Evgenij Barsoukov
http://sudy_zhenja.tripod.com
>I believe it is important to resolve
>paradoxes and eliminate singularities
>or infinities, such as the infinite vacuum
>energy required by the theory of virtual
>particles.
It is quite interesting that you equal "infinity" and "paradox".
Did it ever occure to you that "realy existing" infinity can be as real as any
other hypothesis?
Why whould you reject it from the begining - just because it is not part of everyday experience? - but then, Heisenberg principle and QM formalizm fundamentaly contradicts everyday experience too, and so does relativity. It would be only consequent that a theory which results in conterintuitive formalizm (which however correctly described experimental data) is based on conterintuitive basic axiomes, such as nothingness which posesses infinite energy.
Even more, for the teaching purposes and general piece of mind of would be instructive to derive all QM from this singularity - nothingness/infinity, and not another way around. Interestingly, I found an interesting mathematical paradox related to this question:
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Imagine absolute nothingness - no time, no matter, nothing at all.
Concept of causality requires some cause for something to happen, therefore it can be stated that probability of an event is proportional to number of possible causes which exist. Now, in nothingness there is no causes, so probability of single event is 0. But that is of single event at one try. What about probability that event _ever_ happens? There is a problem here, because we have nothing at all (so also no time) - no restrictions of number of tries. Practically the problem can be mathematically formulated as such:
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what is the probablity that an event ever happens if number of tries approaches infinity if probability at single try is approaching 0?
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First, how to calculate the probability P that an event with "one-try probability" p will happen during N tries _at least once_? P is quite easy to derive from the theorem of multiplication of independent probabilities. I give only the answer:
P = 1-(1-p)^N
It can be seen that at any finite p<1 with increasing N P will be approaching 1, which means "certainty". However, what will happen, if p is "infinitely small" or in the limit = 0, but the number of tries in also infinite? Logic says it will give 0. But who needs logic here, let the mathematic speak!
It is known that infinity can be described as 1/0, so we considering N=1/p (here p is approaching 0) we can write equation for P as one limit
Pinf= Lim {p->0} (1-(1-p)^(1/p))
What is the answer? Using rules of finding limits, we find the answer, but it is not zero!
Pinf = 1-exp(-1)=0.6....
We find a magic number! Probability in not only not 0, but actually more then 1/2!
While all this considerations can be seen as a joke (through mathematically correct) I see more philosophical meaning to it. It is a proof that given infinite time (or number of tries), _something_ can and must arise out of _nothing_. It is a solution for the paradox of causality, which usually was resolved by notion of God as a first-cause!
Regards,
Evgenij