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A Body in Motion
Setting up the equations of motion for a body composed of N particles at rest is very similar to the development of the Klein-Gordon Equation shown in some previous notes. The result is: p 2 2 2 2 2 c b b D |ψi = M 1 − β + βN P c − i (βN ~b ω ) |ψi (1) n = β= N
V c
2
(2)
where n is the number of matter-wave states, N is the number of particlestates when n = 0, and V is the apparent speed of the body in motion. p ~ 1 − β 2 is the mass vector of the stationary partiIn equation (1): M cles, β P~ c is the momentum vector of the body in motion, and β~~ω is the conserved kinetic energy of the body in motion, a vector. Both the momentum matter-wave state and the photons from the conserved kinetic energy which at any particular moment in time are in photon-particle interaction are in an imaginary dimension of Reality.
In equation(2): the middle term represents an ensemble of matter-wave photons, and in the last tern V, the apparent speed of the body, is a time average. This is the equation that determines the equivalence of ensemble and time averages in Statistical Mechanics., which has never been proven until now.
From equation (1) we can write the following Quantum Mechanical equations: 2 p ~ |ωi c 1 − β 2 |ωi = M M βN Pbc |iλi = P~ |iλi 1
(3) (4)
p 2 ω b 1 − β 2 |ωi = ~ω |ωi
(5)
~ is the mass of the stationary particles, P~ is the body’s momentum where M (in imaginary dimensions of Reality), matter-waves, and ~ω is the rate of time of the stationary particle-state, adjusted for time dilation. The kinetic energy is represented by, T~ = −iβN ~~ω
(6)
where the −i imaginary unit factor in equation (6) shows that the emergent photon-waves have a spin opposite to the spin of the matter-waves.
n N
in the above equations represents the motion of a 2 is how we view a body in motion. body as Reality sees it; using β = Vc This is the difference between ensemble and time averages: the difference between Reality and human perception. Of note: using β =
Ron Poteet 11/26/10
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