Solution of my fundamental equation

[Doctordick]

Hi Russell,

New thread because I think it deserves a new thread!

I think Chapter I of "Foundations" is very difficult for people to follow. Hopefully, you now understand the essence of what I was trying to communicate there. I think that, under the definitions I listed for the concepts convenient to talking about my equation, that equation is both true by definition and totally capable of representing any possible circumstance. I would really appreciate serious criticism of that logic and, until I get some serious rational complaints, I have no option but to assume I am correct.

So, let us look at the solutions. To begin with, a direct solution is simply impossible, at least for any worthwhile problem, as the number of arguments in the equation is almost beyond comprehension. The argument [itex]\vec{x}[/itex] represents the entire set [itex](\vec{x}_1,\vec{x}_2,\vec{x}_3\cdots,\vec{x}_n),[/itex] where n runs over the entire universe, knowables and unknowables combined (remember, NO information outside that referred to by [itex]\vec{x}[/itex] can be available to any examination of the problem)!

For your convenience, Chapter II is at:

http://home.jam.rr.com/dicksfiles/reality/CHAP_II.htm

My first step is to separate the equation into two parts. In my book, I make this first separation between "knowables" and "unknowables". I will follow that exact attack as I have the equations in the book to refer to (saves me a lot of latex problems). Note that, in equation 2.3, although the arguments of [itex]\vec{\Psi}_1[/itex] are only the finite set called "knowables", the arguments of [itex]\vec{\Psi}_2[/itex] span the entire set [itex]\vec{x}[/itex]. If you do not understand why this must be so let me know.

There is one strange notation I use which needs to be understood to avoid confusion. When I want to indicate an integral over all the [itex]\vec{x}_i[/itex] in a defined set, instead of writing down

[tex]\int_{x_a=-\infty}^{x_a=+\infty}\int_{\tau_a=-\infty}^{\tau_a=+\infty}\int_{x_{a+1}=-\infty}^{x_{a+1}=+\infty}}\int_{\tau_{a+1}=-\infty}^{\tau_{a+1}=+\infty}\cdots\int_{x_{a+n}=-\infty}^{x_{a+n}=+\infty}}\int_{\tau_{a+n}=-\infty}^{\tau_{a+n}=+\infty}\cdots [/tex]
​

I use a single integral sign with a circle in the middle (the sign commonly used for a "line integral"). I use the circle to imply the closed completeness of the integration. I also use the symbol dv to indicate the rest of the integration notation which should have been written

[tex]dx_a d\tau_a dx_{a+1} d\tau_{a+1}\cdots dx_{a+n} d\tau_{a+n} \cdots[/tex]
​

I do this simply to save space as all integrals are over exactly the same ranges and the complete notation serves no purpose. Secondly, I do not use any "line integrals" so confusion should not arise.

The only serious issue between equation 2.4 and equation 2.5 is the need for [itex]\vec{\Psi}_1[/itex] to be asymmetric with respect to exchange of variables. The [itex]\tau[/itex] axis was introduced to make sure that the fact of multiple occurrences of the same reference in B was not lost when those references were mapped into points on the x axis. If [itex]\vec{x}_i[/itex] can equal [itex]\vec{x}_j[/itex] then that purpose is defeated. That necessary constraint does not appear in any of the mathematical constraints I have written down.

Asymmetric with respect to exchange is a phrase which means that the result of the algorithm changes sign if two arguments are exchanged. If you wish to make an algorithm asymmetric, all you have to do is add to that algorithm the exactly the same algorithm with two arguments exchanged and multiplied by -1. That sum will be an algorithm asymmetric with respect to exchange of those two arguments which is still a solution to the same differential equation as the original. Now choose a second pair and do the same thing. Eventually you will use up all the arguments (their number is finite) and the result will be asymmetric with respect to any exchange. Notice that I will often propose doing things which are fundamentally impossible to do in fact, the real issue is, are they doable in principal!

This leads to a very important consequence. If the two arguments being exchanged are identical, then the result of the algorithm can not change. There exists but one number which does not change when it's sign changes: that is zero! It follows that [itex]\vec{\Psi}_1[/itex] must vanish if any two arguments are the same and the constraint is now implicit in the representation. Likewise the second term can not contribute as the argument of the Dirac delta function is never zero.

If you do not agree with "theoretical possibility" of actually calculating [itex]\frac{\longleftrightarrow}{f}[/itex] let me know and we can discuss it. The last point is the equivalence of the explicit set defined by the collection of [itex]\vec{x}_i[/itex] and the explicit set defined by the collection of arguments ([itex]\vec{x}_i -\vec{x}_j[/itex]). Again, if you have any question about proving the collections are equivalent, let me know and I will explain it in detail.

Thus we arrive at equation 2.5 which displays explicitly the form of equation any collections of knowables must obey. Now I haven't said I can tell you what [itex]\frac{\longleftrightarrow}{f}[/itex] looks like; I have merely said that such a thing could be known if [itex]\vec{\Psi}_2[/itex] were known.

Think about all that and let me know if you have any questions -- Dick

PS Sorry about the [itex]\frac{\longleftrightarrow}{f}[/itex] notation, it's the best latex notation I can come up with at the moment.

 

[Russell E. Rierson]

This leads to a very important consequence. If the two arguments being exchanged are identical, then the result of the algorithm can not change. There exists but one number which does not change when it's sign changes: that is zero! It follows that [itex]\vec{\Psi}_1[/itex] must vanish if any two arguments are the same and the constraint is now implicit in the representation. Likewise the second term can not contribute as the argument of the Dirac delta function is never zero.

Symmetry = Invariance under Change

http://en.wikipedia.org/wiki/Symmetry

Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it.

http://home.jam.rr.com/dicksfiles/reality/CHAP_II.htm

Doctor D:

A very strange thing has happened: that the above approximation is necessary is not surprising; that it arose the way it did is astonishing. We have arrived at the expression E=mc2 without even mentioning the concept of relativity.

Please elaborate.

 

[Doctordick]

Hi Russell,

You seem to have jumped quite a way beyond where I stopped last time. There are some serious issues which arise between where I left off and the quote you ask about. I'll get to that but before I do, I would like to make sure you picked up on some important points on the way there.

First, you gave a reference on symmetry. I don't understand why that reference is there. I can think of a number of reasons but I sure would like to know what moved you to insert it. You could have put it there for others who might not understand symmetry, you could be trying to let me know you understand symmetry by showing me a reference you have read or you might be suggesting that most everything I get is a consequence of symmetry. If it is the third, believe me I know that; physics uses symmetries all the time (they are the most powerful arguments which can be made). What is different about my approach is that the symmetries are not assumed but a direct consequence of the modeling procedure and are absolutely required.

One of the reasons I made my first separation of the fundamental equation between "knowables" and "unknowables" was the fact that it turns out to require all "knowables" to be Fermions (entities described by asymmetric wave functions obey "Fermi" statistics whereas entities described by symmetric wave functions obey "Bose Einstein" statistics): an interesting result in its own right. There is considerable discussion in the literature concerning a rational for the existence of Fermions and Bosons. That all "knowable" entities must be Fermions puts a rather different picture on what is real and what is actually a mechanism created to explain what we see.

Of course, equation 2.8 was arrived at by exactly the same procedure used to achieve 2.5. Between equations 2.8 and 2.9 I use that same mathematical mechanism I used back in Chapter I

Finally, general differential with respect to t may need to be a constant but there is no constraint that the constant be related to the left side of the fundamental equation. However, once again, if [itex]\vec{\Psi}_0[/itex] is a solution with m=0, simple substitution will confirm that

[tex]\vec{\Psi}_1\,=\,e^{i Mt}\vec{\Psi}_0[/tex]
​

is a solution to for m=M no matter what M may be desired.

Anyone familiar with quantum mechanics will relate time differentials to Energy. In many respects the removal of constants in this equation amount to adjusting the energy calculations by a constant. I only mention this because use of the exponential factor to adjust time derivatives is a common procedure used in quantum. In this case, it will soon become evident that this is exactly what is going on here. It is the total background energy of the Universe which is being removed by adjusting for the (S₂+S_r) factor.

I am hoping that you understand the step from equation 2.12 to 2.13. If you have no problems with that step I won't worry about it; but, if there is any doubt in your mind at all concerning its validity, we need to discuss it.

Some very important things are happening between equation 2.14 and 2.15 which you have not mentioned at all. First there is the fact that I multiply through the entire equation by Plank's constant (times [itex]\frac{c}{2\pi}[/itex]). Note that the actual value of Plank's constant has no bearing on the result at all. The other point, that the "definitions" given for m, c and V(x) do not constitute free parameters is a very serious statement. Check my collection of definitions at this point. In my approach, I have not defined "mass", I have not defined "speed" (much less the speed of light) nor have I defined "energy" (much less "potential" energy). Here, these things are implicitly defined by the solutions of the equation.

At this point, my equation is as general as it ever was (except for the three specific approximations I have made). Absolutely no theory about A has been introduced: i.e., any conceivable universe can be interpreted in a way which will make that equation valid. It follows that any conceivable universe can be interpreted in a way which will make Schrödinger's equation valid (at least in a circumstance where those approximations are reasonable).

It cannot be denied that my attack has laid a solid foundation for Schrödinger's equation and, by doing so, has pointed to physical meanings for the various terms in that fundamental equation. Thus I am led to add three new terms (the definitions given in expressions 2.19 through 2.21: energy, mass and momentum) to my list of defined concepts. Note that these definitions are consistent with the implicit definitions in expressions 2.16. (Just as an aside, notice that I have not defined how to measure these things, I have only defined them as aspects of the solutions to my fundamental equation.)

I was moved to make these definitions by the fact that they lead to Schrödinger's equation as an approximate solution to my fundamental equation. These are "my" definitions of energy, mass and momentum. They are clearly defined concepts in any and all possible universes and, by virtue of the fact that Schrödinger's equation is an approximate solution to my fundamental equation, they map perfectly into the standard concepts used in physics. (This also implies that the measurement of these things can be accomplished by the standard accepted processes.)

To anyone who is familiar with the range of applicability and complexity of Schrödinger's equation, it should be clear that my first two approximations were not really necessary at all but just made life simple. The third approximation was absolutely necessary as it made it possible for the second order differential with respect to x to appear linearly with a first order differential with respect to t, a central feature of Schrödinger's equation.

Interpreted in terms of my definition of "energy" (given in equation 2.19), the third approximation was that the energy had to be approximately equal to mc² (see equation 2.18). This means that the exponential adjustment performed between from equations 2.15 and 2.16 was removal of a constant background energy of exactly mc².

Two things have to be held in mind here. First, if you look at my definitions of mass, energy and momentum you will find that E=[itex]\sqrt{p^2c^2+m^2c^4}[/itex] which implies that my m is Einstein's m₀. So, from Einstein's perspective, the approximation necessary to make Schrödinger's equation an approximation to my fundamental equation is that E, the correct energy, must be approximately equal to m₀c²:i.e., the momentum term must be small compared to the mass term and we cannot be dealing with a relativistic problem. It is well known that Schrödinger's equation is invalid in a relativistic circumstance.

For the moment I won't comment on Part II (Expanding our Horizons). If you can follow it fine; if you have trouble, let me know what bothers you and I will try to clear it up.

If you get all this under your belt, the next step is to increase the dimensionality of my model. Basically, I do that by changing my labeling procedure and deducing the fundamental equation which will result with the new labeling procedure. The derivation essentially goes exactly as it has gone here. You should be able to follow it fairly easily and that should complete Chapter II.

Have fun -- Dick

 

[DrChinese]

DoctorDick,

I think your idea of t (time) and tau (proper time) are very interesting. I would like to discuss some of the implications of this. Assuming that we set t=tau=0 as a starting point (the big bang), I would conclude the following:

That since every proton in my body has traced a different world path before arriving at this location - now more or less a local reference frame - that these protons are of different ages - no two actually the same age. Is that a correct understanding?

 

[Doctordick]

DoctorDick,

I think your idea of t (time) and tau (proper time) are very interesting. I would like to discuss some of the implications of this. Assuming that we set t=tau=0 as a starting point (the big bang), I would conclude the following:

That since every proton in my body has traced a different world path before arriving at this location - now more or less a local reference frame - that these protons are of different ages - no two actually the same age. Is that a correct understanding?

Certainly that is true. My perspective is not necessary to hold that fact as true. Every competent physicist would agree with you. When it comes to decay periods, the only valid measurement of such a thing must be done in the rest frame of the entity and thus it is the integral of d[itex]\tau[/itex] along the path of the proton which is the only possible reference even under standard Einsteinian relativity.

On the other hand, although setting t=tau=0 as a starting point is possible, the definition of t into the future from that point is a very difficult thing to accomplish. Allow any movement of any kind between any reference objects and the integral of dt and the integral of d[itex]\tau[/itex] are no longer the same. If, when a and b are in the same place, t of object a and t of object b are different, a and b cannot directly interact as t is an interaction parameter (i.e., you cannot interact with events in the past nor can you interact with events in the future at the present).

This leads to a very complex problem in generating a mental image. Clearly every entity in existence has its own time frame. Likewise, since the laws of physics cannot be a function of who deduces them, they can neither be a function of time nor can they depend on which of this infinite number of frames is actually used. That is the essence of the idea of relativity. What a lot of people miss is the fact that the idea that the laws of physics cannot depend on which frame you use also explicitly allows one to choose a particular frame within which to work out the laws of physics.

Relativity is the mechanism by which one transforms the laws of physics from one frame of reference to another. Special relativity is absolutely correct and defended right down to the last specific issue. Now Einstein's general relativity is another issue entirely. That is a theory and it is not necessarily true.

Have fun -- Dick

 

[Russell E. Rierson]

Now Einstein's general relativity is another issue entirely. That is a theory and it is not necessarily true.

Have fun -- Dick

http://csep10.phys.utk.edu/astr162/lect/cosmology/equivalence.html

There are several ways to formulate the Principle of Equivalence, but one of the simplest is Einstein's original insight: he suddenly realized, while sitting in his office in Bern, Switzerland, in 1907, that if he were to fall freely in a gravitational field (think of a sky diver before she opens her parachute, or an unfortunate elevator if its cable breaks), he would be unable to feel his own weight. Einstein later recounted that this realization was the "happiest moment in his life", for he understood that this idea was the key to how to extend the Special Theory of Relativity to include the effect of gravitation. We are used to seeing astrononauts in free fall as their spacecraft circles the Earth these days, but we should appreciate that in 1907 this was a rather remarkable insight.

What is ...space?

 

[force5]

What is ...space?

My definition;

1. masses mirror twin with electron as transitional phase.
2. EMR production/consumption and merging/converging.