The Commutator of Vector Fields: Explained & Examples

In summary, the commutator, denoted as [X,Y], is a mathematical operation that measures the difference between following two vector fields, X and Y, in different orders. It has both a computational and geometric interpretation, as shown in section 6.2 of Wald's book. If [X,Y]=0, the two vector fields commute and the resulting points will be the same.
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Zhang Bei
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Hi,

I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?

Thanks!
 
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  • #2
Zhang Bei said:
Hi,

I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?

Thanks!
Here is an easy example in section 6.2: https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

As a geometric intuition, you can imagine to start at a certain point ##p## and follow a flow along vector field ##X## for a small distance, from there along vector field ##Y## for a while and reach point ##s##. If you now start again at ##p## but follow first along ##Y## and then ##X##, you will usually end up at a different point ##t \neq s##. That difference is measured by ##[X,Y]=X \circ Y - Y \circ X##. If ##[X,Y]=0## then ##t=s##.
 
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1. What is a commutator of vector fields?

A commutator of vector fields is a mathematical operation that determines the extent to which two vector fields do not commute. In other words, it measures the difference between the order in which two vector fields are applied to a point in space. It is denoted by [X, Y] and is defined as the vector field [X, Y] = X(Y) - Y(X), where X and Y are two vector fields.

2. How is the commutator of vector fields calculated?

The commutator of vector fields is calculated by taking the difference between the two vector fields, where the first vector field is applied to the second vector field and the second vector field is applied to the first vector field. This calculation results in a new vector field, which is the commutator of the original two vector fields.

3. What is the significance of the commutator of vector fields in physics?

The commutator of vector fields plays a significant role in physics, particularly in the study of quantum mechanics. It is used to determine the uncertainty between two physical quantities, as the commutator of two operators represents the degree to which those operators do not commute. This is known as the Heisenberg uncertainty principle, which states that the more accurately one quantity is known, the less accurately the other can be known.

4. Can you provide an example of the commutator of vector fields?

Consider two vector fields X = x∂/∂x and Y = y∂/∂y, where ∂/∂x and ∂/∂y are partial derivative operators. The commutator of these two vector fields is given by [X, Y] = xy(∂/∂x - ∂/∂y). This example shows that the commutator of two vector fields is itself a vector field, and its value at any point (x,y) is determined by the difference between the values of the two vector fields at that point.

5. How is the commutator of vector fields related to Lie brackets?

The commutator of vector fields is closely related to Lie brackets, as both operations measure the extent to which two vector fields do not commute. In fact, the commutator of two vector fields is also known as the Lie bracket of those two vector fields. The Lie bracket is a fundamental concept in the field of Lie algebras and has many applications in mathematics and physics.

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