Some Additional Notes on The Klein-Gordon Equation
The following are some ideas that occurred to me after I wrote my notes on the K-G E: Mass is a mathematical construct from the human perspective on the particle. From this perspective the particle state is real and can be represented by,
c |ψi = m M ~ |~ω i
where |ψi is a general state vector which, when operated on by a real operator, will produce a specific real state vector (e.g.|~ω i,) where the vector sign,in this equation, is shown for clarity, but is not necessary since the state vector is a vector anyway. |ωi is the real particle state vector representing a particle having a rate of time, ~ω , a vector which makes the mass (and it’s energy) a vector too. Representing ω ~ in it’s operator form gives,
the time operator. This time operator operates in all dimensions of reality, rotating any vector by π2 . This time operator operating on the particle state vector gives,
ω b |ωi = −i~ ω |λ, ωi
an imaginary vector state residing in an imaginary dimension of Reality. This vector state can be written as,
~r |Îť, Ď‰i = exp i âˆ’ ~Ď‰ t Îť
From equation (3) we can see that the imaginary state vector comes from the real state vector:
|Îť, Ď‰i = i |Ď‰i
Equation (4) represents the stationary particle wave-state when the following photon orbital relationship applies: ~ ~r. ~c = Ď‰ ~Ă—
Equation (4) also represents a traveling wave whose states vector is also imaginary.
It should be clear that the particle state is stationary in space and time since the orbiting photons orbit around a fixed point in space, and that the rate of time for the particle is referenced to that fixed point in space.
It should also be clear that the Copenhagen Conventionâ€™s ideas about relative reference frames are debunked. All matter and their motions are referenced to a fixed frame of reference in both time and space.
Equation (5) shows that the displacement of a particle occurs only in imaginary dimensions of Reality. This means that the Human Perception of motion is an illusion. We only see motion as a time average of a body’s particle displacements. Particle displacements are: ~r |ωi = |0i
~r |λ, ωi = ~λ |λ, ωi .
where the displacement of the particle is the wavelength of the particle’s photon wave-state: ~ ~λ = 2πr ω . (9) ω
Ron Poteet 11/25/10
Published on Dec 4, 2010