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Appendix

COSMIC QUANTUM MECHANICS: De Broglie described a vibration inherent in sub-atomic particles which determine their possible trajectories, the rule being that the possible rays of its wave are the same as its possible trajectories and

(1) d S ind dL = an extremum, and

(2) S dL/l = an integer for stable orbits.

IND = N(X,Y,Z) = the index of refraction of de Broglie's wave. Thales describes the vibration inherent in cosmic bodies with the formula:

(3) F = ( .072 * sqrt(NUM)) / R , R measured in AU,

where NUM= the average atomic number of constituent atoms. This formula is approximate, based on the assumption that NUM=1 for Jupiter.

Making use of equation (2) and of de Broglie's ideas presented in his Nobel Prize address, Thales gets

(4) IND = C / V = (C /F) * 1/l = ( C * INT)/( 2*p * .072 * sqrt(NUM))

INT = integer = one for each planet in the solar system, but is undoubtedly greater than one for stars in a galaxy, particularly an elliptical.

When we use this formula for IND in equation (1), we must make use of general relativity for dL, the metric of the continuum. Cosmic quantum mechanics and general relativity must be part of the same subject. Relativity provides a formula for dL², not dL:

(5) dL² = dR²/g - R² dq² + g dT²

(5a) g = (1 - 2M)/R = b²

In order to 'find the root,' Thales uses Dirac's procedure, as given in Rojansky's INTRODUCTORY QUANTUM MECHANICS. Assume the existence of invisible factors:

(5b) a_R, a_q, a_T are all equal to one, with squares equal to one and commutative products producing a change in sign. In other words, multiply any two of these three in one order produces the negative of them multiplied in reverse order.

(5c) dL² = [ a_Rb dR - a_qR dq + a_TdT/b² ]²

These invisible factors cannot be ordinary numbers but they can be matrices. One choice:

	(	0 0 0 1	)		(	1 0 0 0	)		(	0 0 1 0	)
		0 0 1 0				0 1 0 0				0 0 0-1
(5d) aR =		0 1 0 0		, aq =		0 0-1 0		, aT =		1 0 0 0
		1 0 0 0				0 0 0-1				0-1 0 0

Plug (5d) into (5c) (I will leave that for you to do), and we have a formula for dL². Find the square root, i.e., find dL, and dL can be plugged into formulas (1) and (2). Thus,

(5e) dL = a_Rb dR - a_q R dq + a_Tdt / b which can be plugged into equation (1).

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