Maxwell's Equations: A central subset in relativistic quantum equation
The idea for the following relativistic equation is an impuls - energy - quantum formulation on the basis of
Einstein's energy law, Newton's impuls law and Faraday's induction law considering relativistic and arbitrary
movements of an electron -> object / body. In all derivations no conventional field terms D, E, B, H, J are
explicitely used but only the superior magnetic vector potential A and a special formulation of an extended
scalar potential PHIs. In the following relativistic equation for quantum electrodynamics
Maxwell's equations, Einstein's relativistic energy formula, Klein-Gordon's respectively relativistic
Schrödinger's equation, Proca equations, Newton's impuls mechanics etc
are embedded in one equation " Re + i Im = 0 " as subsets.
.
.
In the case of maxwell's field theory term 5) equals to zero yielding the light field wave gauge A = PHIs / c
( c is speed of light, for v << c & PHIs = PHI the Lienard-Wiechert's potential A = v∙ PHI / c^2 is derivable).
This real part gauge A = PHIs / c is also the condition for achieving the
-> relativistic Schrödinger's equation respectively -> Klein-Gordon's equation equals to term 4).
These equations are based on Einstein's special relativity theory and with A = 0 und js = 0
directly from real part < Re > -> Einstein's relativistic energy -> W = √ [(m0 c²)² + (c p)²],
where m0 is the mass in rest and p = m v the relativistic impuls of a mass moving with speed v.
This above shown formula re-formulated as a square root term -> m = m0 /√ [ 1 - ( v / c )² ]
implies Einstein's equivalence of mass and energy W = m c² [ -> " E = m c² " ]
and is the basis of Lorentz' length contraction and Einstein's time dilatation.
As to imaginary part < Im > of this above show equation
the Lorentz' gauge (including Coulomb's gauge with PHIs = const) is directly seen,
if term 6) = 0. With these gauges we can simply derive all maxwell's equations
in conventional field terms using also Maxwell's transformation B = curl A
and the known constitutive relations. Prerequisite for all mentioned re-formulations is an
experienced handling with operators in vector analysis i.e. PHI (A) (line based fields) and
quantum mechanics representated i.e. by impuls operator p and energy Hamilton operator H :
unified theory operators |  | (10) |
.
1. Real part Re of this equation shows the DUALITY of classic waves (left side)
and quantum particles (right side).
.
2. Using the light field gauge term 5) = 0 φS = c ∙ A) and applying quantum flux (hbar / charge q)
to the imaginary part Imthe re-formulated term 6)
yields the total electric field strength E' = - ∂ A / ∂ t - gradφS (9a)
with all influences (i.e. arbitrary translation, rotation, distortion movements) in classic electrodynamics:
Total electric field strength | E' = - ∂ A / ∂ t - grad j + ( v x curl A ) + [( A x curl v ) + further terms (II+III)] | (9) |
. By this simple re-formulation of the imaginary part Im we've automatically
changed the term 7) so that we also see the DUALITY of classic electric field strength E
and quantum flux based alternative formulation in quantum electrodynamics.
The 2. Maxwell equation = Faraday-Lorentz' law you get from (9) applying the operator "curl".
A good excercise for you: Where are the other Maxwell equations "hidden" in Re + i Im = 0 ?
.
3. Furthermore we have 6 possible gauges for electrodynamic field defined by
div A = ... (eq. Ib) as combination i.e. the simplest
gauges are Coulomb's gauge div A = constant and Lorentz' gauge div A = -[dφ/dt] / c^2
The important 7th gauge (in literature often not mentioned) is the light field gauge term 5) = 0.
This light field gauge is a basis for classic electrodynamics.
But a further conclusion is, that PHOTONS (and GRAVITONS)
must have a rest mass m0 = 0, propagating with the velocity of light in vacuum.
.
4. From real part < Re > we see a surprising (but measured) phenomenon, that electrodynamics
can influence the mass of rigid bodies! The speed of rigid bodies v
is always less then speed of light c. From eq. (1a) we can conclude
that the gravitational masses of atoms in a material can be changed, especially reduced
or nullified or even inverted under special electrodynamic field conditions !
5. Using no classic field terms, i.e. E, H, D, B or mixed terms like in Proca's equations
the equation Re + i Im = 0 in quantum electrodynamics shows a
general result: magnetic vector potential A and scalar potential PHIs are superior,
classic field terms i.e. E, H, D, B are secondary because those are
simply to derive from primarily A and PHIs equations.
.
6. Term 6) = 0 in imaginary part < Im > in above shown equation Re + i Im = 0 implies not
only the Lorentz' gauge but also - choosing light field gauge term 5) = 0 -
the central basis for classical interdisciplinary physics.
.
Some examples for interdisciplinary evaluations of imaginary part < Im >
.
a) Directly from eq.(Ib) yields the special Lorentz gauge: div A = - { d φS / dt/ } / c^2 ___(11a)___
or from eq.(Ib) unified equations i.e. using naturics UNIT checks: div J = - d ρ / dt _______(11b)___
or directly from (1b) with J = vρ (with electric charge density RHO ρ in electrodynamics)
div ( v∙ RHO ) = - d (RHO) / dt | _(=11b) |
div curl H = 0 = div J + div [ d D /dt ] with electric Gauss' law div D = RHO:
Because of J = (current) FLOW and RHO ρ= (electrical charge) DENSITY = charge q / m^3
(analogous quick derivation as for the basic equation in mechanics in the following lines)
eq. (11a) or (11b) can be written as an universal law:
div (FLOW DENSITY) = d (specific Charge Density) / dt(11)_ => CONTINUITY law
.
FLOW DENSITY is based on electrodynamics, thermodynamics or hydrodynamics etc., too ...
Specific DENSITY means the equivalent medium (charge, mass etc).
Eq. (11) can be re-formulated i.e. with respect to mechanics
div (ρm ∙ v ) = - d (ρm) / dt | _(11c) |
. where RHOm = Mass Density in mechanics as formal equivalence with charge density in electrodynamics.
A separate quick development of new formulas can be related to naturics based UNIT checks, i.e.
directly from eq. (Ib) for electrodynamics leading to eq. (11c) governing mechanics:
RHO. A [As/m^3 Vs/m = Ws/m^3 s/m = Nm/m^3 s/m = kg/m^3 m/s] = ρm. v __(11d)
ρ. PHIs / c^2 [As/m^3 V s^2/m^2 = Nm/m^3 s^2/m^2 =
kg m / s^2 m /m^3 s^2/m^2 = kg / m^3] = ρm_(11e)_
Eq. (11d) and (11e) in eq. (1b) yields eq. (11c)
. .
b) additionally applying light field gauge: d (q ∙ A) / dt = - grad (q ∙ PHIs)___(12a)
because of q ∙ A = Impuls and q ∙ PHIs = (potential) energy eq. (12a) can be written as an universal law:
.
d (IMPULS) / dt = - grad (ENERGY) (12)_ => FORCE law__
.
.
eq. (12) or directly (11c) can be re-formulated i.e. with respect to mechanics:
d (m ∙ v ) / dt = m ∙ dv / dt + v∙ dm / dt = - grad ( Wpot )___(12b)__
where m = Mass, v = speed of body, Wpot = m ∙ g ∙ h (i.e. and/or other force-"sources")
.
NOTE: Regarding (11b), (11c) and (12) we can formally switch between Newton's and Maxwell's relations
using Hamiltonian vector gradient formulation eq. (II), (III) shown above in unified equation Re + i Im = 0
... always thinking in analogies.
HINTS: Start from grad (A ∙ v) = (v Nabla) A + (A Nabla) v + v x curlA + A x curlv _____(12c) = ( ** )
or simplify with A = v as needed i.e. for NAVIER-STOKES equations in hydrodynamics you directly get :
d v / dt - ∂ v / ∂ t= (v Nabla) v = grad ( v^2 / 2 ) - v x curlv | _(12d) |
. _________Using (12c), (12d) we can compare important features of Maxwell equations and Newton's law, too.
... it's a good excercise for you to test your capabilities in handling vector analytic operations!
.
NOTE: 1. Though the physicist Heinrich Hertz thought that Maxwell's equations are not
derivable from Newton's equations, you can prove it ... at least in a formal analogy
... useful for multiphysics applications in engineering.
2. But never forget thinking in analogies: In real mechanics nothing is identical
with electric charge in electrodynamics.
.
c) Integrating eq. (12) yields the well known universal energy law in general form:
.
W total = W kinetic + W potential= constant (13)_ => ENERGY law
.
NOTE: Kinetic energy derived from relativistic Energy with Taylor approximation:
W (kinetic) = W (total) - W (restmass) = m ∙ c^2 - m0 ∙ c^2 = 0.5 ∙ m0 ∙ v^2 + ... tiny terms (x)
(x) can be neglected in non-relativistic applications
.
Further details: discussion about maxwell's equations combined with quantum mechanics
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