1 CHAPTER SIX

THE ECE2 VACUUM

In ECE2 theory the vacuum is considered as the geometry of spacetime, so is richly

structured and has physical effects such as the radiative corrections and the Aharonov Bohm

(AB) effects (the subject of UFT336). The AB vacuum is defined as regions in which electric

and magnetic fields are zero but in which the ECE2 vacuum four-potential is non zero and

may cause observable effects. The opening of thsu chapter summarizes UFT336, in which it

is shown that the vacuum potential causes effects in electron spin resonance in the absence of

a magnetic field. The well known Chambers experiment can be adopted for experiments

designed to look for this effect.

The AB effects are well known {1 - 12} to be due to potentials in the absence of fields.

Consider the ECE2 definition of magnetic flux density used in previous chapters:

where

Here and are respectively the spin connection and tetrad vectors. Therefore the

AB vacuum is defined by the geometry:

and

Using the identity:

the AB vacuum geometry becomes:

i.e.

Now consider the electric field strength E and magnetic flux density of ECE2 theory as

defined by the spin and orbital curvature vectors as in previous chapters:

and

So the AB effects in this type of theory are defined by a Cartan geometry in which torsion

and curvature vanish but in which the tetrad and spin connection are finite. In minimal

notation

{1 - 12}:

so the AB vacuum geometry is:

with:

The well known Chambers experiment {1 - 12} shows that the AB vacuum is a

physical vacuum because the Young diffraction of electron matter waves is affected by

potentials in the absence of fields. The AB vacuum is defined in a different way from the

traditional definition in electromagnetic theory, one based on the absence of charge current

density. In this case the ECE2 electrodynamical field equations are:

with:

where:

Note 336(4) on www.aias.us shows that the solution:

means that E and B vanish. The simplest solution of Eqs. ( ) to ( ) is:

in which case the traditional vacuum, in which the charge current four density vanishes,

reduces to the AB vacuum.

Note 336(4) shows that if the traditional vacuum is accepted, and plane wave

solutions used for Eqs. ( ), the result is:

Under condition ( ), ECE2 theory allows vacuum electric and magnetic fields to exist

in the absence of charge current density - the traditional vacuum of electrodynamics. The AB

vacuum on the other hand is is defined by Eq. ( ). The interaction of the AB vacuum

with one electron allows leads to the possibility of NMR and ESR (Chapter five) in the

absence of a magnetic field. This type of interaction is considered in Note 336(5) and is based

on the minimal prescription:

where the relevant four potential is defined in previous chapters:

Therefore the Einstein energy equation becomes:

The total relativistic energy E and the relativistic momentum p are defined by the Einstein /

de Broglie equations as in previous chapters:

and

The Lorentz factor is therefore:

and it follows that:

As in chapter five this reduces to the Dirac theory if

i.e. by the de Broglie equation:

where is the rest angular frequency. The Dirac theory is therefore self contradictory

because the electron is not moving:

In the approximation:

Eq. ( ) becomes:

The ESR term is contained in the first term on the right hand side, and spin orbit efefcts in the

second term. Relativistic quantization is defined by:

i.e.:

and

This procedure cannot be proven ab initio, it is an empirical rule. The required ESR term is

given by:

Using the Pauli algebra:

so the real and physical part of Eq. ( ) is:

The spin angular momentum of the electron is:

so:

in which:

where:

Therefore:

in which electron spin resonance is defined by:

with resonance frequency:

In the AB effects A is non zero when B is zero, so from Eq. ( ) the AB vacuum is

defined by:

and under this condition the AB vacuum causes the resonance defined by Eq. ( ).

A computational and graphical analysis of this theory is given later on in this

chapter.

As in UFT337 the ECE2 theory that describes the AB vacuum can be used to

describe the radiative corrections, notably the Lamb shift. In order to do this, the minimal

prescription used earlier in this chapter for ESR effects of the vacuum is replaced by a new

type of minimal prescription using the W potential. This can be developed in terms of a

relativistic particle flux and the Tesla vacuum. The theory of UFT337 defines the ECE2

vacuum particles, which are identified as particles of the Tesla vacuum. Therefore there is

particulate energy momentum in the ECE2 vacuum that can be transferred to matter using

well known theoretical methods.

Consider the ECE2 minimal prescription:

where:

In ECE2 theory:

where the spin connection four vector is:

It follows that:

and:

The units of W are those of magnetic flux:

A summary of S. I. Units is given as follows:

The ECE2 magnetic flux density B (in units of tesla) is defined by Eq. ( ), and the

ECE2 electric field strength E in volts per metre is:

The elementary quantum of magnetic flux is {1 - 12}:

where h is the reduced Planck constant, the quantum of angular momentum in J s. Therefore:

The AB spacetime can therefore be defined in terms of the vacuum potential:

and on the most fundamental level:

So the AB spacetime is defined by the spin connection vector within the fluxon

The latter is negative under charge conjugation symmetry. In the absence of electric and

magnetic fields the AB spacetime (or vacuum or aether) is defined by Eq. ( ). The

fields E and B on the other hand are defined by curvature as in Eqs. ( ) and ( ).

The latter is zero in the AB vacuum, and so is the torsion:

Consider now the Einstein energy equation:

Using the minimal prescription ( ) the effect of the AB spacetime on material matter

such as an electron is:

If the electron is at rest:

so:

The AB spacetime contains an angular frequency:

so Eq. ( ) becomes:

from which it is clear that the rest frequency of a particle of material matter is increased by:

due to the presence of the AB vacuum. So the mechanism of energy from spacetime becomes

clear.

The AB spacetime imparts energy momentum to material matter as follows:

where:

The angular frequency of the AB spacetime is:

and the wave vector of the AB spacetime is:

The Einstein / de Broglie equations of the AB spacetime (or vacuum) are:

where the vacuum Lorentz factor is:

Therefore the existence of a vacuum particle of mass m( ) has been inferred

via the Einstein / de Broglie equations. There is a statistical ensemble of such particles. The

AB vacuum is quantized using:

and:

where is the wavefunction of the vacuum wave / particle. The wavefunction obeys

the ECE wave equation {1 - 12} in the limit:

where:

The vacuum wavefunction is therefore:

and the ECE wave equation of the vacuum is:

Eq. ( ) is the quantized version of the Einstein energy equation of the vacuum:

The process of taking energy from the vacuum becomes simple to understand:

and is observed as an energy shift in spectra, notably the Lamb shift of atomic hydrogen, and

also in the anomalous g of the electron.

It appears that such a particle vacuum was proposed but not proven by Tesla.

The Lamb shift and anomalous g factor can be defined in terms of emergy /

momentum transfer. The conventional theory of the Lamb shift assumes that the electron in

the H atom fluctuates in the presence of the vacuum - this phenomenon is known as

jitterbugging. It can be shown as follows that this is due to vacuum energy of ECE2 theory,

and the observed Lamb shift can be used to calculate a mean vacuum angular frequency. The

AB vacuum and the B(3) field {1 - 12} can be defined in terms of ECE2 theory.

By considerations of the Einstein energy equation in ECE2 theory, and by use of

the minimal prescription, it can be shown as in Notes 340(1) and 340(2) that the anomalous g

factor of the electron is defined by:

where H is the hamiltonian of ECE2 relativity:

For a static electron for which the de Broglie equation ( ) holds:

In general, the anomalous g factor of the electron is:

where is the angular frequency of the electron wave, and is the

angular frequency of the ECE2 vacuum wave particle. In Note 340(2) on www.aias.us It is

shown in complete detail that the process of momentum transfer from the vacuum wave

particle results in the observable energy shift:

Various methods of calculating this shift are described in Note 340(3). Therefore momentum

as well as energy can be transferred from the ECE2 vacuum.

In Note 340(4) the ECE2 vacuum potential is defined as:

It follows that the Coulomb potential U between an electron and a proton in the H atom

is augmented by:

In the well known Bethe theory {1 - 12} of the Lamb shift, it is assumed that

jitterbugs as described in Note 340(3):

in which denotes the fluctuation in position of the electron due to the vacuum, in

this case the ECE2 vacuum. This idea implies that the vacuum potential is:

in which the change in potential energy due to the ECE vacuum is, self consistently:

If it is assumed that:

Eq ( ) can be written as:

and averaging over the ensemble of vacuum particles:

Using the Bethe assumption:

it follows that:

where is the fine structure constant.

So the mean square fluctuations result in a mean vacuum angular frequency.

By using a Maclaurin series expansion of the equation:

it can be shown that

For the orbital of the H atom {1 - 12}:

where is the value of the wavefunction of H at the origin:

where is the Bohr radius. From Eqs. ( ) and ( ) the Lamb shift in the

energy level of the H atom is:

The measured Lamb shift is:

where:

Computing expectation values from the hydrogenic wavefunctions it is found that:

The relevant value for is:

This gives the mean vacuum angular frequency of an ensemble of vacuum wave particles:

The de Broglie frequency of one vacuum particle is:

The ensemble averaged frequency is much lower than the de Broglie frequency,

and the former is responsible for the Lamb shift in . This means that there is a tiny

universal anisotropy in the vacuum, and this is a tiny anisotropy of the universe itself, for

example in the microwave background radiation. The wavefunction of atomic H

vanishes at the origin and so there is no Lamb shift in this case.