I wonder how to apply the uncertinty principle to the big bang?
I don't see that there is any relevance. Why do you think there is?
It's an interesting question. There might be relevance at the intuitive level. You know how intuitively the principle of uncertainty in a way epitomizes the quantum nature of matter---that you ultimately cannot "pin nature down".
John Archibald Wheeler (Feynman's mentor at Princeton) had the idea that it you could quantize nature's geometry it would resolve the big bang singularity. (Also the black hole collapse singularity would be resolved, he had the idea of a "bounce" due to quantum effects.)
However Wheeler's attempt to quantize the equation of the big bang failed to resolve the singularity. The Wheeler-DeWitt equation. It was, I guess, mankind's first QG equation. (quantum geometry/gravity, same thing). It was hoped to resolve the bang and hole singularities but did not.
Lee Smolin did a postdoc at Princeton, where Wheeler was, and as I recall he said got the idea of a QG bounce from Wheeler. A black hole collapse might bounce and result in a new expanding region of spacetime. Our big bang might be the result of a prior collapse. Intuitively because nature resists final absolute definition.
Now of course Wheeler is gone but his program has been pursued along many lines. There are several climbing parties trying to get up the QG mountain. A "big bounce" resolution of the initial singularity has been derived in several different ways, including in a string context. And that is just one way that a theory of QG might resolve the classical singularities of GR.
Someone has written a review paper of all the different "big bounce" cosmologies people have worked on. I'll hunt it down if anyone is interested. The best-known is probably the Loop cosmology one, it has had the most work done on it and has been developed in considerable detail. Computer modeling, equation models, many cases considered including with positive cosmological constant.
A recent paper derives "deSitter space" (a nonsingular bouncing solution to Gen Rel), but that is just one case of many that have been studied.
If curious, google "bianchi cosmic constant spinfoam" and get http://arxiv.org/abs/1101.4049
Cosmological constant in spinfoam cosmology
Eugenio Bianchi, Thomas Krajewski, Carlo Rovelli, Francesca Vidotto
(Submitted on 20 Jan 2011)
We consider a simple modification of the amplitude defining the dynamics of loop quantum gravity, corresponding to the introduction of the cosmological constant, and possibly related to the SL(2,C)q extension of the theory recently considered by Fairbairn-Meusburger and Han. We show that in the context of spinfoam cosmology, this modification yields the de Sitter cosmological solution.
[ Google is smart enough to search "cosmological" if you say "cosmic", so if you feel lazy and want to save typing extra letters you can tell it "cosmic constant" even though the officially correct term is cosmological constant.]
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