Whatever you want!
Antonio Lao said:
Quantum mechanics, due to its probabilistic nature cannot truly be more than 3 dimensions (2 of space and 1 of time).
First, I will comment that you are confused and obviously do not understand quantum mechanics. Standard, old fashioned non-relativistic quantum mechanics, is a three dimensional Euclidian theory. In that presentation, time is an interaction parameter and, whenever an interaction yielding a known result occurs (under the standard old fashioned perspective); the wave function collapses everywhere "simultaneously", and one begins with a new wave function reflecting the results of that interaction.
Now, if the outcome of the interaction is not known, then the new state of the system is a linear sum of all the possible outcomes weighted by the probability of each. I won't go into relativistic quantum mechanics as the approach, which was introduced by Dirac, is not simple (I really wish he were still alive as I would like to talk to him); however it is consistent with Einstein's special relativity and it is an extension of the old fashioned non-relativistic interpretation with the exception that time is now considered a dimension. Thus it is a four dimensional theory, not a three dimensional theory.
The reason I would like to talk to Dirac (if he were alive) is that I have examined the consequences of old fashioned non-relativistic quantum mechanics under the assumption that reality is a four dimensional Euclidian space. I have found the results quite astounding in that, not only is special relativity a direct consequence of old fashioned quantum mechanics (Dirac's equation is a direct outcome), but the system also directly displays consequences normally attributed to general relativity.
But that is not why I posted here. I am afraid that I am the only person on Earth who understands exactly why our mental image of the universe is three dimensional. I have hoped for twenty years that someone (educated enough to follow the math) would find the idea that such an idea was interesting enough to look into that they would talk to me seriously. But, of course, I am a crackpot and such a thing will probably never happen.
The explanation of why we see the universe as three dimensional is actually very simple. As, I believe it was Hurkyl has said, analytically, anything explainable in three dimensions is explainable in one (given the right interpretation of the variables). In fact the number of dimensions used to explain anything is more a matter of convenience than it is of necessity.
The issue seriously is as follows:
I. A one dimensional universe displays very few easily expressible relations with complex outcomes. Let me put it this way; I can prove Newton's laws are a good approximation for the interactions of "objects" (collections of events which can be considered a stable unit). But, think about it, that is not a very valuable deduction in a one dimensional universe.
II. A two dimensional universe is somewhat better but is still quite limited. Again, think about the value of the knowledge that Newton's laws are a good approximation to the interactions of objects (as defined above). Particularly in view of the fact that, in order to keep the observer and his laboratory in close proximity, we must have a very weak interaction (weak enough that it does not disturb most of our experiments) pulling these object together and some structure preventing collapse to nothing (I am referring to the planet we stand upon). It follows that most everything essentially comes back to a one dimensional problem discussed above.
III. Ah! But a three dimensional universe who's behavior is constrained to obey Newton's laws is quite another matter! Now we have something complex enough to provide guidance on a great number of complex situations. That is, some very simple rules yield the consequences of quite a great number of situations.
Certainly level three is sufficient for anthropomorphic survival. There is thus no real pressing need for our subconscious to produce a higher dimensional image of our environment. This leads to a problem. Very few of us can mentally visualize what things would look like in a four dimensional universe (our subconscious just won't present the image for us). {I actually suspect the number of people who can actually visualize a four dimensional universe in all it's glory is zero!} However, if our understanding of mathematics is strong enough, we can lay out a mathematical representation of such a universe and visualize some interesting aspects with a little thought.
It turns out that once we can separate ourselves from that three dimensional constraint (through mathematics), we can vastly increase the number of complex situations the consequences of which can be predicted by simple rules. In my humble opinion, that is exactly the reason for the predictive power of string theory.
In fact, I have I have shifted my perspective to a completely alien view. In my view, dimensionality is a characteristic of the interactions we are interested in, not a characteristic of the universe. A three dimensional universe is fine so long as we are only interested in interactions which are expressible in three dimensions (which is most everything on an anthropomorphic level).
The most interesting view is a 3n+1 dimensional view where n is the number of fundamental entities in the universe. Believe it or not, that particular view is solvable in closed form (but it is a very complex solution and clearly not calculatable).
So, what is my position? My position is that the dimensionality of the universe is a consequence of your perspective.
Have fun -- Dick
