Why should the universe appear to be three dimensional?

Doctordick · 02-23-2009

I started this thread because the issue came up in my post to Bombadil earlier.

Quote:

Originally Posted by Doctordick

By the way, dimensionality of that picture is another issue we have not yet gotten to and perhaps now is the time to discuss that issue. I think I will open another thread concerning the most probable reason we see the universe as having three dimensions. In my head, that is an issue worth some serious thought.

If you carefully examine the opening post of “Deriving Schrödinger's Equation from my Fundamental Equation” you will discover that what I actually derive there is a one dimensional version of Schrödinger's equation. The tau coordinate is removed through integration so that the final picture (and the relevant Schrödinger equation) is expressed via an analysis of apparent point motion in an (x,t) space: i.e., temporal motion in a one dimensional space. There are lots of examples of use of a one dimensional Schrödinger equation to solve real physical problems but certainly our common picture of reality is not such a problem.

Let us look at the mental analysis of a one dimensional universe. One immediate problem which arises is very closely related to the “locality” brought up by Qfwfq quite a while ago. If we suppose the existence of an entity within that universe capable of examining aspects of that universe, we are confronted with a difficulty. We need to divide the system into two parts: the observer and the rest of the universe. The observer needs to be a “local” construct (an entity capable of comprehending such a separation). The problem with a one dimensional universe is that there are only two (point-like) interfaces between such an observer and the rest of the universe. If the entities the observer can interact with are anything more complex than simple elemental entities, we are confronted with only two points of interaction (one at a greater x than the observer and one at a smaller x than the observer. This simply is not an interaction providing much information for analysis; it just does not provide a background amenable to logical analysis.

There is a solution. Go back to my deduction of Schrödinger's equation; only, this time, divide all the labels into two different sets. Give the first set numerical labels x_i, give the second set numerical labels y_i and display the information as points in a Euclidean two dimensional space. Remember here that, in my deduction of my fundamental equation, a bunch of “invalid” (or hypothetical) ontological elements were added (the set D). “It is very important to realize that the set D is not part of A but is rather part of the explanation itself: that is, information presumed to be true and, without which, the explanation is incomplete" (I usually refer to these elements as "invalid" elements in order to partition them from what is being explained: i.e., "valid" elements). This implies that the number of “valid” labels in the set x_i need not match the number of “valid” labels in the set y_i; it is always possible to generate (via addition of "invalid" labels" a complete set which can be displayed as a set of points in the hypothetical (x,y) plane.

If you follow my deduction of that fundamental equation carefully, you will discover that the two dimensional representation of information yields a two dimensional fundamental equation of essentially exactly the same form as the original one dimensional version (the tau axis still needs to be added for exactly the same reason it was originally added). (Essentially the two sets, x and y, independently obey the same relationships deduced for a single axis.) Furthermore, the deduction of Schrödinger's equation goes through exactly the same as it did the first time; only, this time, the result is a two dimensional Schrödinger equation.

Now life is a little more complex. The observer is now a two dimensional entity and the rest of the universe can be seen as a collection of objects (objects being coherent, essentially temporally stable, collections of elemental entities) which approximately obey Newtonian mechanics (they have mass, momentum and interactions defined by those V(x,y) functions). The observer can interact with any and all such objects. This is a circumstance allowing considerable analysis.

However, there is still a troubling difficulty. In order to make a careful analysis of any specific object in this two dimensional universe, the observer and the object must remain in reasonably close proximity over substantial periods of time. If the objects are not bound together by some aspect of those V(x,y) functions then repeatable observations become rather difficult to arrange. Since all elemental entities are identical (their behavior is mathematically equivalent to one another) there needs to be a range of interaction where the forces are sufficient to keep a collection of objects in the same vicinity but are, at the same time, insufficient to distort the objects under examination. In our real world, this roll is very clearly played by what we call “the force of gravity”: it holds objects in the same neighborhood without creating major distortions of those objects (in most of the cases we use for the basis of our world-view).

Once again, since all elemental entities are equivalent (as to their mathematical behavior in our model) any such force must require the coherent impact of a great number of elemental entities: the source of such a force (sufficient to keep a collection of objects in the same vicinity but, at the same time, insufficient to distort those objects) must be physically large object compared to the objects under examination. Essentially, what I am pointing out is that the roll played by a force like gravity is both important (it holds our world together) and arises out of large objects (at least with regard to the details of our world). The problem with such a circumstance should be relatively obvious: the observer is, in many respects, confined to the surface of that large object and the world avalible to be analyzed becomes somewhat one dimensional. At least complex analysis of objects, which must obey Newtonian mechanics, yields a rather limited repertoire.

Thus a two dimensional universe, though not actually as one dimensional as a true one dimensional universe, is nonetheless rather limited as to what kinds of objects can exist and what kind of behavior can occur. The fact that all objects must approximately obey Newtonian mechanics does not really yield a lot of dynamic information. Try to think about the structure and behavior of two dimensional objects approximately confined to the outside of a large circle and I think you should be able to comprehend just how limited such a picture is.

So, let us move up one more dimension. Instead of dividing those undefined numerical label into two sets, let us divide them into three sets: x_i, y_i and z_i. Again the derivation of my fundamental equations goes through exactly as before yielding a four dimensional version (where $\vec{x}_i$ has four independent components: x_i, y_i, z_i and, of course that ubiquitous $\tau_i$ always required). The deduction of Schrödinger's equation once again goes through in exactly the same manner as before but now results in the standard three dimensional version.

Now we have a three dimensional world-view where objects have to obey Newtonian mechanics in three dimensions. This is clearly a very complex picture as it is exactly what the common picture of the universe is. I suggest it is a sufficiently complex picture to provide a very usable model of the important aspects of reality as it is indeed the “material” picture used by almost everyone. Why did we not proceed to the next possibility (mathematically equivalent to dividing the numerical labels into four sets)? Clearly there was no need; macroscopic survival was apparently sufficiently handled by via three dimensional picture.

Several people have suggested that one should take a serious look at the consequences of a higher dimensional analysis. My position on the issue is that such an analysis would be quite difficult to relate to the common perceptions as all we have to determine issues is mathematics. That is, our intuition is totally limited to three dimensions and offers little towards comprehending a four dimensional representation. Nevertheless, higher dimensional analysis may be of substantial value. Certainly the mathematical relationships embedded in higher dimensionality are very interesting; many modern physics concepts seem to be consistent with higher dimensional analysis so I will leave the central question as open.

The consequences of a representation of dimensionality equal to the total number of numerical labels is an entirely different story. One consequence of such a representation is that the addition of the tau axis is no longer necessary as every label plots to its own axis. As all axes are essentially independent, there ends up being no $V(\vec{x},t)$ and the general mathematical solution to the resulting fundamental equation is quite easy to develop. The solution is no more than what one might call simple quantized rotation in an n dimensional universe. That is, in fact, the reason I looked at what could be said of a three dimensional projection of that solution. The result is detailed in my thread ”A simple geometric proof with profound consequences” which seems not to have generated much interest.

Have fun -- Dick

Doctordick · 02-26-2009

Since no one has responded to this thread, I will continue the thought a little further.

Quote:

Originally Posted by Doctordick

The consequences of a representation of dimensionality equal to the total number of numerical labels is an entirely different story. One consequence of such a representation is that the addition of the tau axis is no longer necessary as every label plots to its own axis. As all axes are essentially independent, there ends up being no $V(\vec{x},t)$ and the general mathematical solution to the resulting fundamental equation is quite easy to develop.

There is still the possibility of interactions with those hypothetical entities which make up the set D. The unknowable data (that set D) can be viewed as forming a potential well which constrains our point to the origin: by simple symmetry it cannot depend on any angles. The interesting thing about this model is that the radial function, the only part where any variation is possible, is totally unimportant. Clearly, any solution for the entire universe must be dimensionally scalable and we only have one linear dimension: i.e., "r". In this view, the fundamental equation of the universe becomes

$\left\{\vec{\alpha}\cdot\vec{\nabla}+\beta g(r)\right\}\vec{\Psi}=k\frac{\partial}{\partial t}\vec{\Psi}=ikm\vec{\Psi}$

where I have explicitly inserted conservation of energy via, $i\frac{\partial}{\partial t}\vec{\Psi}=-km\vec{\Psi}$ , $\hbar$ being omitted as not being necessary). Consistent with previous work, we can use the implied operator identity $\vec{\alpha}\cdot\vec{\nabla}+\beta g(r)=ikm$ to generate the n dimensional Laplacian equation

$\left\{\nabla^2+g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}$

From earlier work, we can interpret the term g(r) as the probability density of the unknowable data and the square root of K is the energy of the universe divided by $\hbar c$ (just to give a little meaning to the above equation from our earlier definitions). The curious thing about the equation above is that the angular part, the only significant part, admits of solution in closed form. Note that, converting to pseudo spherical coordinates, the (n+1) dimensional Dalembertian operator may be written

$\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x_i^2}=\left(\frac{1}{r}\right)^2\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{i=1}^{i-1}csc^2\theta_i \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin \theta_i)^{n-i}\frac{\partial}{\partial \theta_i}$

where

$r=\sqrt{\sum_{i=0}^n x_i^2}$ , $\theta_{i+1}= cos^{-1}\left( \frac{x_i}{\sqrt{\sum_{j=i}^n x_j^2}}\right)$ , $x_i=rcos\theta_{i+1}\prod_{j=1}^i \sin \theta_j$ and $\theta_{n+1}=0$ .

Note that I use n+1 because it yields n angular components: i.e., r is really a separate issue here.

It is quite easy to show that, under these definitions, $\vec{\Psi}$ has a rather simple product solution.

$\Psi=\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi(\theta_i)$

Notice that the vector nature of $\vec{\Psi}$ has vanished. It should be quite easy to see that there exists no solution which requires anything beyond one single component.

In analogy to the common Legendre polynomials (solutions to the three dimensional problem) the entire collection of possible solutions can be expressed in terms of a generalized definition of such a function. For those with sufficient mathematics to follow the thing, it is not difficult to show that a generalized version of Rodrigues' formula will generate every possible solution.

$\Phi(\theta_i)=P_{m_{q+1}m_q}^q = \frac{\Gamma(1+\frac{q}{2})\sqrt{(1-x^2)^{m_q}}}{2^{m_q}\Gamma(m_q+\frac{q}{2}+1)} \frac{d^{m_q}}{dx^{m_q}}\frac{1}{\sqrt{(x^2-1)^q}}\frac{d^{m_{q+1}}}{dx^{m_{q+1}}}[x^2-1]^{m_q+\frac{q}{2}}$

where $x=sin^2(\theta)$ .

The final radial equation is

$\frac{1}{U}\frac{d^2U}{dr^2}=\frac{\frac{1}{4}n(n-2)+m(m+n-1)}{r^2}-K-g^2(r).$

Side note here: I deduced that result some twenty years ago and am not sure exactly what m stands for. Clearly it is related to the m's and q's in the angular solution. As it has no subscript; in analogy to the generation Legendre polynomials, though my memory is somewhat vague, I have a feeling it is essentially the “m” related to the largest “q” in the specific collection of $\Phi(\theta_i)$ but the issue certainly needs to be straightened out. I may have very well dropped the ball here: I may have omitted a sum. I will have to look at it and see if I can clarify the issue (but not at the moment, I just don't have the time). If I find an error or discover the actual meaning of "m" I will edit the post to correct it. Meanwhile, the $1/r^2$ dependence is correct.

I have looked at my original algebraic deduction and suspect very strongly that the m term should not stand alone as shown but should be replaced by a sum; however, that deduction is so involved that I would have to redo it in its entirety to discover exactly what the correction should be. At my age, I simply do not have the time nor the inclination to do so. If someone were interested, it might be worth the effort but, if no one who ever reads this will be capable of following that algebra, its a pretty worthless endeavor.

At any rate, such a radial equation is certainly possible and it will result in a specific radial function (arising from that 1/r term) plus a constant and that g(r) a function set by our explanation. Maybe there is some value in that equation; however, as it does nothing except change the scale of the system (which, by symmetry is essentially immaterial) it seems to be essentially beside the point except for one subtle issue.

Even if the analysis above is correct, the usefulness of a closed form solution to the universe is highly questionable as that solution is numerically far to complex to analyze (on the order of $10^20$ dimensions; however, there exists an attack which seems to somewhat ameliorate that problem. In much of my earlier work, I took advantage of the case where interactions between two different collections of data can be neglected. The fundamental equation in that case may be separated and that the result is an exact duplicate of the fundamental equation in a universe containing only that subset of data (in this case, the dimensionality of the solution would be smaller). Thus it is that we have significant reason to have interest in universes containing relatively small numbers of events.

A little thought reveals that the above change in perspective yields a fundamental shift in our use of the concept of dimension. From our new perspective, we do not think of dimensionality as a characteristic of the universe but rather, it becomes the characteristic of particular events of interest. Events have dimensionality, not space!

From this new perspective, to say that we live in a 3 dimensional universe is to presume that all events in the universe may be separated into 3 dimensional events: i.e., that the universe can be separated into triplets of data (x_i,y_i,z_i) such that the interaction between triplets may be neglected (each entity of interest can be thought of as existing in a three dimensional space without interacting with “distant” entities). So long as we are dealing with "gravitational" or "electromagnetic" forces and with macroscopic objects, that presumption may very well be valid; however, to extend that presumption to the entire universe of events is an insupportable jump of faith.

In addition, if we do solve the problem of representing such “higher” dimensional interactions, the scale factor might very well become an important issue when solving the problem of macroscopic interactions between two of these isolated “higher” dimensionality objects. Such a problem might require solving that radial equation (or the “correct” radial equation) and might very well give meaning to that scale: i.e., just as relativity requires a certain scale relationship, this higher dimensional case might set some parameters for those higher dimensional entities (which I suspect may be representations of the family of sub nuclear particles).

At any rate, I think the whole issue raises some very interesting problems and it would be nice to talk to someone who wouldn't find the math outside their abilities.

Have fun -- Dick

Pyrotex · 02-27-2009

Why does the universe appear to have three dimensions?

Because we have only three eyes!

Duh!!!!!!!

Moontanman · 02-27-2009

Dr. Dick, your math is completely beyond me but could our idea of three dimensions be because our senses are not capable of "seeing" more than three with a tiny peripheral sense of a forth? Could it be because the other dimensions are out side our world view and to be able to "see" them would mean being able to see outside what we perceive as "the world"? One of my first threads was about my idea of 11 dimensions or at least the way I see a multidimensional universe with all 11 dimensions being equal and real as the three we perceive. It wasn't well received probably because it made no real sense to anyone but me. However it was and is a representation of how I think of the world. I can imagine 11 spatial dimensions but I could never "see' them directly.

Doctordick · 02-28-2009

Pyrotex, it seems to me that a “moderator” and/or “editor” would have better things to do with his time than posting mindless graffiti in a serious thread! If you want to be humorous, why don't you go post in the lounge or the water cooler. I am sure you would be much more appreciated there!

Quote:

Originally Posted by Moontanman

Dr. Dick, your math is completely beyond me but could our idea of three dimensions be because our senses are not capable of "seeing" more than three with a tiny peripheral sense of a forth?

You seem to miss the central issue of what I am talking about. The concept, “our senses”, requires a world view to be defined. I am talking about what one can say about reality without defining “reality”.

Quote:

Originally Posted by Moontanman

Could it be because the other dimensions are out side our world view and to be able to "see" them would mean being able to see outside what we perceive as "the world"?

You should go read my thread, ”Defining the nature of rational discussion!” as I define two ways of generating conclusions (“logical” thought and “squirrel” thought) together with the strengths and failings of each. Unless you understand that issue, you cannot understand my presentation. One of the failings of “logical” thought is that, sans mathematics, it is far to limited to accomplish anything truly meaningful. The clear failing of “squirrel” thought is that it cannot be proved correct.

Quote:

Originally Posted by Moontanman

One of my first threads was about my idea of 11 dimensions or at least the way I see a multidimensional universe with all 11 dimensions being equal and real as the three we perceive. It wasn't well received probably because it made no real sense to anyone but me. However it was and is a representation of how I think of the world. I can imagine 11 spatial dimensions but I could never "see' them directly.

Eleven dimensions is quite easy to define (it is no more than a mathematical concept consisting of eleven different axes which are each orthogonal to all the others) but analysis of figures represented by such a geometry via “logical” thought is not possible without mathematics and “squirrel” thought is simply inadequate to the task. Human beings have never managed to develop the ability to see their world in an eleven dimensional representation; in fact that very failure is the issue of this thread.

That is to say, a dimension is no more than an ontological concept of standard mathematics used as a method of displaying a number. There is no more to it than that.

Have fun -- Dick

Bombadil · 02-28-2009

Quote:

Originally Posted by Doctordick

There is a solution. Go back to my deduction of Schrödinger's equation; only, this time, divide all the labels into two different sets. Give the first set numerical labels x_i, give the second set numerical labels y_i and display the information as points in a Euclidean two dimensional space. Remember here that, in my deduction of my fundamental equation, a bunch of “invalid” (or hypothetical) ontological elements were added (the set D). “It is very important to realize that the set D is not part of A but is rather part of the explanation itself: that is, information presumed to be true and, without which, the explanation is incomplete" (I usually refer to these elements as "invalid" elements in order to partition them from what is being explained: i.e., "valid" elements). This implies that the number of “valid” labels in the set x_i need not match the number of “valid” labels in the set y_i; it is always possible to generate (via addition of "invalid" labels" a complete set which can be displayed as a set of points in the hypothetical (x,y) plane.

Then is it possible to choose the invalid elements so that the Schrödinger equation still behaves as though it where still one dimensional. That is, is the number of dimensions that are being used defined as much by the remainder of the universe as by our choice of how many dimensions to use in the fundamental equation or will adding dimensions cause the elements to use all of them no matter what the reminder of the universe is defined as. That is will how the remainder of the universe is defined have an effect on how many dimensions are being used.

Also, how will this effect V(x) ? Will it simply add a collection of new terms to it that are a result of the additional dimension, or will it become an all new function?

Quote:

Originally Posted by Doctordick

Once again, since all elemental entities are equivalent (as to their mathematical behavior in our model) any such force must require the coherent impact of a great number of elemental entities: the source of such a force (sufficient to keep a collection of objects in the same vicinity but, at the same time, insufficient to distort those objects) must be physically large object compared to the objects under examination. Essentially, what I am pointing out is that the roll played by a force like gravity is both important (it holds our world together) and arises out of large objects (at least with regard to the details of our world). The problem with such a circumstance should be relatively obvious: the observer is, in many respects, confined to the surface of that large object and the world avalible to be analyzed becomes somewhat one dimensional. At least complex analysis of objects, which must obey Newtonian mechanics, yields a rather limited repertoire.

So then gravity is a result of considering a large number of elements? Then does this mean that at a quantum level that is a level in which just a few elements are of interest that gravity has no effect. Or will the effect of gravity only appear in the equations if a large number of elements are under consideration?

Quote:

Originally Posted by Doctordick

The consequences of a representation of dimensionality equal to the total number of numerical labels is an entirely different story. One consequence of such a representation is that the addition of the tau axis is no longer necessary as every label plots to its own axis. As all axes are essentially independent, there ends up being no $V(\vec{x},t)$ and the general mathematical solution to the resulting fundamental equation is quite easy to develop. The solution is no more than what one might call simple quantized rotation in an n dimensional universe. That is, in fact, the reason I looked at what could be said of a three dimensional projection of that solution. The result is detailed in my thread ”A simple geometric proof with profound consequences” which seems not to have generated much interest.

So in such a representation is any change over t defined by the initial definition of such a arrangement or would such an arrangement even change with a changing t. I thinking that this is only the case if there are no invalid elements to consider which is an issue you seem to be addressing in your second post.

Now for your second post

Quote:

Originally Posted by Doctordick

There is still the possibility of interactions with those hypothetical entities which make up the set D. The unknowable data (that set D) can be viewed as forming a potential well which constrains our point to the origin: by simple symmetry it cannot depend on any angles. The interesting thing about this model is that the radial function, the only part where any variation is possible, is totally unimportant. Clearly, any solution for the entire universe must be dimensionally scalable and we only have one linear dimension: i.e., "r". In this view, the fundamental equation of the universe becomes

$\left\{\vec{\alpha}\cdot\vec{\nabla}+\beta g(r)\right\}\vec{\Psi}=k\frac{\partial}{\partial t}\vec{\Psi}=ikm\vec{\Psi}$

So then you are in a sense using a sort of n dimensional generalization of polar coordinates and since every valid element must take up a different dimension any effect from the angels used in such a coordinate system is removed so that r, that is the distance from the element of interest to the invalid element it is interacting with, is the only thing that has any effect in the fundamental equation?

Also can we solve for the form of g(r) or is it unimportant or unknown in this situation.

Quote:

Originally Posted by Doctordick

where I have explicitly inserted conservation of energy via, $i\frac{\partial}{\partial t}\vec{\Psi}=-km\vec{\Psi}$ , $\hbar$ being omitted as not being necessary). Consistent with previous work, we can use the implied operator identity $\vec{\alpha}\cdot\vec{\nabla}+\beta g(r)=ikm$ to generate the n dimensional Laplacian equation

$\left\{\nabla^2+g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\Psi$

So the last equality $k\frac{\partial}{\partial t}\vec{\Psi}=ikm\vec{\Psi}$ is just the conservation of energy equation?

Now am I correct in understanding that we know that we can use $\vec{\alpha}\cdot\vec{\nabla}+\beta g(r)=ikm$ as a operator because each side is dependent on total different variables and that the application of it to each side can only result in generating a constant on each side of the equation?

Quote:

Originally Posted by Doctordick

From earlier work, we can interpret the term g(r) as the probability density of the unknowable data and the square root of K is the energy of the universe divided by $\hbar c$ (just to give a little meaning to the above equation from our earlier definitions). The curious thing about the equation above is that the angular part, the only significant part, admits of solution in closed form. Note that, converting to pseudo spherical coordinates, the (n+1) dimensional Dalembertian operator may be written

$\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x_i^2}=\left(\frac{1}{r}\right)^2\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{i=1}^{i-1}csc^2\theta_i \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin \theta_i)^{n-i}\frac{\partial}{\partial \theta_i}$

I’m not quite sure how it is that you conclude that g(r) is the probability density of the unknowable data?

As for the equation the first equality seems to make sense. Am I correct in understanding that any elements containing differentials of two different variables is removed due to the alpha operators and the two cross term are removed due to each having a alpha and beta operator in opposite orders?

But I’m having some difficulty understanding just what is going on in your second equality it seems to be just the necessary change in the differentials needed to change to a new coordinate system but I’m not quite sure what a pseudo spherical coordinate system is unless this is what you are defining on the next line. In which case I haven’t had a chance to try and perform the substitution but it looks like it would be a task to substitute it into the equation if I can figure out how such a thing would be done. But it looks to me like you may have a problem in your latex where you are defining r.

Seeing as this post is getting a little bit long and I really don’t understand exactly what the next line is meant to be I will save the remainder of my questions until you answer this post.

HydrogenBond · 02-28-2009

The universe appears 3-D, because humans have 2 eyes and see in 3-D. That is the neural matrix set up for visual observation. The math is set up for this matrix to forfills sensory expectation. If humans had only one eye, and lost their depth perception, it would be harder to convince people of the absolute nature of that nebulous third dimension. Things moving perpendicular to sight would be easier to define, but the depth issue of Z would be subject to uncertainty. The universe would be exactly the same, just our image would shift.

If you cover one eye, since the brain is already wired for 3-D, the one eye will shift back and forth to create an approximation for 3-D. But it is not as perfect as having two eyes. To test this, put a patch on one eye and have someone throw a tomato at your head, to see if one can predict the extra dimension. The math would need to change to make it easier to predict. We would have a fudge factor. The z-axis would look more statistical and that would be how the universe will be defined. The question this thread would be asking, in a one eyed world is, why does Z have this uncertainty?

Say we had two eyes in the front and one in the back. The models of the universe we would develop would need to be consistent with this point of reference. One would see some simultaneity of events, one can't see with only two eyes in front. Something in the front, appearing at 3-D, will also have an overlay with the 2-D from the back, where the events is not. The universe would then be created in this image because the brain is wired that way. The question we would be asking is, why does motion in front get more uncertain in Z when it moves to the back of us?

Here is an interesting universe. Say we had three eyes, with one eye each, pointing x, y and z. We would have three, 3-D zones of observation, at eyes (x,y) (y,z) and (z,x). At the fridge where only one eye can see, we get more 2-D. But at (x,y,z) we get sort of a 4-D affect. It would be a tri-optical overlap of three 3-D points of view. This would give some special affects that would be part of the universe. One would have to tip and tilt the head just right to be able to do these experiments.

Doctordick · 02-28-2009

Quote:

Originally Posted by Bombadil

Then is it possible to choose the invalid elements so that the Schrödinger equation still behaves as though it where still one dimensional.

The invalid elements are no more than additional ontological elements presumed in your explanation (the elements which, together with the rule F=0, constrain the valid ontological elements to what is seen). Your explanation may very well divide the valid ontological elements into two independent sets (x_i,y_i), in which case each of these sets must be constrained by an independent collections of presumed ontological elements

Quote:

Originally Posted by Bombadil

That is will how the remainder of the universe is defined have an effect on how many dimensions are being used.

The “remainder of the universe” is no different than the portion being explained by the reduction to Schrödinger's equation. The dimensionality is a function of your explanation; a characteristic which can be easily embedded in my model as, internal to the model, all valid elements are independent anyway. Think of it this way, the fundamental ontological elements of your explanation can be seen as two dimensional entities. Ok, so you say, “only some ontological elements of my explanation are two dimensional, the others are one dimensional”!

One of your problems is to communicate (or explain) your explanation to me. When I go to interpret that explanation I need to form that explanation in my mind; that process itself requires me to make assumptions: i.e., presume some ontological entities (think facts) not actually communicated to me. Very clearly, I can presume all valid ontological labels are two dimensional simply adding additional invalid labels.

Quote:

Originally Posted by Bombadil

Also, how will this effect V(x) ? Will it simply add a collection of new terms to it that are a result of the additional dimension, or will it become an all new function?

The constraint is still “no two 'valid' points” (valid numerical labels) can be identical. That sets the Dirac constraint to $F=\sum_{i \neq j}\delta(x_i-x_j)\delta(y_i-y_j)\delta(\tau_i-\tau_j)=0$ . The integration over “the remainder of the universe” is an integration over that three dimensional F, i.e., every variable except for x and y. Thus the final result will be a V(x,y) a two dimensional potential.

Quote:

Originally Posted by Bombadil

So then gravity is a result of considering a large number of elements? Then does this mean that at a quantum level that is a level in which just a few elements are of interest that gravity has no effect. Or will the effect of gravity only appear in the equations if a large number of elements are under consideration?

These questions will be answered later; when I get to the issue of my explanation of gravity. It does not mean that there exists any level such that “that gravity has no effect”; however, it is none the less a fact that there must exist a realm where gravity is strong enough to keep entities together without being strong enough to destroy the existence of those elements. There is an essential scale issue embedded in the question.

Quote:

Originally Posted by Bombadil

So in such a representation is any change over t defined by the initial definition of such a arrangement or would such an arrangement even change with a changing t.

Change and “t” are implicitly linked by definition of “t”, t being an index on “changes in the information being explained”

Quote:

Originally Posted by Bombadil

I thinking that this is only the case if there are no invalid elements to consider which is an issue you seem to be addressing in your second post.

No invalid ontological elements must obey exactly the same rules obeyed by valid ontological elements otherwise your explanation could be proved false: i.e., elements of your explanation do not obey the rules of your explanation thus must be false. The issue here is that it is impossible to tell which elements are valid and which are simply required by your explanation; but, certainly, they must all be consistent with your explanation.

Quote:

Originally Posted by Bombadil

So then you are in a sense using a sort of n dimensional generalization of polar coordinates and since every valid element must take up a different dimension any effect from the angels used in such a coordinate system is removed so that r, that is the distance from the element of interest to the invalid element it is interacting with, is the only thing that has any effect in the fundamental equation?

I think you are oversimplifying the circumstance. First, it is impossible to tell the difference between an invalid and a valid ontological element. Thus it is, once one reaches a dimensionality where the dimensionality equals the number of valid ontological elements, each valid ontological element constitutes a position in one dimension of that n dimensional picture: i.e., the entire set of valid ontological elements constitute one point in that n dimensional universe. But, at the same time, your explanation may require some additional “invalid” ontological elements. It follows that these invalid ontological elements essentially add additional dimensions.

At any rate, we are left with the universe being represented by a single point in an n dimensional universe. By simple symmetry, if this universe is seen as displayed in an n dimensional polar coordinate system, nothing can depend on any angles. All that can be said is that particular angles can have probabilities which depend upon those angles once a particular orientation is chosen to represent a specific time. How these distributions (positions in the n dimensional space) change is defined by your explanation; however, those changes must still be solutions to my fundamental equation.

Quote:

Originally Posted by Bombadil

Also can we solve for the form of g(r) or is it unimportant or unknown in this situation.

We certainly cannot “solve for the form of g(r)” as that function is a result of your explanation: i.e., the distribution of quantum numbers describing the “rest of the universe”: those coordinates outside the ones you are interested in calculating.

Quote:

Originally Posted by Bombadil

So the last equality $k\frac{\partial}{\partial t}\vec{\Psi}=ikm\vec{\Psi}$ is just the conservation of energy equation?

Essentially, yes, that is a thing which must be conserved (by symmetry) and we have defined it to be called “energy”.

Quote:

Originally Posted by Bombadil

Now am I correct in understanding that we know that we can use $\vec{\alpha}\cdot\vec{\nabla}+\beta g(r)=ikm$ as a operator because each side is dependent on total different variables and that the application of it to each side can only result in generating a constant on each side of the equation?

They are both mathematical operators. It is the issue of their equality which is important here. Since, if we have the correct $\Psi$ , that equation must yield equality and one operator yields a constant, so must the other (if and only if they are operating on the correct $\Psi$ ) which is true in all the cases of interest to us).

Quote:

Originally Posted by Bombadil

I’m not quite sure how it is that you conclude that g(r) is the probability density of the unknowable data?

There are some subtle arguments there which are best left to later. Let us say, for the moment, that the term is there because of the possibility some effect may exist. Fundamentally that requires an analysis of the impact of considering subsets of data (essentially objects of lower dimensionality and their interaction with the rest of the universe).

Quote:

Originally Posted by Bombadil

Am I correct in understanding that any elements containing differentials of two different variables is removed due to the alpha operators and the two cross term are removed due to each having a alpha and beta operator in opposite orders?

Yes!

Quote:

Originally Posted by Bombadil

But I’m having some difficulty understanding just what is going on in your second equality it seems to be just the necessary change in the differentials needed to change to a new coordinate system but I’m not quite sure what a pseudo spherical coordinate system is unless this is what you are defining on the next line.

I am not sure which “second equality” you are referring to here. If it is $-2k^2m^2\Psi=-K\Psi$ , I apparently left off the vector on the second $\Psi$ , I have just made that correction. If you are referring to the expansion of the Euclidean representation of the Dalembertian operator to the spherical representation, I have an interesting story behind that equation. In 1965, when I was a graduate student, that representation was utterly obvious to me (I knew it was right without even thinking about it). When I went to publish my discovery (in 1982; almost twenty years later) I wasn't sure that expression was correct so I worked out the algebra to prove it. The proof took me about two hours and it turned out to be right. When I converted that document to html in 2002 (another twenty years later) I got to that expression and worried about it being correct. So I once again worked out the substitution and proved it was correct; however, in 2002 the proof took me almost a month. I suspect if I tried to do it again now, it would probably take me several months. When you get old, your mind is one of the first things to go. (When I was in graduate school I quite often used the Dalembertian in both three and four dimensions so I was quite familiar with the kinds of cancelations which took place.)

About fifteen years ago I happened to be near the university where I got my Ph.D. and walked into the theoretical alcoves where I discovered the physicis library copy of my thesis (a piece of crap by the way) sitting on a graduate students desk. I asked him what he was doing with it and he said that the appendices were an excellent presentation of how to do quantum angular momentum calculations with complex collections of particles. So I guess some parts of my thesis were worth writing after all. Essentially it was an exercise in squeezing theoretical calculations presumed to be correct onto inadaquate computers. "A programing problem!" and nothing more.

Several years ago, I made a comment to someone that I had worked that Dalembertian out and they told me that was nothing, it was common knowledge. He then gave me a url to a reference. The reference agreed with me so I decided it has apparently come to be common knowledge. Just now I googled it and couldn't find the reference he gave me. It is a real algebraic bitch to work it out. Substitute in my definitions and you will get an equation several pages long on an ordinary lined paper tablet; however, a whole slew of terms simply cancel out and the final result is exactly what I have written down. The problem is, keeping tract of what cancels what; make a single error and the whole thing never gets smaller because some pieces cancel only portions of sums and you end up with lots of trash which never cancels. As I said, the last time I did it it took me two months and I don't want to do it again. If you want to, go ahead. If you don't make any algebraic errors, I will pretty well guarantee you will get exactly what I wrote down (after all, I have proved it three times in the last fifty years and got the same thing every time). Plus that, at least one other person has arrived at the same conclusion.

Quote:

Originally Posted by Bombadil

But it looks to me like you may have a problem in your latex where you are defining r.

You are indeed correct. I omitted the {} around the i=0 for the beginning of the sum and got $\sum_i=0^nx_i^2$ instead of $\sum_{i=0}^nx_i^2$ , the correct definition of the sum. I have just made a correction in the post. Thank you for spotting it.

Quote:

Originally Posted by Bombadil

Seeing as this post is getting a little bit long and I really don’t understand exactly what the next line is meant to be I will save the remainder of my questions until you answer this post.

Thank you; I appreciate your attention to details.

Quote:

Originally Posted by HydrogenBond

The universe appears 3-D, because humans have 2 eyes and see in 3-D.

I am afraid you have utterly no comprehension of what I am talking about as your opening sentence implies you are working with a world view presumed to be the only valid solution to the problem. You are totally ignoring the possibility of an infinite number of alternate explanations (other possible world views). I am talking about what we can say “without defining reality” that would be without making any presumptions whatsoever. That is why I call what I am doing “analytical metaphysics”; the idea is “solving for an undefined ontology”.

Have fun -- Dick

Bombadil · 03-04-2009

Quote:

Originally Posted by Doctordick

We certainly cannot “solve for the form of g(r)” as that function is a result of your explanation: i.e., the distribution of quantum numbers describing the “rest of the universe”: those coordinates outside the ones you are interested in calculating.

When I first saw that you had put in g(r) I thought that g(r) was the result of the second sum in the fundamental equation but on closer examination this doesn’t look like the case but rather that it is a result of possible interactions with elements that are not yet known and the second sum in the fundamental equation has been completely removed.

Quote:

Originally Posted by Doctordick

At any rate, we are left with the universe being represented by a single point in an n dimensional universe. By simple symmetry, if this universe is seen as displayed in an n dimensional polar coordinate system, nothing can depend on any angles. All that can be said is that particular angles can have probabilities which depend upon those angles once a particular orientation is chosen to represent a specific time. How these distributions (positions in the n dimensional space) change is defined by your explanation; however, those changes must still be solutions to my fundamental equation.

Then do we now have a rotational symmetry? That is, we no longer have a defined orientation to define the angles off of so that the actual angels can’t have any effect on the possible solutions to the fundamental equation? But that after we assigned an orientation we still have a probability assigned to each angle which combined with the r component would be nothing more then a probability assigned to the location of each element?

Quote:

Originally Posted by Doctordick

Several years ago, I made a comment to someone that I had worked that Dalembertian out and they told me that was nothing, it was common knowledge. He then gave me a url to a reference. The reference agreed with me so I decided it has apparently come to be common knowledge. Just now I googled it and couldn't find the reference he gave me. It is a real algebraic bitch to work it out. Substitute in my definitions and you will get an equation several pages long on an ordinary lined paper tablet; however, a whole slew of terms simply cancel out and the final result is exactly what I have written down. The problem is, keeping tract of what cancels what; make a single error and the whole thing never gets smaller because some pieces cancel only portions of sums and you end up with lots of trash which never cancels. As I said, the last time I did it it took me two months and I don't want to do it again. If you want to, go ahead. If you don't make any algebraic errors, I will pretty well guarantee you will get exactly what I wrote down (after all, I have proved it three times in the last fifty years and got the same thing every time). Plus that, at least one other person has arrived at the same conclusion.

I am going to try to do this although I don’t know how long it will take as I can’t say that I’ve done any substitutions of this complexity and I have some other things that I may try to get done before I get into it too far. When I do mange to get it done I will let you know what result I get, but, I‘m quite sure that I will get the same result. It sounds like it has been thoroughly proven. Just out of curiosity though, do you normally go from Cartesian to pseudo spherical coordinates or from pseudo spherical to Cartesian coordinates. Also do you really mean the D'Alembert operator as what you have looks more like the Laplace operator to me or am I missing something.

Quote:

Originally Posted by Doctordick

It is quite easy to show that, under these definitions, $\vec{\Psi}$ has a rather simple product solution.

$\Psi=\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi(\theta_i)$

Notice that the vector nature of $\vec{\Psi}$ has vanished. It should be quite easy to see that there exists no solution which requires anything beyond one single component.

So is this the solution to the equation

$\left\{\nabla^2+g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}$

after it has been transformed to pseudo spherical coordinates?

Quote:

Originally Posted by Doctordick

The final radial equation is

$\frac{1}{U}\frac{d^2U}{dr^2}=\frac{\frac{1}{4}n(n-2)+m(m+n-1)}{r^2}-K-g^2(r).$

I’m really not sure where this equation came from unless this is the differential equation derived from

$\left\{\nabla^2+g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}$

after changing to pseudo spherical coordinates.

Quote:

Originally Posted by Doctordick

A little thought reveals that the above change in perspective yields a fundamental shift in our use of the concept of dimension. From our new perspective, we do not think of dimensionality as a characteristic of the universe but rather, it becomes the characteristic of particular events of interest. Events have dimensionality, not space!

So, are you saying that any interaction between elements or collections of elements that take place in an explanation are a result of the number of dimensions that are considered?

So, does this mean that by considering more dimensions that it may be that interactions can take place that we can’t observe because it takes more dimensions in order to take place and that such interactions may have effects that result in currently unknown or unexplainable (by conventional physics) consequences due to the necessary dimensions?

Quote:

Originally Posted by Doctordick

In addition, if we do solve the problem of representing such “higher” dimensional interactions, the scale factor might very well become an important issue when solving the problem of macroscopic interactions between two of these isolated “higher” dimensionality objects. Such a problem might require solving that radial equation (or the “correct” radial equation) and might very well give meaning to that scale: i.e., just as relativity requires a certain scale relationship, this higher dimensional case might set some parameters for those higher dimensional entities (which I suspect may be representations of the family of sub nuclear particles).

Are you saying that the actual scale something which is symmetric in the fundamental equation may in fact have some greater meaning? If so, would this make it vary much like what is considered by “modern physics” to be a constant of nature? Also, are all of the constants of “modern physics” in fact a consequence of some symmetry in the fundamental equation?

Also are you saying, that for a small number of dimensions, that you suspect that the solutions to the corresponding n dimensional equation may in fact be equivalent to explanations of how subatomic particles behave?

Doctordick · 03-08-2009

Quote:

Originally Posted by Bombadil

When I first saw that you had put in g(r) I thought that g(r) was the result of the second sum in the fundamental equation but on closer examination this doesn’t look like the case but rather that it is a result of possible interactions with elements that are not yet known and the second sum in the fundamental equation has been completely removed.

I would go along with that. In actual fact, I produced that equation almost thirty years ago and, at the time, I concluded that the radial form was of utterly no significance; however, since that date, I have come to see that there might be some significance required by the interaction of higher dimensional entities.

As far as additional elements goes, they could not have any real influence on the angular part but it might be possible that they could have an impact on the radial form. To tell you the truth, reading the stuff I wrote so long ago is almost like reading someone else's stuff; I would have trouble explaining what was going through my mind back then.

Quote:

Originally Posted by Bombadil

Then do we now have a rotational symmetry? That is, we no longer have a defined orientation to define the angles off of so that the actual angels can’t have any effect on the possible solutions to the fundamental equation? But that after we assigned an orientation we still have a probability assigned to each angle which combined with the r component would be nothing more then a probability assigned to the location of each element?

Essentially yes. If you understood quantum mechanics, particularly the quantization of angular motion, you would understand that the motion of that point in the n+1 dimensional space (that would be n angular specification and one radial specification) would be a function of the rotational velocities in each of those n “planes of rotation” which would end up, when transformed back into a rectangular coordinate system, being velocities of the ith particle: i.e., given a position for every individual entity at t=0, would specify where to expect to see that entity at some later time.

That is why I seriously considered a three dimensional projection of that n+1 dimensional point. Essentially, if I have made no mistakes, the universe can indeed be seen as a rotating n dimensional equilateral polyhedron. The fact that any collection of points in a three dimensional space can be specified by a specific rotation of an n dimensional object was somewhat surprising.

Quote:

Originally Posted by Bombadil

I am going to try to do this although I don’t know how long it will take as I can’t say that I’ve done any substitutions of this complexity and I have some other things that I may try to get done before I get into it too far.

If I were you, I wouldn't try to do it without first doing it in two and three dimensions. You have to make sure you understand exactly how the angular measurements are defined and why they are accurately transformed by the original relationships I provide. For the time being, I wouldn't take the issue too seriously.

Quote:

Originally Posted by Bombadil

Just out of curiosity though, do you normally go from Cartesian to pseudo spherical coordinates or from pseudo spherical to Cartesian coordinates.

One normally goes both ways; which way you are going depends upon the problem you are doing and the answers you are searching for. Such a transformation is quite common in ordinary physics (usually in two or three dimensions).

Quote:

Originally Posted by Bombadil

Also do you really mean the D'Alembert operator as what you have looks more like the Laplace operator to me or am I missing something.

No, I don't think you are missing anything. Language changes (you will learn that when you get old), fifty years ago, gay meant happy and five hundred years ago, piss was an onomatopoeic euphemism used in polite society in place of the vulgar term used by men (which as I understand it was not recorded). When I was a student, there were really no commercial applications for what is currently referred to as general relativity (that was long before satellites and gps) and the subject was pretty esoteric. The term Laplacian was often used to refer to the shorthand notation for three dimensions (a triangle has three points) and the D'Alembertian referred to four dimensions (the square has four points) (where I got my spelling I do not remember, but I didn't make it up on my own; some minor authority had used that spelling somewhere). Apparently it now refers only to Einstein's space time with imaginary time and the dimensionality of the Laplacian is open.

When I followed your link to the D'Alembratian operator I noticed a reference to the “Heaviside step function” something I had never heard of although I had used the Green's function on many occasions. I had heard of Heaviside though so I went to look at his biography and got a big kick about his problems with rigor (see his “Middle years”). This quote seems to express his attitude.

Quote:

Originally Posted by Heaviside

He famously said, "Mathematics is an experimental science, and definitions do not come first, but later on." He was replying to criticism over his use of operators that were not clearly defined.

In many ways he reminds me of myself except that he was considerably more successful.

Not long ago I got into an argument with someone on this forum concerning the meaning of the word “measure” in geometry. Back when I was a student, the measure of a geometry was a differential statement of the definition of how one calculated distances in that geometry; now they seem to have a new meaning which is somewhat outside my ken. So, as I say, the language is a living thing and meanings change; and it appears that is is as true in physics as it is in English. One hell of a lot of the time I suspect the vocabulary of physics today is used as “buzzwords” by people who don't know how to calculate the thing being referred to.

Pretend like I didn't use either word; its the math that is important, not what you call it.

Quote:

Originally Posted by Bombadil

So is this the solution to the equation

$\left\{\nabla^2+g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}$

after it has been transformed to pseudo spherical coordinates?

That is what it is supposed to be; but I originally wrote that down about thirty years ago and I haven't checked it recently; as I said, I never before considered the radial part significant (it could very well be wrong; it certainly needs a definition of m which, at the time, I seem to have thought was obvious). I know it does not refer to the m in the above equation as one arises from a differential with respect to t and the other seems to come from angular algebra.

I have just taken a cursory look at my original algebra and have convinced myself I have made an error. Maybe down the way, I will find time to do the solution again in detail.

Quote:

Originally Posted by Bombadil

I’m really not sure where this equation came from unless this is the differential equation derived from

$\left\{\nabla^2+g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}$

after changing to pseudo spherical coordinates.

That is certainly what it is supposed to be but it clearly is not. As I say, I will look more carefully at it down the road.

Quote:

Originally Posted by Bombadil

So, are you saying that any interaction between elements or collections of elements that take place in an explanation are a result of the number of dimensions that are considered?

No I am not. What I am saying is that, when one examines elemental entities in higher dimensions, one obtains what one may call higher dimensionality pseudo angular momentum effects. That, using such relationships will generated phenomena analogous to spin effects of Electromagnetic interactions. I presume this would lead to those “special unitary groups” used to explain quantum numbers related to heavy particles.

Quote:

Originally Posted by Bombadil

So, does this mean that by considering more dimensions that it may be that interactions can take place that we can’t observe because it takes more dimensions in order to take place and that such interactions may have effects that result in currently unknown or unexplainable (by conventional physics) consequences due to the necessary dimensions?

No; rather, that it might provide a better reason for those heavy particle quantum numbers than, “they are needed to explain nuclear interactions”.

The common physics position that SU(3) lies behind the structure of reality and provides the basis for the dynamics of particle physics always seemed to me to be quite analogous to the ancient position that celestial spheres lay behind the structure of the universe and provided the basis for the dynamics of astronomical physics. Both positions postulate structures pulled from the scientist's hats without any support other than "they work". I would personally put them in the category of "rules of thumb".

Quote:

Originally Posted by Bombadil

Are you saying that the actual scale something which is symmetric in the fundamental equation may in fact have some greater meaning? If so, would this make it vary much like what is considered by “modern physics” to be a constant of nature? Also, are all of the constants of “modern physics” in fact a consequence of some symmetry in the fundamental equation?

I don't know; I haven't figured the whole thing out yet and I probably won't live long enough to find out the answer to that question. What I am thinking at the moment is that the radial solution may yield consequences (due to the need to provide the same answers as expressed in lower dimensionality) which might require the observed strength of nuclear forces as compared to electromagnetic forces. And yes, I am saying that I suspect many of the characteristics of subatomic particles may be directly due to the symmetries we have been talking about here.

Finally, when we get to the point where my deduction of Schrödinger's equation is reasonably understood, I will lay out my proof that Dirac's equation is also an approximation to my fundamental equation. When I do that, Maxwell's equations drop out as a necessity but in a way slightly different from the norm. The new approach to the development of his equations yields some insights into exactly why they fail in some interesting circumstances.

Meanwhile, I am preparing a post which lays out special relativity as it arises in my picture. It is pretty straight forward and requires no more than high school algebra together with a little bit of thought. I have been moved by all the falderal in the “what is space time” thread. It should make relativity a little more understandable by people who can't get out of the Newtonian picture.

Have fun – Dick

Why should the universe appear to be three dimensional?

Hypography?

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