Regarding coordinate and non-coordinate bases

In summary, the conversation discusses the difference between coordinate and non-coordinate bases in GR. It is mentioned that partial derivatives of a coordinate system always commute, but this is not necessarily true for non-coordinate bases. The book "Lie Algebras in Particle Physics" then introduces a Lie group parameterized by a set of real parameters and shows that the generators can be found through a Taylor expansion around the identity. The question is raised if this means that the resulting generators will also commute, but the response states that this is not necessarily the case. It is recommended to learn about Lie groups and algebras from a more specialized mathematics textbook.
  • #1
Kontilera
179
24
Hello PF members!
I have a problem regarding coordinate and non-coordinate bases.

As I understood from my course in GR, the partial derivatives of a coordiante system always commute:
[tex][\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}] = 0 .[/tex]
Which is not necessarly true for non-coordinate bases.

However in Giorgi's book 'Lie Algebras in Particle Physics' he starts out by parameterizing a Lie group G, by a set of N real parameters. (I.e. a coordinate system.)
The he shows that if we taylor expand around the identitiy (for a representation) the we get a set of generators which are the Lie algebra of our Lie group G.



My question then is:
Since he started out by choosing his generators from a coordinate system doesn't this mean that he will find:
[tex][X_a, X_b] = 0 \quad?[/tex]

Thanks in advance!

// Kontilera
 
Physics news on Phys.org
  • #2
Not having read the book I am not sure but usually with Lie groups one looks at the Lie bracket of left invariant vector fields
 
  • #3
Georgi's book is not a tool to learn mathematics. Moreover, if you have a diff. geom. background, by propery going to Lie groups + Lie algebras would soon make you realize that most of what's written in physics book is wrong or at best confusing. My advice is to capture the Lie groups and Lie algebras theory from a more specialized (math-oriented) textbook such as Barut and Raczka.
 

1. What is a coordinate basis?

A coordinate basis is a set of vectors that span a vector space and are used to represent any vector within that space using coordinates. These coordinates are typically expressed as numbers or symbols that represent the magnitude and direction of the vector with respect to the basis.

2. How is a non-coordinate basis different from a coordinate basis?

A non-coordinate basis is a set of vectors that span a vector space but cannot be used to represent vectors using coordinates. This means that the magnitude and direction of a vector in a non-coordinate basis cannot be expressed as numbers or symbols, but rather must be described using the basis vectors themselves.

3. What is the purpose of using a coordinate basis?

The purpose of using a coordinate basis is to simplify calculations and representations of vectors within a vector space. By expressing vectors using coordinates, we can use familiar mathematical operations and equations to manipulate and analyze them.

4. Can a vector be represented differently in different coordinate bases?

Yes, a vector can be represented differently in different coordinate bases. This is because the basis vectors used to represent the vector will be different, resulting in different coordinate values for the vector. However, the vector itself remains the same regardless of the basis used to represent it.

5. How do you convert between coordinate and non-coordinate bases?

To convert between coordinate and non-coordinate bases, you can use a process called change of basis. This involves finding the transformation matrix that maps the coordinates of a vector in one basis to the coordinates of the same vector in the other basis. This transformation matrix can then be used to convert between the two bases.

Similar threads

I Non orthogonal basis and the lines of its coordinate grid
    • Linear and Abstract Algebra
Replies
9
Views
203
Confusion regarding Lie groups
    • Differential Geometry
Replies
12
Views
3K
A Diffeomorphisms & the Lie derivative
    • Differential Geometry
Replies
9
Views
3K
A How Does Noether's Theorem Extend Beyond Conservation in Hamiltonian Systems?
    • Classical Physics
Replies
7
Views
790
B Understanding how coordinates change under the flow of a vector field
    • Special and General Relativity
Replies
1
Views
613
Ricci rotation coefficients and non-coordinate bases
    • Differential Geometry
Replies
10
Views
4K
Lie Groups and Canonical Coordinates
    • Differential Geometry
Replies
9
Views
5K
I Diffeomorphism invariance and contracted Bianchi identity
    • Differential Geometry
Replies
7
Views
3K
A Unitary representations of Lie group from Lie algebra
    • Quantum Physics
Replies
5
Views
1K
Non-coordinate basis for vector fields
    • Differential Geometry
Replies
5
Views
5K

Hot Threads

Recent Insights

Back
Top