Applications of measurement of commutativity

• Oct 27, 2022

• #1

askmathquestions

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Dang this place has a topology and analysis section too, nice.

This is probably a graduate level topic, but I am by no means an expert on these subjects, just things I learn from wikipedia and other people. The commutator is an operation on two linear operators (most often matrices) of the form ##[A,B] = AB - BA.## This is often touted as a measurement of how "badly" two matrices fail to commute, but I don't think it quite is.

I'd like to research what happens when we induce a matrix norm to the commutator, when the matrix norm of the commutator ##|| [A,B] ||^2## actually returns a specific number, I think that's more of a "measurement" of the commutativity of two matrices.

However, why bother? Are there any theoretical or scientific applications for measuring the size of matrix commutators? Possibly in quantum physics, though I don't know that subject in depth, I hope there are more applications than that.

 

Last edited: Oct 27, 2022

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• Oct 27, 2022

• #2

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You have to consider ##A## and ##B## as vector fields. ##[A,B]## measures the gap you get if you flow along ##A## and then ##B##, or the other way around. It doesn't measure how badly matrices commute, it measures how commutative flows along vector fields are.

 

• Oct 28, 2022

• #3

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Okay, that's interesting. What about measuring the commutativity of matrices instead of vectors though? I haven't seen the commutator in the case that ##A## and ##B## are vector fields, but rather matrices.

 

• Oct 28, 2022

• #4

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askmathquestions said:

Okay, that's interesting. What about measuring the commutativity of matrices instead of vectors though? I haven't seen the commutator in the case that ##A## and ##B## are vector fields, but rather matrices.

Yes, but I explained where the comparison comes from, i.e. why people say they measure commutativity.

It is a heuristic rather than a precise definition. It needs a curved space, since ##[A,B]=0## in a flat world. In a flat world, it doesn't matter whether you go left or right in a parallelogram. In a curved world, it does matter.

You have to define a scale before you make a proper definition of what "distance from commutativity" means.

 

• Oct 28, 2022

• #5

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I've never studied differential geometry, but it sounds like this has some kind of fundamental relationship to trajectories and derivatives on various manifolds. Would you be able to illuminate that connection?

 

• Oct 28, 2022

• #6

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askmathquestions said:

I've never studied differential geometry, but it sounds like this has some kind of fundamental relationship to trajectories and derivatives on various manifolds. Would you be able to illuminate that connection?

https://www.physicsforums.com/insig...t-iv/#B-Left-Invariant-Vector-Fields-and-GL-n

 

What is commutativity?

Commutativity is a mathematical property that refers to the ability to change the order of operations without changing the result. In other words, it means that the order in which numbers are added or multiplied does not affect the final outcome.

Why is commutativity important in measurement?

Commutativity is important in measurement because it allows us to simplify calculations and make them more efficient. By being able to change the order of operations, we can avoid unnecessary steps and reduce the chances of making errors in our measurements.

What are some common applications of measuring commutativity?

Some common applications of measuring commutativity include basic arithmetic operations such as addition, subtraction, multiplication, and division. It is also used in more advanced mathematical concepts such as matrix operations, group theory, and abstract algebra.

How is commutativity measured?

Commutativity is typically measured using mathematical equations and proofs. In order to determine if a set of numbers or operations is commutative, scientists and mathematicians use various methods such as substitution, algebraic manipulation, and logical reasoning.

What are the benefits of understanding the concept of commutativity?

Understanding commutativity can help scientists and mathematicians solve problems more efficiently and accurately. It also allows for a deeper understanding of mathematical concepts and can lead to the discovery of new relationships and patterns. Additionally, commutativity is a fundamental concept in many fields of science, making it an essential tool for scientific research and innovation.