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First Quantized Electrodynamics

The parametrized Dirac wave equation represents position and time as operators, and can be formulated for many particles. It thus provides, unlike field-theoretic Quantum Electrodynamics (QED), an elementary and unrestricted representation of electrons entangled in space or time. The parametrized formalism leads directly and without further conjecture to the Bethe-Salpeter equation for bound states. The formalism also yields the Uehling shift of the hydrogenic spectrum, the anomalous magnetic moment of the electron to leading order in the fine structure constant, the Lamb shift and the axial anomaly of QED.


I. INTRODUCTION
Recent observations [1] of photons entangled in time raise the possibility that fermions could be entangled in time. Quantum Field Theory, as recently shown [2], does allow a certain sense of timelike entanglement for a massless vacuum state of unspecified spin. However, Quantum Field Theory (QFT) has only limited ability to represent entangled states. Violations of Bell's inequality in the vacuum state for a massive particle decay exponentially with increasing spacelike separation, the decay scale being the Compton wavelength. If one particle location is subjected to a timelike displacement until the separation is timelike, then violations decay exponentially with that displacement measured again in Compton wavelengths [3].
The purpose of this analysis is to show that the parametrized Dirac equation proposed by Feynman [4][5][6] allows the unrestricted space-time entanglement of electrons. It is also shown that the parametrized formalism leads to the Bethe-Salpeter equation [7][8][9] for bound states. The formalism further predicts, by use of a partial summation of the Born series, several fundamental phenomena previously predicted [10][11][12][13] by field-theoretic or secondquantized Quantum Electrodynamics (QED). Included here are the lowest-order corrections yielding the Uehling potential, the anomalous magnetic moment of the electron, the Lamb shift and the axial anomaly.
The contents of this article are as follows. The parametrized Dirac wave equation is stated in Section II. Free wavefunctions, influence functions and Møller operators for the parametrized Dirac equation are constructed in Section III. The constructions are readily extended to several particles, as outlined in Section IV. Unrestricted entanglement in space-time is shown to be an explicit contingency in the parametrized formalism. The spinstatistics connection has been proved by Jabs [14] using first quantization, and the line of proof is very briefly described here. Two-particle scattering is defined in terms of Møller operators, which lead without further conjecture to the Bethe-Salpeter equation. The construction leading to the Møller operators is given in some detail, even though it parallels those found in classic texts [15,16], since the operators lead to precise agreement with successful predictions of field-theoretic QED. Scattering is then simplified to a single particle in an external potential. The first-order scattering of a single particle is outlined in Section V, with Mott scattering as an example. The standard one-loop corrections for scattering of arbitrary strength are derived in Section VI by use of a substitution that is accurate for beams undergoing weak scattering. The substitution resembles the QED relation between propagators and vacuum-to-vacuum expectations. The standard QED axial anomaly is derived in Section VII for classical fields rather than fields of operators. The summary in Section VIII includes a discussion of the wide utility of semiclassical theory and the further possibilities for parametrized formalisms.

II. THE PARAMETRIZED DIRAC WAVE EQUATION
The parametrized Dirac wave equation is stated, as are its Lagrangian and its discrete symmetries.

A. covariant formulation
For a single spin-1/2 particle the wavefunction is a four-spinor ψ(x, τ ). The event x is in R 4 , while the parameter τ is an independent variable in R. The event x is also denoted by x µ having indices µ = 0, 1, 2, 3, with x 0 = ct where c is the speed of light and t is coordinate time. The Lorentz metric g µν on R 4 has signature (− + ++). The position x is denoted by x j having indices j = 1, 2, 3. Thus x = (ct, x).

The parametrized Dirac wave equation for
where e is the charge of the particle, c is the speed of light and is the reduced Planck's constant. The γ µ are the four Dirac matrices, while the Maxwell electromagnetic potential A µ (x) is independent of the parameter τ . The covariance of the theory with respect to the homogeneous Lorentz transformation (x µ ) ′ = Λ µ ν x ν and ψ ′ (x, τ ) = S(Λ)ψ(Λ −1 x, τ ) follows for S(Λ) generated in the standard way [15]. No mass constant appears in (1), but masses are introduced through boundary conditions as τ → ±∞. Feynman's development of QED using (1) has been reviewed by Garcia Alvarez and Gaioli [4]. A simple consequence of (1) is the identity where ψ ≡ ψ † γ 0 , and the τ -dependent current is j µ = cψγ µ ψ. The indefiniteness of the invariant bilinear form ψψ has impeded [17,18] the development of the parametrized Dirac formalism as a relativistic extension of quantum mechanics.
The commutator of the operators x µ and p ν is Henceforth the units are chosen such that c = = 1. The Maxwell field strength tensor The summation convention is assumed with respect to repetitions of Greek indices such as µ = 0, 1, 2, 3 . The covariant and contravariant indices µ, ν, . . . will be omitted wherever convenient, as in The Newton-Wigner position operator [19] can only localize the position x of, say, an where m e is the conventional rest mass. The operator is extendable in the parametrized formalism to an event operator x µ with a footprint of the same order. However, the formalism does not restrict the state to a single or 'sharp' rest mass. Consider a wavefunction ψ(x, τ ) = exp(imτ )λ(x). The mass m can have infinite range and so perfect localization of x µ and of p µ is attainable [20,21].

B. Lagrangian and discrete symmetries
The single-particle Lagrangian is where d 4 x is the Lorentz-invariant measure on R 4 . The Feynman slash notation is for example / p = γ µ p µ , and in particular / p / p = −p · p . Variation of L with respect to ψ yields (1), while variation of L with respect to A µ in the Lorenz gauge, where ∂ · A = 0 , yields which is Maxwell's equation for A µ . The 'concatenated' current in (5) is divergenceless, satisfying ∂ · J = 0 .
The discrete symmetries of charge conjugation C, parity P and time reversal T act on the single-particle wavefunctions as follows.

III. WAVE FUNCTIONS AND MØLLER OPERATORS
The essential constituents of scattering theory are developed in detail for one particle.
The details are routine, being close analogs of those in standard quantum mechanics. The major difference is the existence of states having 'negative mass' but they do not participate, even virtually, in the scattering process.

A. free wavefunctions
The free wavefunctions are the solutions of (1) in the absence of the electromagnetic potential A µ . They have the form and where p µ is a constant energy-momentum vector, while ϕ p = θ(p 0 ) − θ(−p 0 ) is the sign of The mass m p , also denoted m(p) , is the positive square root of −p · p. It is assumed that the states are not superluminal, and so p · p ≤ 0.
The phases of the complex exponentials are p · x ± ϕ p m p τ , hence There are two linearly independent solutions for the four-spinor u(p) and two for the four-spinor v(p). In the rest frame where p = (p 0 , 0) they are In an arbitrary frame the solutions for u and v have the 4 × 2 block forms where K = [2m p (m p + E p )] −1/2 , and the σ j are the Pauli matrices (j = 1, 2, 3). The symbol I n denotes the n × n unit matrix. The blocks u and v are eigenspinors of the 'forward' and 'backward' projections respectively, and they obey the orthonormality conditions The free solutions having the same sense of propagation are orthonormal over R 4 in the continuum normalization, with block form while those having the opposite sense are orthogonal, with block form where f The spin projection operators in an arbitrary frame are In the rest frame, s = (0, s) where |s| = 1 . The operators Λ u (p) , Λ v (p) and P (s) commute for all s and p , since / p/ s = −p · s = 0 . The chirality projection operators are P ± = (1 ± γ 5 )/2, yielding for example the projections u ± = P ± u . The helicity operator for the unit momentump = p/|p| isp In the ultrarelativistic limit as m p /E p → 0, the projections u ± are eigenstates of the helicity operator with eigenvalues ±ϕ p respectively. In the same limit, the projections v ± = P ± v are helicity eigenstates also with eigenvalues ±ϕ p respectively.
Defining the free wavefunctions h it follows that Moreover, the T PC conjugate of an eigenstate of the forward projection Λ u (p)P (+s) is an eigenstate of Λ v (−p)P (−s) . Thus, defining the antiparticle to be the T PC conjugate of a particle, it follows that a positron is an electron having the reversed sense of propagation in time t as τ increases, the reversed energy-momentum and spin, but the same coordinate in S + , for example, can be scattered into S + states of negative energy without lower bound and so hole theory must be invoked. Equivalently, the negative-energy electron states in S + are regarded as states of different particles, namely positive-energy positrons [22]. Pair creation and annihilation are real processes, but the formalism of second quantization is avoided here by adopting the Feynman-Stueckelberg interpretation. That is, a positron in S + is formally represented by the T PC-conjugate of an electron in S + .

B. free influence functions
The two free influence functions Γ 0 and are given by where 0 < ǫ << 1. The small positive number +ǫ serves to remind that the inversion path is below the pole at +m p in the complex plane of m, and above the pole at −m p . Contour integration in the m-plane yields where θ is again the Heaviside unit step function and The two free influence functions are related by As τ ′ − τ → +∞, Γ 0 + evolves forward-propagating waves of positive energy and backwardpropagating waves of negative energy. In terms of the forward-propagating waves and their backward-propagating T PC conjugates waves, Γ 0 + becomes, for τ ′ − τ > 0, Thus as τ ′ − τ → +∞, Γ 0 + evolves only states in S + , that is the positive-energy, forwardpropagating particles (for example, electrons) and the positive-energy, backward propagating antiparticles (positrons).

C. Møller operators
The influence functions Γ ± (x ′ , τ ′ ; x, τ ) are defined as solutions of They depend upon the charge e and potential A µ , but in the interest of clarity the argument lists here for Γ ± include only the events x, x ′ and parameters τ, τ ′ . The boundary conditions Expanding (39) in the Born series shows that The semigroup property (35) holds also for Γ ± .
For φ i ∈ S + , define ω + φ i by If the support of the potential A µ is bounded in E 4 , and if the support It may be seen from (42) that ω + φ i (x, τ ) − φ i (x, τ ) ∈ S ± as τ → ±∞, that is, there is 'τ -increasing' scattering into positive-energy, forward-propagating particles and positiveenergy, backward-propagating antiparticles, and also 'τ -decreasing' scattering into negativeenergy, forward-propagating particles and negative-energy, backward-propagating antipar- Similarly Then It may be seen from (44) that In summary, the Møller operators ω ± are defined by for all φ ∈ S + . Again, ω ± φ = 0 for all φ ∈ S − .
The potential A µ involved in the Møller operators is calculated as the particular integral of (5) for the concatenation of the Møller current assuming that A µ = O(e) .

IV. TWO DIRAC PARTICLES
The single-particle formalism of the preceding sections is extended to two particles.
Space-time entanglement is displayed in an elementary and unrestricted way. A proof of the spin-statistics theorem is very briefly described, and the Bethe-Salpeter equation for bound states is derived from the two-particle formalism without further conjecture. The mutual scattering problem for two particles is reduced to that of a single particle in the presence of an 'external' potential.
Consider first the case of externally-sourced electromagnetic potentials A µ (x) and A µ (y) .
Two indistinguishable spin-1/2 particles entangled at the events x = (t, x) and y = (s, y) may be represented in the standard way with antisymmetric combinations of tensor products of single-particle 4-spinor wavefunctions, such as This representation of entanglement, in space or time, is simpler and more general than the QFT representations.
Consider now interaction potentials and self potentials acting on both particles. The potentials are constructed by integrating over the dependences of the two-particle currents upon one of the particles. That is, where e 1 and e 2 are the charges of the two particles. The currents are, after concatenation and marginalization, and In particular, a particle may experience the potential arising from its own current. If the two particles are indistinguishable, the wavefunction Ψ is chosen to be antisymmetric as in (48) . Then the corresponding quantum currents J µ 1 , J µ 2 and hence the semiclassical field A µ are unaltered by permutation of the two single-particle states at the same parameter τ .
That is, the currents and the potential are appropriately bosonic.
If the particles are distinguishable and the wavefunction is a simple tensor product Ψ(x, y, τ ) = ψ(x, τ ) ⊗ χ(y, τ ), where both ψ and χ are normalized beams or packets, then the concatenated currents reduce to and

B. spin and statistics
The spin-statistics connection is usually presented [10][11][12][13] as a theorem in relativistic QFT. On the other hand, the unimodular and unitary group SU(2) spanned by the Pauli spin matrices is a spin-1/2 representation of the unimodular and orthogonal group SO (3) of spatial rotations [23]. That is, spin is not intrinsically a relativistic phenomenon. Indeed, the single-particle, spin-1/2 Pauli wave equation [15] is covariant with respect to SO(3) provided that the electric potential A and the magnetic field B are functions of |x| alone. The spin-statistics connection has furthermore been proved for both nonrelativistic and relativistic quantum mechanics of arbitrary spin [14]. The proof takes into account the phases of spin eigenstates. The phases are indeterminate since only a spin axis is specified, rather than a spin frame. Homotopically consistent permutations of the arbitrary phases between numerous otherwise indistinguishable particles, along with permutations of the conventionally observable positions and spins, lead to the standard spin-statistics connection.
The proof holds in particular for relativistic spin-1/2 quantum mechanics. The phase is not a standard observable, but the universal observance of the exclusion principle impresses physical significance upon the phase indeterminacy.

C. two-particle Møller operators
There are four two-particle free influence functions. Of particular interest are Γ 0 ++ and Γ 0 −− given by Proceeding as in Section III C leads to the two-particle Møller operators Ω ++ and Ω −− acting on S + ⊗ S + , such that for all Φ ∈ S + ⊗ S + . The dependence of the influence functions Γ ±± and the Møller operators Ω ±± upon the two charges e 1 and e 2 , and upon the potential A µ at x and at y, is of fundamental importance but for clarity is not made explicit here. It follows that Ω ±± Φ = 0 for all Φ ∈ S ⊗ S \ S + ⊗ S + . The influence functions Γ ±± satisfy The scattering potential in (56) is The semigroup property (35) extends to Γ ±± . When the potential A µ is external, it may be shown that and that The separability property (58) implies that unentanglement is conserved with the passage of τ . The Møller currents are and

D. bound states
It is now convenient to combine the influence functions in the form and similarly there is the combined free influence function Γ 0 (x ′ − x, y ′ − y, τ ′ − τ ) . Then for all τ ′ and τ , any two-particle wavefunction Ψ(x, y, τ ) evolves as or by virtue of (56) as where the freely propagated wavefunction Ψ 0 (x ′ , y ′ , τ ′ ) is The Fourier transform of (64) with respect to τ ′ is the inhomogeneous Bethe-Salpeter equation [7][8][9] Ψ(x ′ , y ′ , m) = Ψ 0 (x ′ , y ′ , m) in the casual notation where The interaction potential V (x, y) as given in (57) is determined semi-classically, that is, in terms of potentials A µ (x) satisfying Maxwell's equation (49) for the Dirac currents (50) and (51). Bound states are defined as eigenstates of the homogeneous equation [9], that is, (66) for Ψ 0 = 0 . Indeed, if there is binding energy in the two-particle state Ψ then the freely-propagating Ψ 0 is a kinematical impossibility [8,9]. The Bethe-Salpeter equation is in general nonlinear but may be expanded at least formally in powers of the charges e 1 and e 2 , yielding a series of linear equations. A method of partial summation is described in §VI below.

E. two-particle scattering matrix
Let Φ i and Φ f be two free wavefunctions in S + ⊗ S + . Then the scattering matrix S f i is defined by The incident and final two-particle states for indistinguishable particles are independently fermionic. That is, the incident single-particle states may be permuted independently of the final states and vice versa, leading in each case to a reversal of the sign of S f i .
It is readily shown that ∂S f i /∂τ = 0 , since / π † γ 0 = γ 0 / π . Alternatively, it may be shown that which is explicitly independent of τ . It follows from (69) that where which is independent of τ assuming that Φ i and Φ f are free particle pairs. If the particles are distinguishable then the incident and final free wavefunctions are of the form Φ i = φ i ⊗ξ i and Φ f = φ f ⊗ ξ f respectively, while the Møller operators are of the form Ω ±± = ω ± ⊗ ω ± .
The Møller currents are calculated, following (52) and (53), as and If the two particles are indistinguishable then both the incident and final two-particle free states Φ i and Φ f in (68) must be fermionic as in (48). However, both two-particle free states in the Møller currents (60) and (61) must be replaced with bosonic states of the form The Møller currents are then bosonic, prior to marginalization. It follows eventually that the scattering matrix is appropriately fermionic with respect to the incident two-particle state, and also with respect to the final two-particle state.
The integrand for nontrivial scattering in (70) is, in the case of distinguishable particles for simplicity, Iteration on (71) by expansion in powers of the charges e 1 and e 2 proceeds the most efficiently if the Møller operators in the factors not explicitly displaying scattering, that is in the factors ξ f ω + ξ i and φ f ω + φ i , are not so iterated. These factors reduce to unity, after forming the scattering cross-section and averaging over the incident and final states. It suffices therefore to consider single-particle scattering off the semiclassical potential owing to any source including the current of the particle itself. The incident and final energy-momenta p µ i and p µ f are assumed timelike, hence the ostensibly 'external' photon is spacelike. The potential owing to an 'external' source Z µ (x) is now denoted A µ (x), with If the external source Z µ is in fact the current e 2 J µ 2 of a second and possibly distinguishable spin-1/2 particle then, owing to the parity symmetry of the D'Alembertian , it is readily shown that the single-particle scattering matrix is symmetric with respect to the two particles. That is, the scattering of the second particle by the first is the same as the scattering of the first particle by the second.

V. FIRST-ORDER SCATTERING OF A SINGLE PARTICLE
The scattering matrix for a single particle resembles (68) or equivalently (70). To leading order in powers of the charge e , the nontrivial contribution to the analog of (70) is Assume that both φ i and φ f have the form (11) with both p 0 i and p 0 f positive, hence both wavefunctions are in S + . It follows that where ∆p = p f − p i and ∆m p = m(p f ) − m(p i ), while u i = u(p i )a i and u f = u(p f )a f . The coefficients a i and a f are complex 2 × 1 matrices. Finally, A µ (∆p) is the Fourier transform of A µ (x) defined by Note that the mass of the particle is conserved, that is, m f = m(p f ) = m(p i ) = m i which may as well be assigned the notation m e for, say, an electron.
Conservation of energy p 0 as well as mass m(p) implies that the magnitude of momentum p is also conserved, and so a scattering parameter κ may be defined by p f · p i = |p| 2 cos κ .
It follows from (78) in the standard way [15,24] that the effect of spin is to modify the κdependence of the Rutherford cross section by the Mott factor M = 1 − |p/m e | 2 sin 2 (κ/2) .
It may be remarked in passing that, for the purposes of calculating cross-sections, the continuum normalizations (11) and (12) are replaced with 4-cube normalizations rather than the standard 3-cube. Specifically, (2π) 4 is replaced with L 4 where L is the edge of the 4-cube. The cross-section as calculated is then a volume rather than the standard area. In the case of Coulomb scattering, where the potential is independent of coordinate time x 0 , the cross-sectional volume is proportional to T 0 = 2πδ(p 0 i − p 0 i ) which is interpreted in the standard way [15] as a large interval in x 0 during which scattering is observed. The incident wavefunction is in practice a beam having finite energy spread. The beam accordingly takes a finite time to sweep through an external scattering potential, although it must be conceded that such a time is undefined for an inverse-distance potential. Thus it is in general appropriate when the potential is time-independent to divide the cross-sectional volume by T 0 , yielding a cross-sectional area [25].

VI. ONE-LOOP CORRECTIONS
Assuming as in Coulomb scattering that A is O(e), the first-order external scattering S while M 21 differs from M 12 only by exchanging the potentials / A and / A in (80). The self potential A µ in (80) satisfies and is obtained as where the Feynman photon propagator D F is [15] Again, the metric here has the signature (− + ++).
A change to the O(e 4 ) scattering matrix contribution M 4 = M 12 + M 21 is now made, by the substitution where m = (m f + m i )/2, but the substitution (84) is made without regard to the τ -ordering.
The standard additional minus sign is introduced for a closed electron loop [15]. and (e) of Bjorken and Drell [15]. In order of renormalization the diagrams are the vacuum polarization of the external field (e), the mass-renormalization counter term (d) and selfmass (c) on both the incident and final fermion lines, and the vertex correction (b). The four diagrams are different, even though they arise here from two duplicated integrals, hence the symmetry factor for each diagram is S = 1. Each of the four additive contributions to the modification factor F is independent of the parameter scale T τ . The contributions are, relative to S (1) f i , precisely the standard wavenumber integrals in Eqs (8.8), (8.34) and (8.49) of Bjorken and Drell [15]. All the integrals derived here have the signs as deduced from QFT [10,12,13]. That is, there is no need to introduce a sign correction by fiat, as is done by Bjorken and Drell [15] who assume the connection between spin and statistics. It is concluded that the Uehling potential, the anomalous magnetic moment of the electron and the Lamb shift have the respective leading-order values as eventually deduced by Bjorken and Drell [15] and as accurately observed [15].
The above-mentioned diagrams, integrals and resulting one-loop corrections are obtained by Bjorken and Drell [15] from the standard Dirac wave equation for a single particle. Instead of making the substitution (84), those authors replace the potential product A µ (x)A ν (y) in the second-order scattering matrix with a superposition of products of leading-order Møller currents. The resulting matrix is then symmetric in its dependence upon the two particles.
Bjorken and Drell refer to their text on QFT [26] for strict justification.

VII. AXIAL ANOMALY
The parametrized Dirac equation (1) leads to the vector current j µ = ψγ µ ψ obeying the identity (2), and also to the axial current j µ 5 = ψγ µ γ 5 ψ obeying The concatenated vector current J µ is again divergenceless, that is, while for the sharp mass value m e the divergence of the concatenated axial current J µ 5 is The preceding identities also hold for Møller charges and currents, that is, if the replacements ψ → ω − φ f and ψ → ω + φ i are made. In particular the Møller identities must hold at all orders in e, although verification order by order is tedious. It follows immediately from the Møller analog of (2) that the concatenated vector Møller current ω − φ f γ µ ω + φ i dτ is divergenceless.
The divergence of the axial Møller current ω − φ f γ µ γ 5 ω + φ i dτ is next recalculated correctly to O(e 2 ) for a particle in an external field. The particle is assumed for simplicity to be massless, that is, m e → 0 . Fermion loops and triangles are again closed using the substitution (84). It is straightforward to show that the concatenated axial Møller divergence vanishes at O(1), and also at O(e) . At O(e 2 ) the divergence is the sum of three terms, each of which is an integral over the energy-momenta of photons at all the vertices of a triangle.
The three terms are found to be identical, and so a symmetry factor of S = 3 is assigned.
The assignment is not S = 3! since the factor of 2 is already incorporated into each of the three original integrals, following symmetrization with respect to the photons at the two vector vertices. Hence the required divergence is the value of just one such term, which is precisely equal to the divergence of the QED coordinate-time-ordered, vacuum-to-vacuum amplitude of the product of an axial current and two vector currents [12]. The 'anomalous' divergence is accordingly found to have the standard value [10,12,13] Here, however, the fields are classically valued rather than operator valued.  Radiation (QTR) predicts [10][11][12][13] that the vacuum is a zero-point quantum field of energy density ω/2 per mode of circular frequency ω, leading in particular to the Planck law or blackbody spectrum. Consider, however, Maxwell's equation subject to a boundary condition which is a Lorentz-invariant classical free potential, which is statistically stationary and isotropic, and which has the same energy density ω/2 per mode. The classical solution can [29][30][31] then account not only for the blackbody spectrum, but also for the Van der Waals forces, the Casimir effect and the Einstein 'A' coefficient for spontaneous emission.
A random classical potential cannot [27] , however, explain antibunching for light beams passing through the two arms of an interferometer. That is, owing to the Cauchy-Schwarz inequality, the correlation of intensity for classical beams must be greater than unity. Correlations less than unity are routinely observed [32] at very low levels of illumination, and such antibunching can be explained [27] with QTR. It appears that only QTR can explain the sub-Poisson statistics which are detected again in very weak light beams [27]. Yet there is no immediate prospect of observing antibunching or sub-Poisson statistics for the W or Z gauge bosons, while a weak beam of free gluons is thus far a theoretical impossibility.
It remains to consider the wider utility of parametrized relativistic quantum mechanics.
The Higgs mechanism [10], for example, is essentially unaffected. The semiclassical massless gauge potential is independent of the parameter τ , while the scalar Higgs wavefunction obeys the parametrized wave equation of Stueckelberg [33][34][35] with the standard 'sombrero' selfinteraction.