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the Bohr compactification of the real line
RBohr
is essential (it seems) to Loop Quantum Cosmology.
Here are some links illustrating that:
https://www.physicsforums.com/showthread.php?s=&postid=147432#post147432
I want to understand the Bohr compactification better.
It is putting a different topology on the Reals
(or the reals embedded as a dense subset of a larger
collection of numbers) so that they become compact.
But it is not the "one point" compactification which
is a widely known trick of adjoining a "point at infinity".
It is a different and probably more cool compactification.
The younger brother of Niels thought of it. His name was Harald.
Well, why shouldn't we just keep on using the line of real numbers
that we know and love, but have a special topology handy for
use on Sundays and other special occasions that makes them compact.
Sounds OK to me.
To understand the construction we probably need to know the idea of the "dual group" or (a droll synonym) the "group of characters" of a commutative group G.
The characters of a commut. topol. group G are just the continuous homomorphisms from G to the Unit Circle---the complex numbers with |z|=1, under multiplication.
[tex]\chi : G \longrightarrow S^1[/tex]
[tex]\chi(gh) = \chi(g)\chi(h)[/tex]
a familiar example would be χ(x) = exp(ix)
notice (please) that the characters form a group themselves
you can multiply two of them and it is still continuous
and it still satisfies that multiplicativity condition
I'll post this and get back to it later.
Oh, the bohr cmptfn of the Reals is the dual group of the Reals equipped with a wacko topology called the "discrete" topology in which the real line is totally atomized as if by a giant sneeze and no point is near any other point.
Viqar Husain and Oliver Winkler
"On singularity resolution in quantum gravity"
http://arxiv.org/gr-qc/0312094
Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski
"Mathematical Structure of Loop Quantum Cosmology"
http://arxiv.org/gr-qc/0304074
I see the Ashtekar/Bojowald/Lewandowski paper came out
in the 2003 edition of "Advances in Theoretical and Mathematical Physics" 7, 233-268
RBohr
is essential (it seems) to Loop Quantum Cosmology.
Here are some links illustrating that:
https://www.physicsforums.com/showthread.php?s=&postid=147432#post147432
I want to understand the Bohr compactification better.
It is putting a different topology on the Reals
(or the reals embedded as a dense subset of a larger
collection of numbers) so that they become compact.
But it is not the "one point" compactification which
is a widely known trick of adjoining a "point at infinity".
It is a different and probably more cool compactification.
The younger brother of Niels thought of it. His name was Harald.
Well, why shouldn't we just keep on using the line of real numbers
that we know and love, but have a special topology handy for
use on Sundays and other special occasions that makes them compact.
Sounds OK to me.
To understand the construction we probably need to know the idea of the "dual group" or (a droll synonym) the "group of characters" of a commutative group G.
The characters of a commut. topol. group G are just the continuous homomorphisms from G to the Unit Circle---the complex numbers with |z|=1, under multiplication.
[tex]\chi : G \longrightarrow S^1[/tex]
[tex]\chi(gh) = \chi(g)\chi(h)[/tex]
a familiar example would be χ(x) = exp(ix)
notice (please) that the characters form a group themselves
you can multiply two of them and it is still continuous
and it still satisfies that multiplicativity condition
I'll post this and get back to it later.
Oh, the bohr cmptfn of the Reals is the dual group of the Reals equipped with a wacko topology called the "discrete" topology in which the real line is totally atomized as if by a giant sneeze and no point is near any other point.
Viqar Husain and Oliver Winkler
"On singularity resolution in quantum gravity"
http://arxiv.org/gr-qc/0312094
Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski
"Mathematical Structure of Loop Quantum Cosmology"
http://arxiv.org/gr-qc/0304074
I see the Ashtekar/Bojowald/Lewandowski paper came out
in the 2003 edition of "Advances in Theoretical and Mathematical Physics" 7, 233-268
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