_{1}

^{*}

A source of the divergences in QED is proposed, and a theory in which the Lamb shift and electron’s anomalous magnetic moment are calculated free of divergences is reviewed. It is shown that Dirac’s equation for a relativistic electron can be inferred from a Lorentz invariant having the form of the Lorentz gauge equation,
, on identifying a carrier-wave energy
with the electron’s rest mass energy mc^{2}, the vector potential’s polarization vector with Pauli’s vector σ, and the envelops of the scalar and vector potentials with the four components of Dirac’s vector wave function. The same methodology is used to infer relativistic equations of motion having the Dirac form for a neutrino accompanied by ab initio neutrino-matter interaction terms. Then it is shown that the theory, which comprises Dirac’s equation plus the relativistic equations of motion for the neutrino, supports binding on a nuclear scale of energy and length. The experimentally observed weakness of the interaction energy of free neutrinos and matter is due to the smallness of the rate of tunnelling of free neutrinos through a potential barrier which exists in the interaction of free neutrinos and matter. Models are also proposed for the proton and neutron, and good agreement is obtained for the neutron-proton rest mass energy difference in view of the approximations made to solve the appropriate equations of motion.

In previous work [

I pursue a mixed matter-light concept using a quantum-optical methodology firstly by inferring Dirac’s equa- tion for a relativistic electron from the Lorentz gauge equation, which is the Lorentz invariant found from the scalar product of the 4-gradient and the electromagnetic 4-potential,

as I show in the next section. Dirac’s desiderata for his equation is that it should satisfy the energy-momentum relationship of special relativity and that it should be compatible with the equation of continuity, which is the Lorentz invariant found from the scalar product of the 4-gradient and the 4-current,

Notice that matter and light are not separated in classical field theories. For example Maxwell’s wave equa- tion follows by requiring that Ampere’s equation should be compatible with the equation of continuity given by Equation (2).

equation and using Coulomb’s equation,

To be sure light and matter are inextricably interwoven in classical electromagnetic theory. The matter con- tributions were originally understood using classical mechanics and then were later updated using quantum me- chanics to arrive at an overall semiclassical theory (classical theory of radiation combined with a quantum theory of matter). This situation changed when Dirac quantized the free radiation field to calculate the Einstein A and B coefficients for emission and absorption of radiation as discrete photons by matter [

It is possible to interpret Lamb’s experiments not as the result of the interaction of free photons and matter, which is the standard interpretation following the success of renormalization theory, but rather as an observation that radiation is a permanent part of the structure of matter, which of course is not compatible with the prior de- velopment of the quantum theory of matter as a radiation-free theory. It is possible to criticize QED, notwith- standing its success, on the basis not only that theory must be augmented by physical argument and ad hoc ma- thematical procedures to obtain agreement with experiment but that its pioneers failed to confront Lamb’s expe- riments in the first place as revolutionary for the structure of matter and decided to use the matter-free photon theory and the photon-free matter theory which were on hand [

The removal of divergences in QED requires renormalization schemes based the notion that a photon is al- ways present in the structure of a free electron [

These radiant-aspect equations, which are given explicitly in the next section, are inferred from the Lorentz invariant found from the scalar product of a renormalized 4-gradient and a postulated 4-potential in analogy to Equation (1),

(where the notation

normalized 4-gradient,

gives the electromagnetic equation of continuity,

since the scalar product of the

energy density and

aspect equations inferred from Equation (3) are used to calculate the Lamb shift [

The details of this theory are given in the next section. Then in the following sections the theory is applied in an exploratory manner to nuclear binding. Thus it appears that a theory free of divergences in electron and atomic phenomena also includes a theory of nuclear phenomena.

We may use Lorentz’ equation [Equation (1)] to elucidate the structural relationship of Dirac’s equation for a relativistic electron with the spinorial form of Maxwell’s equation, a subject which has been studied conti- nuously [

On substituting Equation (5) into Equation (1) and separately setting the coefficients of the exponential fac- tors equal to zero, one obtains,

which are identically Dirac’s pair of first-order equations for a free electron on setting

functions are the Dirac two-component spinors. Hence one has a Lorentz-invariant relativistic equation of mo- tion for a material particle (MEOM) if the carrier-wave frequency belonging to the posited 4-potential is equal to the rest-mass energy of the material particle divided by

will be given below for its interaction with the familiar electromagnetic 4-potential,

Lorentz factor

should be compatible with the material equation of continuity given by Equation (2). The latter demand is satis- fied by Dirac’s equation, giving a current,

where the superscripts denote Hermitian conjugates.

As with the electron the posited 4-potential given in Equation (3) can be written in the form of carrier-wave ex- pansions,

from which on substituting Equation (8) into Equation (3) and separately setting the coefficients of the exponen- tial factors equal to zero, we obtain,

On setting

Writing

and

equations for the electric and magnetic radiant-particle wave functions which have the Helmholtz form,

where we have used the identity,

in physical applications to calculate a divergence-free Lamb shift [

Notice that in Equation (11) external electromagnetic fields and not external electromagnetic potentials occur such there is no question of a gauge dependence of matter-light interactions in the electron’s REOM. The suc- cess of the use Equation (11b) to calculate divergence-free radiative properties of matter [

As shown below in the electric-field and magnetic-field equations of motion with second-order Helmholtz

form [Equation (11)], the same-parity,

contribute, among other terms, all four of Maxwell’s equations as interaction terms, the same parity addition

vector contributing

The radiation-matter interaction terms are guaranteed to be gauge invariant by depending on the electromag- netic fields rather on the the electromagnetic potentials. Our approach here has been to face Lamb’s observation that the structure of matter has a permanent radiative component by finding first-quantized equations of motion for the radiant aspect of matter, in analogy to Dirac’s first-quantized equation of motion for the material aspect of matter. It seems remarkable that an established REOM does not already exist in the literature. The omission seems to follow from the neglect of a requirement that a complete relativistic-electron theory should be compa- tible with the electromagnetic equation of continuity [Equation (4)] as well as with the material equation of con- tinuity [Equation (2)]. Dirac required only that his equation be compatible with the material equation of continuity.

In summary the renormalization of the 4-gradient using

in which the radiant particle can be identified with the photon since the REOM gives a divergence-free theory of the Lamb shift [

of the 4-gradient using

section.

Finally it remains to follow through with the “quantum optical” derivation of Dirac’s equation for the material particle (MEOM) in the presence of external electromagnetic potentials, which is given by

Notice that Equation (12) follow from Equation (1) on renormalizing the 4-gradient as follows,

4-potential has been dropped.

The equations of motion for the radiant and material properties of matter are given by Equation (11) and Equa- tion (12)] respectively. Equation (11b) and Equation (12) were used in earlier work [

REOMEF:

MEOM (Dirac’s equation in second-order form):

where

appropriate for a mass-0 particle.

Models for a proton and neutron (Figures 1-3) are constructed by summing the electric field in the REOMEF and the potential in the MEOM over the electric fields and potentials respectively arising from two positrons and one electron (proton model) and over two positrons and two electrons (neutron model). In the MEOM the short-range repulsion due to the spin-orbit interaction is overcome by short-range attraction due to the magnetic interaction,

The REOMEF with MEOM interaction terms in the REOMEF are given by the summed electric fields arising from the quantum charge densities of bare electrons and bare positrons.

for the interaction of particles with the same (+) or opposite (?) charges and where the electronic density is that inferred from the MEOM using the large and small components of the MEOM wave function for a bare electron

(or bare positron)

the right sides of the minima in

The MEOM with REOMEF interaction terms in the MEOM are given by the A^{2} and Pauli ^{2 }contribution is found to be neg- ligible compared to the Pauli contribution such that only the scalar potential, V, and the

The vector potential is calculated from Maxwell’s equation,

where the current arises from the radiant particle’s spin,

Equation (10) on using an equation of continuity for the electron’s (or positron’s) radiant aspect

conjugate. Hence the spinor analysis for a REOMEF particle, for which

The Cartesian components of the current for

The magnetic field is found by taking the curl of both sides of Equation (15),

Only the diagonal or z-component of

tion (16),

Proceeding heuristically the radial equations (for

trial forms

ticle respectively, where w = w' and in Equation (18a) below the contribution given by

luated approximately by replacing it with

“small” components are calculated using the trial forms,

diant and bare electron respectively, where m_{p} is the proton mass, which is the only empirical parameter in the calculation. The minimum energy, as we shall see below, occurs for w approximately equal to the reciprocal of

the proton Compton wavelength,

tional form for

which the large denominator m_{p}c^{2} is cancelled by the numerator

force-carrier picture of QCD.

Once the derivative operation indicated in the interaction contribution,

ly evaluated as outlined above Equation (11a) has the Schroedinger form,

where in Equation (18b)

I posit that negative imaginary energy states are physically-interpretable in the sense that they are temporally bound and are therefore “confined” since transition rates vanish by destructive interference of contributions to

transition-rate matrix elements between temporally bound and temporally harmonic states. (Notice that imagi-

nary-energy states of the Dirac Coulomb problem, for which

tion.

Standard WKB theory [

where

square-root sign is positive.

In the following discussion the following nomenclature will be used. The bound solutions emerging from the REOMEF, which is coupled to the MEOM, will be referred to as radiant particles (charge-0, mass-0, spin-1/2 particles). The bound solutions emerging from the MEOM, which is coupled to the REOMEF, will be referred to as radiant electrons and radiant positrons as distinguished from bare electrons and bare positrons, which are

the solutions of the MEOM uncoupled to the REOMEF. The radiant electrons and radiant positrons have totally different binding characteristics than bare electrons and bare positrons as I discuss in detail as follows.

It is possible to find bound states for two radiant positrons and one radiant electron to give a model for the proton by finding the variational parameter w' for the REOMEF trial wave functions to give the three lower MEOM energy minima shown in

The scaling of squared energies in each MEOM to give a balance between spin-orbit repulsion and magnetic

attraction has the form

Energy (GeV) | w' | Lifetime | |
---|---|---|---|

Radiant positron 1 | 0.315 | 3.5w | ∞ |

Radiant positron 2 | 0.520 | 2.7w | ∞ |

Radiant electron | 0.103 | 2.2w | ∞ |

Total | 0.934 |

for N radiant particles determining the strength of the magnetic field given by Equation (17) due to the radiant- particle current and spin-orbit repulsion where A, B, and D are dimensionless constants from the calculation. Notice the w^{2 }scaling of the magnetic attraction (in contrast to Coulombic terms which scale as 2 mc^{2}e^{2}w) mak- ing it competitive with the kinetic energy term scaling as^{3} eventually becomes dominant with increasing w. If N is too large, the net result with- out minimum diverges in the negative region; if N is too small, the net result without minimum diverges in the positive region. The net spin of the three MEOM fermions is 1/2 due to the combination of two spin-up radiant positrons and one spin-down radiant electron. The large number of radiant particles (2130) were needed in order to overcome the repulsive spin-orbit interaction of the radiant electron with the two radiant positrons, as com- pared with the repulsive spin-orbit interaction of either radiant positron with the radiant electron and the attrac- tive spin-orbit result of one radiant positron with the other radiant positron, such that the radiant electron energy minimum occurs at smaller values of w than the radiant positron energy minima (

The upper four MEOM minima shown in

Standard WKB theory [

where the integration limits in the exponential term are over the barrier width.

For the value of w giving the minimum Dirac energy in ^{?4} s^{?1}, whose reciprocal gives a lifetime of 1.15 × 10^{3} s, which is close to the observed neutron lifetime of nearly fifteen minutes. This rate is calculated for a positive REOMEF particle energy of^{?29} and the integral in the deno- minator is 3.32 × 10^{?10}, such that the quotient is 2.10 × 10^{?20}, which, converted to cgs units by division by the atomic unit of time, 2.42 × 10^{?17}, gives the rate and lifetime cited above and in

In the proton model (

Energy (GeV) | w' | Lifetime | |
---|---|---|---|

Radiant positron 1 | 0.315 | 1.2w | ∞ |

Radiant positron 2 | 0.520 | 1.0w | ∞ |

Radiant electron 1 | 0.103 | 2.2w | ∞ |

Radiant electron 2 | 0 | 2.3w | 1.15 × 10^{3} s |

Total | 0.934 |

energy of free neutrinos and matter is due to the smallness of the rate of tunnelling of free neutrinos through a potential barrier which exists in the interaction of free neutrinos and matter such that the weak interaction energy of neutrinos and matter can be estimated from the uncertainty principle

I should comment here on the scaling of the interaction terms in the two coupled equations so that readers feel comfortable with the nuclear-scale binding found in these calculations.

can support bound states of the REOMEF. The interaction scales according to the quantum expectation value of

the attractive interaction term,

with increasing w the kinetic energy (first term in the above operator) scaling as

Compton wavelength

Finally ^{−}, model elastic scattering cross section in the Born approximation as compared with measured results for e^{?}, p^{+} elastic scattering [^{13 }cm^{?1} for the scattering electron’s neutrino interaction. The form factor has contributions scal-

ing as

tum transfer, such that, as shown in

Equations of motion for the radiant properties of matter (REOM) have been presented based on Lamb’s experi- ments showing that the energy levels of an atom are permanently shifted by radiation. The REOM are posited as relativistic equations of motion which are compatible with the electromagnetic equation of continuity in analogy to Dirac’s equation for the material properties of matter (MEOM), which is compatible with the material equa- tion of continuity. The REOM with electric-field interaction (REOMEF) supports the binding of radiant matter on a GeV energy scale and Fermi-unit length scale. Bound states of the REOMEF which are stable against dis- sociation occur in a region of negative squared energy, for which the energy is negative imaginary, such that binding occurs both in time (or in the 4-space scaled time ct) and space. Since transitions rates vanish between temporally bound and temporally harmonic states, bound matter may not be broken up into individual constitu- ents and may be considered to describe a form of confinement found in QCD Bound states of the REOMEF which occur in a region of positive squared energy are unstable against dissociation into a bare electron or bare positron and a mass-0, spin-1/2, neutral particle which I have posited to be a neutrino. The theory is used to con- struct a model for the proton which shows good agreement with the proton rest-mass energy and measured e^{?}, p^{+} elastic-scattering cross section. A model for the neutron is also constructed and the neutron-proton rest-mass difference agrees well with the observed value in view of the approximations made to solve the appropriate equ- ations. The neutron model invites new experiments to look for stimulated beta decay as given by the reaction,

The author is grateful to T. Scott Carman for supporting this work. He is grateful to Professor John Knoblock of the University of Miami for seminal discussion. This work was performed under the auspices of the Lawrence Livermore National Security, LLC (LLNS) under Contract No. DE-AC52-07NA27344.