flux in the spin direction we get: observables: ~ ~  e  T (ˆs)= ρ β. (123) L = ψiγ3ψ − Aψγ0ψ − mecρ cos β − ρs β 2 c D e  E = ρ(P −e A) · v + (ev ∧ ) · (ρS) Combining (122) with (119) we get c − mecρ cos β − ρs · β. (129) pc = (p · v)v + S˙ · v − sβ.˙ (124) The mass term < mecΨΨ > = mecρ cos β has always As we shall see, it is especially important to note that in been problematic in QED. Indeed, it has been eliminated these equations both p and pc are defined independently from the Standard Model,e which aims to derive the mass of ρ, and physical interpretation of the strange parameter from fundamental theory. β appears to be tied up with spin. When the β-factor eiβ is constant, (47) can be used to Having thus identified the canonical momentum P as factor it out of the Dirac equation (46) to exhibit its role a local observable, we can express the Dirac equation as explicitly: a constitutive equation relating observables. Thus, from e (114) and (115) we derive the expression {~ψiγ γ − Aψ}ψeiβ = m cψγ ψ = m cρv. (130) 3 0 c e 0 e ~   iβ −iβ ( Ψ)iγ3Ψ = [ρP + [ (ρe S)]e ]v, (125) Note that setting eiβ =e −1 amountse to reversing orien- tation of the bivector i = iγ3γ0 that generates rotations which we inserte into the Dirac equation (93) to get it in in the phase plane along with reversing the sign of the the form charge, as required for antiparticle conjugation according e to the chirality hypothesis. Accordingly, the Dirac equa- ρ(P − A)eiβ = m cρv − (ρeiβS) (126) c e tion is resolved into separate equations for electron and positron. Its vector part is a constitutive equation involving the We have seen how plane wave solutions of the Dirac Dirac current: equation suggest that the β distinguishes particle from e antiparticle states. Let me call that suggestion the “chi- ρ(P − A)cos β = m cρv −  · (ρeiβS). (127) c e rality hypothesis.” Some credence to this hypothesis is given by the fact that unitary spinors R and Ri are The right side of this equation has vanishing divergence, distinct spin representations of the Lorentz group, so and we identify it as the well known Gordon current. it is natural to associate them with distinct particles. Unlike the vector part, the trivector part of (126) does However, the Dirac spinor Reiβ/2 is a continuous con- not have any evident physical meaning, though it does nection between both representations, suggesting that β serve as a constraint among the variables. parametrizes an admixture of particle/antiparticle states. This completes our exact reformulation of Dirac The- After I discovered that cos β = ±1 solves the problem ory in terms of local observables. We have found clear of negative energies for plane waves and thereby sepa- physical interpretations for all components of the Dirac rates electron and positron plane wave states [7], I set wave function except the parameter β. The strangeness about studying the physical significance of β in the gen- of β is most explicit in equation (127) for the Gordon iβ eral case. I got great help from my graduate student current, where the factor e generates a duality rotation Richard Gurtler, who thoroughly examined the behavior without obvious physical significance. And that equation of β in the Darwin solutions of the Dirac equation for implies the conservation law Hydrogen [16]. The results do not seem to support the e chirality hypothesis, for the parameter β varies with posi-  · [ρ(P − A)cos β]=  · (ρp cos β)=0, (128) c tion in peculiar ways. The values cos β = ±1 appear only 11 in the azimuthal plane, which suggests that 2d solutions where F =  ∧ A is the external electromagnetic field. might satisfy the chirality hypothesis, but the chirality Dotting with velocity v we get jumps in sign across nodes in the plane in an unphysical e way. At about the same time I began a systematic study p˙ = F · v + v · ( ∧ P )+ f (v), (137) c s of local observables in Dirac theory [17] and their roles in Pauli and Schr¨odinger theories [18–20] , but I was unable which looks like an equation of motion with Lorentz force, to make sense of the peculiar behavior I found for β. though functional dependence of P and fs(v) ≡ ` p` · v This problem of interpreting β has never been recog- remain to be determined. nized in standard QED. Indeed, it is commonly claimed Now, to relate the Gordon current to the conjugate that second quantization solves Dirac’s problem of nega- momentum, we note that v · S = 0 implies tive energies. My suspicion is that cos β = ±1 has been  ·  · tacitly assumed in QED when it begins by quantizing (v ∧ ) (ρS)= ρ( ∧ v) S. plane wave states. Consistency of that procedure with Hence the momentum equation (134) gives us a density- the Darwin solutions has never been proved to my knowl- free expression for generalized mass: edge. It seems that a perceived need for such a proof is avoided by claiming that the Darwin case is concerned v · p = mec + S · ( ∧ v). (138) with one-particle quantum mechanics, whereas QED is a many-particle theory. Whence (122) gives us an explcit expression for the con- To nail down the interpretation of β, we need to study jugate momentum: its role in specific solutions of the Dirac equation. A step ·  ˙ · in that direction will be taken in the next Section. In pc = [mec + S ( ∧ v)]v + S v. (139) the meantime, it is worth looking at simplifications when To understand this better, we examine what the Dirac cos β = ±1. equation says about the velocity curl. With cos β = 1 the Dirac equation (126) reduces to Since the Dirac current ρv is conserved, it determines e a congruence of velocity streamlines, which can be re- ρ(P − A)= m cρv − (ρS). (131) c e garded as timelike paths. We are interested in how local observables evolve along each path. The evolution of the To reiterate an important remark: the physical content comoving frame eµ = RγµR along a path is determined of this equation resides entirely in its vector part. The by the equation of motion trivector part e e˙µ = v ·  eµ =Ω · eµ, (140)  ∧ (ρS)=  · (ρsv)i =0. (132) with proper angular velocity appears to be a constraint on density flow beyond the µ ·  conservation laws Ω ≡ v Ωµ = 2(v R)R. (141) However, the proper definition of the time derivative for  · (ρv) = 0 and  · (ρs)=0. (133) e rotor R is tricky. We must look to the Dirac equation to Obviously, these are kinematic constraints on local ob- see how to do that and to express Ω in terms of observ- servables independent of external interactions. ables. The Dirac equation (130) with cos β = 1 can be Comparison of (131) with (122) shows that the vec- put in the form tor part equates the momentum density to the Gordon 2(ψ)ψ−1 = 2(R)R +  ln ρ current: e −1 e = (mecv + A)S . (142) ρp = ρ(P − A)= m cρv −  · (ρS). (134) e c c e We can eliminate R with the identity The conservation law for the Dirac current and the iden- tity (R)Rv + v(R)R = v + 2(v · R)R, (143)

 · [ · (ρS)] = ( ∧ ) · (ρS)=0 and then separatee scalare and bivector parts.e From the scalar part of (143) we get the familiar con- then gives us the Gordon conservation law servation law for the Dirac current:  ·  · (ρp)=0. (135) (ρv)=0. (144)

Further, note that the momentum p = P − (e/c)A is But note now that this law plays an essential role in independent of ρ. Its curl has the strikingly simple form defining an invariant time derivative for local observables as the directional derivative along a particle path. Thus, e  ∧ p = − F +  ∧ P, (136) ·   · c v ln ρ =ρ/ρ ˙ = − v. (145) 12

To be invariant, the time derivative of rotor R must be Subsequently, physical interpretation of Schr¨odinger defined differently. theory has been hotly debated, while, ironically, relevant From the bivector part of (143) we get implications of the more precise Dirac theory have been overlooked. To correct this deficiency, our first task here e −1  ∧ v + 2(v · R)R = (mec + A · v)S . (146) is to update Born-Dirac theory with recent insights on c interpretation of Schr¨odinger theory. Then we can con- In other words, e sider enhancements from our study of local observables in Dirac theory. e  ∧ v +Ω=(m c + A · v)S−1. (147) After decades of debate and clarifications, it seems safe e c to declare that de Broglie–Bohm “Pilot Wave” theory is well established as a viable interpretation of quantum This is the expression for Ω that we were looking for. It mechanics, though that may still be a minority opinion is eminently reasonable that rotation along Dirac stream- among physicists. Current accounts suitable for our pur- lines be determined by the curl of velocity. And the term poses are given in [21, 22]. The point to be emphasized generating rotation in the spin plane specifies time evolu- here can be regarded as a refinement of the Born rule, tion of the electron clock, including the effect of external which says the wave function for a single electron speci- potentials. Thus, for constant velocity it reduces to fies its probable position at a given time. The Pilot Wave rule extends that to regarding the wave function as spec- ·  2 −1 cΩ=2c(v R)R = mec S = −ωeI, (148) ifying an ensemble of possible particle paths, with the electron traversing exactly one of those paths, but with a where ω = 2m c2/~ is the familiar free particle zitter e e e certain probability for each path. So to speak, the wave frequency and I = e e =2S/~ is the unit spin bivector. 2 1 function serves to guide the electron along a definite path, For the comoving frame (146) gives us but with a specified probability. Hence the name “pilot wave” for the wave function. In his “theory of the double v˙ =Ω · v = v · ( ∧ v)=Ω · v, (149) solution,” de Broglie argued for a physical mechanism to select precisely one of those paths, but that alternative is s˙ =Ω · s = s · ( ∧ v)=Ω · s. (150) not available in conventional Pilot Wave theory. Instead, path selection is said to require an act of observation, and which continues to be a subject of contentious debate and will not be discussed here. Ω · I = e1 · Ω · e2 = e1 · e˙2 = −e2 · e˙1 Strictly speaking the Pilot-Wave rule requires only an assignment of particle paths to interpret the wave func- = −I · ( ∧ v) + (ωe + eA · v)/c. (151) tion; whence, ρ(x,t) can be interpreted as a density This last result describes more explicitly the evolution of of paths. However, for agreement with the Born rule electron phase or, if you will, time on the electron clock. it allows assignment of probabilities to the wave func- This completes our analysis of geometric structure in tion in its initial conditions, which then propagate to Dirac Theory. In the following we apply it to physical probabilities at any subsequent time. Accordingly, these interpretation, with special emphasis on particle proper- probabilities should not be interpreted as expressions of ties and the roles of phase (in zitter), “density” ρ and the randomness inherent in Nature as commonly claimed for overlooked parameter β. Schr¨odinger theory. Rather, consistent with its realist perspective, Pilot Wave theory regards probabilities in quantum mechanics as expressing limitations in knowl- IV. PILOT WAVE THEORY WITH THE DIRAC edge of specific particle states (0r paths). This view- EQUATION point is best described by Bayesian probability theory, as most trenchantly expounded by Jaynes [23]. Accord- Let me coin the name Born–Dirac for standard Dirac ingly, we regard the Born-Dirac wave function as specify- theory with the Born rule for interpreting the Dirac wave ing Bayesian conditional probabilities for electron paths. function as a probability amplitude. The Schr¨odinger wave function in Pilot Wave theory is The Born rule was initially adopted for Schr¨odinger a many particle wave function. Here we confine attention theory and subsequently extended to Dirac theory with- to the single particle theory, and we review some well known specifics [22] to focus on crucial points. out much discussion — in fact, without even establish- i~ ing the correct relation between Dirac and Schr¨odinger With wave function ψ = ρ1/2eS/ , Schr¨odinger’s equa- wave functions. The latter is supposed to describe a tion can be split into a pair of coupled equations for real particle without spin. However, a correct derivation functions ρ = ρ(x,t) and S = S(x,t) with scalar poten- from the Dirac equation [19, 20] implies instead that tial V = V (x): the Schr¨odinger equation describes an electron in a spin eigenstate, and its imaginary unit must be identified with (∇S)2 ~2 ∇2ρ1/2 ∂ S + − + V =0, (152) the spin bivector i~ =2is . t 2m 2m ρ1/2 13

ρ∇S ∂ ρ + ∇ · =0. (153) When the spin vector s is constant, the stress tensor t 2m term in (159) reduces to the “Quantum force” −∇Q in   Schr¨odinger theory. Thus we see that the ~2 factor in Equation (152) can be written Q comes from squaring the spin vector, and the Quan- 1 tum force is actually a momentum flux. All this puts (∂ + (∇S) · ∇)∇S = −∇(V + Q), (154) the diffraction problem in new light. Indeed, we shall see t m that spin dependence of the quantum force is even more where obvious in Dirac theory. Derivation of Pauli and Schr¨odinger equations as non- ~2 ∇2ρ1/2 Q = Q(x,t)= . (155) relativistic approximations to the Dirac equation in [19] 2m ρ1/2 also traces corresponding changes in local observables. That brings to light many inconsistencies and omissions Identifying in standard treatments of those approximations. The m−1∇S = v = x˙ (156) most egregious error is failure to recognize that the Schr¨odinger equation describes the electron in an eigen- as the velocity of a curve x(t) normal to surfaces of con- state of spin. Implications of that fact are discussed at stant S, from (154) we get an equation of motion for the length in [20]. curve: Another surprising result from [19, 20] is proof that β makes an indisputable contribution to the energy in 1 (∂t + x˙ · ∇)mx˙ = mx¨ = −∇(V + Q). (157) Pauli-Schr¨odinger theory, even though it has been ban- m ished from the wave function. It arises from the spin This has the form of a classical equation of motion, but density divergence (96), which in the non-relativistic ap- with the classical potential V augmented by the quantity proximation takes the form Q, commonly called the Quantum Potential to emphasize ∇· its distinctive origin. mecρβ = − (ρs). (162) A striking fact about Q is its influence on electron mo- This deepens the mystery of β. More clues come from tion even in the absence of external forces. Its noteworthy solutions to the Dirac equation. use in [24] to compute particle paths in electron diffrac- tion stimulated a resurgence of interest in Pilot Wave theory. That computation supported interpretation of A. Pilot Waves in Dirac Theory Q as a ”causal agent” in diffraction, but identification of a plausible ”physical mechanism” to explain it has Extension of the Pilot Wave interpretation for nonrel- remained elusive. So interpretation of Q as an intrin- ativistic wave functions [21] to Dirac theory with STA sic property of the wave function that does not require has been critically examined at lenght in [27], where it is further explanation has remained the default position in demonstrated with many examples that calculations and Pilot Wave theory. analysis with the Real Dirac equation is no more com- The Pauli equation has been used to analyze the effect plicated than with the Pauli equation. Indeed, the first of spin on electron paths in 2-slit diffraction [25]. The order form of the Dirac equation makes some of it decid- authors identify the correct generalization of the Pilot edly easier. The treatment of scattering at potential steps Wave guidance law (156) as is generalized to include both spin and oblique incidence, x˙ = ∇S + ρ−1∇ × (ρs) (158) with STA simplifications not to be found elsewhere. The analysis of evanescent waves exhibits the flow of Dirac However, they failed to note the more fundamental fact streamlines (without commitment to their interpretation that, even in Schr¨odinger theory, the “quantum force” is as particle paths). The study of tunneling times shows spin dependent, though that was spelled out in one of how part of the wave packet passes through the bar- their references [26]. Indeed that reference derived the rier while part slows down and turns back. No notion equation of motion of wave function collapse is needed to interpret observa- tions. It is also shown that the distribution of tunneling ρmx¨ = ρf + T` (∇` ), (159) times observed experimentally can be attributed entirely to structure of the initial wave packet, thus making it where the accent indicates differentiation of the stress clear that, contrary to claims in the literature, no super- tensor T(n), and the applied force has the general form luminal effects are involved. The general conclusion is e that interpretation of Dirac streamlines as particle paths f = e[E + v × B/c]+ ∇` B` · s, (160) mc is consistent with the Dirac equation and helpful in phys- ical interpretation. while components of the stress tensor are Indeed, the fundamental momentum balance equation ρ (134) gives us a complete and straightforward relativistic σ · T(σ )= s · [∂ ∂ s + s ∂ ∂ ln ρ]= T . (161) i j m i j i j ji generalization of Pilot Wave theory that seems not to 14

−1 have been recognized heretofore. One needs only to apply ∇Φ= mec r˙ + (c ∂t ln ρ)s × r˙ − s⊥ ×∇ ln ρ. (170) it to a single streamline z = z(τ) with proper velocity v = z˙ and spin bivector S = S(z(τ)). Then the equation can A detailed proof that the term (s×r˙)·∇ ln ρ does indeed be put in the form of a generalized Pilot Wave guidance reduce to Bohm’s quantum potential in the nonrelativis- equation: tic limit is not needed here. Suffice it to say that both have been derived from Dirac’s equation. The term S˙ · z˙ Φ= mecz˙ + S ·  ln ρ + S˙ · z,˙ (163) has been ignored in these equations, because it has no analogue in the nonrelativistic theory. It’s implications where are studied in the following Sections. e Φ= P − A. (164) c B. Stationary states with β is the gradient of a generalized electron phase expressed in action units. This gradient expression may have im- To solve the Cauchy problem for an electron, it is con- portant implications for electron diffraction. For a free venient to perform a spacetime split of operators in the particle, the generalized momentum P is necessarily a Dirac equation (44) to put it in the form phase gradient. However, electron motion in diffraction might also be influenced through a vector potential gen- (∂ + c∇Ψi~ = m c2Ψ∗ + e(A − A)Ψ, (171) erated by material in the guiding slits. Since the curl t e 0  ∧ A must vanish in the vacuum near the slits, the ∗ where Ψ = γ0Ψγ0. This equation is readily re-expressed vector potential is necessarily a gradient, so it can be in standard Hamiltonian form combined with P as in (164). This possibility has been overlooked in the literature on diffraction. It may be ∂tΨi~ = HΨ, (172) crucial for explaining how the slits transfer momentum to each electron in diffraction. We will have more to say though the structure of the Hamiltonian operator H may about that in the Section on many particle theory. look unfamiliar at first.. The remaining piece of Pilot Wave theory is given by Boudet has applied this approach to a thorough treat- the conservation law for the Dirac current as expressed ment of the Darwin solutions for Hydrogen and their ap- by (135). Evaluated on the particle path it gives us plication to basic state transitions [28]. (See also [27] for a somewhat different STA treatment.) For a stationary 2 ·  Φ= −mecz˙ ln ρ, (165) state with constant energy E and central potential V (r), the wave function has the form which describes the evolution of path density. σ ~ The relativistic guidance law (163) not only combines Ψ(r,t)= ψ(r)e−i 3 Et/ . (173) the the two basic equations (152) and (158) of nonrela- tivistic theory into one, it generalizes the scalar Quantum And Boudet puts equation (171) in the form Potential into a vector S ·  ln ρ and makes its spin de- pendence explicit. 1 ∇ψ = [−E ψ∗ + (E + V )ψ]iσ , (174) To compare the two versions, we must perform a space- ~c 0 3 time split of (163), taking due account of their different 2 notations. For brevity, we draw on the more complete where E0 = mec . He then splits the wave function into treatment of spacetime splits in Section VIIB. From (247) even and odd parts defined by we have the velocity split ∗ ψ = ψe + iψo ψ = ψe − iψo (175) zγ˙ 0 = γ + r˙ with γ = ct.˙ (166) to split (174) into a pair of coupled equations for quater- and, for the spin bivector S = isv, from (254) we have nionic spinors: the split 1 ∇ψe = [−E0 − E + V )ψoσ3, (176) S = s × r˙ + is⊥. (167) ~c where s⊥ is given by (254). Writing a =  ln ρ with the a 1 split aγ0 = a0 + and using (32), we get the split of the ∇ψo = [−E0 + E + V )ψeσ3. (177) “Quantum Vector Potential:” ~c

(S · a)γ = (s × r˙) · a + a s × r˙ + a × s . (168) These he solves to get the Darwin solutions. 0 0 ⊥ The same even-odd split was used in [19] to get non- Putting it all together, for the split of the guidance law relativistic approximations to the Dirac equation. The (163), we get the generalization of (152) and (158): split there mixes β and boost factors in a peculiar way with no evident meaning. Indeed, the peculiar behav- −1 c ∂tΦ= mecγ + (s × r˙) ·∇ ln ρ, (169) ior of β and local velocity in the Darwin solutions defies 15 any sensible physical interpretation in terms of local ob- of the packet at a given time, which seems to escape the servables, with nodes separating positive and negative spreading theorem. energy components in strange ways [16]. These facts Now here is the informative part. Inspection of eqns. are not even recognized in the standard literature, let (182) and (183) shows us that β and θ determine in- alone regarded as problematic. Nevertheless, they pose a dependent components of α. More to the point, for challenge to associating particle properties with the wave a given a density function with the phase gradient θ function. balancing α in the spin plane, the function β can be To address this challenge, we first look to clarify the always adjusted to balance the component of α in the interpretation of β in the simpler case of a free particle spin direction. So to speak, it appears that the role of wave packet. Then we consider a new approach to relat- β is to adjust the probability density to the spin. Thus, ing particle properties to the wave function in the next we have considerable freedom in adjusting the functional Section. form of the probability density to given initial conditions, We look for wave packet solutions of the free particle such as spherical symmetry for example. wave equation This fact raises a question about the β-factor in Born- Dirac theory. Should it be lumped with probability den- ∇ σ ~ 2 ∗ iβ 1/2 (∂t + c )Ψi 3 = mec Ψ (178) sity ρ, so (ρe ) is the full probability component of the wave function? Or should it be lumped with the ro- that generalize the plane wave solutions with zitter in tor R to describe the kinematics of motion? A surprising Section IVC. Accordingly, we look for solutions of the answer is given below. However, it does not preclude form the possibility that β might also play the role of chirality 1 parameter. iβ 1/2 (α+iβ)−iσ3ϕ −iσ3θ Ψ(t, r) = (ρe ) R = e 2 V0 e , (179) σ · ~ where V0 = a + b 2 is a constant boost and ϕ = p x/ C. Scattering and QED with zitter with p = mecγ0. For the simplest case with V0 = 1, substitution into the wave equation gives us the complex The link between standard quantum mechanics (QM) equation of constraint: and quantum electrodynamics (QED) passes through the Dirac equation. It is commonly claimed that the link α˙ + iβ˙ + ∇α+i∇β − 2(θ˙ + ∇θ)iσ3 requires second quantization with quantum field the- −1 −iβ σ = λe [1 − e ]i 3, (180) ory (QFT). But Feynman vehemently denied that claim. ~ When the issue arose in a QED course I attended, I recall where λe = /2mec. Separating out scalar and bivector him dramatically remonstrating that, if anyone dares to parts, we get the equations defend axioms of QFT, “I will defeat him. I will CUT HIS α˙ =0, (181) FEET OFF!” (with a violent cutting gesture for empha- sis). Indeed, the famous formula [p, q]= i~, which Born proposed as a foundation of QM (and had engraved on ∇ ˙σ −1 σ his tombstone), cannot be as general as he thought. For β − 2 θ 3 = λe [1 − cos β] 3, (182) there is no explanation why Planck’s constant here is re- while vector and pseudoscalar parts give us lated to electron spin or the Dirac equation. Also, one can argue that QFT commutation relations for particle ∇ σ ×∇ −1σ α =2 3 θ + λe 3 sin β, (183) creation and annihilation operators are merely bookkeep- ing devices for multiparticle physics without introducing new physics. Lets look at how Feynman got along with- ˙ σ ·∇ β =2 3 θ. (184) out it. A reformulation of Feynman’s approach to QED with These four constraints on components of the wave func- STA is laid out in [29, 30], with explicit demonstrations tion are part of the constraints on any solution of the of its advantages in Coulomb and Compton scattering Dirac equation, so what they tell us is of general inter- calculations. For example, the S-matrix is replaced by a est. scattering operator S that rotates and dilates the initial First, equation (181) tells us that the density ρ is con- fi state to the final state, as expressed by stant in time, so α = ln ρ = α(r) is a function of position only. Its shape is determined by initial conditions consis- ψf = Sfiψi (185) tent with the other constraints. But wait, it is a theorem that any Dirac wave packet must expand in time. Indeed, with no closed wave packet solutions of the free particle Dirac 1/2 Sfi = ρ Rfi, (186) equation are known [11]. However, conventional wave fi packets are superpositions of plane wave solutions with where Rfi is a rotor determining the change in direction 2 momenta in different directions causing the spread. And of spin as well as momentum, while ρfi = |Sfi| is a in this case the momentum is the same at every point scalar dilation factor determining the cross section. 16

Feynman linked QM to QED by reformulating the spacetime path for each point on an initial plane wave. Dirac equation as an integral equation coupled to The conservation law for the Dirac current implies that Maxwell theory through the vector potential: these paths do not intersect, though they may converge or separate. Accordingly, if we assign uniform density to 4 ′ ′ ′ ′ ψ(x)= ψi(x) − e d x SF (x − x )A(x )ψ(x ). (187) paths beginning on the initial plane wave, then the scat- Z tering operator determines the density of particle paths This solves the Dirac equation (46) with p0 = mecγ0 if intersecting a surface surrounding the scattering center. ′ the Green’s function SF (x − x ) satisfies the equation In other words. the squared modulus ρ of the Dirac wave function specifies the density of particle paths! This is  ′ ′ ′ ′ SF (x − x )M(x )i − SF (x − x )M(x )p0 a completely geometric result, independent of any asso- = ϕ4(x − x′)M(x′), (188) ciation with probabilities. Of course, for experimental purposes the density of paths can be interpreted as a where M = M(x) is an arbitrary multivector valued func- particle probability density, but no inherent randomness tion of x. It has the causal solution in nature is thereby implied. ′ 3 The bottom line is that QED scattering is fundamen- ′ Θ(t − t ) d p −ip·(x−x′) SF (x − x )Mi = − (pM + Mp0)ie tally about paths. (2π)3 2E Z Our STA formulation reveals another aspect of QED ′ 3 Θ(t − t ) d p ip·(x−x′) that has been generally overlooked and may be funda- + (pM + Mp0)ie , (2π)3 2E mental; namely, the existence of zitter solutions and the Z (189) possibility that they may describe a fundamental feature of the electron. As we have seen, zitter wave functions where E = p · γ > 0. Note that S (x − x′) is a linear 0 F with opposite chirality can be obtained from a general operator on M here. In general M does not commute wave function Ψ by projection with a lightlike “zitter with p, p , or the bivector i = iσ , so it cannot be pulled 0 3 boost” from under the integral. We can draw several important conclusions from the Σ± = γ±γ0 = (γ0 ± γ2)γ0 =1 ± σ2. (190) present approach to QED. One advantage of the inte- gral form (187) for the Dirac equation is that the causal Thus we obtain boundary condition (189) explicitly enforces the associ- Ψ (x)=Ψ(x)Σ = (ρeiβ)1/2R Σ , (191) ation of electron/positron states with positive/negative ± ± ± energy states respectively. As noted in Section IVB, these where, as before, R = R(x) is a general spacetime ro- states can be switched by multiplication with the pseu- tor, though we often wish to make the phase explicit by doscalar i. writing R = Ve−iϕ. Actually, the β-factor can also be At this point, permit me to insert a relevant anecdote incorporated into the rotor R, to give us that I heard Feynman tell on himself. One day, when he iβ/2 iϕ iσ3ϕ iσ2β/2 was demonstrating his spectacular prowess at complex e Ve Σ+ = Ve e Σ+, (192) QED calculations, a brave student objected: “You can’t because the Σ factor converts it to a rotation: normalize negative energy states to plus one, you must + iβ/2 iσ2β/2 iσ2β/2 use negative one.” “O yes I can!” retorted Feynman e Σ+ = e Σ+ =Σ+e . (193) with the confidence of one who had won a Nobel prize with his calculations and demonstrated them repeatedly Note that the β-rotation will occur before the phase- over more than a decade in QED courses and lectures. rotation in expressions for local observables given below. Then he proceeded to prove that the student was right! Thus, the β-factor tilts the spin vector before the phase Sure enough, check out eqn. (62) to see that the minus rotation in the spin plane. In other words, it is a “tilting sign comes from squaring the unit pseudoscalar (which, factor.” Here, at last, we have a clear geometric meaning of course, Feynman never did learn)! for the parameter β! It suggests that the “β problem” for Returning to the main point, we note that the absence Hydrogen can be solved by simply multiplying Boudet’s of a β-factor eiβ in the scattering operator (186) shows Darwin solutions by a zitter boost to get zitter solutions. that positive and negative energy states are not mixed in The physical significance of zitter in QED remains an scattering. Indeed, the question of a β-factor never arises open question. But later we will note a possible role in in QED, because all calculations are based on plane waves weak interactions. without it, and it is not generated by conventional wave Zitter boosts possess the reversion, idempotence and packet construction. orthogonality properties

Of course, the Born rule is not an intrinsic feature of 2 the Dirac equation, but is imposed only for purposes of Σ± =Σ∓, (Σ±) = 2Σ±, Σ±Σ± =0. (194) interpretation. It is important, therefore, to recognize Consequently, we have lightlike local observables for elec- e e that results of plane wave scattering have a straight for- tron current : ward geometric interpretation without appeal to prob- 1 ability: Indeed, the Dirac equation generates a unique 2 Ψ+γ0Ψ+ = Ψ(γ0 + γ2)Ψ= ρu, (195)

e e 17 and for spin: V. PILOT PARTICLE MODEL 1 Ψ+iσ3Ψ+ = Ψ(γ0 + γ2)γ1Ψ= ρue1, (196) 2 We assume that the lightlike helical path of a fiber in or the wave function has a well-defined center of curvature e e ρS = ~Ψ iσ Ψ = 1 ~Ψγ γ Ψ= ρsu. (197) with a timelike path that we can identify as a particle + 3 + 2 3 + Center of Mass (CM). Accordingly, we regard the CM as Now, to adapt the Dirac equation (46) to zitter and a particle with intrinsic spin and internal clock. Then we e e relate it to local observables, we try projecting it into can describe it by observables and equations of motion the form derived from the Dirac theory in the preceding Section. e (~Ψ iσ − AΨ )Ψ = m cΨ γ Ψ . (198) With deference to de Broglie, let’s call the resulting par- + 3 c + + e + 0 + ticle model a Pilot Particle embedded in a Pilot Wave solution of the Dirac equation. This fails, however, becausee Ψ+Ψ+ = 0. Onee way to cor- Our first task in this Section is to derive and study rect that is to modify the left side to the gauge invariant equations of motion for a well-defined and self-consistent form e e Pilot Particle Model (PPM). Our second task is to (~ + AI)Ψ iσ Ψ , (199) apply the PPM to description of quantized stationary c + 3 + states. Remarkably, our derived quantization conditions with I = Viσ3V , so e are identical to those of Sommerfeld using Old Quan- tum Theory. However, inclusion of spin in our particle −IΨ iσ Ψ =Ψ Ψ =2ρ(1 − e e ). (200) + 3e + − + 2 0 model enables resolution of famous discrepancies in Som- With a similar adaptation of (125), we get merfeld’s energy spectrum for Hydrogen when compared e e ~ with the standard result derived from the Dirac equation. (Ψ )iσ Ψ = ρP (1 − e e )+ (ρS). (201) 2 + 3 + 2 0 Putting it all togethere in analogy to (126), we get the A. Equations for Particle Motion with Clock equivalent of the zitter Dirac equation in terms of local observables: Let z = z(τ) designate the particle path with proper e velocity v =z ˙ = e0. We have already identified this path ρ(P − A)(1 − e2e0)= mecρu − (ρS), (202) c with the CM of an electron with zitter and derived an ex- This can be separated into a trivector part act equation for it (134). That equation is so important, it bears repeating for comparison with the approach in e  ρ(P − A) ∧ (e2e0)= ∧ (ρS), (203) this Section: c e and a vector part p = mecv + S˙ · v = P − A. (205) e e c ρ(P − A)−ρ(P − A)·(e e )= m cρu−·(ρS), (204) c c 2 0 e Here we begin anew to derive and analyze stand-alone wherein we recognize the zitter version of the Gordon particle equations of motion from the treatment of local current. observables in the preceding Section. Averaging over a zitter period reduces (202) to the The electron’s local observables are now restricted to Gordon current in (131), which we have already identified a comoving frame attached to the path: as of prime physical importance. And that has the clear physical meaning of averaging out zitter fluctuations, in eµ = eµ(τ)= RγµR, (206) contrast to simply dropping the β-factor as done before. ~ Furthermore, the trivector equation (203), like its zit- where R = R(τ) is a Rotor with spine vector s = ( /2)e3 ter average (132), puts a constraint on the observables and spin bivector S = isv defined as before. with no evident physical significance. So we have good Our problem is to ascertain dynamical consequences reason to drop it altogether. Fortunately, that can done of the Dirac equation for the Pilot Particle. We seek in a principled way, simply by symmetrizing equation equations of motion consistent with the conservation laws (202) with its reverse, which eliminates the trivector part. of Dirac theory formulated in terms of local observables, This is tantamount to declaring the Gordon current (with for the most part under the assumption cos β = 1. Asour zitter) as physically more significant than the Dirac equa- analysis is not a straightforward derivation, our results tion. may not be completely general, but we take care to make The bottom line is a claim that observables of the wave them self consistent. On the particle path, the particle momenta p and pc function Ψ+(x) describe a congruence (or fibration, if you will) of lightlike helical paths with the circular period of retain their forms in (122) and (124), so we can simply an electron clock. Then we aim to extract individual repeat them here for reference: fibrations from the wave function to create a well-defined e p = P − A, (207) particle model of electron motion. c 18

pc = (p · v)v + S˙ · v − sβ,˙ (208) Or equivalently, where S˙ = i(s ∧ v˙). (218)

p · v =Ω · S = pc · v. (209) Thus, the spin bivector is constant if and only if the particle acceleration is collinear with the spin vector. can be regarded as a (possibly variable) dynamical mass. Instead of appealing to conservation laws for particle The problem remains to relate Ω to particle dynamics. equations of motion, we can look directly to the Dirac Also, it should be understood that the “spin momentum” equation. Analysis is simplified by assuming that for any term S˙ ·v =v ˙ ·S describes linear momentum due to inter- function f = f(z(τ)) defined on the particle path, nal angular momentum, like a flywheel in a macroscopic moving body. The term sβ˙ has been retained for gener- v · f = f˙ and f = vf.˙ (219) ality, though we shall mostly ignore it in the following. Ascertaining equations for momentum and angular In particular, for momentum on the particle path, from momentum conservation on the particle path is simpli- (136) we get fied by the assumption that particle density is constant e v ∧ p˙ = − F +  ∧ P, (220) on the path, so we can factor it out of the exact equa- c tions. One way to derive conservation laws for the parti- cle motion is to surround the particle path with a spher- Hence, ical tube. Then, integrating the equation for momentum e p˙ = F · v + v(v · p˙) − v · ( ∧ P ) (221) conservation (100) over the tube and, assuming no radi- c ation through the spherical walls, shrinking the tube to the particle position z(τ) gives the familiar Lorentz force This agrees with our force law (210) only if equation v · ( ∧ P )=0. (222) e p˙ =p ˙ = F · v, (210) c c Perhaps this is a more general condition for radiationless motion than ∧P = 0, which we have already associated where F = F (z(τ)) is the external field acting on the with stationary states. Recall that standard interpreta- particle. tions of Dirac theory exclude radiation, regarding it as a Similarly, equations (103) and (107) for angular mo- special province of QED. No such exclusion is assumed mentum conservation give us here, though questions about radiation are deferred for ˙ ˙ another time. J =p ˙ ∧ z and Jc =p ˙c ∧ z, (211) Now note that we can solve (213) for where, in accordance with (105) and (108), the total an- p = (p · v + p ∧ v)v = (p · v)v + S˙ · v (223) gular momentum is defined by And if we apply (219) to restrict the Gordon current J = p ∧ z + S or Jc = pc ∧ z − S. (212) (134) to the particle path we get: Inserting (212) into (211) yields alternative equations for e p = P − A = m cv − v · S.˙ (224) angular momentum conservation: c e

S˙ = p ∧ v = v ∧ pc. (213) Thus the Gordon current tells us that the dynamical mass p · v can be reduced to the rest mass, and p (rather than An explicit relation between the two kinds of momenta pc) should be regarded as the particle momentum. p and pc can be derived by dotting this double equation While the kinetic momentum mecv in (224) is familiar with v and using (208); whence the spin momentum q ≡ S˙ · v = −v · S˙ is not, so let us examine some of its properties. Using (218) we can put ˙ · · ˙ · p = pc − 2S v = (p v)v − S v. (214) it into the form

Note that equation (213) implies the constraint q = S˙ · v = [i(s ∧ v˙)] · v = i(s ∧ v˙ ∧ v) = (ivv˙ ) · s. (225)

S˙ ∧ v =0. (215) Note that we can drop the wedge in s ∧ v˙ ∧ v = (s ∧ v˙)v because v is orthogonal to the other two vectors. Hence, We can use S = isv to put it in the form q2 = (S˙ · v)2 = −(s ∧ v˙)2 = s2v˙2 − (s · v˙)2. (226) (s ˙ ∧ v + s ∧ v˙) · v = (s ˙ ∧ v) · v =0. (216) Spin and kinetic momenta are orthogonal to one another, Hence, because

s˙ = (s ˙ · v)v = −(s · v˙)v. (217) q · v = (S˙ · v) · v = S˙ · (v ∧ v)=0. (227) 19

Hence, where the emphasis on the right indicates which quantity is to be differentiated, and v · ( ∧ A)= A˙ − ` A` · v has 2 2 ˙ · 2 2 2 2 p = (mecv) + (S v) = mec − (s ∧ v˙) . (228) been used. The external potential is said to be static in a reference frame specified by a constant vector γ0 if −1 This suggests that γ0 · A = c ∂tA =0. In that case, we have a conserved total energy given by −(s ∧ v˙)2 = (S˙ · v)2 = −(p ∧ v)2 (229) 2 e cP0 = cP · γ0 = mec v · γ0 + A0 + (S˙ · v) · γ0. (238) is a measure of energy (or mass) stored “in” an acceler- c ated electron, so its time derivative is a measure of radi- With some analysis, the last term can be identified as ated energy. From the Lorentz force we find an equation spin-orbit energy. for its flow: As developed to this point, our Pilot Particle model has much in common with classical models for a “par- −1 d e p˙ · p = (p ∧ v)2 = F · (v ∧ p) (230) ticle with spin” considered by many authors [9], so it is 2c dτ c of interest to see what they can contribute to our anal- For acceleration of spin momentum on its own, we find ysis. It is reassuring to know that the self consistency from (225): of those models was established by derivation from a La- grangian. Since the kinematic details align perfectly with · q˙ = i[(s ∧ v˙) ∧ v] soq ˙ · v =0. (231) our present model, we can restrict our attention to the key kinematical equation studied there. But questions about the physical significance of spin mo- The relevant rotor equation of motion in [9] has the mentum will remain until testible physical implications strange but simple form: are worked out and verified. ~ ˙ Turning now to a different issue, kinematics of the co- Rγ2γ1 = pRγ0 + βiR, (239) moving particle frame (206) is determined by an angular where a (suggestive) dummy parameter β has been intro- velocity bivector duced as placeholder for a simple identity derived below. Using (232) we find an informative expression relating Ω ≡ Ω(z(τ))=2R˙ R, (232) local observables: so e ΩS = pv + iβ. (240) We get several relations by decomposing it into pseu- e˙ =Ω · e (233) µ µ doscalar, bivector and scalar parts:  on the electron path. Using ∧ v = vv˙ in (146) we get Ω ∧ S = i Ω · (s ∧ v)= i(s ˙ · v), (241) Ω(z(τ)) =vv ˙ + (P · v)S−1, (234) S˙ =Ω × S = p ∧ v, (242)

Hence, for motion of the “electron clock hand” e1 we have Ω · S = p · v. (243) e˙ = (vv ˙ ) · e + (P · v)S−1 · e 1 1 1 The pseudoscalar part is a mere identity. However, the · 2P v bivector part reproduces the spinor equation of motion = −(v ˙ · e1)v + e2. (235) ~ (213), while the scalar part gives us the same expression for dynamical mass as (209). This shows the role of the canonical momentum P in Of course, the dynamical mass p · v = Ω · S must be evolution of the electron clock explicitly. We aver that positive, which fixes the relative orientation of the two this mechanism determines quantization of atomic states, bivectors Ω and S. Suppose that is also true in a corre- though we will apply it indirectly. sponding equation for a positron: We have seen good reasons throughout this paper to re- gard the Canonical Momentum P in Dirac theory as the p · v = −Ω · S, (244) Total Momentum of the electron plus external sources, or, if you will, ρP as momentum density of the vacuum. The opposite sign on the right indicates an opposite sense Accordingly, for for rotational velocity Ω projected on the spin plane, in other words, the opposite chirality that distinguishes e e positron from electron. P = p + A = m cv + A + S˙ · v, (236) c e c This completes our formulation of Pilot Particle dy- namics and kinematics. Its significance will be tested from (210) we get the vacuum conservation law on the by application to a central problem of quantum mechan- electron path ics, the description of quantized electron states. Physical e interpretation of the results is facilitated by employing P˙ = ` A` · v, (237) c spacetime splits, which we turn to next. 20

B. Particle Spacetime Splits Now −~2 Spacetime splits with respect to a timelike unit vec- S2 = s2 = = s2 − s2 = (s × r˙)2 − s2 . (255) 4 0 ⊥ tor γ0 were defined in Section II. Here we are concerned with spacetime splits of particle equations where γ0 de- Hence, fines an inertial reference frame, called the “Lab frame” 1 for convenience. These splits are important for two rea- 2 2 1 ~ ˆs · v sons. First, physical interpretation is enhanced, because |s| = [s2 − s2] 2 = 1+ . (256) 0 2 c of our experience with physics in inertial systems. Sec- "   # ond, as we shall see, because quantization of stationary ~ · states is defined with respect to an inertial system. Here, Thus we have |s| = |s| = /2 when “helicity” s vˆ = 0. we lay out the splits of relevant physical quantities and Then the spin is said to be “transverse” and s⊥ = s. × · equations for convenient reference. When s v = 0, we have s vˆ = ±|s|, and the spin With the split of an arbitrary spacetime point x de- is said to be “longitudinal.” We shall see that spin is fined by (15), the split of the particle path z = z(τ) with transverse in quantized states. The split of the proper momentum is given by respect to any arbitrary fixed origin x0 is given by

pγ0 = p0 + p where p = p ∧ γ0. (257) (z − x0)γ0 = ct + r, (245) Whence the orbital angular momentum split is where ct = (z − x0) · γ0 and p ∧ z = p0r − ctp + il r = (z − x ) ∧ γ = z − x . (246) 0 0 0 with il = r ∧ p = i(r × p). (258) Without loss of generality, we set x = 0 hereafter. Split 0 So the split of the total angular momentum J = p∧z +S of the proper velocity v =z ˙ is then given by can be put in the form J = j0 + ij, where

vγ0 = ct˙ + r˙ = γ(1 + v/c), (247) j0 = p0r − ctp + s × r˙ (259) where and γ dr dt r˙ = v, v = , γ = ct˙ = . (248) j = l + s⊥ = r × p + s⊥. (260) c dt dτ Now we turn to splits for equations of motion. These splits define the variables for particle kinematics. Given the familiar split F = E + iB, from (32) we get As they suggest, it is often simpler to use r˙ rather than the split for the Lorentz force: v as velocity variable. Splits of the spin vector and bivector are especially (F · v)γ0 = E · r˙ + γE + r˙ × B. (261) tricky. The split Accordingly, split of the momentum equation gives us sγ = s + s (249) 0 0 e e p˙0 = E · r˙ with p˙ = (γE + r˙ × B). (262) is simple enough. But c c The most helpful angular momentum equation is s · v =0= γs0 − r˙ · s (250) e v j˙ = l˙ + s˙ = r × (E + × B). (263) implies ⊥ c c s · v s · r˙ Alternatively, with s = s · γ = = . (251) 0 0 c γ vp = (γ + r˙)(p0 − p)= p0r˙ − γp + i(r˙ × p)+ mec Hence, and the split (254), we get

sv = s ∧ v = (s0 + s)(γ − r˙)= s⊥ + r˙ ∧ s, (252) . (s × r˙) = p0r˙ − γp. (264) where and s s r ⊥ = γ − s0 ˙. (253) s˙⊥ = r˙ × p. (265) So the split of the spin bivector S = isv is given by Finally, we will be interested in the split

S = s × r˙ + is⊥. (254) qγ0 = (S˙ · v)γ0 = (¨r × r˙) · s +γ ˙ s × r˙ + γ ¨r × s⊥. (266) 21

The scalar part is especially interesting, because it con- where n is a positive integer. It follows that temporal tributes a “spin-orbit” coupling to the total energy cP0 quantization is given by given by (238). Thus, noting (225) as well, we have Tn e P T = (p + A )dt = (n + 1 )h. (276) q = S˙ · (v ∧ γ ) = (ivv˙ ) · (s ∧ γ ) = (¨r × r˙) · s. (267) 0 n 0 c 0 2 0 0 0 Z0 And for the vector part we get The originator of this equation, E. J. Post [31], called it Electroflux quantization. However, we see it as setting ˙ q = (S · v) ∧ γ0 = i(s ∧ v˙ ∧ v) · γ0 the period of the electron clock, which is then coordinated =γ ˙ s × r˙ + γ ¨r × s⊥. (268) with quantization of orbit integrals in (273). Equation (275) is well known in standard quantum me- This concludes our synopsis of spacetime splits. chanics as a WKB approximation, but here it is exact, and, as we shall see, sufficient for exact results from the PPM. There has been some dispute about significance C. Quantization of Stationary States 1 of the term 2 h, which is sometimes identified as a zero point energy. We have identified it more specifically as In this section we examine the quantization of station- a consequence of electron zitter, so we can consistently ary states for a Pilot Particle, that is, a point particle ignore it when we ignore zitter. But we should not forget with spin and clock. As defined by (246) the particle or- its likely role in the Hall effect. bit r = r(τ) is a projection of the particle path z = z(τ) Now note that into an inertial system. The orbit is bounded and peri- q · dz = (S˙ · v) · dz = (S˙ · v) · v dτ =0. (277) odic if r(τ) = r(τ + τn) for some period τn. For a sta- tionary state, we also require periodicity of the particle’s Hence, the spin momentum q does not contribute to the comoving frame: action integral on the complete particle path. However, the split eµ(τ + τn)= eµ(τ), (269) q · v =< qγ γ v >= γq − q · r˙ =0. (278) and constant energy 0 0 0 shows that there could be equal and opposite contribu- · 2 E = cP γ0 − mec . (270) tions to the spacelike and timelike integrals (275) and A somewhat stronger periodicity condition that implies (276). Their integrals need not cancel, however. Indeed, (269) is compactly expressed in terms of the rotor we shall see good reason to assume

i − ϕ(τn) q · r Ψ(ˆ τn)= R(τn)e = Ψ(0)ˆ . (271) d =0 (279) I This implies periodicity of the wave function phase, while q0 survives in the timelike integral. which we now consider in detail. Further conditions on the action integrals reflect sym- Here we employ quantization conditions restricted to metries and conservation laws. Central symmetry in an the particle path z = z(τ), so we have the so-called Ac- atom can be expressed by locating the origin of the elec- tion integral tron position vector r = z − x0 at the nucleus. Then conservation of angular momentum L = r ∧ p = i(r × p) τn e P · dz = (p + A) · dz =0 (272) is expressed by separating radial and angular motions. c I I Accordingly, we introduce the split ˙ −1 where, from (236), p = mecv + S · v is the particle mo- p · dr =< prr dr >= (p ∧ r) · (ˆr ∧ dˆr)+ p · ˆr dr mentum and A is the external vector potential. = −L · (i dφ)+ p · ˆr dr = |L|dθ + prdr, (280) As before, the split Pγ0 = P0 + P gives us where dr = ˆr · dr and ˆr ∧ dr = idθ. Then we have τn Tn P · v dτ = P0 dt − P · dr =0, (273) p · dr = |L|dθ + prdr = nh, (281) I Z0 I I I I The constant energy condition for a stationary state re- with duces this to |L|dθ =2π|L| = ℓh, (282) P · r P0Tn = d where Tn = γdτ. (274) I I I where the angular momentum quantum number ℓ

p = p⊥ + pk, (286) a perennial puzzle that the hydrogen energy spectrum derived from the Dirac equation [34] is identical in form where to Sommerfeld’s formula, with only a small difference in e specification of the angular quantum number. The chief p = r × B = m c r × ω (287) ⊥ c e c discrepancy has been resolved recently by Bucher [35, 36] who restored solutions with zero angular momentum that and the integration constant p is parallel to the mag- k Sommerfeld had dismissed on grounds that they would netic field. The vector potential for constant B supplies collide with the atomic nucleus. Bucher called these solu- a momentum tions “Coulomb oscillator” states and demonstrated that e e mec 1 they have potential for deepening our understanding of A = B × r = ω × r = − p (288) c 2c 2 c 2 ⊥ chemical bonds [37]. The problem remained to account Hence, inserting the canonical momentum for spin, as that had not been considered in OQT. The pilot particle model (PPM), regarded as an extension e 1 of OQT to include spin, is offered here as the solution. P = p + A = p + p (289) c 2 ⊥ k Since quantization rules for the PPM are derived from the Dirac equation, we should be able to resolve any dis- into the quantization condition (275) gives us crepancies between Old and New Quantum Theory. Let’s 1 see how. P · dr = p · dr = nh, (290) 2 ⊥ The theoretical apotheosis of OQT was Max Born’s I I book: The Mechanics of the Atom [38]. Ironically, it Using (287), we evaluate the integral was hardly off the press when Born collaborated with Heisenberg and Jordan to eclipse OQT completely with 1 p · dr = ω · (dr × r)=2πr2ω . (291) the new “Matrix Mechanics.” Even so, Born boldly pro- m c ⊥ c c e I I jected a second edition that would bring the theory to a Hence satisfactory conclusion. That never happened, although he published quite a different kind of book with Jordan 2 2 2 2 ~ p⊥ = (mec) ωc r =2mec ωcn. (292) in its place. Even so, the present work can be regarded as revitalizing the original OQT program by linking it Thus we obtain the quantized energy states for an elec- to the ascendant Dirac equation. The OQT method for tron in a magnetic field: solving the Hydrogen atom in Born’s book applies equally 1 well here, so details leading to the Sommerfeld formula 2 2 2 ~ 2 P0 = ± mec + pk +2mec ωcn (293) need not be repeated (See also [33]). A new approach to h i solving the equations of motion is presented in the next for (n =1, 2, ...). These are the relativistic Landau levels subsection. Here we concentrate on issues of interpreta- first found by Rabi [32] by solving the Dirac equation. tion, especially concerning the introduction of spin in the 23 conservation laws for momentum the particle traverses a path the “clock hand” e1 rotates e in the spin plane, and the integer n is the number of p = m cv + S˙ · v = P − A, (295) e c complete cycles it makes in an orbital period, while the integer nr is the number of cycles in a complete radial and angular momentum. oscillation. But, for a given orbit, there are two distinct Applying (238) and (266), a spacetime split of the mo- angular cycles for the clock hand: “spin up,” where j = 1 mentum gives us an expression for the total energy ℓ+ 2 so nℓ = ℓ+1, where orbital motion adds a half cycle 1 2 2 to the clock hand; and “spin down,” where j = ℓ − 2 so cP0 = E + mec = mec γ − V + c(¨r × r˙) · s, (296) nℓ = ℓ, where orbital motion subtracts a half cycle from the clock hand. In each case there is an integral number where V = eA · γ is the Coulomb potential and the last 0 of clock cycles in an orbital period. term looks like a spin-orbit energy. Spin-orbit coupling is This completes our reinterpretation of the Sommerfeld treated as a perturbation even in standard Dirac theory, formula. But there is more, as we have not yet taken an- so let us omit it for the time being. Without that explicit gular momentum conservation into account. According spin contribution, the energy form used by Sommerfeld to (263), for an electron moving with transverse spin in is identical to the one used in Dirac theory. Our problem a Coulomb field the total angular momentum j = l + s therefore is to resolve any discrepancies in interpretation. is conserved, so we take that as a fixed axis defined by For constant energy, Sommerfeld solved the equations j = |j|σ . It follows from the quantization conditions of motion for the particle orbit, which turns out to be a 3 (299) and (300) that l and s must precess around j with precessing ellipse. Then he made the orbit periodic by a constant angular frequency ω = ω σ , and inclined at imposing the quantization condition (275), and decom- p p 3 a fixed angle given by posed it into angular and radial quantum conditions us- 2 2 2 2 2 ing spherical coordinates (r, θ, ϕ) with coordinate frame l · j = (j + ℓ − (1/2) )~ /2= jmℓ~ , (302) (ˆr, eθ, eϕ). Thus, where mℓ = 0, ±1, ±2, ..., ±(ℓ − 1), ±ℓ. Though mℓ is commonly known as the magnetic quantum number be- · p dr = nh = (nr + nθ + nϕ)h, (297) cause it describes splitting of spectral lines in a magnetic I field, it shows up spectroscopically even in the absence of with radial quantization a detectable magnetic field. Here we see it arising from quantization of free precession of orbital angular momen-

p · ˆr dr = prdr = nrh, (298) tum balanced by spin precession. Though its effect is not I I implicit in the Sommerfeld formula (294), it does appear in spectra, which are statistical averages of light emitted angular momentum quantization from an ensemble of atoms in different orientations. This insight helps us reconcile our particle model with · p (eθ dθ + eϕ dϕ)= nθh + nϕh (299) the strange expectation values for angular momentum I operators in wave mechanics, such as with nθ + nϕ = nℓ, and space quantization 1 hLi =0, but L2 2 = ℓ(ℓ + 1)~2. (303) · p eϕ dϕ = pϕdϕ = nϕh (300) As number of authors [39–41] havep noted, they can be de- I I rived from expectation values of the standard operators with nϕ = mℓ, in agreement with the conditions following Lz with eigenvalues ℓ~ taken to be outcomes of measure- (275). ment. While others present this fact as a mere curiosity, Following Bucher [35], we replace Sommerfeld’s as- Post [41] has argued that the derivation can be regarded sumption that nr = 0 by nr = 1 in the ground state, as a statistical average over an ensemble of equally prob- and note from (297) that the range of angular quantum able measurement outcomes. numbers nℓ = ℓ must be changed to ℓ =0, 1, 2, ..., n − 1. Thus, for a given atom, the angular momentum aligned This change is indicated by referring to ℓ as the orbital along the z−direction can have any one of the 2ℓ + 1 val- angular momentum quantum number. As a consequence, ues: +ℓ~, +(ℓ − 1)~, ..., −(ℓ − 1)~, −ℓ~. Regarding all val- circular orbits (for which nℓ = n) are excluded, but the ues as equally probable, the average value for the squared ground state is a linear oscillator passing through the angular momentum is therefore nucleus. ~2 ℓ ℓ(ℓ + 1)(2ℓ + 1)/6 Now, in agreement with Dirac theory, for a pilot par- L2 = n2 = ticle we have z 2ℓ +1 2ℓ +1 n=−ℓ X 1 = ℓ(ℓ + 1)~2/3 (304) nℓ = (j + 2 ), (301)

1 Since no direction is preferred, we have where j = ℓ± 2 is the total angular momentum quantum 2 2 2 2 2 ~2 number. Pilot theory gives this a clear meaning: As L = L = Lx + Ly + Lz = ℓ(ℓ + 1) , (305)

24 where L = r ∧ p is as defined in the PPM. For the total ignore q0 to see what role q might play in Sommerfeld’s angular momentum with half integral quantum numbers original formulation: the same argument gives a result with the same form. 2 2 This argument supports the view of Post [41] and oth- cP0 = E + mec = mec γ − V (308) ers that standard quantum mechanics is about statistical We look for it in the γ = ct˙ term. That term is related ensembles rather than individual physical systems. For a to spatial momentum by assuming more general argument, we need to explain how the PPM 2 2 2 relates to the probability interpretation of the Dirac wave v = γ − r˙ = 1 and p = mec r˙ (309) function. A definitive relation must exist, because both are grounded in the Dirac equation. It should be re- to get membered that the probability interpretation, for which m cγ = m2c2 + p2. (310) Born received the Nobel prize in 1954 [42, 43], solved the e e problem of explaining spectral intensities, which OQT Then quantization is imposedp by inserting had failed to do. So we cannot dispense with probabili- l2 ℓ2 ties, but we need to relate them to ensembles of particle p2 = p2 + = n2 + ~2 (311) r r2 r r2 states. How to do that, is left here as an open problem,   but with suggestions for attacking it below. It should be With these assumptions, the energy (308) becomes an noted that the quantization conditions we “derived” from explicit function of the radius r which Sommerfeld solved an integrability condition have been identified with those to get an equation for the orbit. of OQT derived from Hamilton-Jacobi theory. This pur- The point to be emphasized here is: though we know ported equivalence should be turned into a theorem. It that the quantized momentum in (311) could include suggests that H-J theory can be applied to rigorously de- spin momentum, that possibility is excluded tacitly in fine the approach to wave function singularities, thereby (310) and explicitly in (309). So, to help resolve our deriving particle equations of motion for them. It seems spin-momentum quandary, we turn to a more geometri- that a good start has already been made by Synge [44], cally perspicuous solution of Sommerfeld’s problem that who used relativistic H-J theory in a unique way to solve shows how spin can be incorporated to implement space the Hydrogen atom without Dirac theory. On reformu- quantization. lating that approach with STA, it should be evident how to incorporate spin and relate it to the PPM. This could be a helpful step toward relating the PPM to statistical 3. The Kepler-Sommerfeld problem ensembles. Related remarks will be made at the end of this paper. We still have issues to address concerning the role of Sommerfeld solved the relativistic Kepler problem for spin momentum in the hydrogen atom solutions. Since Hydrogen to find closed electron orbits with quantized the spin momentum q was derived from Dirac theory, it is energies. Let’s call this the Kepler-Sommerfeld (K-S) presumably embedded in solutions of the Hydrogen atom, problem. With the rise of standard Quantum Mechan- though that is not at all evident in the standard Darwin ics it was dismissed as a historical artifact. However, solutions [45] of the Dirac equation or the Sommerfeld the insight that Sommerfeld orbits are inherent in sin- energy spectrum (294). To study its role, we begin with gular solutions of the Dirac equation revitalizes the K-S problem and restores it to a central place in relativistic the explicit appearance of the spin-orbit term q0 = c(¨r × r˙)·s of the total energy (296). To solve for the energy, we particle physics. need to express this term as a function of position using Here we reconsider the K-S problem with a new spinor the equation of motion method enabled by Geometric Algebra. We reconstruct Sommerfeld’s solution with the aim of generalizing it to ∂ V include electron spin and a foundation for perturbation p˙ = m c¨r + q˙ = −γ∇V = −γr r (306) e r theory. To emphasize geometric structure, we adopt c = 1 (in q For = 0 we get a well known expression for the spin- this Section only) and write the equation of motion (306) orbit energy: in the form

∂rV ∂rV r × r · s s · r × r l · s ¨r = −t˙ˆr ∂rV + f, (312) (¨ ˙) = γ ( ˙) = ⊥ 2 2 . (307) mecr mec r 2 2 where V = k/r, with k = e /mec , is the Coulomb po- Including q 6= 0 adds unfamiliar terms without precedent tential. A (scaled) perturbing force f has been included or evident physical meaning. But (278) tells us that we for generality, but we shall ignore it until final remarks. cannot dispense with q entirely without killing the spin- A first integral for the equation of motion can be deter- orbit energy q0, so we are put in a quandary. mined from constants of motion. Accordingly, we rescale One way out of the quandary might be to assume that the energy constant (308) to q does not contribute to quantizing orbital motion by 2 2 asserting the integral condition (279). That leads us to W ≡ cP0/mec = E/mec +1= t˙ − V. (313) 25

Another constant of motion is the angular momentum To transform our equations of motion from vector vari- (per unit mass) L = il, defined by ables to spinors, we use (319) to put the energy constant of motion (313) in the form L = r ∧ ¨r = r2 ˆr ˆr˙. (314) W = t˙ − ks.˙ (324) This can be solved for This integrates immediately to W τ = t + ks. Instead, we ˆr L L ˙2 r2 ˙ ˆr˙ = and r˙ = r˙ + ˆr. (315) use it along with t − ˙ = 1 to eliminate the t variable r2 r from the equation of motion (312). Thus, with a little   algebra to put it in a form for comparison with (323), we Hence, obtain

2 1 2 2 1 2 2 2 l ¨rr + (r˙ − V )= (W − 1). (325) r˙ =r ˙ + , (316) 2 2 r2 Using (316), we see that where l = |l| = |L|. Finally, on multiplying (312) by L and using (315) we get a first integral in the form l2 − k2 r˙ 2 − V 2 =r ˙2 + , (326) r2 L r˙ = k(ˆr + e). (317) which tells us that the V 2 term can be absorbed into the This is an equation relating velocity and focus radius angular momentum by shifting to a precessing system. vector for a precessing ellipse with eccentricity vector e Accordingly, we factor the position spinor into of constant magnitude. Rather than integrating it for the 1 orbit, we turn now to a different method. − iωpτ U = UpUr, where Up = e 2 (327)

generates precession with a fixed angular velocity 4. Spinor Particle Mechanics l2 − k2 ω = σ . (328) We follow the method in [46] to represent the electron’s p l2 3 particle orbit r = r(τ) by a spinor U = U(τ) defined by And substituting (325) into (323) gives us r σ r = U 1U = rˆ (318) d2U 1 r = (W 2 − 1)U . (329) ds2 4 r and introduce a new parametere s = s(τ) defined by This is the desired spinor equation of motion for the or- dτ r = |U|2 = sos ˙ =1/r. (319) bit. It describes an ellipse expressed as a 2-d harmonic ds oscillator. Thus, the complete position spinor U = UpUr The proper time derivative of the “position spinor” U describes a precessing ellipse. can be parmeterized in general form The solution of (329) has the familiar general form: U = α cos 1 ϕ + i β sin 1 ϕ, (330) dU 1 −1 ω r 2 2 = 2 (rr ˙ − i r)U. (320) ds 1 where ϕ = ωs with angular frequency ω = (W 2 − 1) 2 . Then we can choose the angular velocity ωr so that r · ω It can be substituted into (318) to get an explicit har- r = 0 and monic equation for the radius vector r = r(s), but it is simpler to use as is. At least that substitution relates the 2U˙ σ U = (rr ˙ −1 − iω )r = r˙ (321) 1 r coefficients to the ellipse’s semi-major, minor axes (a,b), semilatus rectum Λ, and eccentricity |e| : Thus we find a simplee relation between the spinor deriva- tive and the particle velocity: α2 + β2 a = α2, b = αβ, Λ= β2, e2 = . (331) α2 dU ˙ 1 σ = rU = 2 r˙ U 1. (322) ds This completes our spinor solution of the relativistic Ke- Differentiating once more, we obtain pler problem. Now the solution can be quantized as before by apply- d2U ing the Sommerfeld quantization conditions. The result- = 1 (¨rr + 1 r˙ 2)U. (323) ds2 2 2 ing orbit parameters (labeled by the quantum numbers) are This completes our formulation of spinor kinematics by 2 2 relating it to vector kinematics. an = n rB, bnℓ = nℓrB, Λℓ = ℓ rB , (332) 26

2 2 where rB = ~ /mec is the Bohr radius. The orbital VI. ELECTRON ZITTER frequency is given by Dirac’s strong endorsement [51] of Schr¨odinger’s zitter- ω = W 2 − 1, (333) nℓ nℓ bewegung [52] as a fundamental property of the electron q has remained unchallenged to this day, though it plays with quantized values determined by the energy Enℓ in Sommerfeld’s formula (294). little more than a metaphorical role in standard quan- It must be understood that these quantized values can- tum mechanics and QED. However, evidence is mounting not be directly compared to observed values in atomic that zitterbewegung is a real physical effect, observable, spectra. Instead, one must compare the averages over an for example, in Bose-Einstein condensates [53] and semi- ensemble of atoms with different spatial orientations de- conductors [54]. Analysis with a variant of the model rived in (304) and (305). Bucher [47] has pointed out that proposed here [9] even suggests that zitterbewegung has an ellipse’s geometric properties of size and shape have in- been observed already as a resonance in electron channel- dependent quantitative representations by its major axis ing [55]. That experiment should be repeated at higher and semilatus rectum respectively, or, for a particle orbit, resolution to confirm the result and identify possible fine by total energy and angular momentum. Consequently, structure in the resonance [56]. the major axis can be held constant while the angular Theoretical analysis of zitterbewegung, or just zitter, momentum in (332) is averaged to get requires a formulation in terms of local observables. We have already noted that the zitter frequency is inherent 2 an = n rB , bnℓ = n ℓ(ℓ + 1)rB, in the phase of the Dirac wave function. But Schr¨odinger Λℓ = ℓ(ℓp+ 1)rB, (334) claimed more, namely, that it is to be interpreted as a frequency of position oscillations at the speed of light in agreement with Bucher [35]. Corresponding averages about a mean velocity, and it has been further claimed can by obtained by simple substitution in Eqs. (311), that association of electron spin with circular zitter was (316), (326) and (328). implicit in his analysis [57]. We can easily generalize the solution to include Som- In Section IIIC we proved the existence of plane wave merfeld’s space quantization as a freely precessing ellipse solutions of the Dirac equation that are fibrated by light- with obital angular momentum balanced by spin in ac- like helical paths fixed at the zitter frequency ω . This cordance with (302). The elliptical orbit is tilted and pre- e is a limiting case of conventional plane wave solutions cesses with quantized projected area exactly as described ˆ · σ of the Dirac equation, so it does not arise in in the con- by Bucher [35]. For a tilt angle with cosine mℓ = l 3, struction of wave packets and other solutions of the Dirac the tilt of the angular momentum is given by equation, or the flow of observables in Section IIID. ˆ σ 2 σ Fortunately, the adjustment required to incorporate l = Umℓ 3 Umℓ = Umℓ 3. (335) zitter into standard Dirac theory is fairly straightforward, Hence the tilt rotor is e so we can brief. We suppose that the Pilot Particle Model 1 (PPM) studied in the previous Section can be derived U = (ˆl σ ) 2 . (336) mℓ 3 by averaging out high frequency zitter fluctuations. Ac- And the position spinor (327) generalizes to cordingly, we define the “Zitter Particle Model” (ZPM) to restore those fluctuations. U = UpUm Ur. (337) ℓ More generally, we see that lightlike zitter velocity This appears to be a satisfactory solution to the space factors the Dirac Lagrangian into separate electron and quantization problem. However, our quandary about in- positron parts. This has implications for QED, but that cluding spin momentum to derive it remains unresolved is outside the scope of this paper. — at least for the time being. Finally, incorporating the perturbing force f in (312) into our spinor equation, we have A. Zitter Particle Model 2 d U ω 1 2 2 σ 2 + i pU − 2 (W − 1)U = frU = |U| fU 1. (338) ds2 It should be recognized that the Dirac equation by it- Our spinor equation of motion for the relativistic Ke- self does not imply any relation of the wave function to pler problem is almost identical to the one for the non- electron velocity. Hence, a fundamental question in Dirac relativistic Kepler problem, where it is known as the Theory is how to relate observables in the Dirac equation Kustanheimo-Stiefel equation [46, 48]. The latter has to particle position or path. That is sometimes cast as been applied extensively to perturbations in celestial me- the problem of defining a position operator. The PPM chanics by Vrbik [49] with great effect. We should expect offers an implicit answer by modeling the electron as a it to be equally effective for treating perturbations in particle with a timelike velocity v = e0 =z ˙. Instead, the relativistic quantum mechanics. It might be further en- ZPM in this Section models electron velocity as a light- hanced by spinor formulations of perturbations such as like vector and defines a complete set of local observables given in [50]. consistent with that. 27

We have seen that the hand of the electron clock ro- And the scalar part gives us an expression for particle tates with the zitter frequency, so it is natural to identify energy: the velocity of circulation with the vector e2 while e1 is the direction of the zitter radius vector. Since there p · u =Ω · S > 0. (348) are two senses to the circulation corresponding to elec- tron/positron, we have two null vector particle velocities: We cannot divide (346) by the null vector u to get an analogue of equation (223), but we can divide by v to get e± = v ± e2 = Rγ±R, with γ± = γ0 ± γ2. (339) a comparable expression for momentum We restrict our attention to the electron case and redefine e p = (p · v)u + S˙ · v. (349) the local observables to incorporate zitter. Our choice of sign here is a convention in agreement with [9]. And (345) also gives us Accordingly, we define the electron’s “chiral velocity” u and “chiral spin” S by u˙ =Ω · u = p · S. (350)

u = Rγ+R = v + e2 (340) ZPM dynamics is driven by the obvious Lorentz force and e (210) analogue: ~ ~ e S = Rγ+γ1R = ud, where d = e1. (341) p˙ = F · u, (351) 2 2 c Note that the null velocitye u2 = 0 implies null spin bivec- which can be related directly to the rotor equation (345) tor S2 =0. Using the identities by derivation from a common Lagrangian, as shown in [9]. ue1 = e0e1 + ie0e3 = iue3, (342) The ZPM is not complete until we specify its kinemat- we can write S in the several equivalent forms: ics relating the particle velocity to its spacetime path. Accordingly, we define S = ud = v(d + is)= vd + S = ius. (343) ~ To designate the vector d, let me coin the term “spinet” re = λee1 with λe = = c/ωe (352) 2mec (that which spins) as counterpart of the “spin” (vector) s, since they generate electric and magnetic moments to- as the radius vector for circular zitter at the speed of gether. The overbar designates a “zitter average,” that light. The zitter center follows a timelike path z = z(τ) is, an average over the zitter period τe =2π/ωe. So the with velocity v =z ˙. Hence, the particle path z+ = z+(τ) “linear velocity” v =u ¯ is an average of the chiral velocity with lightlike velocity u, and, since d¯= 0, the spin bivector S = isv is the zitter average of the chiral spin S. u =z ˙+ = v(τ) +r ˙e(τ) (353) The spinor kinematics for the ZPM is a straightfor- ward generalization of the PPM, and has been thoroughly integrates to studied in [9], so the results are simply summarized here. As before, the rotational velocity for local observables is z (τ)= z(τ − τ )+ r (τ), (354) specified by + c e

Ω=2R˙ R. (344) where the time shift τc amounts to an integration con- stant to be determined below. Note that the time vari- Generalizing (239) but ignoringe the dummy parameter able is the proper time of the zitter center. we assume the rotor equation of motion One satisfying feature of the ZPM is the physical inter- pretation it gives to e and e , which are merely rotating ~ 1 2 vectors in the PPM. From (352) see that e1 =r ˆe is the ~Rγ˙ +γ1 = ΩRγ+γ1 = pRγ+. (345) 2 unit radius vector of the zitter, and from (340) we see that e =r ˙ is the zitter velocity. Hence, 2 e Remarkably, our model of the electron as a particle ΩS = pu. (346) with circular zitter was proposed by Slater [58] well be- fore the Dirac equation and Schr¨odinger’s zitterbewe- As in the analogous PPM case, the bivector part of this gung. His argument linking it to the null Poynting vector expression gives us the spin equation of motion: of the photon may also prove prophetic. Of course, we get much more than Slater could by embedding the model in S˙ =Ω × S = p ∧ u. (347) Dirac theory. 28

B. Zitter Field Theory expressed in terms of zitter frequency instead of electron mass to emphasize that the two signs refer to the opposite We have seen that introduction of zitter into Dirac chiralites of electron and positron. theory requires replacing the timelike velocity v in the Obviously, the chiral Dirac equations (360) differ from Dirac current with the lightlike velocity u. For decades the original Dirac equation only by the simple projections I thought that, for consistency with Dirac field theory, (357). There is a significant difference, though, in the this requires modification of Dirac’s equation. I realized appropriate choice of observables for each solution, so only recently that it is achieved more simply by a change our analysis will be facilitated by carefully defining the in the wave function. That suggests that radical surgery key observables. We concentrate on the electron, as the to excise the mass term from the Dirac equation, as re- positron analog is obvious. quired by the Standard Model, is unnecessary at best. Chiral projection produces a small but physically sig- Here we introduce a new surgical procedure based on nificant difference in the energymomentum tensor (132), identification of electron and positron with zitter states so it is worth spelling out. As before, from translational of opposite chirality. The surgery is best done on the La- invariance of the chiral Dirac Lagrangian (359) we derive grangian, because that is an invariant of the procedure. the chiral energymomentum tensor We shall see that it appears to vindicate Dirac’s original idea of hole theory: the “cut” of the wave function to sep- µ T +(n)= γ ~ DµΨ+iγ3ψn . (361) arate electron and positron is merely done in a different way. D E e Since the surgery must be gauge invariant, it is con- As before, the flux in the direction of v is especially sig- venient to express the Lagrangian in terms of the gauge nificant for two reasons: First, because it is related to invariant derivative the electron clock by

µ µ e ~ DΨ= γ DµΨ= γ (∂µΨ+ ~ AµΨiγ3γ0), (355) T (v)= γµ~ D Ψ iσ Ψ = − γµρ(u · D e ). (362) c + µ + 3 2 µ 1 D E with due attention to the caveat about the gauge invari- e ant derivative expressed in connection with (199). Second, because its values are determined by the chiral Whence the Dirac Lagrangian (84) takes the compact Dirac equation, with the result: form T +(v)= mecρu − D · (ρS), (363) L = ~ DΨiγ3Ψ − mecΨΨ . (356) where D E Now, note that the two nulle vector velocitiese introduced ~ σ 1 ~ in (339) correspond to distinct Majorana states: ρS = Ψ+i 3Ψ+ = 2 Ψγ3γ+Ψ. (364) σ Ψ± = Ψ(1 ± 2)=Ψγ±γ0, (357) The zitter average of thise tensor is obviouslye the tensor T (v) given by (86), where the “linear momentum” mecv which are related by charge conjugation:. is the zitter mean of the “chiral momentum” mecu. For the energy density we get Ψ(x) → ΨC (x)=Ψ(x)σ1 (358) · · Accordingly, we identify Ψ+ with the electron and Ψ− v T +(v)= mecρ − (v ∧ D) (ρS). (365) with the positron. 1 The last term on the right can be written Now, a “Majorana split” L = 2 (L+ + L−) of the La- grangian (356) gives us separate Lagrangians for positron eρ and electron: (v ∧ D) · (ρS)= D · (ρd)= ∇· (ρd) − A · d, (366) ~c ~ L± = [ DΨ±iγ3 − mecΨ±]Ψ where d = dv. The divergence does not contribute to D E the total energy by Gauss’s theorem. Locally, of course, = [~DΨiγ − m cΨ]γ γ Ψ . (359) 3 e ± e0 the whole term fluctuates with the zitter frequency as D E d rotates, so its zitter average vanishes. This completes Variation with respect to Ψ givese us charge conjugate our characterization of local observables in chiral Dirac Dirac equations theory. e More details are needed to flesh out the picture. How- σ ωe DΨ±i 3 − Ψ±γ0 =0, (360) ever, the bottom line is, whether or not zitter arises from 2c an orbital motion, zitter is still manifested in the phase To emphasize the incorporation of chiral zitter, let’s call of the electron wave function and, thereby, in the quan- them “chiral Dirac equations.” The last term has been tization of electron states. 29

VII. ONTOLOGY CUM EPISTEMOLOGY completes the description of the electron in Born-Dirac theory as a particle with intrinsic spin and zitter in its When long-standing scientific debates are finally re- motion. solved, it invariably turns out that both sides are correct Concerning the Born rule for interpreting the wave in positive assertions about their own position but incor- function in Quantum Mechanics: There is no doubt that rect in negative assertions about the opposing position. probability is essential for interpreting experiments. In- The Great Debate over the interpretation of quantum deed, overcoming the failure of Old Quantum Theory to mechanics can be cast as a dialectic between Einstein’s account for intensities of spectral lines was one of the first emphasis on ontology and Bohr’s emphasis on epistemol- great victories for Quantum Mechanics and Born’s rule ogy [59]. This paper offers a new perspective on the for statistical interpretation. But that did not seal the debate by focusing on the local observables determined demise of Old QT as commonly believed. For Old QT by the Dirac wave function. is resurrected in Section V and revitalized as a particle As we saw in Section IV, the spacetime flow of local model for the Hydrogen atom. Moreover, it comes not to observables is governed by the equation destroy QM but to fulfill! For the particle model offers e an ontic interpretation to QM, while QM offers an em- ρ(P − A)= mecρu −  · (ρS), (367) pirically significant way to assign probabilities to particle c states. The particle model provides electron states with which, with some provisos, can be regarded as a reformu- definite position, momentum, spin and (zitter) phase. lation of the Dirac equation. When restricted to a Dirac The Born rule offers a way to assign probabilities to streamline, as an electron path z = z(τ) with proper ve- these states. However, such probabilities do not im- locity u =z ˙, it becomes a relativistic generalization of the ply uncertainties inherent in Nature as often claimed. de Broglie-Bohm guidance equation. Thus it provides a Rather, they express limitations in our knowledge and secure foundation for a Pilot Wave interpretation of the control of specific states, best described by Bayesian Dirac wave function, wherein the scalar ρ is interpreted probability theory so ably expounded by E. T. Jaynes as a density of particle paths. [23, 59]. Since the Dirac equation is a linear differential equa- tion, the superposition principle can be used to intro- Of course, the present approach calls for reconsider- duce probabilities in initial conditions and construct the ation of many arguments and applications of standard wave packets of standard quantum mechanics with the quantum mechanics, especially those involving zitter and Born interpretation of ρ as probability density. This es- the Heisenberg Uncertainty Relations, which are already tablishes full compatibility between the Pilot Wave and burdened by many conflicting interpretations [60]. Born interpretations of Dirac wave functions. A new perspective on the Great Debate on the inter- A definitive analysis of zitter in the Dirac wave func- pretation of QM will be introduced in a sequel to the tion was given in Section IVC and proposed as a defin- present paper [1]. ing property of the electron. That shows that zitter can be regarded as electron phase incorporated in particle Acknowledgment. I am indebted to Richard Clawson motion. It is included in the Pilot Wave guidance law and Steve Gull for critical give and take about Real Dirac with a slight modification of (367) given by (204). That Theory over many years.

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