Electron internal energy and internal motion (Zitterbewegung) as consequence of local U(1) gauge invariance in two-spinor language
aa r X i v : . [ phy s i c s . g e n - ph ] S e p Electron internal energy and internal motion(Zitterbewegung) as consequence of local U(1)gauge invariance in two-spinor language
September 15, 2020
J. BuitragoUniversity of La Laguna, Faculty of Physics38205, La Laguna, Tenerife, [email protected]
Abstract
Starting with the results obtained in a previous paper in whichclassical local U(1) gauge invariance in terms of the electromagneticfield strenghts instead of the usual formulation mediated by the fourpotential was introduced it is shown that using the gauge freedomassociated with the third component of the magnetic field, previouslyobtained spinor equations of motion describe the internal dynamics of afree 1/2 spin particle suggesting a kinematic origin of its rest mass andhelicity. The controversial Zitterbewegung (trembling motion) appearsin a natural way as internal motion with the velocity of light. Suchan interpretation is in contrast with the usual quantum mechanicalexplanation of transitions between positive and negative energy states.
The main purpose of this article is to present the Zitterbewegung (tremblingmotion) phenomena from a different perspective and other outlook than theusual quantum mechanical interpretation based in the Dirac equation. Asfar as I know, the only study leading to the same results and similar inter-pretation dates back to 1990 (see [2]). Curiously enough both approachesbeing conceptually and technically quite different converge to the same re-sults. From the technical and conceptual standpoint, the present work is1ased on a previous paper [1] presenting a classical U (1) local gauge in-variance formulation via a lagrangian with an interaction in terms of theelectric and magnetic field strengths. This approach is certainly differentto the usual lagrangian leading to the Dirac Equation interacting with anexternal electromagnetic field described by the four potential A µ . As thepresent study is strongly based on the mentioned article, I have tried tofind a compromise between continuous referencing to [1] and a certain selfconsistency while avoiding too much duplication.The starting point of [1] (see also [5]) are the following linear first orderdifferential spinor equations (the derivative is taken respect to the propertime τ ): dη A dτ = em φ AB η Bdπ A dτ = − em φ AB π B (1)(A short derivation of these equations is included at the beginning of nextsection) These coupled spinor equations (natural units ¯ h = c = 1 will beused) are equivalent to the Lorentz Force describing the motion of a 1/2 spinparticle of mass m and charge e (typically an electron) under an electromag-netic field described by the symmetric second-rank spinor φ AB , explicitelygiven by φ AB = 12 [ E − iB ] − i [ E − iB ] − E + iB − E + iB [ − E + iB ] − i [ E − iB ] ! . (2)In turn, φ AB and its complex conjugate ¯ φ A ′ B ′ form the antisymmetric four-rank electromagnetic field spinor in its standard form that can be found in[7] and any other book dealing with this subject. F ABA ′ B ′ = ǫ AB ¯ φ A ′ B ′ + ǫ A ′ B ′ φ AB , (3)where ǫ AB = ǫ AB is the spinor metric ǫ AB = − ! . (4)(Capital indices are lowered (raised) by means of the metric spinor ǫ AB already defined above. See Appendix 1)The solution of equations (1) determine the four momentum of the par-ticle given by the hermitian spinor defined as superposition of the two null2irections π A ¯ π A ′ and η A ¯ η A ′ as [3] p AA ′ = 1 √ h π A ¯ π A ′ + η A ¯ η A ′ i . (5)Since p AA ′ is to represent the four-momentum of a massive particle, mustbe time-like and certainly fulfill the condition: p AA ′ p AA ′ = m . (6)On the other hand, following the standard representation, the differentcomponents of p AA ′ are labeled according to p AA ′ = p ′ p ′ p ′ p ′ ! = 1 √ p + p p + ip p − ip p − p ! . (7)This last expression must be used when solving an specific case, via thespinor equations, to identify the components of p AA ′ in the solution. U (1) Local gauge transformation of field strengthquantities and Zitterbewegung
As I have mentioned in the introduction, the particle field interaction, willnot involve the four potential A µ as in the traditional approach. Instead,the coupling will be to the electric and magnetic field strengths. To thisend and to emphasize the geometrical origin of this formulation (see also[5]) consider an infinitesimal transformation of η A mediated by an elementof the group SL(2,C): δη A = exp (cid:20)
12 ( ~δω~σ + i~δθ~σ ) (cid:21) η A , (8)being ~δω and ~δθ the infinitesimal parameters of boosts and rotations respec-tively and ~σ the usual Pauli matrices. Since the transformation is infinites-imal, we can write δη A = (cid:20) I + 12 ( ~δω~σ + i~δθ~σ ) (cid:21) η A (9)3ccording to the interpretation given in [5] associating the fields ~E and ~B with infinitesimal boosts and rotations, and via a classical detour, wewould say that any change in the dynamic state of the particle should beproportional to the force field acting on the particle and the lapse of propertime. Accordingly ~δω = K~ǫ ( x ) δτ (10) ~δθ = K ~β ( x ) δτ (11)By substitution of ǫ and β by ~E and ~B and expanding the term associatedwith the Pauli matrices we get a second rank spinor (coincident with thesecond rank electromagnetic field spinor φ AB ): φ AB = 12 " E E + iE E − iE − E ! + i B B + iB B − iB − B ! . (12)It is only necessary to lower the first upper index, following the rules givenin the first appendix, to find the symmetric form (2) of the field spinor φ AB .On the other hand, from equation (9) and subsequents equations, making K = e/m , it is immediate to obtain ddτ η A = em φ AB η B . (13)As was done in [1] and repeated here for self-consistency, the lagrangiandensity, for a free particle, along the classical path of the particle (withdimension energy per unit length) is L = π A ˙ η A , (14)(dot denoting derivative respect to proper time τ ) together with the EulerLagrange equation ddτ ∂ L ∂ ˙ η A − ∂ L ∂η A = 0 . (15)With a similar equation for the spinor π A . The former equation leads to˙ π A = 0 ⇒ π A = const. In the definition of the four-momentum p AA ′ , the spinors π A and η A enter on equalfooting. It is then clear that the lagrangian could also be defined swapping both spinors.
4e consider now the consequences of imposing invariance under local ( alongthe classical path parametrized by τ ) phase transformations η A → e iα ( τ ) η A π A → e iξ ( τ ) π A . (16)The phase parameters α ( τ ) and ξ ( τ ) cannot be independent as thespinors η A and π A are also not independent since from (5), they are re-lated by the condition p AA ′ p AA ′ = | π A η A | = m . (17)In consequence η A π A = const. , leading to the constraint ξ ( τ ) = − α ( τ ).As the classical trajectory should not be affected by any phase transfor-mation, it is apparent that local gauge transformations leaves invariant thefour-momentum of the particle: p AA ′ = 1 √ h π A ¯ π A ′ + η A ¯ η A ′ i . However, the free lagrangian (14) transforms to
L → i ˙ αη A π A + ˙ η A π A = i ˙ αǫ AB η B π A + ˙ η A π A . (18)To find a gauge invariant lagrangian we have to add a term − em φ AB η B π A , (19)and impose the condition for the new field φ AB of transforming, under localphase transformations, as φ AB → φ AB + i me ˙ αǫ AB , (20) From a pure mathematical point of view, the validity of transformation (20) is con-sequence of the following theorem applied to valence-2 spinors (see Stewart J.
AdvancedGeneral Relativity . 1991 Cambridge Univ. Press. Page 69): “Any spinor τ A...F is the sumof the totally symmetric spinor τ ( A...F ) and (outer) products of ǫ ′ s with totally symmetricspinors of lower valence L = ˙ η A π A − em φ AB η B π A , (21)is invariant under U (1) local-phase transformations. The transformationthat holds for the conjugate second-rank spinor ¯ φ A ′ B ′ , is given by¯ φ A ′ B ′ → ¯ φ A ′ B ′ − i me ˙ αǫ A ′ B ′ . (22)These kind of transformations leave however invariant the associated four-rank spinor of the Maxwell field strength [7] F ABA ′ B ′ = ǫ AB ¯ φ A ′ B ′ + ǫ A ′ B ′ φ AB . From the Euler Lagrange equations applied to the lagrangian given by (21)it is immediate to obtain ˙ π A = − em φ AB π B . (23)This equation and those of (1) are gauge invariant. If the invariance of thefour-rank spinor F ABA ′ B ′ follows from the transformation rules of the fieldspinors φ AB and ¯ φ A ′ B ′ a simple look at (2) and (4) reveal that only the com-ponents φ and φ (together with their complex conjugates) are affected.Furthermore since the transformation is purely imaginary, there is only onefield quantity (i.e. B ) affected by the transformations. Consequently, thespinor equations are modified in much the same way as a gauge change inthe electromagnetic potential A µ modifies the equations and their solutions(for example: the Coulomb gauge in QED). It is clear that this peculiarityof the third component of the magnetic field deserves further attention. Inwhat follows, we shall examine the consequences of this local gauge invari-ance for B . For simplicity the case of B = const. will be studied. Giventhe equation ˙ η A = − em φ AB η B , (24)and applying the gauge transformations: φ → ˜ φ = φ + i me ˙ αǫ (25)6 → ˜ φ = φ + i me ˙ αǫ (26)The transformed equations for the physical components (upper indices) are˙ η = em i (cid:18) − B me ˙ α (cid:19) η (27)˙ η = em i (cid:18) B me ˙ α (cid:19) η . (28)As in the last equations we cannot gauge away simultaneously both effectivefields, let us choose − B me ˙ α = 0 , (29)then ˙ η = 0 = ⇒ η = const. = √ ˆ m (the integration constant √ ˆ m needs to have the dimension of an energysquared) Under condition (29), (28) reduce to˙ η = em i (cid:18) me ˙ α (cid:19) η = i αη (30)As both charge and mass of the particle as well as magnetic field havedisappeared, the last relation should be valid for any 1/2 spin particle. Doingthe integration: η = √ ˆ me i ατ (31)The spinor η A is η A = √ ˆ m e i ατ ! . (32)As for the other spinor and proceeding in the same way: π A = √ ˆ m e i ατ ! . (33)Denoting ˙ α = ω , from the four-momentum definition (5) and (7) p AA ′ = 1 √ m me i ωτ me − i ωτ m ! . (34)Remembering that Det [ p AA ′ ] equals 1/2 the Lorentz norm, we have a nullpath in momentum spacetime. After some lengthy calculations to solve forthe components: p = E = 2 ˆ m = 0 p = 2 ˆ m cos 2 ωτp = − m sin 2 ωτ. mc / ¯ h (in conventional units) is the so called Zitterbewebung (tremblingmotion), found in the Dirac Theory and subject of many controversial in-terpretations. From the last and following relations, we shall find a classicalinterpretation as an internal circular motion taking place inside of the parti-cle (electron). It makes sense now to identify ˆ m as the electron mass. Since p i = 2 mu i , (i = 1,2)( p ) + ( p ) = 4 m = ⇒ ( u ) + ( u ) = 1 . (35)(Circular motion in the x-y plane with the velocity of light).Let us now take the other alternative (20), namely B me ˙ α = 0 (36)The solutions are now π A = √ m e i ατ ! (37) η A = √ m e i ατ ! . (38)Performing the same calculations as in the other case p = 2 m cos 2 ωτp = 2 m sin 2 ωτ The spatial trajectories in the x-y plane can be immediately obtained : x = 12 ω sin 2 ωτy = − ω cos 2 ωτ and x = 12 ω sin 2 ωτy = 12 ω cos 2 ωτ Being clock and counter clock-wise respectively.The previous results concerning the momentum and position of the elec-tron behavior are in agreement with those obtained in [2] (see equation 64in the cited work). 8
Discussion
A fundamental question that, perhaps, we have some reasons to ask nowis what is the origin of the electric charge. Previously the charge “e” dis-appeared in (23) and did not appear again. However, the results seemsindependent of the charge. Since the neutrinos have a certain mass, andpossibly a magnetic moment, it may be possible to explain the opposite he-licities of neutrinos within the three families. As early as 1930, Schrodinger[4] made the first analysis of what was to be called afterwards “Zitterbewe-gung” (Trembling Motion). The frequency 2 ω also appears in the solutionsof the Dirac Equation for the propagation of a free packet. He interpreted ω = 2 mc / ¯ h as a fluctuation in the position of the electron with radius∆ r = cω = ¯ h mc . Assuming a velocity of the electron equal to the velocity of light about somemean position inducing an spin angular momentum∆ r.mc = ¯ h . A relatively modern interpretation [8] is that there are unavoidably crossterms between the positive and negative energy solutions which oscillaterapidly in time with frequencies2 p c ¯ h ≥ mc ¯ h Perhaps the reader will ask why B ? The answer has to do with thechoice of the third Pauli matrix σ . There is nothing special about the z -axis, but once we choose this axis for σ , the z -axis has, in practice, aspecial relevance. (More technically, magnetic fields are related to rotationsdescribed by the SO(3) group having SU(2) as covering group (see [9] , [1]and [10] ).Finally, and as already mentioned, the results in this work seems to bein agreement with those obtained in [2]. Appendix 1: A Short Introduction to Spinor Calculus
With the aim of making this article accessible to potential readers notfamiliar with two-spinor formalism, I consider that a very simple and basicintroduction to spinor calculus would be particularly helpful for the obvi-ous reason that the formalism originally developed by Penrose and Rindler9n their books “Spinors and space-time”, [7] that I have cited in this andprevious papers, is not widely used or very familiar to a large number ofphysicists.Spinors like η A or π A belong to a simplectic complex two-dimensionalvector space S . The complex conjugate vector space S ′ has elements ¯ η A ′ .We also need to consider the two dual spaces S ∗ , S ∗ ′ with elements ξ A , ¯ ξ A ′ . η A = η η ! ∈ S, ¯ η A ′ = ¯ η ′ ¯ η ′ ∈ S ′ ξ A = ( ξ ξ ) ∈ S ∗ , ¯ ξ A ′ = (cid:0) ¯ ξ ¯ ξ (cid:1) ∈ S ∗ ′ (39)Just as the familiar metric tensor η µν of Minkowski space, in S we alsohave a metric spinor ǫ AB = ǫ A ′ B ′ = ǫ AB = ǫ A ′ B ′ − ! (40)relating any spinor η A with η A according to the rules: η A = ǫ AB η B and ξ A = ǫ BA ξ B , with similar rules for the complex conjugate quantities. It follows that thecomponents of η A are related to the components of η A by η = η , η = − η Accordingly, for any spinor η A η A = η η − η η = 0Care is needed with the index ordering because ǫ AB is skew: η A = ǫ AB η B = − ǫ BA η B . Just as in ordinary tensor calculus, spinors of higher rank, like those usedin this work: p AA ′ , φ AB and F ABA ′ B ′ , can be defined. In particular, thereis an isomorphism between real four vectors in Minkowskian space-time andhermitian second rank spinors: V α −→ V AA ′ = 1 √ V + V V + iV V − iV V − V ! . (41)10he Lorentz norm equals one half the determinant of the above matrix. Inparticular, if the vector is null, like π A ¯ π A ′ , it may be written as the outerproduct of a complex two-dimensional vector and its complex conjugate: π A ¯ π A ′ = π ¯ π ′ π ¯ π ′ π ¯ π ′ π ¯ π ′ ! . (42) Appendix 2: Classical Weyl-spinor lagrangian and quantum Weyland Dirac lagrangians
As we have seen the rather simple choice, in spinor language, of thelagrangian density leading to the spinor equation of motion is L = π A ˙ η A , (14 b )in contrast, by comparison, with the familiar expression of the Dirac quan-tum lagrangian for a free particle L = i ¯ ψγ µ ∂ µ ψ − m ¯ ψψ. (43)One thing to note is that while the spinor lagrangian density has di-mension energy per unit length, the dimension of the classical lagrangian isenergy thus emphasizing the deep difference between the standard approachand the spinorial one (note that the mass is also absent in the spinor def-inition of the four momentum). In any case it is apparent that classicallagrangians for a free particle only contain one term.If we try to find the Weyl version of the Dirac Equation (DE) via theusual lagrangian approach we need to add a mass term. In [1] the Weyl2-spinor version of the DE is found in a simple way starting from p AA ′ = 1 √ π A ¯ π A ′ + η A ¯ η A ′ ](see (22) and subsequent equations in [1]).Following now the standard approach, in the DE case the Euler-Lagrangeequations to obtain the DE from the lagrangian (43) are ∂ µ ∂ L ∂ψ ,µ − ∂ L ∂ψ = 0 , (44)and a similar one for ¯ ψ . As it is standard material found in many books,we shall not develop further the steps leading to the DE. However, in our11eyl two-spinor approach things are, if not substantially more involved,less familiar. Let us start with the free lagrangian (14b). Spinors π A and η A are now no longer defined along any classical path parameterized byproper time but instead should be regarded as functional or distributionsin four dimensional spacetime. In consequence, the derivative with respectto proper time should be replaced by general derivatives in the context oftwo-spinor calculus: ˙ π A → ∇ AA ′ ¯ π A ′ (45)˙¯ η A ′ → ∇ AA ′ η A . (46)The appropriate Euler-Lagrange equations in two-spinor calculus (general-ization of (15)) is ∇ AA ′ ∂ L ∂ ( ∇ AA ′ ¯ π A ′ ) − ∂ L ∂ ¯ π A ′ = 0 (47)and ∇ AA ′ ∂ L ∂ ( ∇ AA ′ η A ) − ∂ L ∂η A = 0 . (48)The two coupled lagrangians are defined as L = π A ∇ AA ′ ¯ π A ′ + m √ η A ′ ¯ π A ′ (49) L = ¯ η A ′ ∇ AA ′ η A − m √ η A π A . (50)Taking the first ∂ L ∂ ( ∇ AA ′ ¯ π A ′ ) = π A L ∂π ′ A = m √ η A ′ , and from (50): ∇ AA ′ π A = m √ η A ′ . Performing the same calculations with the second we arrive at the 2-spinorversion of the DE in Weyl representation as they appear in [7]: ∇ AA ′ π A = m √ η A ′ (51) ∇ AA ′ ¯ η A ′ = − m √ π A . (52)12 eferences [1] Buitrago J. Results in Physics 6 (2016) 346-351[2] Hestenes D. Found. Physics., Vol. 20, No. 10, (1990) 1213[3] A. Bette, J. Math. Phys. , 4617 (1993)[4] Schrodinger E. Sitzungber. Preuss. Akad. Wiss. Phys.-Math. Kl. 24,418 (1930)[5] J. Buitrago and S. Hajjawi Spinor extended Lorentz-force-like equationas a Consequence of a Spinorial Structure of Space-Time , J.Math.Phys.,48¯ 022902 (2007).[6] L.D. Landau and E.M. Lifshitz,
The classical Theory of Fields , Perga-mon Press, 1971.[7] R. Penrose and W. Rindler