### The vacuum field theory electrodynamics equations: Hamiltonian analysis

**Any Lagrangian theory has an equivalent** canonical Hamiltonian representation via the classical Legendre transformation[1; 2; 46; 56; 104]. As we have already formulated our vacuum field theory of a moving charged particle q in Lagrangian form, we proceed now to its Hamiltonian analysis making use of the action functionals (2.39), (2.48) and (2.51).

**Take, first,** the Lagrangian function (2.41) and the momentum expression (2.40) for defining the corresponding Hamiltonian function

Consequently, it is easy to show [1; 2; 56; 104] that the Hamiltonian function (2.62) is a conservation law of the dynamical field equation (2.38); that is, for all t, t £ R

which naturally leads to an energy interpretation of H. Thus, we can represent the particle energy as

Accordingly the Hamiltonian equivalent to the vacuum field equation (2.38) can be written as

and we have the following result.

**Proposition 2.6.** The alternative freely moving point particle electrodynamic model (2.38) allows the canonical Hamiltonian formulation (2.65) with respect to the "rest" reference system variables, where the Hamiltonian function is given by expression (2.62). Its electrodynamics is completely equivalent to the classical relativisticfreely moving point particle electrodynamics described in Section 2.

**In an analogous manner, one can now use the Lagrangian (2.48) to construct the Hamiltonian function for the dynamical field equation (2.46) describing the motion of charged particle q in an external electromagnetic field in the canonical Hamiltonian form:**

where

**Here we took into account that,** owing to definitions (2.45) and (2.49),

or

is the related magnetic vector potential generated by the moving external

is the related magnetic vector potential generated by the moving external

charged particle. Equations (2.67) can be rewritten with respect to the laboratory reference system K in the form

which coincides with the result (2.61).

Whence, we see that the Hamiltonian function (2.67) satisfies the energy conservation conditions

for all

and that the suitable energy expression is

where the generalized momentum P = p + qA. The result (2.72) differs in an essential way from that obtained in [57], which makes use of the Einsteinian Lagrangian for a moving charged point particle q in an external electromagnetic field. Thus, we obtain the following result:

Proposition 2.7. The alternative classical relativistic electrodynamic model (2.70), which is intrinsically compatible with the classical Maxwell equations (2.7), allows the Hamiltonian formulation (2.66) with respect to the rest reference system variables, where the Hamiltonian function is given by expression (2.67).

The inference above is a natural candidate for experimental validation of our theory. It is strongly motivated by the following remark.

**Remark 2.8.** It is necessary to mention here that the Lorentz force expression (2.70) uses the particle momentum p = mu, where the dynamical "mass" m := — WW satisfies condition (2.72). The latter gives rise to the following crucial relationship between the particle energy £0 and its rest mass m0 (at the velocity u : = 0 at the initial time moment t = 0 £ R) :

or, equivalently,

where

which strongly differs from the classical formulation (2.34).

To make this difference more clear, we now analyze the Lorentz force (2.57) from the Hamiltonian point of view based on the Lagrangian function (2.54). Thus, we obtain that the corresponding Hamiltonian function

Since p = P — qA, expression (2.75) assumes the final "no interaction" [12; 57; 67; 80] form

which is conserved with respect to the evolution equations (2.52) and (2.53), that is

for al

These equations latter are equivalent to the following Hamiltonian system

as one can readily check by direct calculations. Actually, the first equation

and definitions holds, owing to the condition

and definitions

postulated from the very beginning. Similarly we obtain that

coincides with equation (2.55) in the evolution parameter t £ R. This can be formulated as the next result.

**Proposition 2.9.** The dual to the classical relativistic electrodynamic model (2.57) allows the canonical Hamiltonian formulation (2.78) with respect to the rest reference system variables, where the Hamiltonian function is given by expression (2.76). Moreover, this formulation circumvents the "mass-potential energy" controversy associated with the classical electrodynamical model (2.32).

The modified Lorentz force expression (2.57) and the related rest energy relationship are characterized by the following remark.

Remark 2.10. If we make use of the modified relativistic Lorentz force expression (2.57) as an alternative to the classical one of (2.35), the corresponding particle energy expression (2.76) also gives rise to a different energy expression (at the velocity u := 0 £ E3 at the initial time t = 0) corresponding to the classical case (2.34); namely, £0 = m0 instead of £0 = m0 + q f0, where f0 := f\t=0.

### Concluding remarks

**All of dynamical field equations discussed above** are canonical Hamiltonian systems with respect to the corresponding proper rest reference systems Kr, parameterized by suitable time parameters t £ R. Upon passing to the basic laboratory reference system K with the time parameter t £ R,naturally the related Hamiltonian structure is lost, giving rise to a new interpretation of the real particle motion. Namely, one that has an absolute sense only with respect to the proper rest reference system, and otherwise completely relative with respect to all other reference systems. As for the Hamiltonian expressions (2.62), (2.67) and (2.76), one observes that they all depend strongly on the vacuum potential field function W : M4 ^ R, thereby avoiding the mass problem of the classical energy expression pointed out by L. Brillouin [59]. It should be noted that the canonical Dirac quantization procedure can be applied only to the corresponding dynamical field systems considered with respect to their proper rest reference systems.

**Remark 2.11.** Some comments are in order concerning the classical relativity principle. We have obtained our results without using the Lorentz transformations of reference systems – relying only on the natural notion of the rest reference system and its suitable parametrization with respect to any other moving reference systems. It seems reasonable then that the true state changes of a moving charged particle q are exactly realized only with respect to its proper rest reference system. Then the only remaining question would be about the physical justification of the corresponding relationship between time parameters of moving and rest reference systems.

The relationship between reference frames that we have used through is expressed as

where

is the velocity with which the rest reference system Kr moves with respect to another arbitrarily chosen reference system K. Expression (2.81) implies, in particular, that

which is identical to the classical infinitesimal Lorentz invariant. This is not a coincidence, since all our dynamical vacuum field equations were derived in turn [53; 54] from the governing equations of the vacuum potential field function W : M4 ^ R in the form

which is a priori Lorentz invariant. Here p £ R is the charge density and v := dr/dt the associated local velocity of the vacuum field potential evolution. Consequently, the dynamical infinitesimal Lorentz invariant (2.82) reflects this intrinsic structure of equations (2.83). If it is rewritten in the nonstandard Euclidean form:

it gives rise to a completely different relationship between the reference systems K and Kr, namely

where r := dr/dT is the related particle velocity with respect to the rest reference system. Thus, we observe that all our Lagrangian analysis in Section 2 is based on the corresponding functional expressions written in these "Euclidean" space-time coordinates and with respect to which the least action principle was applied. So we see that there are two alternatives – the first is to apply the least action principle to the corresponding Lagrangian functions expressed in the Minkowski space-time variables with respect to an arbitrarily chosen reference system K, and the second is to apply the least action principle to the corresponding Lagrangian functions expressed in Euclidean space-time variables with respect to the rest reference system Kr.

**This leads us to a slightly amusing but thought-provoking observation:** It follows from our analysis that all of the results of classical special relativity related to the electrodynamics of charged point particles can be obtained (in a one-to-one correspondence) using our new definitions of the dynamical particle mass and the least action principle with respect to the associated Euclidean space-time variables in the rest reference system.

**An additional remark concerning the quantization** procedure of the proposed electrodynamics models is in order: If the dynamical vacuum field equations are expressed in canonical Hamiltonian form, as we have done here, only straightforward technical details are required to quantize the equations and obtain the corresponding Schrodinger evolution equations in suitable Hilbert spaces of quantum states. There is another striking implication from our approach: the Einsteinian equivalence principle [29; 57; 63; 70; 80] is rendered superfluous for our vacuum field theory of electromagnetism and gravity.

**Using the canonical Hamiltonian formalism devised here for** the alternative charged point particle electrodynamics models, we found it rather easy to treat the Dirac quantization. The results obtained compared favorably with classical quantization, but it must be admitted that we still have not given a compelling physical motivation for our new models. This is something that we plan to revisit in future investigations. Another important aspect of our vacuum field theory no-geometry (geometry-free) approach to combining the electrodynamics with the gravity, is the manner in which it singles out the decisive role of the rest reference system Kr. More precisely, all of our electrodynamics models allow both the Lagrangian and Hamiltonian formulations with respect to the rest reference system evolution parameter T £ R, which are well suited the to canonical quantization. The physical nature of this fact still remains somewhat unclear. In fact, as far as we know [4; 5; 57; 63; 80], there is no physically reasonable explanation of this decisive role of the rest reference system, except for that given by R. Feynman who argued in [70] that the relativistic expression for the classical Lorentz force (2.35) has physical sense only with respect to the rest reference system variables (t, r) £ E4. In future research we plan to analyze the quantization scheme in more detail and begin work on formulating a vacuum quantum field theory of infinitely many particle systems.

## The modified Lorentz force and the radiation theory

### Introductory setting

**Maxwell’s equations may be** represented by means of the electric and magnetic fields or by the electric and magnetic potentials. The latter were once considered as a purely mathematically motivated representation, having no physical significance.

**The situation is actually not so simple now** that evidence of the physical properties of the magnetic potential was demonstrated by Y. Aharonov and D. Bohm [92] in the formulation their "paradox" concerning the measurement of a magnetic field outside a separated region where it is vanishes. Later, similar effects were also revealed in the superconductivity theory of Josephson media. As the existence of any electromagnetic field in an ambient space can be tested only by its interaction with electric charges, the dynamics of the charged particles is very important. Charged particle dynamics was studied in detail by M. Faraday, A. Ampere and H. Lorentz using Newton’s second law. These investigations led to the following representation for the Lorentz force

where E and B £ E3 are, respectively, electric and magnetic fields, acting on a point charged particle q £ R having momentum p = mu. Here m £ R+ is the particle mass and u £ T(R3) is its velocity, measured with respect to a suitably chosen laboratory reference frame. That the Lorentz force (2.86) is not completely correct was known to Lorentz. The defect can be seen from the nonuniform Maxwell equations for electromagnetic fields radiated by any accelerated charged particle, as easily seen from the well-known expressions for the Lienard-Wiechert potentials.

**This fact inspired many physicists to "**improve" the classical Lorentz force expression (2.86), and its modification was soon suggested by M. Abraham and P.A.M. Dirac, who found the so-called "radiation reaction" force induced by the self-interaction of a point charged particle:

The additional force expression

depending on the particle acceleration, immediately raised many questions concerning its physical meaning. For instance, a uniformly accelerated charged particle, owing to the expression (2.88), experiences no radiation reaction, contradicting the fact that any accelerated charged particle always radiates electromagnetic waves. This "paradox" was a challenging problem during the 20th century [96-98; 100; 102] and still has not been completely explained [101]. As there exist different approaches to explanation this reaction radiation phenomenon, we mention here only some of the more popular ones such as the Wheeler-Feynman [99] "absorber radiation" theory, based on a very sophisticated elaboration of the retarded and advanced solutions to the nonuniform Maxwell equations, and Teitelbom’s [95] approach which exploits the intrinsic structure of the electromagnetic energy tensor subject to the advanced and retarded solutions to the nonuniform Maxwell equations. It is also worth mentioning the very nontrivial development of Teitelbom’s theory devised recently by [94] and applied to the non-abelian Yang-Mills equations, which naturally generalize the classical Maxwell equations.