Ebooks Paperback, or Free pdf http:www.lulu.comkiehn or http:www.cartan.pair.com email: rkiehn2352ao

1
Ebooks Paperback, or Free pdf
http//www.lulu.com/kiehn or
http//www.cartan.pair.com email
rkiehn2352_at_aol.com
2
Propagating topological discontinuities The
Photon
As Topological Defects in the Light cone
Prof. R. M. Kiehn, Emeritus Physics, Univ. of
Houston www.cartan.pair.com rkiehn2352_at_aol.com S
PIE, San Diego Aug 2, 2005
This Powerpoint slide show can be downloaded from
http//www22.pair.com/csdc/download/spie2sound.ppt
3
This presentation exploits a Topological
Perspective of Nature
The mathematics is based on Cartans theory of
exterior differential forms and leads to a few
(perhaps) heretical conclusions for those who
have been brainwashed to believe that all science
is geometrical.
4
There exist fundamental topological relationships
between Electromagnetic SIGNALS and
PROPAGATING
Multi-valued SINGULARITIES,
TOPOLOGICAL DEFECTS,
Topologically Coherent SOLITONS,
Surfaces of ZERO MEAN CURVATURE,
SPINORS ,
and PHOTONS !
5
In my opinion, the bottom line is
The PHOTON is a topological defect in a
propagating topological defect. -
Math Example of a topological defect in a
topological defect The Minimal Klein bottle,
(not a Photon) A 2D surface can be a 2D
topological defect in 3D space. The 2D defect can
have holes, twists and intersections that are
additional topological defects in a defect.
The Figure was generated by F. Lopez and in
effect is a Spinor surface of Zero Mean
curvature. http//www.ugr.es/fmartin/dvi/survey.p
df e-mailfjlopez_at_ugr.es
6
Heresy 1
WARNING I plan to deliver a number of somewhat
heretical statements.
As this is a conference for Optical Engineers,
who know that the speed of light can be different
for different states of polarization, let me
start out with the first, little appreciated,
heretical statement
The Speed of light is not necessarily the same
in opposite directions!
7
Heresy 1. The Speed of light is not necessarily
the same in opposite directions!
Experiments conducted by V. Sanders and R. M.
Kiehn in 1977, using dual polarized ring lasers
verified that the speed of light can have a 4
different phase velocities depending upon
direction and polarization. The 4-fold Lorentz
degeneracy can be broken.
Such solutions to the Fresnel Maxwell theory,
subject to a gauge constraint, were published
first in 1979. Then the full theory of singular
solutions to Maxwells equations without gauge
constraints was released for publication in
Physical Review in 1991. R. M. Kiehn, G. P.
Kiehn, and B. Roberds, Parity and time-reversal
symmetry breaking, singular solutions and Fresnel
surfaces, Phys. Rev A 43, pp. 5165-5671, 1991.
An example of the theory is presented in the next
slide, which shows the exact solution for the
Fresnel Kummer singular wave surface for combined
Optical Activity and Faraday Rotation.
8
Heresy 1. The Speed of light is not necessarily
the same in opposite directions!
Fresnel Kummer Wave Surface for Combined
Optical Activity and Faraday Rotation
e 1, m 1, Optical Activity 0.3, Faraday
rotation 0.3
4 distinct Z axis roots -1.698, -0.676, 0.842,
1.138
4 distinct phase velocities, 1 for each direction
and each polarization along Z axis.
9
Heresy 2.
The next somewhat heretical claim is
Maxwell's theory of Electromagnetism is a
Topological Theory not a Geometric theory
10
Heresy 2. Maxwells Electromagnetism is a
Topological (not geometrical) Theory
By using Cartans methods of exterior
differential forms and two topological
constraints, the Maxwell Partial Differential
equations can be logically deduced as a UNIVERSAL
system on any ordered set of 4 or more symbols
(variables), without recourse to experiment
!!!. Maxwells addition of the ?D/?t term is a
topological issue! See extended
writeup http//www22.pair.com/csdc/pdf/spie2ext.pd
f
11
Heresy 2. Maxwells Electromagnetism is a
Topological (not geometrical) Theory
Two Topologically Distinct Field Structures
F(E,B) , G(D,H) No Constitutive constraint
between D,H and E,B
Conservation of Flux From a 1-form of Action
potentials A Ak(xm) dxk With
Topological constraint F dA
0 Yields Maxwell Faraday PDEs dF
0
div B 0, curl E ?B/?t 0
Conservation of charge-current density From a
2-form of Field Excitations G Gjk(xm)
dxj dxk With Topological constraint
J dG 0 Yields Maxwell Ampere PDEs
dJ 0
div D ?, curl H ?D/?t J div J ??/?t
0
12
Cartans Methods of Exterior Differential Forms
Heresy 2. Maxwells Electromagnetism is a
Topological (not geometrical) Theory
Go beyond the methods of Tensor Analysis

• No Metric constraint
• No Connection constraint
• No Diffeomorphic (Tensor) Constraint
• No Gauge constraint
• No statistics required

Can be used to describe
Continuous Topological Evolution of non
equilibrium systems and irreversible processes
!!!
Which is impossible in Hamiltonian mechanics
13
Heresy 3.
The next somewhat heretical claim is An
Electromagnetic Signal is a Propagating
Topological discontinuity in E and B fields,
not a Sinusoidal Wave train. V. Fock, Space
Time Gravitation, MacMillan (1932)
14
V. Focks definition of a Signal in terms
ofSingular, non unique, non analytic solutions
to Maxwells PDEs
Heresy 3. An Electromagnetic signal is a
propagating topological discontinuity

• Degenerate Electromagnetic topological
  singularities propagating (with speed C) satisfy
  the null Eikonal equation (Hadamard)
• (?f/?x)² (?f/?y)² (?f/?z)² - (?f/c?t)²
  0.
• The Eikonal equation is a quadratic form 0 upon
  which the E and B fields are not uniquely defined
  by the Maxwell PDEs.

15
Heresy 3. An Electromagnetic signal is a
propagating topological discontinuity
Focks Propagating Discontinuity of
non-uniqueness separates different topological
domains
Electromagnetic Energy zero on dark side of
propagating singularity
DARK side
E 0, B 0
Focks Propagating (tangential) Discontinuity
Surface
E
BRIGHT side
E2 0 B2 0
Electromagnetic Energy finite on bright side of
propagating singularity
B
4D Discontinuity Surface Light Cone In 3D,
Expanding Sphere parameterized by t.
16
Heresy 3. An Electromagnetic signal is a
propagating topological discontinuity
Solutions to quadratic forms 0, such as the
Eikonal equation, define complex isotropic
vectors of ZERO length.
Impossible for real Euclidean vectors, but OK
for complex Euclidean vectors.
Such complex isotropic vectors were defined as
SPINORS, 0, by E. Cartan (1913).
Bottom Line Eikonal Solutions are in effect
(macroscopic) Spinors, hence Spinors represent EM
signals as propagating topological singularities.
17
Heresy 3. An Electromagnetic signal is a
propagating topological discontinuity
The Spinors of interest to EM theory are
eigenvectors of antisymmetric matrices of
functions, such as that defined by the exterior
differential 2 -form of Field Intensities, F
dA.
F
18
Spinors can represent a propagating topological
discontinuity
Spinor Eigenvectors of anti-symmetric matrices
are artifacts of Zero Mean curvature
hypersurfaces Non Equilibrium Systems Can (and
should) appear in continuum mechanics Have been
ignored in most classical treatments. Are the
source of kinematic fluctuations, temperature and
pressure. Are required to produce irreversible
dissipation.
19
Heresy 4.
The next somewhat heretical claim is Spinor
Singular solutions to Maxwells equations are
Topological Features that do not depend upon
microphysical scales.
Bottom Line Spinors are not necessarily
microscopic or relativistic artifacts.
20
Heresy 4. Spinors are Topological features that
do not depend upon microscopic scales.

• The eigenvectors of any anti-symmetric matrix are
  of two types
• Extremal (Hamiltonian) eigenvectors of eigenvalue
  zero,
• of finite quadratic form (length).
• Spinor eigenvectors with imaginary eigenvalues,
• but with zero quadratic form (length).
• The number of Spinor eigenvectors depends upon
  the
• and the rank of the matrix F dA.

If the determinant of an anti-symmetric matrix F
is NOT zero, then ALL eigenvectors of F are
Spinors. In EM theory, this result implies that E
B is not zero. and the evolutionary process is
dissipative (Bulk Viscosity).
21
Heresy 4. Spinors are Topological features that
do not depend upon microscopic scales.
Thermodynamic Systems can be put into equivalence
classes defined by the Pfaff Topological
Dimension -- PTD(A)
Open Systems (mass and energy exchanged with
the environment) PTD(A) 4 (Implies 4
eigenspinors, and E B is not zero) Closed
Systems (energy but not mass exchanged with the
environment) PTD(A) 3 (Implies 2
eigenspinors, and E B 0) Isolated-Equilibrium
(no exchange with the environment) PTD(A) 2
or less (Pfaff Topological Dimension 2 implies
non-equilibrium)
As defect subspaces in defects PTD(A) 2 ?
PTD(A) 3 ? PTD(A) 4
The Pfaff dimension is a topological property
equal to the irreducible number of functions
required to specify the topological structure of A
22
Heresy 5.
The next somewhat heretical claim is Photon
Quantization is a topological idea that does not
depend upon microphysical scales.
23
Heresy 5. Quantization is a Topological idea
that does not depend upon microscopic scales.
The fundamental quantum feature of a Photon is
that it carries Photon Spin Angular Momentum
1 (h/2p)
A Topological Perspective indicates that there
are TWO important 3-forms
AF TOPOLOGICAL TORSION i(T4) dxdydzdt,
units (h/2p /e)2 AG TOPOLOGICAL SPIN
i(S4) dxdydzdt, units
h/2p
In engineering format on space time, these 4
component objects are given by the expressions
(both involve spinor components) T4 - E
A B f, A B Photon Orbital angular momentum
(helicity) S4 A H D f, A D Photon
intrinsic spin angular momentum
24
Heresy 5. Quantization is a Topological idea
that does not depend upon microscopic scales.
Note that exterior derivative of AF and AG lead
to the Poincare invariants
Poincare II d(AF) ? 4div(T4) 2 E
B Poincare I d(AG) ? 4div(S4) B H-D
E - A J - ? f
Topological Quantization comes from the deRham
theory of Period Integrals of Closed Forms.
?closed (AF ) m (h/2p /e)2 The Torsion
Quantum In regions where d(AF) 0
(E B 0) ?closed (AG) n
(h/2p ) The Spin Quantum In regions
where d(AG) 0 B H-D E - A J -
? f
The values of period integrals are automatically
proportional to the quantum numbers, m and n.
25
Heresy 5. Quantization is a Topological idea
that does not depend upon microscopic scales.
In addition to topologically quantized Torsion
and Spin, there exists the concepts of quantized
Flux and quantized Charge.
Period integrals are topological properties with
rational ratios. After all, you can not have an
irrational fraction of a hole.
Bohm Arahanov effects ?closed (A) m (h/2p /e)
The Flux Quantum In domains where dA F ? 0
(no E or B) ?closed (G) n (e) The
Charge Quantum In domains where dG J ? 0 (no
J or ?)
m and n are integers
26
Heresy 5. Quantization is a Topological idea
that does not depend upon microscopic scales.
Ratios of period integrals form rational
invariants.
Application of the ratios of Period
integrals Type II superconductors Rotating
Bose-Einstein condensates. ?closed (A)/ ?closed
(G) (m/n) (h/2p /e2) cite E.J.Post Fractional
Quantum Hall effect ?closed (AF)/ ?closed (AG)
(m/n) (h/2p /e2) cite R. M. Kiehn Hall
Impedance Z (h/2p /e2)
27
Heresy 6.
The next somewhat heretical claim is Long
Lived propagating states of Topological
coherence (Solitons) can be created in non
equilibrium Electrodynamic systems
28
Heresy 6. Topological Coherent states
(Solitons) can exist in non equilibrium systems
Falaco Solitons in a density discontinuity
A reality example of a Propagating Topologically
Coherent Defect structure.
Photo by Best Boy, David Radabaugh Fall, 1986
29
FALACO SOLITONSOptical and Topological Features
Heresy 6. Topological Coherent states
(Solitons) can exist in non equilibrium systems
See the movie at http//www22.pair.com/csdc/downlo
ad/blackspots.avi
30
Heresy 6. Topological Coherent states
(Solitons) can exist in non equilibrium systems
Spinor Surfaces of Zero Mean Curvature
WHEELER WORM HOLE Euclidean signature. Gauss
Curvature FALACO SOLITON Minkowski signature. Gauss
Curvature 0
Maximal Surface Pair Singular thread between
Disconnected Vertices
Minimal Surface Connected surface Open Throat
31
Heresy 6. Topological Coherent states
(Solitons) can exist in non equilibrium systems
Falaco Soliton deformations
Strings connected to Branes
A paradigm for
Type II Superconductors Rotating Bose Einstein
Condensates A new Fermi spin pairing
mechanism Flat Spiral Arm Galaxies connected
with a thread. Cosmological Strings in a
swimming pool.
Adapted from Strings in the Einsteins
paradigm of matter Vladimir Dzhunushaliev
http//arxiv.org/abs/gr-qc/0205055
32
Cosmic Strings from Hubble ?
33
Heresy 6. Topological Coherent states
(Solitons) can exist in non equilibrium systems
Falaco Soliton deformations
Strings connected to Branes
Brane Submanifolds A discontinuity 3D hypersurface
2D Maximal Surface The Brane deformation
2D Maximal surface deformations in a 3D
discontinuity surface, connected with a 1D string
34
Finally
A Model for the Topological Photon
Projected light cone Shells of propagating E B
field discontinuities (No EM energy outside
shells Finite EM energy within the shells)
The Photon,
Signal on
2D Topological defects of zero mean curvature
generated in the lightcone by isotropic Spinor
eigenvalues of the 2-form, F dA.
as a Falaco Soliton pair between Off and On light
cones.
Signal off
The 1D topological defect where the evolution V
in the direction of the 3-form AF produces zero
Work. Example V x B 0, E 0, B 0. A
Vortex line.
35
A Conjecture

• Realize that the detection of the Photon energy
  involves an interaction with a detector, with an
  exchange of unit angular momentum.
• Recall that the time rate of change of angular
  momentum is energy.
• The exchange of angular momentum implies the
  period integral of AG (a topological property)
  must change during the interaction process. The
  process of topological evolution is like a phase
  transition.
• So E ?L/?t ?d(AG) . Assume ?L h/2p, ?t
  1/?
• Then the Photon energy E (h/2p) ?

This Powerpoint slide show can be downloaded from
http//www22.pair.com/csdc/download/spie2sound.pp
t
36
I invite you to visit Cartans Corner
http//www.cartan.pair.com