Modern Classical Electrodynamics and Electromagnetic Radiation – Vacuum Field Theory Aspects (Electromagnetic Waves) Part 3

Radiation reaction force: the vacuum-field theory approach

In the Section, we shall develop our vacuum field theory approach [6; 52-55] to the electromagnetic Maxwell and Lorentz theories in more detail and show that it is in complete agreement with the classical results. Moreover, it allows some nontrivial generalizations, which may have physical applications.

For the radiation reaction force in the vacuum field theory approach, the modified Lorentz force, which was derived in Section 1, acting on a charged point particle q, is

tmp12-160_thumb[2]_thumb

is the extended electromagnetic 4-vector potential. To take into where tmp12-161_thumb[2]_thumb is the extended electromagnetic 4-vector potential. To take into account the self-interaction of this particle, we make use of the distributed charge density p : M4 ^ R satisfying the condition


tmp12-162_thumb[2]_thumb

Then, owing to for all tmp12-163_thumb[2]_thumb in a laboratory reference frame K with coordinates tmp12-164_thumb[2]_thumb

Then, owing to 2.89 and results in [96], the self-interaction force can be expressed as

tmp12-165_thumb[2]_thumb

where

tmp12-166_thumb[2]_thumb

are the well-known Lienard-Wiechert potentials, which are calculated at the retarded time parameter

tmp12-167_thumb[2]_thumb

Then, taking into account the continuity equation

tmp12-168_thumb[2]_thumb

for the charge q, from (2.91) one finds using calculations similar to those in [96] that

tmp12-169_thumb[2]_thumb

where we defined, respectively, the positive electrostatic self-interaction repulsive energy and force as

tmp12-170_thumb[2]_thumb

Assuming and the force component corresponding to the term

tmp12-171_thumb[2]_thumb 

now that the external electromagnetic field vanishes, from (2.89) one obtains that

tmp12-172_thumb[2]_thumb

where we have made use of the inertial mass definitions

tmp12-173_thumb[2]_thumb

following from the vacuum field theory approach. From (2.97) one computes that the additional force term is

tmp12-174_thumb[2]_thumb

Then we readily infer from (2.97) that the observed charged particle mass satisfies at rest the inequality

tmp12-175_thumb[2]_thumb

This expression means that the real physically observed mass strongly depends both on the intrinsic geometric structure of the particle charge distribution and on the external physical interaction with the ambient vacuum medium.

Conclusion

The charged particle radiation problem, revisited in this section, allows the explanation of the point charged particle mass as that of a compact and stable object, which should have a negative vacuum interaction potential WW £ R3 owing to (2.98). This negativity can be satisfied if and only if the quantity (2.99) holds, thereby imposing certain nontrivial geometric constraints on the intrinsic charged particle structure [103]. Moreover, as follows from the physically observed particle mass expressions (2.98), the electrostatic potential energy comprises the main portion of the full mass.

There exist different relativistic generalizations of the force expression (2.97), all of which suffer the same common physical inconsistency related to the no radiation effect of a charged point particle in uniform motion.

Another problem closely related to the radiation reaction force analyzed above is the search for an explanation to the Wheeler and Feynman reaction radiation mechanism, which is called the absorption radiation theory. This mechanism is strongly dependent upon the Mach type interaction of a charged point particle in an ambient vacuum electromagnetic medium. It is also interesting to observe some of the relationships between this problem and the one devised above in the context of the vacuum field theory approach, but more detailed and extended analyzes will be required to explain the connections.

Maxwell’s equations and the Lorentz force derivation – the legacy of Feynman’s approach

Poissonian analysis preliminaries

In 1948 R. Feynman presented but did not published [127; 128] a very interesting, in some respects "heretical", quantum-mechanical derivation of the classical Lorentz force acting on a charged particle under the influence of an external electromagnetic field. His result was analyzed by many authors [129-137] from different points of view, including its relativistic generalization [138]. As this problem is completely classical, we reanalyze the Feynman’s derivation from the classical Hamiltonian dynamics point of view on the coadjoint space T* (N), N c R3, and construct its nontrivial generalization compatible with results [6; 52; 53] of Section 1, based on a recently devised vacuum field theory approach [52; 55]. Upon obtaining the classical Maxwell electromagnetic equations, we supply the complete legacy of Feynman’s approach to the Lorentz force and demonstrate its compatibility with the relativistic generalization presented in [52-55; 72].

Consider the motion of a charged point particle tmp12-176_thumb[2]_thumb under the influence of an external electromagnetic field. For its description, following [114; 123; 124], it is convenient to introduce a trivial fiber bundle structure

tmp12-177_thumb[2]_thumb

with the abelian structure group G := R\{0} equivariantly acting [1] on the canonically symplectic coadjoint space T* (M). Then we endow the bundle with a connection one-form A :

tmp12-178_thumb[2]_thumb

defined as

tmp12-179_thumb[2]_thumb

is a differential form, on the phase space M, where tmp12-180_thumb[2]_thumb is a differential form, constructed from the magnetic potential tmp12-181_thumb[2]_thumb

This reduced space If l :

tmp12-182_thumb[2]_thumb

is the related momentum mapping, one can construct the reduced phase space

tmp12-183_thumb[2]_thumb

has the symplectic structure is taken to be fixed. This reduced space

tmp12-184_thumb[2]_thumb

From (2.102), one readily where we taken in to account that

tmp12-185_thumb[2]_thumb

From (2.102), one readily computes the respective reduced Poisson brackets on T* ( N) :

tmp12-186_thumb[2]_thumb

for i, / = 1,3 with respect to the reference frame K.{t,q), characterized by the phase space coordinates

tmp12-187_thumb[2]_thumb


If one introduces a new momentum variable tmp12-188_thumb[2]_thumb giving rise to the on tmp12-189_thumb[2]_thumb

it is easy to verify that If one introduces a new momentum variable

tmp12-190_thumb[2]_thumb

giving rise to the following "minimal coupling" canonical Poisson brackets [12; 123; 124]:

tmp12-191_thumb[2]_thumb

for i, i = 1,3 with respect to the reference frame

tmp12-192_thumb[2]_thumb

characterized by the phase space coordinates

tmp12-193_thumb[2]_thumb

if and only if the Maxwell field equations

tmp12-194_thumb[2]_thumb

are satisfied on N for all tmp12-195_thumb[2]_thumb   for the curvature tensor tmp12-196_thumb[2]_thumb are satisfied on N for all tmp12-197_thumb[2]_thumb

The Poisson structure (2.104) makes it possible to describe a charged particle tmp12-198_thumb[2]_thumb located at point

tmp12-199_thumb[2]_thumb moving with a velocity tmp12-200_thumb[2]_thumb with respect to the reference frame K(t,q).The particle is under the electromagnetic influence of an external charged particle point tmp12-201_thumb[2]_thumb located at point tmp12-202_thumb[2]_thumb and moving with respect to the same reference frame K(t, q) with a velocity

tmp12-203_thumb[2]_thumb is the temporal derivative with respect to the temporal parameter tmp12-204_thumb[2]_thumb

More precisely, consider a new reference frame tmp12-205_thumb[2]_thumb moving with respect to the reference frame K(t, q) with velocity u f. With respect to the reference frame tmp12-206_thumb[2]_thumb the charged particle g moves with the velocity tmp12-207_thumb[2]_thumb and, respectively, the charged particle g f stays in rest. Then one can write the standard classical Lagrangian function of the charged particle I with a constant mass tmp12-208_thumb[2]_thumb subject to the reference frame tmp12-209_thumb[2]_thumbtmp12-210_thumb[2]_thumb

is the corresponding potential energy. On the other and the scalar potential tmp12-211_thumb[2]_thumb hand, owing to (2.106) and the Poisson brackets (2.104), the following equation for the charged particle g canonical momentum with respect to the reference frame tmp12-212_thumb[2]_thumb holds: tmp12-213_thumb[2]_thumb

or, equivalently,

tmp12-214_thumb[2]_thumb

expressed in the units when the light speed c = 1. Taking into account that the charged particle g momentum with respect to the reference frame K(t, q) equals

tmp12-215_thumb[2]_thumb

one computes from (2.108) that

tmp12-216_thumb[2]_thumb

which was obtained in [54; 55; 126] using for the magnetic vector potential

tmp12-217_thumb[2]_thumb

which was obtained in [54; 55; 126] using a vacuum field theory approach. Now, it follows from (2.106) and (2.109) one has the Lagrangian equations,

tmp12-218_thumb[2]_thumb

which induce the charged particle g dynamics

tmp12-219_thumb[2]_thumb

As a result of (2.111), we obtain the modified Lorentz type force

tmp12-220_thumb[2]_thumb

obtained in [54; 55], where

tmp12-221_thumb[2]_thumb

This differs from the classical Lorentz force expression

tmp12-222_thumb[2]_thumb

by the gradient component

tmp12-223_thumb[2]_thumb

Remark now that the Lorentz type force expression (2.112) can be naturally generalized to the relativistic case if to take into account that the Lorentz condition

tmp12-224_thumb[2]_thumb

imposed on the electromagnetic potential

tmp12-225_thumb[2][2]

Indeed, from (2.113) one obtains the Lorentz invariant field equation

tmp12-226_thumb[2][2]

is the generalized density function of the external

tmp12-227_thumb[2][2]

is the generalized density function of the external charge distribution g f. Employing calculations from [54; 55], derive readily from (2.117) and the charge conservation law

tmp12-228_thumb[2][2]

the Lorentz invariant equation on the magnetic vector potential tmp12-229_thumb[2][2]tmp12-230_thumb[2][2]

Moreover, relationships (2.113), (2.117) and (2.119) imply the true classical Maxwell equations

tmp12-231_thumb[2][2]

on the electromagnetic field tmp12-232_thumb[2][2]

Consider now the Lorentz condition (2.116) and observe that it is equivalent to the following local conservation law:

tmp12-233_thumb[2][2]

This gives rise to the important relationship for the magnetic potential tmp12-234_thumb[2][2]tmp12-235_thumb[2][2]

with respect to the reference frame K(t, q), where Ot c N is any open domain with a smooth boundary 90t, moving together with the charge distribution gf in the region N c R3 with velocity qf. Taking into account relationship (2.109), one obtains the expression for the charged particle g ‘inertial’ mass as

tmp12-236_thumb[2][2]

coinciding with that obtained in [54; 55; 126]. Her we denoted the corresponding potential energy of the charged particle tmp12-237_thumb[2][2]

The modified least action principle and its Hamiltonian analysis

Using the representations (2.122) and (2.123), one can rewrite the determining Lagrangian equation (2.110) as

tmp12-238_thumb[2][2]

which is completely equivalent to the Lorentz type force expression (2.112) calculated with respect to the reference frame K(t, q).

Remark 4.1. It is interesting to remark here that equation (2.124) does not allow the Lagrangian representation with respect to the reference frame K(t, q) in contrast to that of equation (2.110).

The remark above is a challenging source of our further analysis concerning the relativistic generalization of the Lorentz type force (2.112). Namely, the following proposition holds.

Proposition 4.2. The Lorentz type force (2.112), in the case when the charged particle g momentum is defined asp = — WW u, according to (2.123), is the exact relativistic expression allowing the Lagrangian representation with respect to the charged particle g rest reference frametmp12-239_thumb[2][2], connected with the reference frame K(t, q) by means of the classical relativistic proper time relationship:

tmp12-241_thumb[2][2]

and, by definition,

Here tmp12-242_thumb[2][2] is the proper time parameter in the rest reference frame tmp12-243_thumb[2][2] and, by definition, the derivative tmp12-244_thumb[2][2]

Proof. Take the following action functional with respect to the charged particle g rest reference frame

tmp12-245_thumb[2][2]tmp12-246_thumb[2][2]

are considered to be fixed. In contrast, the where the proper temporal values tmp12-247_thumb[2][2] are considered to be fixed. In contrast, the temporal parameters tmp12-248_thumb[2][2] trajectory in the phase space. The least action condition depend, owing to (2.125), on the charged particle g

tmp12-249_thumb[2][2]

applied to (2.126) yields the dynamical equation (2.124), which is also equivalent to the relativistic Lorentz type force expression (2.112). This completes the proof. □

Making use of the relationships between the reference frames tmp12-250_thumb[2][2] in the case when the external charge particle velocity u f = 0, we can easily deduce the following result.

Corollary 4.3. Let the external charge distribution gf be at rest, that is the velocity uf = 0. Then equation (2.124) reduces to

tmp12-251_thumb[2][2]

which implies the following conservation law:

tmp12-252_thumb[2][2]

Moreover, equation (2.128) is Hamiltonian with respect to the canonical Poisson structure (2.104) with Hamiltonian function (2.129) and the rest reference frame Kr (t, q) :

tmp12-253_thumb[2][2] 

In addition, if the rest particle mass is defined as tmp12-254_thumb[2][2] the "inertial" particle mass quantity

tmp12-255_thumb[2][2] has the well-known classical relativistic form

tmp12-256_thumb[2][2]

which depends on the particle velocity tmp12-257_thumb[2][2]

As for the general case of equation (2.124), analogous results to those above hold as described in detail in [52-55]. We need only mention that the Hamiltonian structure of the general equation (2.124) results naturally from its least action representation (2.126) and (2.127) with respect to the rest reference frame kr (t, q).

Conclusion

We have demonstrated the complete legacy of the Feynman’s approach to the Lorentz force based derivation of Maxwell’s electromagnetic field equations. Moreover, we have succeeded in finding the exact relationship between Feynman’s approach and the vacuum field approach devised in [54; 55]. Thus, the results obtained provide deep physical backgrounds lying in the vacuum field theory approach. Consequently, one can simultaneously describe the origins of the physical phenomena of electromagnetic forces and gravity. Gravity is physically based on the particle "inertial" mass expression (2.123), which follows naturally from both the Feynman approach to the Lorentz type force derivation and the vacuum field approach.

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