Towards the end of the nineteenth century the Italian mathematician Eugenio Beltrami developed a system of equations for the description of hydrodynamic flow in which the curl of a vector is proportional to the vector itself. An example is the use of the velocity vector. For a long time this solution was not used outside the field of hydrodynamics, but in the fifties it started to be used by workers such as Alfven and Chandrasekhar in the area of cosmology, notably whirlpool galaxies. The Beltrami field as it came to be known has been observed in plasma vortices and as argued by Reed  is indicative of a type of electrodynamics such as ECE. Therefore this chapter is concerned with the ways in which ECE electrodynamics reduce to Beltrami electrodynamics, and with other applications of the Beltrami electrodynamics such as a new theory of the parton structure of elementary particles. The ECE theory is based on geometry and is ubiquitous throughout nature on all scales, and so is the Beltrami theory, which can be looked upon as a sub theory of ECE theory.
3.2 Derivation of the Beltrami Equation Consider the Cartan identity in vector notation, derived in Chap. 2:
In the absence of a magnetic monopole:
Assume that the spin connection is an axial vector dual in its index space to an antisymmetric tensor:
where is the totally antisymmetric unit tensor in three dimensions. Then Eq. (3) reduces to:
An example of this in electromagnetism is:
in the complex circular basis ((1), (2), (3)). The vector potential is defined by the ECE hypothesis:
From Chap. 2, Eq, (2.76) the geometrical condition for the absence of a magnetic monopole is:
where the spin curvature Eq. (2.63) is defined by:
and where is the magnetic flux density vector. Using Eq. (4):
In the complex circular basis defined by Eq. (6) the spin curvatures are:
and the magnetic flux density vectors are:
Eq. (8) may be exemplified by:
which may be developed as:
Possible solutions are
and in order to be consistent with the original [1-10] solution of B(3) the negative sign is developed:
From Eq. (3.2):
and the following is an identity of vector analysis:
A possible solution of Eq. (3.17) is:
Now multiply both sides of the basis equations (3.6) to (3.8) of Chap. 2 by
where the electromagnetic phase is:
to find the cyclic equation:
From Eqs. (3.23-3.25):
which are Beltrami equations .
The foregoing analysis may be simplified by considering only one component out of the two conjugate components labelled (1) and (2). This procedure, however, loses information in general. By considering one component, Eq. (3.1) is simplified to:
and the assumption of zero magnetic monopole leads to:
proceeding as in note 257(7) in the UFT section of www.aias.us leads to:
is the simplified format of the spin curvature. From Eqs. (3.31) and (3.32):
However, in ECE theory:
so Eqs. (3.35) and (3.36) imply:
Therefore in this simplified model the spin connection vector is parallel to the vector potential. These results are consistent with [1-10]:
from the minimal prescription. So in this simplified model:
The electric field strength is defined in the simplified model by:
where the scalar potential is
From Eqs. (3.39) and (3.40):
which is the same as the structure given by Heaviside, but these equations have been derived from general relativity and Cartan geometry, whereas the Heaviside structure is empirical. The equations (3.29) to (3.43) are oversimplified however because they are derived by consideration of only one out of two conjugate conjugates (1) and (2) . Therefore they are derived using real algebra instead of complex algebra. They lose the B(3) field and also spin connection resonance, developed later in this book.
In the case of field matter interaction the electric field strength, E is replaced by the electric displacement, D, and the magnetic flux density, B by the magnetic field strength, H:
where P is the polarization, M is the magnetization, is the vacuum permittivity and is the vacuum permeability. The four equations of electrodynamics for each index (1) or (2) are:
where is the charge density and J is the current density.
The Gauss law of magnetism:
implies the magnetic Beltrami equation :
So the magnetic Beltrami equation is a consequence of the absence of a magnetic monopole and the Beltrami solution is always a valid solution. From Eqs. (3.49) and (3.51)
and for magnetostatics or if the Maxwell displacement current is small:
In this case the magnetic flux density is proportional to the current density. From Eq. (3.51):
Eqs. (3.54) and (3.56) imply that the current density must have the structure:
in order to produce a Beltrami equation (3.51) in magnetostatics. Eq. (3.54) suggests that the jet observed from the plane of a whirlpool galaxy is a longitudinal solution of the Beltrami equation, a J(3) current associated with a B(3) field.
In field matter interaction the electric Beltrami equation:
is not valid because it is not consistent with the Coulomb law:
From Eqs. (3.58) and (3.59):
which violates the vector identity:
The electric Beltrami equation:
is valid for the free electromagnetic field.
Consider the four equations of the free electromagnetic field:
for each index of the complex circular basis. It follows from Eqs. (3.64) and (3.66) that:
The transverse plane wave solutions are:
and where is the angular velocity at instant and is the magnitude of the wave vector at Z.
From vector analysis:
and for the free field the divergences vanish, so we obtain the Helmholtz wave equations:
These are the Trkalian equations:
So solutions of the Beltrami equations are also solutions of the Helmholtz wave equations. From Eqs. (3.64), (3.67) and (3,76):
which is the d’Alembert equation:
For finite photon mass, implied by the longitudinal solutions of the free electromagnetic field:
in which case:
which is the Proca equation. This was first derived in ECE theory from the tetrad postulate of Cartan geometry and is discussed later in this book. From Eqs. (3.67) and (3.68):
so the general solution of the Beltrami equation
will also be a general solution of the equations (3.63) to (3.66) multiplied by the phase factor, .
ECE theory can be used to show that the magnetic flux density, vector potential and spin connection vector are always Beltrami vectors with intricate structures in general, solutions of the Beltrami equation. The Beltrami structure of the vector potential is proven in ECE physics from the Beltrami structure of the magnetic flux density B. The space part of the Cartan identity also has a Beltrami structure. If real algebra is use, the Beltrami structure of B immediately refutes U(1) gauge invariance because B becomes directly proportional to A. It follows that the photon mass is identically non-zero, however tiny in magnitude. Therefore there is no Higgs boson in nature because the latter is the result of U(1) gauge invariance. The Beltrami structure of B is the direct result of the Gauss law of magnetism and the absence of a magnetic monopole. It is difficult to conceive why U(1) gauge invariance should ever have been adopted as a theory, because its refutation is trivial. Once U(1) gauge invariance is discarded a rich panoply of new ideas and results emerge.
The Beltrami equation for magnetic flux density in ECE physics is:
In the simplest case is a wave-vector but it can become very intricate. Combining Eq.
(3.87) with the Ampere Maxwell law of ECE physics:
the magnetic flux density is given directly by:
Using the Coulomb law of ECE physics:
it is found that:
a result which follows from:
where c is the universal constant known as the vacuum speed of light. The conservation of charge current density in ECE physics is:
so is always a Beltrami vector.
In the simplified physics with real algebra:
where A is the vector potential. Eqs. (3.94) and (3.95) show immediately that in U(1) physics the vector potential also obeys a Beltrami equation:
so in this simplified theory the magnetic flux density is directly proportional to the vector potential A. It follows immediately that A cannot be U(1) gauge invariant because U(1) gauge invariance means :
and if A is changed, B is changed. The obsolete dogma of U(1) physics asserted that Eq. (3.98) does not change any physical quantity. This dogma is obviously incorrect because B is a physical quantity and Eq. (3.97) changes it. Therefore there is finite photon mass and no Higgs boson.
Finite photon mass and the Proca equation are developed later in this book, and the theory is summarized here for ease of reference. The Proca equation [1-10] can be developed as:
where the 4-current density is:
and where the 4-potential is:
Proca theory asserts that:
where m is the finite photon mass and is the reduced Planck constant. Therefore:
The Proca equation was inferred in the mid-thirties but is almost entirely absent from the textbooks. This is an unfortunate result of incorrect dogma, that the photon mass, is zero despite being postulated by Einstein in about 1905 to be a particle or corpuscle, as did Newton before him. The U(1) Proca theory in S. I. Units is:
It follows immediately that:
Eq. (3.110a) is conservation of charge current density and Eq. (3.110b) is the Lorenz condition. In the Proca equation the Lorenz condition has nothing to do with gauge invariance. The U(1) gauge invariance means that:
and from Eq. (3.108) it is trivially apparent that the Proca field and charge current density change under transformation (3.111), so are not gauge invariant, QED. The entire edifice of U(1) electrodynamics collapses as soon as photon mass is considered.
In vector notation Eq. (3.109) is:
and the Coulomb law of this simplified theory (without index a):
to find that:
which is the equation of charge current conservation:
In the Proca theory, Eq. (3.110) implies the Lorenz gauge as it is known in standard physics:
The Proca wave equation in the usual development  is obtained from the U(1) definition of the field tensor:
Eq. (3.121) follows from Eq. (3.108) in Proca physics, but in standard U(1) physics with identically zero photon mass the Lorenz gauge has to be assumed, and is arbitrary. So the Proca wave equation in the usual development  is:
In ECE physics [1-10] Eq. (3.122) is derived from the tetrad postulate of Cartan geometry and becomes:
In ECE physics the conservation of charge current density is:
and is consistent with Eqs. (3.48) and (3.49).
In ECE physics the electric charge density is geometrical in origin and is:
and the electric current density is:
Here and are the spin and orbital components of the curvature tensor [1-10]. So Eqs. (3.93), (3.125) and (3.126) give many new equations of physics which can be developed systematically in future work. In magnetostatics for example the relevant equations are:
so it follows from charge current conservation that:
If it is assumed that the scalar potential is zero in magnetostatics, the usual assumption, then:
because there is no electric field present. It follows from Eqs. (3.129) and (3.131) that
in ECE magnetostatics.
In UFT258 and immediately preceding papers of this series it has been shown that in the absence of a magnetic monopole:
and that the space part of the Cartan identity in the absence of a magnetic monopole gives the two equations:
In ECE physics the magnetic flux density is:
so the Beltrami equation gives:
Eq. (3.134) from the space part of the Cartan identity is also a Beltrami equation, as is any non-divergent equation:
From Eq. (3.137):
Using Eq. (3.138):
which implies that the vector potential is also defined in general by a Beltrami equation:
QED This is a generally valid result of ECE physics which implies that:
From Eq. (3.110) it follows that:
is a general result of ECE physics.
From Eqs. (3.135) and (3.141):
so the spin connection vector of ECE physics is also defined in general by a Beltrami equation. This important result can be cross checked for internal consistency using note 258(4) on www.aias.us, starting from Eq. (3.50) of this paper. Considering the X component for example:
and it follows that:
and similarly for the X and Z components. In order for this is to be a Beltrami equation, Eqs.
(3.141) and (3,144) must be true, QED.
In magnetostatics there are additional results which emerge as follows. From vector analysis:
It is immediately clear that Eqs. (3.87) and (3.144) give Eq. (3.147) self consistently, QED Eq. (3.148) gives
and using Eq. (3.148):
so the spin curvature is defined by a Beltrami equation in magnetostatics. Also in magnetostatics:
so it follows that the current density of magnetostatics is also defined by a Beltrami equation:
All these Beltrami equations in general have intricate flow structures graphed following sections of this chapter and animated on www.aias.us. As discussed in Eqs. (3.31) to (3.35) of Note 258(5) on www.aias.us, plane wave structures and O(3) electrodynamics
[1-10] are also defined by Beltrami equations. The latter give simple solutions for vacuum plane waves. In other cases the solutions become intricate. The B(3) field is defined by the simplest type of Beltrami equation
In photon mass theory therefore:
It follows from Eq. (3.154) that:
produces the Helmholtz wave equation:
Eq. (3.155) is
to find that Eq. (3.160) is the Einstein energy equation for the photon of mass, so the analysis is rigorously self-consistent, QED
In ECE physics the Lorenz gauge is:
with the solution:
This is again a general result of ECE physics applicable under any circumstances. Also in ECE physics in general the spin connection vector has no divergence:
Another rigorous test for self-consistency is given by the definition of the magnetic field in ECE physics:
By vector analysis:
In the absence of a magnetic monopole Eq. (84) also follows from the space part of the Cartan identity. So the entire analysis is rigorously self-consistent. The cross consistency of the Beltrami and ECE equations can be checked using:
as in note 258(1) on www.aias.us. Eq. (3.175) follows from Eqs. (3.168) and (3.172). Multiply Eq. (3.175) by and use Eq. (3.133) to find:
and relabel summation indices to find:
It follows that:
QED. The analysis correctly and self consistently produces the correct definition of the spin curvature.
Finally, on the U(1) level for the sake of illustration, consider the Beltrami equations of note 258(3) on www.aias.us:
In the Ampere Maxwell law
It follows that:
and using the Lorenz condition:
it follows that:
Eq. (3.185) becomes the d’Alembert equation in the presence of current density:
The solutions of the d’Alembert equation (3.189) may be found from:
showing in another way that as soon as the Beltrami equation (3.87) is used, U(1) gauge invariance is refuted.
3.3 Elecrostatics, Spin Connection Resonance and Beltrami Structures As argued already the first Cartan structure equation defines the electric field strength as:
where the four potential of ECE electrodynamics is defined by:
Here is the scalar potential. If it is assumed that the subject of electrostatics is defined by:
then the Coulomb law in ECE theory is given by:
The electric current in ECE theory is defined by:
where is the spin part of the curvature vector and where is the magnetic flux density. From Eqs. (3.193) and (3.195):
so in ECE electrostatics:
From Eqs. (3.198) and (3.199)
so we obtain the constraint:
The magnetic charge density in ECE theory is given by:
and the magnetic current density by:
In ECE electrostatics Eq. (3.204) is true automatically because:
and Eq. (3.203) becomes:
So the equations of ECE electrostatics are:
Later on in this chapter it is shown that these equations lead to a solution in terms of Bessel functions, but not to Euler Bernoulli resonance.
In order to obtain spin connection resonance Eq. (3.208) must be extended to:
Where is the Eckardt Lindstrom vacuum potential [1-10]. The static electric field is defined by:
so from Eqs. (3.212) and (3.213):
By the ECE antisymmetry law:
leading to the Euler Bernoulli resonance equation:
and spin connection resonance [1-10]. The left hand side contains the Hooke law term and the right hand side the driving term originating in the vacuum potential. Denote:
then the equation becomes:
The left hand side of Eq. (3.218) is a field property and the right hand side a property of the ECE vacuum. In the simplest case:
and produces undamped resonance if:
where A is a constant. The particular integral of Eq. (3.219) is:
and spin connection resonance occurs at:
and there is a resonance peak of electric field strength from the vacuum.
Later in this chapter solutions of Eq. (3.218) are given in terms of a combination of Bessel functions, and also an analysis using the Eckardt Lindstrom vacuum potential as a driving term.
In the absence of a magnetic monopole the Cartan identity is, as argued already:
A possible solution of this equation is:
leading as argued already to a rigorous justification for O(3) electrodynamics. The Cartan identity (3.224) is itself a Beltrami equation:
From Eqs. (3.226) and (3.227):
In the complex circular basis:
so from Eqs. (3.228) and (3.229):
which are Beltrami equations as argued earlier in this chapter.
This result can be obtained self consistently using the Gauss law:
which as argued already implies the Beltrami equation:
From Eqs. (3.168) and (3.232)
Using Eq. (3.227) gives:
which implies Eqs. (3.230a) to (3.230c) QED As shown earlier in this chapter the Beltrami structure also governs the spin connection vector:
It follows that the equations:
produce O(3) electrodynamics [1-10]:
As shown in Note 259(3) on www.aias.us there are many inter-related equations of O(3) electrodynamics which all originate in geometry.
Later in this chapter it is argued a consequence of these conclusions is that the spin connection and orbital curvature vectors also obey a Beltrami structure.
The fact that ECE is a unified field theory also allows the development and interrelation of several basic equations, including the definition of B(3):
It can be written as:
Although B(3) is a radiated and propagating field as is well-known [1-10] Eq. (3.241) can be used as a general definition of the magnetic flux density for a choice of potentials. This is important for the subject of magnetostatics and the development [1-10] of the fermion equation with:
Eq. (3.241) gives the transition from classical to quantum mechanics. In ECE electrodynamics A must always be a Beltrami field and this is the result of the Cartan identity as already argued. So it is necessary to solve the following equations simultaneously:
This can be done using the principles of general relativity, so that the electromagnetic field is a rotating and translating frame of reference. The position vector is therefore:
It follows that:
The results (3.246) for plane waves can be generalized to any Beltrami solutions, so it follows that spacetime itself has a Beltrami structure.
From Eqs. (3.242) and (3.244):
and from Eq. (3.250):
QED. Therefore it is always possible to write the vector potential in the form (3.242) provided that spacetime itself has a Beltrami structure. This conclusion ties together several branches of physics because Eq. (3.242) is used to produce the Lande factor, ESR, NMR and so on from the Dirac equation, which becomes the fermion equation [1-10] in ECE physics.
As argued already the tetrad postulate and ECE postulate give:
and the fermion or chiral Dirac equation is a factorization of Eq. (3.253). As shown in Chap. 1:
where is the inverse tetrad, defined by :
In generally covariant format Eq. (3.253) is:
it follows that:
which gives Eq. (3.254) QED The d’Alembertian is defined by:
The Beltrami condition:
gives the Helmholtz wave equation:
From Eq. (3.259):
which is the equation for the time dependence of A. The Helmholtz and Beltrami equations are for the space dependence of A. Eq. (3.267) is satisfied by:
Eq. (3.267) is a generalization of the Einstein energy equation for a free particle:
So mass in ECE theory is defined by geometry.
The general solution of Eq (3.256) is therefore:
It follows that there exist the equations:
where is the scalar potential in ECE physics. For each a:
where m is mass. The relativistic wave equation for each a is:
which is the quantized format of:
Eq. (3.278) is:
where the Lorentz factor is:
and where the relativistic momentum is:
Define the relativistic energy as:
and it follows that:
which is the non-relativistic limit of the kinetic energy, i.e.:
Eq. () quantizes to the free particle Schrӧdinger equation:
which is the Helmholtz equation:
It follows that the free particle Schrӧdinger equation is a Beltrami equation but with the vector potential replaced by the scalar potential , which plays the role of the wavefunction. It also follows in the non-relativistic limit that:
The Helmholtz equation () can be written as:
which is an Euler Bernoulli equation without a driving term on the right hand side. In the presence of potential energy V Eq. () becomes:
where H is the Hamiltonian and E the total energy:
Eq. () is:
which is an inhomogeneous Helmholtz equation similar to an Euler Bernoulli resonance equation with a driving term on the right hand side. However Eq. () is an eigenequation rather than an Euler Bernoulli equation as conventionally defined, but Eq. () has very well-known resonance solutions in quantum mechanics. Eq. () may be written as:
and in UFT226 ff. on www.aias.us was used in the theory of low energy nuclear reactors (LENR). Eq. () is well known to be a linear oscillator equation which can be used to define the structure of the atom and nucleus. It can be transformed into an Euler Bernoulli equation as follows:
where the right hand side represents a vacuum potential. It is exactly the structure obtained from the ECE Coulomb law as argued already.
3.4 The Beltrami Equation for Linear Momentum The free particle Schrӧdinger equation can be obtained from the Beltrami equation for momentum:
which can be developed into the Helmholtz equation:
if it is assumed that:
If p is a linear momentum in the classical straight line then:
In general however p has intricate Beltrami solutions, some of which are animated in UFT258 on www.aias.us and its animation section.
Now quantize Eq. ():
to arrive at:
A possible solution is:
which is the Helmholtz equation for the scalar , the wave function of quantum mechanics. The Schrӧdinger equation for a free particle is obtained by applying Eq. () to:
Eqs. () and () are the same if:
QED Using the de Broglie relation:
which is Eq. (), QED Therefore the free particle Schrӧdinger equation is the Beltrami equation:
The free particle Schrӧdinger equation originates in the Beltrami equation.
This method can be extended to the general Schrӧdinger equation in which the potential energy V is present. Consider the momentum Beltrami equation () in the general case where depends on coordinates. Taking the curl of both sides of Eq. ():
By vector analysis Eq. () can be developed as:
One possible solution is:
Eq. () is Eq. (), QED Eq. () can be written as:
A possible solution of Eq. () is the Schrӧdinger equation:
So the Schrӧdinger equation is compatible with Eq. ().
Eq. () gives:
which is consistent with Eq. ( ) only if:
Eq. ( ) gives:
From a comparison of Eqs. ( ) and ( ) we obtain the subsidiary condition:
giving a cubic constraint in V - E :
this can be written as a cubic equation in E, which is a constant. E is expressed in terms of V,
, and . Using:
gives a differential equation in V which can be solved numerically, giving an expression for V. Finally this expression for V is used in the Schrӧdinger equation:
to find the energy levels of E and the wavefunctions . These are energy levels and wavefunctions of the interior parton structure of an elementary particle such as an electron, proton or neutron. The well-developed methods of computational quantum mechanics can be used to find the expectation values of any property and can be applied to scattering theory, notably deep inelastic electron-electron, electron-proton and electron-neutron scattering. The data are claimed conventionally to provide evidence for quark structure, but the quark model depends on the validity of the U(1) and electroweak sectors of the standard model. In this book these sector theories are refuted in many ways.