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Klein-Gordon Equation
The following are my notes and views concerning the Klein-Gordon Equation:
The Relativistic Hamiltonian: H 2 = m2 c4 + p2 c2 .
(1)
The Hamiltonian (Energy) Operator: b |ψi = i~ ∂ |ψi . H ∂t The Momentum Operator:
(2)
∂ |ψi . (3) Pb |ψi = i~ ∂x Applying equations (2) and (3) into equation (1) provides the The KleinGordon Equation: " 2 2 # mc 1 ∂2 |ψi = |0i . (4) − ∇2 + 2 2 c ∂t ~
My views concerning equation (4):
Equation (4) is a wave-equation which represents a stationary particle of energy, mc2 . Energy is always in motion at the speed of light, c. Therefore, the first two terms of equation(4) represent a stationary orbit of photons about a fixed point in space (and time.) The energy of these photons (fixed in number) is, ξ = ~ω.
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(5)
Equation (5) was given to us by Dirac, where ~ is called the Dirac constant, and ω is the angular frequency of the orbiting photons about a fixed point in space. There is a vector relation that links the radius of this orbit, r, and the speed of the photons, c, with this angular frequency,
~ ~r. ~c = ω ~×
(6)
Equations (5) and (6) show us that the energy of a particle is a vector. And equation (5) is also a vector-equation,
ξ~ = ~~ω .
(7)
~ it’s rate of time, ~ω , Equation (7) defines a photon state by it’s energy, ξ, and it’s fixed location in space. The particle state and the photon state are equivalent. The photon state, by the nature of our perception, is a time average of the particle’s position in space; by this same perception, the photon state is an ensemble of photon motion, and that during this moment in time where we perceive the time average of the particle state (an illusion) the photon state becomes an ensemble average (in reality).The position of the particle is fixed because the only energy that the particle has, by definition, ~ is the orbital energy of the photons, mc ~ 2 = ξ.
1.1
Is Mass a Vector?
The question now arises: Is the mass, m, of a particle a vector?
ξ~ = mc ~ 2? 2
(8)
The energy of the particle is not only conserved but, by the nature of it’s photon state, it has a spin, which has a direction equivalent to the direction of the photon state’s angular velocity,
m ~ =
~ c2
~ω.
(9)
We know that the photon has two states: the particle-state and the wavestate, where the difference between these two states is that the particle-state is defined by a standing wave and the wave-state is defined by a traveling wave. Both of these states are represented by the Klein-Gordon Equation, equation (4). I have just shown that the standing wave defines a fixed point in space, where the particle has a stationary energy, mc2 . The traveling wave defines the momentum of the particle as it displaces in space. WE define the momentum operator as,
∂ Pb = i~ . ∂x
(10)
The particle-state is defined by the Mass Operator operating on a stationarystate in space:
c |~xi = m |~xi . M
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(11)
The wave-state is defined by the Mass Operator operating on a traveling wave-state:
E
E ~ = i ~ ∂
iλ ~ . c
iλ M c ∂x
(12)
where ~λ represents the
Ewavelength and the direction of the matter-wave in
~ the imaginary state iλ .
I have read that Dirac had some ideas about matter-waves, and a sea of particles, but I don’t know what happened with these ideas. I believe that I have touched on them here. His ideas has certainly led me to have some ideas of my own.
1.2
The i-Factor
It is now appropriate to address the factor i in the differential form of the above operator definitions. The imaginary unit, i, rotates the real state on which it ”operates”, producing an imaginary state which is orthogonal to that real state. So, the energy operator and the momentum operator (including it’s cousin, the mass operator) produce wave-states in both Real and Imaginary dimensions of Reality (they operate on both real and imaginary states). When an imaginary operator operates on a real state-vector it collapses that vector state and creates an imaginary state-vector which exists in an Imaginary dimension of Reality. Equations (3) and (2), respectively, show that these imaginary dimensions are dimensions of space and time, and within Imaginary dimensions or Reality travel matter-waves and photon waves! ”Matter” is mass, energy and time ... in all their forms and states. When mass travels as a matter-wave it takes it’s energy and ”rate of time” into an Imaginary dimension of Reality and, after it’s displacement, it returns with these quantities to their real state in tact: MATTER IS CONSERVED. ENERGY IS CONSERVED. AND RATE OF TIME IS CONSERVED. 4
I would like to note: When a ”body of matter” is in motion a ”conserved” number of it’s particles exist in Imaginary dimensions of Reality at any moment of time (our real perception of time ... it’s time average). As such, the body’s mass and rate of time (in real time) are reduced accordingly. Specifically, this is the source and the nature of what we call ”Time Dilation.” ω is the particle’s rate of time. A body’s stationary rate of time is the sum of the rates of time of all the particles in the body which are not in wave-state. The rate of time for a body in motion can be represented by,
ω = ω0
np
o 1 − β 2 + iβ ,
β=
n N0
(13)
(14)
where n is the number of matter-wave states, and N0 is the number of stationary particle wave-states when n equals zero. In equation (14), n includes the interactions from all conserved energies (photons), no matter their form or states.
1.3
Photon-Particle Interaction
What causes a stationary particle to change into a matter-wave is it’s interaction with an external emerging photon of the same energy, and having opposite spins. This interaction causes a particle to displace one wave-length in the direction from which came the emerging photon.
What the Klein-Gordon Equation completely neglects is this emergent photon and it’s interaction with the stationary particle. K-G E neglects any 5
development of the matter-wave state. It is this interaction which creates a third, and until now an unknown, state of matter. Consider the following equation, where I make use of Dirac’s notion of Creation and Annihilation Operators:
cc2 − iPbc M cc2 + iPbc D2 = M h i cc2 , Pb c D 2 = M 2 c4 + P 2 c2 + i M
(15)
(16)
The term on the right of equation (16) is the Poisson Bracket of the particle wave-state, and the matter wave-state, energies. These two terms are the same matter existing in different dimensions of reality during consecutive moments of time. This Poison Bracket has the value,
h
i cc2 , Pbc = i (~b M ω )2
(17)
and equation (15) becomes,
i h D 2 |ψi = M 2 c4 + P 2 c2 − (~b ω )2 |ψi
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(18)
Equation (18) now includes the interaction of the emergent photon with both the particle’s stationary wave-state and the consecutive matter wavestate.
ω b is the Time Operator:
ω b=i
∂ . ∂t
(19)
When this Time Operator, combined with ~, operates one the stationary wave function, which represents the stationary particle-state, it collapses the stationary wave-state and removes the particle from real space-time; but when it operates on the matter-wave’s wave function it produces a wave-state photon and the matter-wave combine, having opposite spins to each other, and traveling through each other producing a state of matter which only exists for a moment in imaginary time. At the conclusion of this photonparticle interaction the photon diverges, continuing in it’s direction along it’s original path. This divergence causes the traveling wave-state to collapse, and the particle wave-state re-emerges with the particle being displaced one wavelength in the direction of the photon’s past. This process shows the ”constraints” of time, equation(19), on the motion of matter. No work has been done on the particle during it’s displacement, No energies have been exchanged. And all quantities have emerged from the interaction with their properties conserved. This is the true nature of the quantum state of the Material Universe.
1.4
In Conclusion
I think that I have shown that by the use of the Dirac Creation and Annihilation Operators the Hamiltonian can provide much more information than just the standing wave-states of the particle. The Poisson Brackets are much more than just a number; they represent how the divergence of photons cause particles to displace and, maybe mor importantly, how a new state of matter is created and annihilated through the photon-particle interaction. Although I didn’t define a photon (I’ll leave that for another time,)
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I showed how photons exist in different states of matter, and how these varried states exist in their own dimensions of Reality. I also showed how nature uses multidimensional space and time to conserve matter during it motion from, and back to, the particle state. I defined the particle-state and the matter-wave-state - which results from the photon-particle interaction, creating a previously unknown state of matter. I also showed how the Quantum Level uses time to constrain the motion of matter.
�Notes on the Klein-Gordon Equation� is just a quick peak into the vastness that Reality hides from Human Perception.
Ron Poteet
11/23/2010
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