Dirac Equation Redux by Direct Quantization of the 4-Momentum Vector

. The Dirac Equation (DE) is a cornerstone of quantum physics. We prove that direct quantization of the 4-momentum vector


Introduction
The 93 years old Dirac Equation (DE) [1,2] is one of the most far-reaching equations in physics: it is Lorentz covariant, electron spin springs from the Equation and it predicted the first instance of antiparticle, the positron, discovered four years later by Anderson [3].DE for a free electron is ( = 1) [4,5]:   −  ψ = 0 (sum);  = ℏ ≡ ℏ   ⁄ ; { ,  } =   +   = 2 ; ,  = 0,1,2,3 is the rest mass of the electron and  = diag(1, −1, −1, −1) is the time-space signature.Dirac argued that  have to be 4 × 4 complex matrices "…describing some new degrees of freedom belonging to some internal motion in the electron" [1,2].This remains the standard position today [4,5].Dirac matrices and their generalizations are considered a fundamental representation to DE and to any domain of theoretical physics where spin 1/2 is relevant [4].In the following we show that these extra degrees of freedom are redundant.
Based on the isomorphism of the Clifford algebras for Dirac  matrices and spacetime vectors, (see Eq. ( 3) below) Hestenes [6] proposed a new version of DE in the formalism of his spacetime algebra STA [7].There is no matrix and no imaginary number in STA DE.However, the spin has been put 'by hand' in the equation, thereby diminishing its predictive power and symmetry compared to standard DE.The real vector space of STA has dimension 16, which is half of the equivalent real dimension for the space of 4 × 4 complex matrices.The Clifford algebra of STA is ℓ ( , ) , the subscript (1,3) standing for the signature.
The Clifford algebras for spacetime frame vectors {x } and Dirac matrices { } being the same, the motivation for the present work has been to show that direct quantization of the 4-momentum vector p of modulus equal to the rest energy  ( = 1), provides the simplest form of Dirac Equation: p ψ = ψ.This is a realization of the "square root" for the relativistic invariant p =  , with the spinor ψ 'intermediating' between a vectorial operator p and a scalar m.As shown below, 'internal degrees of freedom' of the electron follow from the Equation, extras, like matrices unneeded.This minimal approach together with the manifest covariance of the Equation mark immediate improvements over the standard DE, affirming that in order to derive all properties of the Dirac electron / positron we only need the special relativistic momentum 4-vector and the postulate of quantization.After a swift introduction to the Clifford algebra ℓ ( , ) over spacetime, we expand to ℓ ( , ) over spacetime-reflection (STR) in order to accommodate the momentum operator in a real vector space.The STR formalism takes shape by proving how it works.Detailed derivation of some additional results in STR appear in the Appendices A-D.
This report comprises new material and changes relative to deposited preprints [8] with the same title.Apart from rearrangements and enhanced section structure, an expanded analysis of time-reversal symmetry and its geometric meaning appears in Section 4.3 and in App.D. A new Section (4.4) on Lorentz transformations of STR DE comprises significant improvements and corrections compared to previous versions; the covariant transformation is the same as before, now standing on firmer grounds.Two tables have been added, Table 1 giving an overview of the notation (as suggested by one reviewer) and

Clifford product in spacetime.
The Clifford or geometric product [6,7] of two vectors u, v combines the symmetric Hamilton's scalar product with the antisymmetric Grassmann's wedge product [8,9]; in coordinate-free form it is: The two parts relate naturally to the anticommutator { , } and commutator From (3) permutations of given orthogonal vectors can at most change the sign, not adding new basis elements.Similarly to the geometric interpretation of basis bivectors as oriented area elements, trivectors (tetra-vectors) in ( 4) are oriented 3-volume (resp.4-volume) elements in spacetime, all unitless.Multivectors in STA correspond to tensors, represented by matrices in the standard formalism.Multivectors formed by wedge product alone, known as blades, are anti-symmetric relative to the exchange of any two indices.e squares to −1 and defines the pseudo-scalar of STA.However, it commutes only with even grade elements of STA (scalars and bivectors) and anti-commutes with vectors and trivectors.We will define another real vector space in the following, therefore will not use here a specific symbol for e .The STA basis (4) is defined in terms of upper indices, but one can lift indices up and down with the help of the appropriate form of spacetime signature and define a reciprocal basis relative to (4).Another operation is that of reversing ( ~) the order of indices of a multivector, which corresponds to taking the transpose of a matrix.In STA the Hermite conjugate ( †) of a basis vector is defined by the parity transformation (see Eq. ( 35)) e = e e e ; it works without shift to the reciprocal basis.The Hermite conjugate of any elements A, B of STA is: A ≡ e A e , (A + B) = A + B ; e. g.  e =  e (sum over , );  ∈ ℝ Alternatively, one could have defined Hermite conjugate by the combination of index lift and reversal, which changes a given basis vector to its reciprocal (co)vector.This is convenient when proving e.g. that the STA basis ( 4) is orthonormal, i.e. that each basis element has modulus 1 and the product of any pair of different A relativistic quantum vector Equation of first order in spacetime derivatives for a particle of mass  is then: p ψ = ψ with p = ℏ∇= ℏe  ;  ≡   ⁄ =    ⁄ ; sum over repeated indices (8) In most cases we will drop the hat and depict the operators p , ̂ by p,  .Due to the imaginary  entering with the canonical momentum operator, STA's field of scalars has to expand from real to complex numbers.
commutes with all the elements of STA in (4), therefore e (like  ) is Hermitian, as shown.One can certainly build a formalism expanding the STA basis (4) on complex scalars, which as in (9) would mix the complex structures arising from scalars and vectors & multivectors.However, inspired from STA, we prefer taking an alternative path leading to a real vector space with the complex structure springing from vectors & multivectors alone.
3. The real vector space of spacetime-reflection, STR.
Using the same symbol as in ( 9) we assume a Hermitian vector e to be a basis frame vector, on equal footing with {e }, and name the quintet {e , e } an orthonormal vector basis for spacetime-reflection, STR; the reason for the name will become clear in the following.Let X depict the real vector space generated by the action of the Clifford product onto {e , e }.An orthonormal basis for X (dim 32) is: The first 16 elements in (10) are the basis elements of STA in (4) [7].The other 16 elements are obtained by multiplying with the frame vector e .Symbolically we could write STR = STA(1 + e ).16 elements in (10) square to -1, allowing for a rich complex structure in X.Of these only the element of highest grade, the pentavector is both isotropic (i.e.not privileging any spacetime director) and commutes with all elements of X; it constitutes the (geometric) pseudoscalar of STR, depicted by (compare with ( 9)): A general element y ∈ X is y = e ( + I ) + e ( + I ); ,  = 0,1,2,3,5;  ,  ,  ,  ∈ ℝ (there is redundancy; summation over dummy indices is on).The first two terms stand for vectors and tetravectors, respectively of grade 1 and 4. The last two terms stand for scalars & pseudoscalars ( = ) and bivectors & trivectors ( ≠ ), respectively of grade 0, 5, 2, 3. Table 1 summarizes the STR notation; the position vector is x =  e with the unit of length attached to the scalar components.We adapt (5) to define the Hermite conjugate of STR elements in (10) taking care of e factors, as we did for I in (12); for a pure multivector: A = e … : A ≡ (−1) e A e ;  = nr. of e in A; e. g.: (Ie ) = (−1) e Ie e = −Ie (13)

Subspaces of 3D relative vectors.
Two subspaces of relative vectors in X as well as their product will be of special interest (,  = 1,2,3): The subspaces ,  share their even grade members, i.e. the scalar 1 (grade 0) and the three bivectors  =  = e ( ≠  grade 2).We will present the parity transformation further down (Eq.(34-35)), but remind that it changes sign to 3D polar vectors (parity-odd) and leaves unchanged axial vectors (parity-even).As shown in Eq. ( 35) the vectors  behave as polar vectors (parity-odd), while  behave as axial vectors (parity-even).The subspace of axial vectors, spin and rotor generators  is isomorphic to the space of Pauli matrices [7].The subspace of polar vectors and boost generators  is the same as the even subspace of Hestenes' STA, in our notation having  = −Ie as 'local' pseudoscalar [6,7,10].The subspace  in (14') comprises the Lorentz group and is home to Dirac spinors, as explained further down.Notice that the vectors  ,  are Hermitian.In passing, the bivectors e =  e are also Hermitian and parity-even, see (35).(,  = 0,1,

Momentum quantization in
(,  = 0,1,2,3)  (a) From the definition of the orthonormal basis for the full STR vector space X in Equation ( 10); (b) From Equations (3, 4); (c) From Equation ( 14); (f) The generalized rotor R is relevant e.g. in passing from the Dirac to the Weyl basis (see Eq. (7A) in Appendix B).The invariant bivector Faraday F is shown in two forms, see Eq. (30, 30').In (∇ ∧ A) the action of the operator ∇ is confined within the brackets.
Due to Lorentz covariance (Eq.(49) further down) the STR Dirac spinor must fulfil ψ ∈  with  already presented in (14').One realization of this requirement is to split ψ into two independent parts φ, χ by the action of the two orthogonal projectors (1 ± e ), with the additional condition that φ, e χ be Pauli spinors: Similarly to the treatment in the standard formalism [4], depending on the physical problem at hand, one may choose to explicate in STR a scalar + pseudoscalar (grade 0&5) factor in ψ.E.g., in the case of free field, one expands ψ in plane waves of positive and negative energy and a constant spinor (with p the 4-momentum vector and  the spin degrees of freedom): φ and χ not being limited to the slow particle regime are more general than φ , χ .Each of the two Pauli spinors φ and e χ splits into two independent components by the action of two orthogonal projectors in .
We illustrate the projection of φ relative to the  axis, chosen by convention as reference direction: The  in  φ can be traded with −I by 'extracting' a  from the projector (1 −  ) in (20).This renders the representations of spin up and spin down formally the same as in STA [7], however remember that  in STA are not parity-even (see also [8]).The same expressions as in (20,21) One finds in a similar fashion the 'negative energy' solutions from the third pair of equations in (24), in this case remembering that − +  > 0. We just show the final result here:

STR at work
With the above preliminaries in place, it is instructive to start by showing how in the presence of external EM field the spin magnetic moment of the electron arises from spacetime-reflection.    (c)  ≡ ( −  ) and  ≡   ; (d) As anticipated in Eq. ( 14),  are axial, therefore they appear naturally here.For  = 0, the axial currents ψ e ψ are conserved.
Note that the final operators in (37) and (39) are identical, while that in (40) has opposite sign at the spatial part (see below).Let list in bullet form few salient properties of the transformations (37), (39), (40).
-All the three transformations invert the pseudoscalar I and are antiunitary [12] in STR.
-The vector operators for 3-momentum and position transform opposite (according) to the expectation for , ′ (resp.′′), which is clear by inspection of the transformed operators.Similarly to the cases of -The vector potential  =   does not invert under the proper t-reversal (40).An additional parity transformation is required, as in (37), (39) in order to invert .In the following, keeping with tradition we will identify t-reversal with (37), (39), which both invert  and preserve the Hamiltonian (41).
-The transformed operator parts being the same in (37) and (39) one expects ψ and ψ to represent the same state.Indeed, the  = † transformation is equivalent to a reversal of the spatial directors alone -Kramers' degeneracy [13] for a system consisting of an odd number of spin ½ particles in an electric field is more directly proved for the transformation  in (39) with  = −1, just as in the standard case [4].However, it is of course valid for in (37), which as argued above, leads to the same state.
The discussion above illustrates the power of STR to elucidate the geometric meaning of t-reversal, a subject that is quite involved in the standard formalism [4].Two remarks before closing the paragraph.Combinations of different forms of ,  in  conserve the Lagrangian and reveal some of its symmetries.Relations (45-50) demonstrate how boosts and rotations actually take place in the physical spacetime.In the discussion above we did not mention any (linearized) infinitesimal transformations [2,4,5] to prove the covariance of STR DE and the Lorentz invariance of the STR Dirac Lagrangian.STR stands at an advantage point when infinitesimal transformations are demanded, e.g. in curved spacetime [4].
How do we change basis in STR? (47) hints to the answer: by two sided general rotor transformations, i.e. The

3 )
[ , ].The bivector u ∧ v = −v ∧ u represents the oriented area encompassed by the two vectors.The geometric product (2) is linear and if not zero, it is invertible.When normalized it renders rotors or boosts, the generators of Lorentz transformations.Examples of both will appear in the following.The geometric product of orthonormal frame vectors is: e e ≡ e = e ⋅ e + e ∧ e =  + e ∧ e ; {e , e } = 2 ; u =  e ; ,  = 0,1,2,3 (Upright letters depict vectors; italics depict scalars.From (3) e ⋅ e =  defines the timespace signature.Signature can 'lift' indices of frame vectors up and down, thus connecting the reciprocal bases.The coincidence of Clifford algebras for e and Dirac matrices leads to STA -spacetime algebra [6, 7].The real vector space arising from the action of Clifford product onto spacetime vectors (2, 3) has dimension 16 with basis elements comprising: the real scalar unit, four vectors e , six bivectors e = e ∧ e ( ≠ ), four trivectors e ( ≠  ≠  ≠ ) and one tetravector e : Basis of STA vector space: 1, e , e , e , e ; , ,  = 0,1,2,3 ; generators: {e }, dim16 elements has zero scalar part 〈 〉 .E.g. for fixed indices (no sum over repeated indices, so that e e = 1): 〈 e e 〉 = 〈e e 〉 = 〈e e 〉 =  , or 〈 e e 〉 = 〈e  〉 + 〈 e −  e 〉 = 0 (6) Now, the relativistic 4-momentum vector p of modulus  and different forms of its square, coordinate-free and coordinate-bound, are given in STA by ( = 1) (sum over repeated indices by default): p = e  ; pp = p ⋅ p + p ∧ p = e   =    = p ⋅ p = Then, in analogy to the definition of Dirac's  matrix, we can define e by (note that e = e ): e = e ; e e = e e = 1 ⇒ (e ) = −(e ) = −e e e = e

( a )
The angle  is standard in the STA literature.In STR it is defined by the relative moduli of the two Pauli spinors, where  =   = 〈ψ e ψ〉 = 〈ψ ψ〉 =   +   =  +  .All -s are sums of a scalar and a pseudoscalar!See also footnote(e) below.(b)The standard antisymmetric traceless tensor is defined by the commutator of Dirac matrices  ≡ [ ,  ];

( 4 . 3 .
e) In STR 〈ψ e ψ〉 = 〈φ e χ − χ e φ〉 = 〈  〉   −   ~I sin , which changes sign under Hermite conjugation, as a pseudoscalar should.The corresponding expression in STA is in our notation  sin , which is a scalar and hints to a simplistic definition of ψ in STA.〈  〉 extracts the scalar part of   .Symmetries of STR DE.Let us look now at the symmetries parity , time reversal and charge conjugation  for the STR DE and for the STR Dirac spinor ψ.In the following we will present three forms ,  ,  ′ for time-reversal symmetry in STR as well as discuss the relation between the three (see also Appendix D).Parity : e → e e e = e  .As in the standard treatment left-multiply STR DE by e (below ℏ = 1): : (P − )ψ = 0 → e (P − )ψ = (e Pe − )e ψ ≡ (P  − )ψ  = e (I +  ) − e I +  −  ψ  = e  I +  −  ψ  ⇒ P  =  e (I +  ); ψ  = e ψ = ( ) φ − χ (34) It is clear from (34) that both 3-momentum and 3-position, therefore also the vector potential, change sign in STR DE under parity.Applying the parity transformation (34) to the 3D vectors from (14) we see that:   → e  e = − are ;   → e  e =  and e  → e e e = e  e e = e are  (35) The last equation in (34) shows that under parity the Pauli spinors φ and χ are respectively even and odd.Time reversal :  → − ⇔  →  = − .We write first DE in terms of the time-reversed quantities: DE: e  +  −  ψ = e (− +  ) + e − −  −  ψ = 0 (36) In order to render the energy term positive in the time-reversed system we need to invert the sign of − in (36) without changing  .This can be achieved in different ways, e.g. by Hermite conjugation & reversal ( † ) (), by  -or  -conjugations (leading respectively to  , ′′) of DE as now shown.We first present the three transformations then discuss and compare them.The Hermite conjugate of DE was shown in (32).: DE ⎯ e e −ℏI +  −  ψ = e (I +  ) + e I −  −  e ψ = 0 ⇒ ⇒ ψ = e ψ = ψ ; I = I = −I;  = 1 (37) Alternatively, we introduce  -conjugation that inverts only the e (below we look at  = 3 or 5):  : {e →  e  = (1 − 2 )e , or e → −e ; e → e ;  = 0,1,2,3,5}
e → −e , leaving e , e unchanged; we call it  -conjugation ( =    ).The property  e  = e distinguishes it from the parity transformation (34). ,  preserve the Clifford algebra {e , e } = 2 .Both reverse handedness of spatial coordinates, therefore the resulting bases from the two conjugations differ by a proper rotation, as illustrated in Figure1.In particular, , ′ both flip spin.
Table 2 summarizing the bilinears of STR DE.In addition, a more symmetric form of the STR Dirac spinor as expressed by two Pauli spinors has been introduced, which is though equivalent to the previous form.Following the suggestion of another reviewer, the presentation of the STR spinor appear earlier in the report, see Eqs. (17-22) and Table 2.The structure of the STR spinor is further elucidated by showing the STR free field solutions in Eqs.(23-29).
The vector-operator ∇ obeys to the chain rule ∇A = e   = e   +   and we need generalize the commutator in (2), as illustrated in Appendix A for [x, p] (position-momentum operators).The generators of the Lorentz group appear naturally in [x, p].Table1summarizes the notation in the STR formalism.In the remaining part of this Section we will define the Dirac form of the spinor ψ in STR and further elucidate its structure by giving the free field solutions of STR DE.
STR -the STR DE.Now we can write the momentum quantization Equation in STR (compare with (8)), obtaining the STR DE:∇ is both coordinate-free and shorthand for e  , therefore one can avoid the Feynman slash notation!ψmust have four components, i.e. it is a 4-spinor because of the four spacetime dimensions leading to four pairwise orthogonal projectors, two relative to the time axis e and two relative to one space axis in , per convention  (see Eqs. (17-22) below).p= Iℏ∇ is a tetravector operator in X containing e .For the electron with charge e, in the presence of an electromagnetic 4-potential A, (15) generalizes to: Multivector Space X(a)

Table 1 .
Summary of notation in STR.Upright letters depict vectorial quantities, letters in italics depict real scalars.
Table 2 summarizes the bilinears of STR DE.

Table 2 .
Dirac bilinears in standard and STR formalisms.Expanded forms of the STR Dirac bilinears appear in the last column, in terms of the Pauli spinors (21, 22).The angled brackets 〈 〉 in STR yield the expectation value.