I would like to present the example of the Harmonic Oscillator as a means to

demonstrate the importance of matter waves. This is a continuation of MEaT,

the theory of Matter, Energy and Time. In this theory matter has two states,

particles which are stationary and exist in real time, and matter waves

which travel at the speed of light and exist in imaginary time. The

multidimensionality of time is necessary for the conservation of Matter,

Energy and Time. Time dilation of the particle state is the natural

evolution of multidimensionality of time. The motion of matter occurs only

as waves of matter in imaginary time.

An object of mass travels at an apparent speed v by having a percentage of

its mass in wave state. The normalized velocity is given by b = v/c, and the

amount of mass in wave state is given by the relation m = M*b, where M* is

the rest mass. The amount of mass remaining in the particle state is M*(1 -

b^2)^.5. This relation for the particle mass is a result of the

multidimensionality of time. An expression of this mass as a complex number

is

M = M*{(1-b^2)^.5 - ib}. The -i in this equation implies that the wave state

lags the particle state by a phase of pi/2 of its conversion cycle in time,

and that the wave state exists in imaginary time. Time has two factors, rate

and duration. Time dilation implies that it is the rate of time which is

conserved, not its duration: w = w*{(1-b^2)^.5 - ib}, where w is defined by

the relation

w*h-bar = M*c^2, where h-bar is Plank's constant divided by 2pi. It is seen

that M^2 and w^2 are constants of the motion. When a particle is converted

to a wave its mass is transferred to imaginary time, and its rate of time is

transferred with it. The rate of time for the remaining particles is thereby

reduced. This is the cause for time dilation, and the reduced mass in the

particle state for a body in motion. I would also like to point out that the

term b is also b = n/N, where n is the number of particles which have been

converted to waves, and where N is the number of particles for the rest

mass. It can be seen that b = v/c represents a time average for the apparent

speed of the body, while b = n/N is an ensemble average. MEaT is the only

representation for Matter, Energy and Time that allows for the proof that

time averages are equal to ensemble averages; until now this equality has

only been assumed, and never proved. We are now at a point where an analysis

of the Harmonic Oscillator can be understood.

The Hamiltonian for the Harmonic Oscillator is given by

H = M*((1-b^2)^.5)(w^2)(x^2)/2 + (p^2)/2M*

where the mass term reflects the stationary particle state and the momentum

refers to the wave momentum, p=mc, where m is the mass of the wave state. It

is instructive at this point to factor the Hamiltonian into Creation and

Annihilation operators as follows:

H = {((M*((1-b^2)^.5)/2)^.5)wX + iP/(2M*)^.5}{((M*((1-b^2)^.5)/2)^.5)wX -

iP/(2M*)^.5}

+ i{X,P}(w/2)((1-b^2)^.25)

H = aa* - (wh-bar/2)((1-b^2)^.25)

where a is the Annihilation operator and a* is the Creation operator. The

Hamiltonian operator and its commutator operate on a wave function on its

right (not shown). The Creation operator operates first, raising the energy

state of the system by one energy level, wh-bar, by causing a particle to

change state into a wave, then the Annihilation operator reduces the energy

state of the system by one level, by causing a wave to change state into a

particle, obtaining the original system energy. The commutator represents

the energy emitted by the wave state when it is converted to a particle; the

term (1-b^2)^.25 represents the time dilation of the energy rate due to the

apparent motion of the body. This commutator has always previously been

considered to be the ground state energy of the system. This is seen to be

incorrect. The ground state energy of the system is the energy of the

particle state, (M*((1-b^2)^.5)c^2)/2. Note that this isn't Einstein's

relation E=M*c^2. Einstein's relation for E is the energy of the wave state

for the body traveling at the speed of light; E=mc^2 is the energy of the

wave state. Also note that the term for the ground state energy is divided

by 2; the other half of the energy for the wave state comes for the

potential energy field. The conservation of the energy of the wave state

comes from the application of the commutator energy being put back into the

system by causing another particle to convert to a wave, etc. This is the

mechanism by which energy is conserved. A change of the energy state of the

system is conducted by the potential energy field. The observant reader

should realize that the motion of the body can be altered by shining photons

with the frequency given by the commutator onto the body ( you might call

this the principle for artificial gravity, or a new principle for mass

engines.) The reader should also note that MEaT provides an explanation for

the quantization of energy (i.e. the energy of the wave states are

quantized.) MEaT describes reality accurately with one theory, instead of

requiring many theories which don't agree with each other.

I would like to include in this paper another example of the wave nature of

matter. It is known that the two states of charge, the stationary particle

state and its wave state, emit fields whose characteristics are particular

to its state of motion; Particles of charge emit E fields and their waves

emit B fields. It is rather naive to think that other types of matter don't

have similar characteristics. Particles of mass attract other particles of

mass, and waves of mass attract other waves of mass; this is called Gravity.

Until now the wave state of matter has been ignored and about 96% of the

gravity within a Galaxy has gone unaccounted for. The part of gravity which

has gone missing is the Lorentz force. It is obvious that the gravity

generated by the Black Hole at the center of a Galaxy emits an axially

symmetric gravitational field; this should lead someone to think that the

Black Hole just might be made of matter waves, and not by a singularity of

particles. The E and B fields of mass are solved by Maxwell's equations with

a negative source of mass for the E field, and a positive source of mass for

the B field. Some questions about Faraday's Law for mass should arise which

can be settled measurements of gravity waves. Do you remember Joe Weber? I

think that if he had used gyros instead of a block mass he might have had

better luck in finding gravity waves. Anyway, there are some nice pictures

from the Hubbell telescope of gravity waves which should be enlightening. To

continue, consider the orbit of stars, planets and other matter about the

center of a galaxy. The motion of these objects imply that they have a

percentage of their matter in wave state. The closer you get to the center

of the galaxy the faster this matter moves (conservation of angular

momentum) and the greater will be the percentage of its mass in wave state.

There will be a radius from the galactic center where all objects will be

completely in wave state. This is the radius where the Black Hole begins. A

Black Hole is made entirely of matter waves which exist in imaginary time. A

Black Hole is black (invisible) because it exists in imaginary time. Dark

matter (matter waves) is dark because it also exists in imaginary time.

Matter waves constitute a vector potential, and the curl of this vector

potential defines a matter (mass) wave B field centered at the galaxy's

center, the center of the Black Hole. You can imagine that this is a B field

of incredible magnitude. Strong enough to influence alignment and motion of

any rotating (spinning) matter within its domain. The Black Hole and all the

spinning stars and planets within the Galaxy act like magnets of matter

(mass) whose structure and order are dictated by the Black Hole at its

center and the motion of the rest of the Galaxy. The Sun and all the planets

of the Solar system, and all other star systems, have their axis of rotation

aligned to the Black Hole, and not to the Sun. For the three planets which

are in question one must consider the possibility that they have rotating

cores, which we know do exist. One can also imagine that the B fields of the

Galaxy and its Black Hole might have an effect on the motion of weather, and

in particular Hurricanes. I would like to note that the Lorentz force does

not tell us anything about the forces between particles and waves, if they

do exist. Also, the mutual attraction of particles is a potential energy

field which affects the transition of particles into waves, which then come

under the influence of other waves.

So, Physics has gone wrong somewhere, sometime, when Physicists decided to

ignore the wave nature of matter. Modern Physics will only begin when the

wave nature of matter waves is finally recognized.

Ron Poteet

7/15/05