Electromagnetic Phenomena Not Explained by Maxwell's Equations - Barrett (1993).pdf

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1
Electromagnetic Phenomena Not
Explained by Maxwell’s
Equations 260
Overview
The conventional Maxwell theory is a classical linear theory in which
the scalar and vector potentials appear to be arbitrary and defined by
boundary conditions and choice of gauge. The conventional wisdom
in engineering is that potentials have only mathematical, not phys-
ical, significance. However, besides the case of quantum theory, in
which it is well known that the potentials are physical constructs, there
are a number of physical phenomena — both classical and quantum-
mechanical —which indicate that the A µ fields, µ =
1
Chapter
0 , 1 , 2 , 3 , do pos-
sess physical significance as global-to-local operators or gauge fields,
in precisely constrained topologies.
Maxwell’s linear theory is of U(1) symmetry form, with Abelian
commutation relations. It can be extended to include physically
meaningful A µ effects by its reformulation in SU(2) and higher
symmetry forms. The commutation relations of the conventional clas-
sical Maxwell theory are Abelian. When extended to SU(2) or higher
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2 Topological Foundations to Electromagnetism
symmetry forms, Maxwell’s theory possesses non-Abelian commuta-
tion relations, and addresses global, i.e. nonlocal in space, as well as
local phenomena with the potentials used as local-to-global operators.
An adapted Yang–Mills interpretation of low energy fields is
applied in the following pages — an adaptation previously applied
only to high energy fields . This adaptation is permitted by precise defi-
nition of the topological boundary conditions of those low energy elec-
tromagnetic fields. The Wu–Yang interpretation of Maxwell’s theory
implies the existence of magnetic monopoles and magnetic charge. As
the classical fields considered here are low energy fields, these theoreti-
cal constructs are pseudoparticle, or instanton, low energy monopoles
and charges , rather than high energy monopoles and magnetic charge
(cf. Refs. 1 and 2).
Although the term “classical Maxwell theory” has a conven-
tional meaning, this meaning actually refers to the interpretations
of Maxwell’s original writings by Heaviside, Fitzgerald, Lodge and
Hertz. These later interpretations of Maxwell actually depart in
a number of significant ways from Maxwell’s original intention.
In Maxwell’s original formulation, Faraday’s electrotonic state, the
A field, was central , making this prior-to-interpretation, original
Maxwell formulation compatible with Yang–Mills theory, and nat-
urally extendable.
The interpreted classical Maxwell theory is, as stated, a linear
theory of U(1) gauge symmetry. The mathematical dynamic entities
called solitons can be either classical or quantum-mechanical, linear
or nonlinear (cf. Refs. 3 and 4), and describe electromagnetic waves
propagating through media. However, solitons are of SU(2) symmetry
form. 259 In order for the conventional interpreted classical Maxwell
theory of U(1) symmetry to describe such entities, the theory must be
extended to SU(2) form.
This recent extension of soliton theory to linear equations of
motion, together with the recent demonstration that the nonlinear
Schrödinger equation and the Korteweg–de-Vries equation — equa-
tions of motion with soliton solutions — are reductions of the self-
dual Yang–Mills equation (SDYM), 5 are pivotal in understanding the
extension of Maxwell’s U(1) theory to higher order symmetry forms
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Electromagnetic Phenomena Not Explained by Maxwell’s Equations 3
such as SU(2). Instantons are solutions to SDYM equations which
have minimum action. The use of Ward’s SDYM twistor correspon-
dence for universal integrable systems means that instantons, twistor
forms, magnetic monopole constructs and soliton forms all have a
pseudoparticle SU(2) correspondence.
Prolegomena A: Physical Effects Challenging a Maxwell
Interpretation
A number of physical effects strongly suggest that the Maxwell field
theory of electromagnetism is incomplete. Representing the influence
of the independent variable, x , and the dependent variable, y ,as
x
y , these effects address: field ( s )
→
free electron ( F
→
FE ) ,
field ( s ) →
conducting electron ( F
→
CE ) , field ( s ) →
particle ( F
→
P ) , wave guide
→
field ( WG
→
F ) , conducting electron
→
field ( s )
F ) interactions.
A nonexhaustive list of these experimentally observed effects, all of
which involve the A µ four-potentials (vector and scalar potentials) in
a physically effective role, includes:
→
F ) and rotating frame
→
field ( s )( RF
→
CE). Ehrenberg and Siday, Aharonov and Bohm,
and Altshuler, Aronov and Spivak predicted experimental results
by attributing physical effects to the A µ potentials. Most com-
mentaries in classical field theory still show these potentials as
mathematical conveniences without gauge invariance and with no
physical significance.
2. The topological phase effects of Berry, Aharonov, Anandan, Pan-
charatnam, Chiao and Wu (WG
→
FE and F
→
F
version, the polarization of light is changed by changing the spa-
tial trajectory adiabatically. The Berry–Aharonov–Anandan phase
has also been demonstrated at the quantum as well as the classi-
cal level. This phase effect in parameter (momentum) space is the
correlate of the Aharonov–Bohm effect in metric (ordinary) space,
both involving adiabatic transport.
→
F and F
→
P). In the WG
→
→
( CE
1. The Aharonov–Bohm and Altshuler–Aronov–Spivak effects
(F
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4 Topological Foundations to Electromagnetism
F). At both the quantum and the
macrophysical level, the free energy of the barrier is defined with
respect to an A µ potential variable (phase).
4. The quantum Hall effect ( F
→
CE ) . Gauge invariance of the A µ
vector potential, being an exact symmetry, forces the addition of a
flux quantum to result in an excitation without dependence on the
electron density.
5. The De Haas–Van Alphen effect ( F
→
CE ) . The periodicity of
oscillations in this effect is determined by A µ potential dependency
and gauge invariance.
6. The Sagnac effect ( RF
→
F ) . Exhibited in the well-known and well-
used ring laser gyro, this effect demonstrates that the Maxwell
theory, as presently formulated, does not make explicit the con-
stitutive relations of free space, and does not have a built-
in Lorentz invariance as its field equations are independent of
the metric.
→
F cases (effects 1, 3–5), the effect is
limited by the temperature-dependent electron coherence length with
respect to the device/antenna length.
The Wu–Yang theory attempted the completion of Maxwell’s the-
ory of electromagnetism by the introduction of a nonintegrable (path-
dependent) phase factor (NIP) as a physically meaningful quantity.
The introduction of this construct permitted the demonstration of A µ
potential gauge invariance and gave an explanation of the Aharonov–
Bohm effect. The NIP is implied by the magnetic monopole and
magnetic charge theoretical constructs viewed as pseudoparticles or
instantons . a
The recently formulatedHarmuth ansatz also addresses the incom-
pleteness of Maxwell’s theory: an amended version of Maxwell’s
CE and CE
→
a The term “instanton” or “pseudoparticle” is defined as the minimum action solutions of SU(2)
Yang–Mills fields in Euclidean four-space R 4 . 32
3. The Josephson effect ( CE
The A µ potentials have been demonstrated to be physically mean-
ingful constructs at the quantum level (effects 1–5), at the classical
level (effects 2, 3 and 6), and at a relatively long range in the case of
effect 2. In the F
→
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Electromagnetic Phenomena Not Explained by Maxwell’s Equations 5
equations can be used to calculate e.m. signal velocities provided that
(a) a magnetic current density and (b) a magnetic monopole theoretical
construct are assumed.
Formerly, treatment of the A µ potentials as anything more than
mathematical conveniences was prevented by their obvious lack of
gauge invariance. 251 , 252 However, gauge invariance for the A µ poten-
tials results from situations in which fields, firstly, have a history
of separate spatiotemporal conditioning and, secondly, are mapped
in a many-to-one, or global-to-local, fashion (in holonomy). Such
conditions are satisfied by A µ potentials with boundary conditions,
i.e. the usual empirically encountered situation. Thus, with the cor-
rect geometry and topology (i.e. with stated boundary conditions)
the A µ potentials always have physical meaning. This indicates that
Maxwell’s theory can be extended by the appropriate use of topolog-
ical and gauge-symmetrical concepts.
The A µ potentials are local operators mapping global spatiotem-
poral conditions onto the local e.m. fields. The effect of this operation
is measurable as a phase change, if there is a second, comparative map-
ping of differentially conditioned fields in a many-to-one (global-to-
local) summation. With coherent fields, the possibility of measurement
(detection) after the second mapping is maximized.
The conventional Maxwell theory is incomplete due to the neglect
of (1) a definition of the A µ potentials as operators on the local inten-
sity fields dependent on gauge, topology, geometry and global bound-
ary conditions; and of (2) a definition of the constitutive relations
between medium-independent fields and the topology of the medium. b
Addressing these issues extends the conventional Maxwell theory to
cover physical phenomena which cannot be presently explained by
that theory.
b The paper by Konopinski 253 provides a notable exception to the general lack of appreciation
of the central role of the A potentials in electromagnetism. Konopinski shows that the equations
from which the Lorentz potentials A ν ( A ,φ) arising from the sources j n ( j ,ρ) are derived are
∂ µ A ν =−
0, and can displace the Maxwell equations as the basis of electromag-
netic theory. The Maxwell equations follow from these equations whenever the antisymmetric
field tensor F µν ( E , B ) = ∂ µ A ν − ∂ ν A µ is defined.
4 π j ν /c, ∂ µ A µ =
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