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KRONOTRAN

Harry I Ringermacher 1/22/2009

Topological Quantization of Electric ChargeHarry I. Ringermacher, G.E. Research Center, Schenectady, NY

Definition of Electrodynamic Connection (H. Ringermacher, CQG, 11 , 2383 (1994) also seeSchrodinger, Space-Time Structure and J.A. Schouten, Ricci-calculus ):

S S S

= + +% contorsion tensor

The effective torsion tensor of this connection is:1

( )2 2

S v F

= =% %

2

emc

= (1)

Generates correct Lorentz force Eqn. of motion

0dv dv

v v v v v F ds ds

+ = + =

% (2)

0 Dv a =where D is the absolute derivative under the Electrodynamic Connection and v is the 4-velocity of test charge, mass m, charge e and 1v v = .

Given, EM field tensor as torsion along test charge path.

2 F v S =

(3)

Definition of dual tensor :

Let dx dx dx dx be area basis 2-form.

1*2

F F g = e.g. flat23

01

* F F = or

1

1* E B= and0 1

2 3*( )dx dx dx dx=

Integrate over dual 2-D surface .

1 1* *( ) ( )2M M

v S g dx dx F g dx dx

= (4)Define g dx dx d where g g = , signature (+ - - -), and = 0,1,2,3.

The RHS is Gausss law:

2 3

01 4M M

F g dx dx E dA Q

= = (5)Use Stokes theorem on LHS of (4):

1 1* *( )M M

v S d D v S d (6)

8/14/2019 e-QuantV4

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KRONOTRAN

Harry I Ringermacher 1/22/2009

Expanding RHS of (6), 0 Dv = from (2). That is, v passes through the exterior derivative leaving:1 * 4

M

v S dx d Q = (7)The Rund structure equation relates the covariant derivative of the torsion to curvature:

12

S dx dx dx R dx dx dx (8)

Eq. (7) becomes:1 * 42 M

v R dx d Q (9)Choose the test charge rest frame: 0 so that 0 1v .Requiring ( , ) = (0,1) corresponding to an E -field only in S (see eqn. (8)), yields for (9):

0 1 01

0101

1 * 4M

v R dx d Q

=

(10)Expanding the dual area element:

1 2 30101

00 11

14

R g dx dx dx Q

g g

=

(11)The sectional curvature (Gaussian curvature )of the dual surface in the rest frame of the test chargeis (Ringermacher, Mead - JMP 05):

01011

00 11

R K

g g =

Let Q ne= be the source charge in terms of the fundamental charge e .Let e be the positive definite fundamental unit of length.Defining the unit sphere through the r-integration, Eq. (11) becomes :

1 2 3

10

14

e

dx K g dx dx ne

= (12)

The integral2 3

1 4 (1 ) K g dx dx P

= is the Gauss-Bonnet invariant. (13)P is the Genus or number of handles of the surface. Thus, from (12), 1n P = is an integer.

Thus, charge has been identified as a topological property, the Genus of the surface surrounding thesingularity at the origin.

(1 )Q P e=

P = 0 P = 1 P = 2 Q = e Q = 0 Q = -e