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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
(1)
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,
ω b=i
∂ ∂t
(2)
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
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(3)
an imaginary vector state residing in an imaginary dimension of Reality. This vector state can be written as,
~r |Îť, ωi = exp i − ~ω t Îť
(4)
From equation (3) we can see that the imaginary state vector comes from the real state vector:
|Ν, ωi = i |ωi
(5)
Equation (4) represents the stationary particle wave-state when the following photon orbital relationship applies: ~ ~r. ~c = ω ~×
(6)
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 Convention’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.
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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
(7)
~r |λ, ωi = ~λ |λ, ωi .
(8)
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
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