# Background independent framework

What are the basics of non commutative geometry and where is a good place to learn more about it?

It is geometry with non-commutative elements involved.

It is geometry with non-commutative elements involved.

Okay do you have any recommendations for good sites on Google that explain it well. I know it involves non communative elements from the title, but why does it include non commutative elements?

wabbit
Gold Member
A starting point is the wikipedia article on noncommutative geommetry. Not that easy a read though, maybe there's something better out there.

ohwilleke
Gold Member
The basic concept of non-commutative geometry is that important aspects of it are path dependent. The shortest path from point A to point B may not necessarily be the shortest path from point B to point A.

While this seems odd and obscure, it isn't hard to imagine every day systems that display that property. For example, if you are in a city with a mix of one way streets, the fastest path from my house to yours may be different from the fastest path from your house to mine.

Non-commutative geometry matters in physics among other reasons, because due to special and general relativity, the time that elapses along a path from point A to point B is observer dependent (due to velocity) and path dependent (due to gravity which also impacts the passage of time), which makes many of the assumptions of non-relativistic Euclidian space-time invalid. Also note that since quantum mechanics includes special (but not general) relativity, non-commutative geometry matters for both the Standard Model and GR, the two most fundamental theories in physics.

Alas, I don't have any better references for you than have been mentioned above.

The basic concept of non-commutative geometry is that important aspects of it are path dependent. The shortest path from point A to point B may not necessarily be the shortest path from point B to point A.

While this seems odd and obscure, it isn't hard to imagine every day systems that display that property. For example, if you are in a city with a mix of one way streets, the fastest path from my house to yours may be different from the fastest path from your house to mine.

Non-commutative geometry matters in physics among other reasons, because due to special and general relativity, the time that elapses along a path from point A to point B is observer dependent (due to velocity) and path dependent (due to gravity which also impacts the passage of time), which makes many of the assumptions of non-relativistic Euclidian space-time invalid. Also note that since quantum mechanics includes special (but not general) relativity, non-commutative geometry matters for both the Standard Model and GR, the two most fundamental theories in physics.

Alas, I don't have any better references for you than have been mentioned above.
THANK YOU! That was really helpful.

wabbit
Gold Member
The basic concept of non-commutative geometry is that important aspects of it are path dependent. The shortest path from point A to point B may not necessarily be the shortest path from point B to point A.

While this seems odd and obscure, it isn't hard to imagine every day systems that display that property. For example, if you are in a city with a mix of one way streets, the fastest path from my house to yours may be different from the fastest path from your house to mine..

This seems completely different from the meaning of "noncommutative geometry" I am (very vaguely) familiar with (as in Connes' NCG). Can you point to some reference ?

ohwilleke
Gold Member
This seems completely different from the meaning of "noncommutative geometry" I am (very vaguely) familiar with (as in Connes' NCG). Can you point to some reference ?

I will look for some references when I get a chance. My description parrots a couple of other descriptions that I've seen in print, but isn't the sort of thing I have well indexed. Obviously, the description I have provided is a heuristic one, rather than a technical mathematically rigorous one, that is focused on conveying the gist of what is really going on, rather than abstract algebra that one deals with mechanistically.

wabbit
Gold Member
OK never mind the references, could you elaborate on what you said ?
The basic concept of non-commutative geometry is that important aspects of it are path dependent. The shortest path from point A to point B may not necessarily be the shortest path from point B to point A.
I don't really understand NCG and while I vaguely get the connection with discreteness and with QM I wasn't aware that NCG involved a non-commutative distance, nor with the connection with classical SR and GR you also mention. How does this work ?

ohwilleke
Gold Member
OK never mind the references, could you elaborate on what you said ?

I don't really understand NCG and while I vaguely get the connection with discreteness and with QM I wasn't aware that NCG involved a non-commutative distance, nor with the connection with classical SR and GR you also mention. How does this work ?

I've made a couple of starts to answering you question (and there is one), but for some reason people expect me to work for \$ too, so I haven't gotten a good response together, but want you to know that I'm not blowing you off either.