- #1
Topolfractal
- 76
- 7
What are the basics of non commutative geometry and where is a good place to learn more about it?
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?JorisL said:It is geometry with non-commutative elements involved.
And you can learn about it in your local maths department.
Also, try google once in while. It will lead you to more information than you can handle in a year
THANK YOU! That was really helpful.ohwilleke 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.
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.
ohwilleke 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.
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..
wabbit said: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 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 ?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.
wabbit said: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 ?
A background independent framework is a theoretical framework used in physics that does not rely on a specific background structure or reference frame. This means that the laws of physics are applicable in any type of space or environment, without the need for a fixed background or coordinate system.
In contrast to background dependent theories, a background independent framework does not require a pre-existing reference frame or background structure. It is also not tied to any specific geometric or mathematical structure, allowing for more flexibility in describing the dynamics of the universe.
Some examples of background independent frameworks include loop quantum gravity, causal set theory, and string field theory. These theories aim to unify the laws of quantum mechanics and general relativity, and do not rely on a fixed background structure.
One of the main implications of a background independent framework is that it allows for a more comprehensive understanding of the universe, as it does not rely on a fixed background or reference frame. It also has the potential to resolve some of the current issues in physics, such as the problem of quantum gravity.
Developing and testing a background independent framework is a complex and ongoing process. Some of the challenges include finding a consistent and mathematically rigorous framework, as well as developing experimental tests to validate the theory. Additionally, these theories often require high levels of mathematical and conceptual understanding, making them difficult to fully comprehend and test.