Is Space-Time Non-Commutative? Understanding Non-Commutative Geometry in Physics

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In summary, non-commutative geometry is a concept where distance is dependent on the path taken and may differ depending on the direction. This idea is often used in physics, but can also be seen in everyday examples such as navigating through a city with one-way streets. However, understanding this concept at a fundamental level can be challenging.
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black hole 123
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ive been reading about people thinking sub Planck scale spacetime is "non commutative". i have a vague idea of non commutative geometry mathematically, but there's no "space" in non commutative geometry so how can SPACEtime be a non commutative ring?

i thought non commutative geometry was just a mathematical generalization of normal (commutative) algebraic geometry. I am not a physics students so I am not in a position to judge, but i find many things in math applied to physics with absolutely NO physical manifestation/interpretation

sorry if this sounds stupid
 
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Non-commutative geometry, in a nutshell, means that distance is a path dependent quantity and that for example, the distance from X to Y via a path, and the trip back may have different distances.

A non-fundamental example would be the geography of getting from one place to another in a city with lots of one way streets and irregular loops (e.g. Boston). The distance from your house to your office may be different from the distance from your office to your house.

Obviously, when this is operating at a fundamental level as a basic element of the structure of space-time, getting your head around the idea is much harder. But, this is the idea at its most basis level.
 

FAQ: Is Space-Time Non-Commutative? Understanding Non-Commutative Geometry in Physics

What is non-commutative geometry?

Non-commutative geometry is a branch of mathematics that studies spaces that do not satisfy the commutative property, meaning that the order in which operations are performed affects the outcome. It is a generalization of classical geometry that includes systems such as quantum mechanics and string theory.

What are some applications of non-commutative geometry?

Non-commutative geometry has applications in various fields such as theoretical physics, differential equations, and number theory. It has been used to study the behavior of particles in quantum mechanics, to solve problems in algebraic geometry, and to develop new approaches to understanding the structure of space and time.

How is non-commutative geometry different from classical geometry?

In classical geometry, the commutative property holds and the order of operations does not affect the outcome. Non-commutative geometry, on the other hand, considers spaces where the commutative property does not hold and the order of operations matters. This allows for a more flexible and generalized approach to studying geometric systems.

What are some challenges in studying non-commutative geometry?

One of the main challenges in non-commutative geometry is developing tools and techniques to study and understand these non-commutative spaces. This involves developing new mathematical structures and methods, as well as finding connections to other fields of mathematics and physics.

What are some open questions in non-commutative geometry?

Some open questions in non-commutative geometry include finding a unified framework for studying non-commutative spaces, developing a better understanding of the relationship between non-commutative and classical geometry, and exploring the applications of non-commutative geometry in various fields.

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