# Background independent framework

• Topolfractal
In summary, Non-commutative geometry involves geometry with non-commutative elements and is a path-dependent concept. It is commonly used in physics, particularly in the study of special and general relativity, to account for the effects of velocity and gravity on the passage of time. Learning more about it can be done through online resources such as Wikipedia and through local math departments.
Topolfractal
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.

JorisL said:
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?

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

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.

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.

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..

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 ?

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 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.

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 ?

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 ?

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.

## 1. What is a background independent framework?

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.

## 2. How does a background independent framework differ from other theoretical frameworks?

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.

## 3. What are some examples of background independent frameworks?

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.

## 4. What are the implications of a background independent framework?

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.

## 5. What are the challenges in developing and testing a background independent framework?

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.

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