What Do Hamiltonians and Hamilton's Equations Tell Us About Dimensions?

[hivesaeed4]

I wanted to know what is the definition and what actually is a dimension. I mean I get that length, width and height story, but how does time fit in as a dimension. Then what about higher dimensions. Like if a certain system was defined in 7 dimensions what exactly would those dimensions be? 4 could be time, length width and height but what about the other 3?

 

[gonegahgah]

Actually, I have a similar question (that I hope won't interfere with the original OPs question) so I thought I'd add it here.

I can see us dividing our space up into dimensions of x,y,z and for myself - though probably not for others - I can see myself adding time as a dimension of movement and also acceleration as a dimension of force or reaction.

My question is about the extra dimensions too. Obviously we can refer to our x,y,z dimensions without absolutely needing a preference ie. x can describe any direction in space and the others are then just 90deg to this and each other. You can rotate your graph any revolution in space and it can still be used to plot points in space.

So what then prevents the simple movement through those other dimensions; the movement that is so easily observed to remain in the 3 dimensions that we do see? What gives them a preference of exclusion?

Obviously, if you referred to mathematical dimensions; as I have to space, movement and acceleration, then you get obvious mechanisms of exclusion ie x,y,z can not represent moving without time as a dimension nor direction changing without reaction as a dimension.

Are there any similar established mechanisms of exclusion for those other dimensions that keep them separate from x,y,z? If there are what are they?

 

[Integral]

The number of variables required to completely specify the state of your system is the number of dimensions of the system.

It takes 3 numbers to specify the location of a object in space, therefore we speak of 3 spatial dimensions. If you throw in time you get a 4th dimension.

If you are tracking the fluid level of 100 different tanks you have a system with 100 dimensions.

 

[gonegahgah]

Thanks Integral. By this it then perhaps becomes more logical to refer to extra dimensions. Maybe it is okay then for theories like string theory to have extra dimensions without needing to resort to physical 'braines' and such.

I would wonder if the real ultimate trick is in trying to reduce a model down to the fewest needed dimensions for its representation then?

I guess this leaves the question of what can propagate through those dimensions as the limiting factor. If for example it were that there were four spatial dimensions (one hidden from us) then I would have to wonder what prevents us smoothly rotating through the fourth direction as easily as any of the other three directions. What provides the barrier to this movement?

 

[Nabeshin]

Thanks Integral. By this it then perhaps becomes more logical to refer to extra dimensions. Maybe it is okay then for theories like string theory to have extra dimensions without needing to resort to physical 'braines' and such.

I don't think this is sound logic. In string theory, the extra dimensions look, feel, and smell just like the other ones, which is why we assume they're on exactly the same footing (well, of course subject to compactification. But they're still genuine spatial dimensions).

 

[gonegahgah]

Thanks Nabeshin. This then raises the question, which I mentioned above, as to what ties us inside our three dimensions? What excludes us from moving through those other dimensions as easily.

The three space dimensions we attribute ourselves x,y,z don't have any preferred direction, as I said, so you can claim any direction as being the x dimension and the other dimensions then only need to be 90deg to this and each other. We can freely rotate within this without restriction.

What stops us from as easily rotating into those other 'hidden' dimensions if they are all equal with ours?

 

[phinds]

Thanks Nabeshin. This then raises the question, which I mentioned above, as to what ties us inside our three dimensions? What excludes us from moving through those other dimensions as easily.

First, it is not proven that they even EXIST.

The three space dimensions we attribute ourselves x,y,z don't have any preferred direction, as I said, so you can claim any direction as being the x dimension and the other dimensions then only need to be 90deg to this and each other. We can freely rotate within this without restriction.

What stops us from as easily rotating into those other 'hidden' dimensions if they are all equal with ours?

As posited by string theory they are TINY. For all we know, we DO move through them.

 

[Naty1]

I mean I get that length, width and height story, but how does time fit in as a dimension.

You can think of it in different ways...and it does require some different thinking...

for a start think about plotting d = vt...what do you do?? typically you'll plot time as one of your 'variables', right??...just think of that as a 'dimension'. It's a simple 'spacetime' diagram with ONE dimension of space. In these 'low speed' physics, space and time remain separate entities...they do not mix.

YOU can also plot two or three dimensions...no need to stop with three...but visualizing beyond that is tough for our senses...

I've found this visual helpful: consider an x,y plane with time t perpendicular to the plane[two space dimensions for simplicity] : then the general plot of a particle moving at constant velocity is a straight line away from the x,y plane at some angle [it'smoving through space and time in a straight line]; circular motion [a merry go round] becomes a corkscrew plot since motion is both rotational in x,y and also moves thru time...say roughly parallel to the constant velocity plot; and a linearly accelerating particle moves away from the x,y plane thru time, but now is a curved plot. Have you ever plotted, say, energy versus say height...as in potential energy? There typically as height increases relative to a reference, so does PE. So here you have PE as a 'dimension'...but we don't think of that one as a physical 'direction' merely an increasing 'value' of some measurement...analogous to conventional time.

Another perspective is this :

A reference frame is composed of origin, a set of axes along with some convenient coordinates [reference points]. Reference frames are not a property of our universe; they are mathematical devices helpful to us for analyzing physics, not physical features themselves. The word "Euclidean" applies to 3 spatial dimensions; The 3+1 geometry (3 space, one time) is Minkowski spacetime used in special relativity.

Things get more involved in special relativity...where neither distance nor time remain the same to different observers...there we talk about 'length contraction' and 'time dilation' due to relative speeds...what different observers do all see the same is, oddly, the speed of light. In essence, space and time mix together ...and became more involved than even Einstein first realized... so as to keep observations of light speed constant...THAT is not part of classical [Newtonian] physics...

so my view is "If Einstein's math professor changed Einstein's view from space and time to 'spacetime' then it's not so dumb that I have some questions,too."

 

[Khashishi]

There are totally different usages of the term dimension. Which one needs to be figured from the context. Here are a few examples:
1) any degree of freedom. This usage is more common in classical mechanics with generalized coordinates, thermodynamics, robotics
2) number of independent basis vectors in a vector space. related to 1). Usual definition in quantum mechanics.
3) fractal dimension, especially Hausdorff dimension. If you are messing with fractals.
4) space-time-like dimensions in the sense that all of our vectors and tensors need to be extended for all of these extra dimensions. It makes sense to rotate from one dimension into another. Fields exist over all these dimensions, and certain force laws follow a 1/r^(#spacedimensions-1) relationship. This is used by some grand unified theories. Since there is no evidence for extra dimensions, they are presumed to be small.

 

[gonegahgah]

As posited by string theory they are TINY. For all we know, we DO move through them.

I imagine that we would have to move in such a way as to maintain our overall rigid or elastic shape shape in our space with maybe some fluctuation at the miniature level or something in and out of these 'tiny' dimensions?

I wonder if the string theory dimensions are supposed to be 90deg to our space and to each other?

As an example, a 2D person looking at a 2D-circle would only see a shaded line. If the circle were actually a cylinder with a hidden 3rd dimension then somehow the cylinder must maintain orientation to that 2D world through exactly the same slice. So some mechanism must be preventing the cylinder from rotating in the 3 dimensions and only being able to move in relation to the 2 dimensions although when it moves it moves in all dimensions.

The other alternative is where the 2D-circle bubbles in and out of the 2D plane into the extra hidden dimension while maintaining it's 2D-circle appearance. The extra dimensions then somehow become like overflow or spillage buffers.

The later seems unlikely and the former begs a mechanism to prevent free rotation in all the dimensions...

If the extra dimensions are tiny are they tiny in respect of having limited depth or are they tiny in respect of severely compacting anything that moves into them. Another sort of question: like our space dimensions; do the extra dimensions of string theory have 'long' distances in them?

 

[gonegahgah]

Thinking about it mathematically I wonder if it is possible to consider that Einstein added a second time dimension when he developed his relativity?
Things no longer move with straight forward relative movement but now can move through 'time' at different speeds due to this added mathematical 'dimension'.
Is it possible to think of it this way?

 

[phinds]

Thinking about it mathematically I wonder if it is possible to consider that Einstein added a second time dimension when he developed his relativity?
Things no longer move with straight forward relative movement but now can move through 'time' at different speeds due to this added mathematical 'dimension'.
Is it possible to think of it this way?

A SECOND time dimension? I don't see how you get that, and it certainly is not how anyone else that I'm aware of views it.

 

[Drakkith]

Thinking about it mathematically I wonder if it is possible to consider that Einstein added a second time dimension when he developed his relativity?
Things no longer move with straight forward relative movement but now can move through 'time' at different speeds due to this added mathematical 'dimension'.
Is it possible to think of it this way?

One can move at different speeds along any single dimension in space too, but that does not require us adding an extra dimension for each existing one.

 

[gonegahgah]

Well, GR and SR actually could be considered to add two extra dimensions to equations couldn't they? I'm not talking about physical dimensions; though science does treat physical aspects mathematically. I'm talking about adding an extra dimension to the classical models which treated things in a direct fashion. They certainly add some extra dimensions to how we think about simple movement.
I'm not saying that anyone has thought of it that way. I'm saying it could be thought of in that way.
I can't see that having two time dimensions means that you are in one or the other.
To me a second (or more dimension) simply affects how you would be thought to move through the spatial dimensions. And GR and SR certainly affect how we animate through the spatial dimensions.
We don't have a fixed x dimension nor y dimension nor z dimension. They are all mutable.
But mathematically we can describe them that way.
In the same mathematical sense I can't see why we can't say that GR & SR add an extra dimension each to our equations?

One can move at different speeds along any single dimension in space too, but that does not require us adding an extra dimension for each existing one.

To move at different speeds along a direction in space you need to be acted upon which I'm wondering if this reaction also adds a dimension itself to our equations as well?

A SECOND time dimension? I don't see how you get that, and it certainly is not how anyone else that I'm aware of views it.

New ways of thinking are not a new thing in science as long as they fit within their logical context...
Take our current debate over big bang, crunch, bounce, expansion, string theory, etc.All I'm saying is that they can be thought to add extra dimensions to our formula that we didn't think were there before Einstein.

 

[Drakkith]

Gonegahgah, your extra dimensions don't fit the same definition as the classic 4 we talk about. Inventing new definitions won't get us anywhere.

 

[gonegahgah]

Gonegahgah, your extra dimensions don't fit the same definition as the classic 4 we talk about. Inventing new definitions won't get us anywhere.

Which is exactly what string theory is attempting to do?
And again as I say our cosmologists are coming up with new definitions of our universe.
Blocking all new definitions isn't generally the way forward in physics. Resistance is certainly fine.

The question then comes back to whether what I'm saying can have it's own context.
And then the next question is can that context be of any use...

Even if it ends up not being useful to anyone it still may have validity within its own framework.
Again, look at string theory.

 

[Drakkith]

Which is exactly what string theory is attempting to do?

The dimensions in string theory are all defined within that theory to work together and make sense. They aren't exactly the same, just as our space and time dimensions aren't, but they all fit together.

And again as I say our cosmologists are coming up with new definitions of our universe.
Blocking all new definitions isn't generally the way forward in physics. Resistance is certainly fine.

I have no problems with new definitions. If you develop a full fledged theory of spacetime and gravity and all that, feel free to create as many dimensions as you want to make it work. Until then, please stick to current mainstream theories and their definitions, otherwise the conversations on the forum break down as people won't be able to follow them and they stop making sense.

 

[Nabeshin]

Clarification for gonegahgah:
In GR, we are certainly not modifying anything about the dimensionality of spacetime. In normal special relativity, we have the invariant interval given by
ds^2=\eta_{\mu \nu} dx^\mu dx^\nu = -dt^2+dx^2+dy^2+dz^2
That's really all there is to the dimensionality story here, Einstein didn't do anything special on that front.

All that happens when we move to GR is that in general we promote \eta to a metric tensor which is in general a function of the spacetime coordinates
ds^2=g_{\mu \nu} (x ^\alpha) dx^\mu dx^\nu
But the tensor is still of dimension four, we really haven't altered the dimensionality at all.

In short, no it is not possible that we've somehow added an extra dimension. Keep in mind that the coordinate vectors we choose to represent our dimensions have to be linearly independent. So it's not as through defining a new vector in normal 4-space promotes us into the fifth dimension, since such a vector is spanned by the existing space.

With re: to the rest of what you've written trying to explain why this makes sense to you, I cannot make heads or tails of it, but I hope my explanation has clarified a bit.

Also, as a general note, it doesn't make any sense to lean on string theory when trying to state that currently established theories somehow add extra dimensions nobody else seems to be aware of. String Theory is an untested proposal, while something like relativity is (as close as can be) scientific fact.

 

[andrewmh]

and for the less educated people like myself, this makes sense using string theory lol http://www.tenthdimension.com/medialinks.php

i understand that string theory is, as stated above, an "untested proposal" but when people start talking about "dimensions" i find that this video is an easy way out

 

[Nabeshin]

and for the less educated people like myself, this makes sense using string theory lol http://www.tenthdimension.com/medialinks.php

Just to clarify, I've seen this video before and while it's an interesting exercise, everything he talks about dimensions above the usual four dimensions is unrelated to string theory. Specifically, you can define these extra 'dimensions' in any way you like, but they're unrelated to any 'dimensions' we are familiar with in the natural sense. So really it's a purely MATHEMATICAL exercise, and his attempt to link it to string theory at the end is very flawed, for it gives the impression to the viewer that this is a PHYSICAL rather than mathematical conclusion.

 

[Max™]

On one hand I'd hate to scare someone off by suggesting they read up on tensors, on the other hand it seems like I'd be doing a disservice to not offer a direction which provides unique insight into the mathematical structure of spacetime.

If nothing else, look through the http://en.wikipedia.org/wiki/Tensor page, ask questions based on what you do and don't get there.

It might help make sense of some terms that get bandied about which are key parts of relativity, and help explain the difference between say, a tensor with certain dimensions, and a space with certain dimensions.

 

[BadBrain]

The number of variables required to completely specify the state of your system is the number of dimensions of the system.

It takes 3 numbers to specify the location of a object in space, therefore we speak of 3 spatial dimensions. If you throw in time you get a 4th dimension.

If you are tracking the fluid level of 100 different tanks you have a system with 100 dimensions.

I absolutely disagree.

The fluid level of 100 different tanks is a function of one dimension (i.e., height) in a universe which contains that dimension, as well as others.

Each tank is not a universe unto itself.

The level of fluid contained within the 100 tanks is a function of the single dimension of height. The volume of fluid contained within a tank is a function of three dimensions: height, depth, and breadth. Add the fluctuation in the fluid levels within the various tanks over time and you have the three spatial dimensions plus the only observed temporal dimension.

I don't know exactly what dimensions are, but I know some of their properties. Dimensions within our universe are the framework which contain eletroweakness and gravitation. They constitute the box which contains stuff. That the box is independent of the stuff it contains is obvious from the fact that space expands faster than stuff (i.e., faster than the speed of light in a vacuum, as per last year's Nobel prize winners), and the fact that gravitation, although motivated by mass, only acts upon dimensions by warping dimensions, as per general relativity. Thus gravitational lensing, in which massless photons, themselves immune to the immediate effects of gravitation, nevertheless travel circuitous paths along spatial dimensions bent by the gravitational force motivated by objects which possesses mass.

Dark Matter and Dark Energy may well be enclosed within alternative spatial dimensions or even alternative temporal dimensions, or even within dimensions which are neither spatial nor temporal within our understanding.

Anyways, here's a fun video showing the difference between the box and the stuff within, in that the unhappy children receive beautifully wrapped boxes which contain nothing except dimensions and air. (It's actually a commentary upon the politics of a certain province within a certain country very near where I live, hut it also just so happens to illustrate my physics point, even as it shows off a couple of lovely young ladies (Hey! I'm a bachelor, so I get to say that!))

http://www.youtube.com/watch?v=k1WQ-srZX8k

So, that's my point. Dimensions are neither matter nor energy, but they contain matter and energy, and gravitation acts upon them.

Acceleration is not a dimension, but is simply a function of the three observed spatial dimensions plus the single observed temporal dimension, and requires the positing of no additional dimensions to explain it. Newton described acceleration quite well without the slightest knowledge of strings.

 

[Max™]

In order to describe the level of fluid in 100 different tanks you need 100 points of data, thus that system has 100 dimensions.
Doesn't matter if you agree.

 

[BadBrain]

In order to describe the level of fluid in 100 different tanks you need 100 points of data, thus that system has 100 dimensions.



Doesn't matter if you agree.

100 points of data along the same single dimension of height do not make 100 dimensions, they make 100 different values for the single dimension of height which our universe contains.

If every difference in values of dimensions constitutes a new dimension, then there are no dimensions at all, just a universe of values lacking a framework of reference.

 

[Drakkith]

100 points of data along the same single dimension of height do not make 100 dimensions, they make 100 different values for the single dimension of height which our universe contains.

If every difference in values of dimensions constitutes a new dimension, then there are no dimensions at all, just a universe of values lacking a framework of reference.

No, in math we can describe a system using numbers. The more numbers you have the more "dimensions" you have. 100 tanks will require 100 different numbers to describe the heights. This is purely a mathematical dimension though, not a "real" one.

 

[BadBrain]

No, in math we can describe a system using numbers. The more numbers you have the more "dimensions" you have. 100 tanks will require 100 different numbers to describe the heights. This is purely a mathematical dimension though, not a "real" one.

Physics is NOT math!

Physics IS reality!

Besides, dimensions in math, per my understanding, constitute values which are to be multiplied against one another to describe the system (i.e., zero power (number 1, point in space/time), first power (one dimension, length of a line connecting two points (as in the fluid level of the contents of a tank), second power (two dimensions; multiple lines within a plane or a single-dimensional line whose length varies over time), third power (height, depth, breadth, or height and breadth changing shape over time, as in a television cartoon, or height or breadth and depth changing shape over time, which would make for a very boring cartoon much of whose motion would be invisible to you), fourth power (reality: the three observed spatial dimensions plus the single observed temporal dimension), whereas the differing levels of fluid within 100 tanks is purely an additive process, not involving the exponential mathematics which describe true dimensions.

By the way, gasses would tend to expand to fill any space which they occupy (losing heat in the process), so what is really meant in this case by "fluid levels" actually means "liquid levels", unless you're talking about some super-heavy gas lying on top of a much lighter gas.

***

Look, I know that I'm going to be banned from this forum for having dared to challenge a Mentor via my intervention in this thread, but I"m just doing what I'm doing in the hope that I might get through to someone who might cary on along the lines I've here proposed.Sheesh!

 

[Drakkith]

Physics is NOT math!

Physics IS reality!

Incorrect. A general definition of physics is: Physics is the general analysis of nature, conducted in order to understand how the universe behaves.

Physics is a branch of science, which has many different subfields and has many many tools to use to analyze the universe and try to accurately describe it. One of these tools is mathematics. Math has very specific rules and definitions that are required to make it function correctly. The dimensions of the water level in the 100 tanks is purely a mathematical way of representing them. It only corresponds to reality in that the numbers each represent the measured value of the height of the water in each tank in our 3 spatial dimensions. If I were to take more measurements later on and compare them, I could then also use time.

Besides, dimensions in math, per my understanding, constitute values which are to be multiplied against one another to describe the system (i.e., zero power (number 1, point in space/time), first power (one dimension, length of a line connecting two points (as in the fluid level the contents of a tank), second power (two dimensions; multiple lines within a plane or a single-dimensional line whose length varies over time), third power (height, depth, breadth, or height and breadth changing shape over time, as in a television cartoon, or height or breadth and depth changing shape over time, which would make for a very boring cartoon much of whose motion would be invisible to you), fourth power (reality: the three observed spatial dimensions plus the single observed temporal dimension), whereas the differing levels of fluid within 100 tans is purely an additive process, not involving the exponential mathematics which describe true dimensions..

Yes, this is pretty much how the 4 dimensions of spacetime are explained.

Look, I know that I'm going to be banned from this forum for having dared to challenge a Mentor via my intervention in this thread, but I"m just doing what I'm doing in the hope that I might get through to someone who might cary on along the lines I've here proposed.


Sheesh!

There's nothing wrong with helping someone understand, but you need to be absolutely sure that what someone said was actually incorrect before calling them out on it. If you are unsure simply ask instead.

 

[Nabeshin]

Physics is NOT math!

Physics IS reality!
[...]

I don't know what your physics background is, but perhaps you've heard of phase space? If not, read up the wiki article. This is the general kind of dimension we're talking about, and the one that applies to a system with 100 different tanks with 100 (independent) levels.

 

[Max™]

Physics is NOT math!

Physics IS reality!

This reminds me of a Moral Orel quote.

"Well, Orel, drunk... IS... nature!"

Physics is the application of mathematical structures in an attempt to model observables.

Besides, dimensions in math, per my understanding, constitute values which are to be multiplied against one another to describe the system (i.e., zero power (number 1, point in space/time), first power (one dimension, length of a line connecting two points (as in the fluid level of the contents of a tank), second power (two dimensions; multiple lines within a plane or a single-dimensional line whose length varies over time), third power (height, depth, breadth, or height and breadth changing shape over time, as in a television cartoon, or height or breadth and depth changing shape over time, which would make for a very boring cartoon much of whose motion would be invisible to you), fourth power (reality: the three observed spatial dimensions plus the single observed temporal dimension), whereas the differing levels of fluid within 100 tanks is purely an additive process, not involving the exponential mathematics which describe true dimensions.

Dimensions don't have to be multiplied against each other.

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

A dimension is simply a value which can be measured independently from other values.

The level of one tank which is not connected to any other tanks is an independent value, that value ranges from some level to some other level, and that constitutes a dimension within the system of fluid filled tanks.

 

[BadBrain]

I don't know what your physics background is, but perhaps you've heard of phase space? If not, read up the wiki article. This is the general kind of dimension we're talking about, and the one that applies to a system with 100 different tanks with 100 (independent) levels.

Yes, I've heard of Phase Space, and I've applied the concept, and it requires no more dimensions than the three observed spatial and the one observed temporal to describe the energy state of any particle within a system.

Each particle with a given energy state within phase space is not a dimension; it is a particle moving within our observed dimensions, and, possibly (but not necessarily) others.

The Hamiltonians, although they foreshadow Quantum Mechanics (and I frankly think that Hamilton suspected that there was something else out there when he arrived at his equations), within themselves, require no dimensions beyond those of which he was aware.

If each particle is a dimension, then how can space be expanding faster than the maximum speed of the fastest known particle? How can gravitation act to warp dimensions only in response to mass, despite the fact that not all particles in our universe possesses mass?

 

[BadBrain]

One more question:

Why are you working so hard to pound me down when you know I'm soon to be permanently banned from this forum for my having challenged a Mentor?

 

[Max™]

Why do you think challenging a mentor gets you banned?

I haven't seen that rule anywhere, as far as I can tell they've just got moderator privileges on the board software, they're not sanctified or anything. Don't be a jerk and you probably won't get banned.Note, a particle can not be completely described with 3+1 dimensions.

You can say where it is, where it is going, but not what kind of particle it is, it's mass, it's spin, it's charge, all are dimensions in themselves.

 

[BadBrain]

Why do you think challenging a mentor gets you banned?

I haven't seen that rule anywhere, as far as I can tell they've just got moderator privileges on the board software, they're not sanctified or anything. Don't be a jerk and you probably won't get banned.

Note, a particle can not be completely described with 3+1 dimensions.

You can say where it is, where it is going, but not what kind of particle it is, it's mass, it's spin, it's charge, all are dimensions in themselves.

Aaauw, man!

Why don't you read what I wrote?

What I write above was: "Each particle with a given energy state within phase space is not a dimension; it is a particle moving within our observed dimensions, and, possibly (but not necessarily) others."

Jeeze, you're pounding me now to the point of attributing arguments to me which I did not make.

And, I probably won't get banned? I challenged a Mentor, so bye bye Charley!

 

[Neandethal00]

I wanted to know what is the definition and what actually is a dimension. I mean I get that length, width and height story, but how does time fit in as a dimension. Then what about higher dimensions. Like if a certain system was defined in 7 dimensions what exactly would those dimensions be? 4 could be time, length width and height but what about the other 3?

When everyone is explaining dimension using Math, particles, motion. etc, let's try another way to interpret dimension.

Dimensions prove existence. Zero dimension means true nothingness.
If anything exists, it has a dimension and vice versa.
Interesting, this also proves empty space is not nothingness as some people think.

 

[BadBrain]

OK, OK, I give up!

Each particle, together with its motion, is a dimension unto itself.

That means that, as gravitation affects the paths of photons, that photons are, in fact, massive particles, capable of response to immediate influence from gravitation.

That, in turn, means that, as each particle in the universe is a dimension unto itself, that the expansion of the observed spaciotemporal dimensions of the universe proceeds at precisely the speed of light in a vacuum.

And, in particular, that means that General Relativity is **** all.

***

Maybe you're right. Maybe I won't get banned after all, now that I've confessed myself back into the mould.

 

[Nabeshin]

Hamiltonians, and in general Hamilton's equations, make no reference to dimensionality at all. You can apply them to an arbitrary system.

What we're trying to say is as simple as this: When calculating integrals in a phase space, for example of one particle with momentum, you integrate over three spatial coordinates and three momenta to perform the integral over all of phase space. This is a six-dimensional integral, everyone calls it that, and these are the 'Dimensions' that we're familiar with. They might not share the same status as the 3+1 that we know about, but they certainly serve a similar mathematical function, and that's why we use the same word for both.

Your final comment about space expanding and gravitation makes little sense, everyone (who understands GR) knows that spacetime is warped by stress/energy/momentum, not simply mass. And the phrase 'each particle is a dimension' was never uttered anywhere except for in your post, and I can not make anything of it.

Finally, indeed challenging a mentor is not grounds for immediate banning, as you seem to think. Being obnoxious and adversarial, however, is not going to contribute any brownie points. I suggest you spend less time complaining about your imminent ban and more time trying to understand.

 

[Max™]

^Trufax.

 

[BadBrain]

Hamiltonians, and in general Hamilton's equations, make no reference to dimensionality at all. You can apply them to an arbitrary system.

What we're trying to say is as simple as this: When calculating integrals in a phase space, for example of one particle with momentum, you integrate over three spatial coordinates and three momenta to perform the integral over all of phase space. This is a six-dimensional integral, everyone calls it that, and these are the 'Dimensions' that we're familiar with. They might not share the same status as the 3+1 that we know about, but they certainly serve a similar mathematical function, and that's why we use the same word for both.

Your final comment about space expanding and gravitation makes little sense, everyone (who understands GR) knows that spacetime is warped by stress/energy/momentum, not simply mass. And the phrase 'each particle is a dimension' was never uttered anywhere except for in your post, and I can not make anything of it.

Finally, indeed challenging a mentor is not grounds for immediate banning, as you seem to think. Being obnoxious and adversarial, however, is not going to contribute any brownie points. I suggest you spend less time complaining about your imminent ban and more time trying to understand.

What I'm saying is that Hamilton was definitely ahead of his time, and that he suspected that something else was out there beyond the physics of his own understanding.

The three spatial dimensions are obvious. The three moments I see as functions of the three spatial dimensions with respect to time. In other words, his calculations respected time as a dimension which interacts with space, even though he never phrased it in that manner. Moments necessarily imply motion, and, in the absence of time, there is no motion, only a dead stasis which makes nonsense of the Hamiltonians, as kinetic energy is impossible and potential energy is irrelevant. He was, in fact, describing space-time eighty years before Einstein, having access to more advanced knowledge than that available to Hamilton, formalized the concept.