5.3 To the Subquantum Law of Motion

5.3  To the Subquantum Law of Motion


All the previous experience convinces us that the nature represents the realization of the elementary mathematically conceivable elements. I am convinced that with the help of pure mathematical constructions we can find those concepts and regular connections between them which will give us the key to understanding of natural phenomenon. Experience can suggest us the appropriate mathematical constructions of physics. But the real creative beginning is peculiar namely to mathematics. Therefore, I consider the belief of ancients that pure thinking is capable of perceiving a reality is, in a sense, rather justified.

Albert Einstein

Within the scope of relativistic paradigm, the law of motion in vector field A:=(φ,A) might be obtained in the point approximation with the help of the variation principle from LaGrange function L with linear in respect of the field invariant summands

Ll = – (gA + m + eAU), (1.Ll)

where: U=(l,lv) is 4-velocity; l is Lorentz-factor (ratio of differentials dt/ds); A is positive value of the square root of field 4-potential Lorentz norm AA; g, m, e are constants. The following equivalent (1.Ll) forms of writing of the LaGrange function are normally used:

Ldt = – (gAds + mds + eAdR), (1.Ldt)
    L = – (gA/l + m/l + e(φAv)). (1.L)

Development of fundamental physics goes along the course of efforts to replace non-field summand with empirical rest mass with the first summand or with another invariant scalar obtained out of components of one or several fields. Poincaré scalar pressure and endeavors to implement the programme of electron's field electromagnetic mass became landmarks on this way. Non-linear generalizations of electrodynamics and several-fields theory stand separately in this rank.

Einstein's disappointment about the subject of his first great love – Maxwell equations – due to their apparent at that moment (1905–1907) inability to describe electron and light quanta and at one stroke to satisfy maximalists' expectations, turned out to be fateful to the 20'th century physics. New choice fell upon gravitation – but now entirely and to the last breath. Einstein's Genius inconceivably fascinated and carried away one scientist after another. Goettingen could not resist either… Analysing this phenomenon, one can hardly refrain from looking for some upper force. Too a smooth chain of events… It required only a decade to form overwhelming conviction (belief) that the gravitation (geometry in the small, inseparable from it) bears responsibility for the fundamental nature of particles and their mass. And all this – regardless of the absence of even theoretical corroborations of that and despite the difference of constants of electromagnetic and gravitational interactions by a factor of 1040 in distance range mastered by the physicists. Only Einstein's Genius was up to it, only He had the power to do it!

When describing inertia of Joule heat in relativistic thermodynamics [1], Max Abraham carried out a transition to a relativistic law of motion with variable rest mass:

d(mU) = KAds,        dm = KAUds (2)

(formulas (294) and (295) [1]). In the same place Pauli reminds of a discussion on this topic between Abraham and Nordström and makes references to their publications on the same theme. Probably it pushed Nordström on to create his first version of scalar gravitation theory (1912), where m depends on scalar potential of gravitational field.

Transition from the Newton law (mdv=Fdt) to the relativistic (mdU=Kds) motion law utilizes new Lorentz-invariant motions: time ds, 4-velocity U

mdU := m(dl, d(lv)) = (Fv, F)dt (3)

and Minkowski 4-forces (M-forces)

K := (kv, k) := (lFv, lF),    KU = 0. (4)

Herewith true is the equation of mixed (transformational) type

mldv = (F – (Fv)v)dt, (5)

which gave reason for not quite a successful definition of the so-called relativistic mass ml. The Abraham–Nordström equation (2) may be written as

mdU = (KA – (KAU)U)ds, (6)

which resembles the equation (5). AN-force and rest mass in the models of Abraham and Nordström are of the same structure:

KA = KN = K + (0, gradm),      dm = – (gradm)vdt. (7)

They differ by physical interpretation of a variable component in rest mass. Abraham puts it as a Joulean heat scalar field, whereas it is a gravitation scalar field with Nordström. A puzzle arises. Why is it so that a vector field A represented by M-force does not give rise to a variable component in m, whereas a scalar field does as an extra component to AN-force as well as to rest mass? Why is Lorentz norm scalar field AA of the vector field A not displayed in m, if this feature is characteristic of an extra scalar field?

Most simple and elegant solution for subcurrents is already prepared and lies on the surface.

An extra scalar field is identically equal to the Lorentz norm scalar field of a Faraday–Maxwell vector field.

In the case of the Abraham model such a solution is almost impossible. The Nordström model is already closer, if we put the primary goal apart – description of gravitational field. However, according to a statement by Dirac (in a broader understanding) – each Elegant mathematical model (structure) is sure to work for the Universe, and sooner or later we are to see it.

Having completed the preliminary work in search of giants on whose shoulders we could raise ourselves to have a better panorama of the perspective and the means to achieve the set goal, we may now return to the never abandoned path of the Faraday–Maxwell field program.

In K0, earlier accepted (5.2) to describe Z-kinematics vectors, i.e. in rest system of electron's symmetry point, its field a fortiori has spherically symmetric structure: A=(φ(r),0). By substituting Z-kinematics vectors into the motion of infinitesimal «point» subcharge equation δq:=±|δq|:

d(φ0U) = (±l(Ev, E) – (0, gradφ0))ds, (8)
E := – gradφ,     φ0 := |φ|,     ± := δq/|δq|, (8.0)

related to Lagrangian

Ll = – |δq|(φ0 ± φl) = – (|δq|φ0 + δqφl), (9)

we will make certain those are true in case the scalar potential of an electron field is subjected to condition

rφ = –1. (10)

The following Z-kinematics conditions may be useful for verification:

W = (t, r) – (0, l(rvt)/r),      lvr = tr. (11)

• Absence of free parameters (empirical constants)!
• Non-field mass constant is identically equal to zero and its place in the motion law takes the invariant scalar field!
• An electron coulomb field with a single effective charge has been obtained as a compatibility condition!
• No more and no fewer than two Z-motions sets of the required sources of an electron field have been obtained, and the number of those equals to the number of subcurrent charge signs!

This is but an incomplete report of incredibly beautiful results lying on the surface (at the basis of verification) of the arising subquantum field theory.

Quantity |δq|φ0 plays the role of a field's rest mass of an infinitesimal «point» subcharge δq plays the role of a field's rest mass of an infinitesimal «point» subcharge φ. This is consistent with conservation of 3-vector of angular momentum along the Z-path

[δpr] = |δq|lφ0[vr] = (0, 0, |δq|o),    o = v0l0 = tgα, (12)

equal to the vector product of 3-momentum |δq|φ0u into r. This dynamic invariant as well as the equation of motion is written in K0. Before we consider their transformation properties let's make a preparatory suggestion.

Cornelius Lanczos in [6,Ch.IX,Art.9] reminds about correlation between canonical transformations of analytical mechanics and the Lorentz transformation: – «Equations (9.9.14) [of electron motion

mdU = e(E,H)Uds ] (13)

allow interesting geometrical interpretation. Ch.VII,Art.8 shows that the motion of phase fluid may be regarded as continuous performance of infinitesimal canonical transformations. Let's focus our attention on the velocity vector U and write down the equation (9.9.14) as

mU(s+δs) = mU(s) + e(E,H)(s)U(s)δs. (14)

All the four equations written in detail, we see that they fully coincide with the equations (9.4.58) that set the infinitesimal Lorentz transformation. There with the electric vector E plays the role of a, and the magnetic vector Hthe role of b. Consequently, motion of the velocity vector of an electron in an outer electromagnetic field may be regarded as continuous sequence of the infinitesimal Lorentz transformation, with the components of this transformation set by electromagnetic tensor (E,H)

For H-motions function of the action S may be written in the following three forms

DS = Ll = – (A ± AU) = (15.1)
          = – (B ± BW). (15.2)

Here A:=B – are positive square root values of quantities:

A2 = AA,    B2 = – BB. (15.3)

Dual towards A field vector B is prescribed by conditions of duality:

AA = – BB,    AB = 0 (15.4)

In the first line (15.1) a subquantum analogue of the principle of least action is prepared; in the second (15.2) – an analogue of the Gauss principle of least constraint.

Appearance of dual pair of the field vectors (A,B) instead of one is predetermined by existence of corresponding symmetry of pair of the fundamental H-kinematics vectors (U,W) describing the motion of field sources. Evolution of pair of the vectors (U(s),W(s)) (correspondingly – (A(s),B(s))) along H-path can be described (interpreted) as a continuous sequence of special (single-parameter) Lorentz transformations L(0,s) or hyperbolic (dual) turns acting on an arbitrarily selected «initial point» (U(0),W(0)) (correspondingly – pair of the field vectors (A(0),B(0)) at this «point») of this H-path.

In this procedure one can perceive an analogue with determination (outcome) of displacement currents when writing the field equations by Maxwell. Symmetrization of the field description with the help of the dual pair of the vectors (A,B) will result in change (symmetrization) of notation of the field equations and conditions of its calibration. The field equations will include additional components that until now were interpreted in terms of «currents» of magnetic monopoles. This natural symmetrization of the field and its equations is strictly necessary to put the motion of field sources into correspondence with hyperbolic (dual) symmetry of H-kinematics. Symmetries of Maxwell equations require their further symmetrization for the sake of correct description (existence) of electron in the field theory!

 The translation from Russian was made by Yuri Nezhentsev
Last modifications: March 29 2005
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The Literature Quoted:
1. Pauli W. Theory of Relativity. Pergamon Press, 1958
6. Lanczos C. The variational principles of mechanics. University of Toronto Press, 1962
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