How to Calculate Structure Constants for Reparametrized Rotation Group?

[turin]

I'm putting this question here because I can't get any help from the HW forum (It's actually not a HW, but it looks a lot like a HW, so I won't be surprised if it gets moved there).



Source: Anderson, Principles of Relativity Physics

p. 13, prob. 1.4

"Reparametrize the rotation group by taking, as new infinitesimal parameters, ε¹ = ε²³, ε² = ε³¹, and ε³ = ε¹² and calculate the structure constants for these parameters."

My assumptions:

(1)
The ε^ij mentioned in the problem are the infinitesimal Cartesian parameters of the 3-D rotation group such that ε^ij = -ε^ji, and yⁱ = xⁱ + Σ_jε^ijx^j, where x is the original point and y is the transformed point.

(2)
To generalize this to non-Cartesian coordinates and still maintain the Lie group-ness, the transformation takes the general form:

yⁱ = xⁱ + Σ_kε^kf_kⁱ(x)

where the f_kⁱ(x) satisfy the following condition.

(3)
The request for structure constants is a request for constants c^k_mn such that:

yⁱ = xⁱ + Σ_kΣ_mΣ_n(ε_B^mε_Aⁿ - ε_A^mε_Bⁿ)c^k_mnf_kⁱ(x)

(4)
The parameters ε^k are the non-Cartesian parameters, and so, they should multiply some functions f_kⁱ(x), and these functions determine the structure constants.

My problem with understanding:

I don't know how to find the f_kⁱ(x). I have:

Σ_jε^ijx^j = Σ_kε^kf_kⁱ(x)

but I don't see how this tells me f_kⁱ(x). Am I supposed to assume some kind of orthogonality or something?

 

[jvicens]

Hello, thank you for reaching out for help with this problem. I can understand your confusion, as it seems like you have a good understanding of the problem and its assumptions. However, I believe you may be overthinking it a bit. Let me try to break it down for you.

First, let's start with the given transformation:

yi = xi + Σkεkfki(x)

This is a general transformation in non-Cartesian coordinates, as you correctly pointed out. The fki(x) functions are the components of the transformation, and they satisfy the condition that Σkεkfki(x) is a linear transformation. This means that the fki(x) functions can be written as linear combinations of the non-Cartesian parameters εk. For example, f12(x) = aε1 + bε2 + cε3.

Now, let's look at the requested structure constants:

yi = xi + ΣkΣmΣn(εBmεAn - εAmεBn)ckmnfki(x)

These constants, ckmn, are just coefficients that relate the non-Cartesian parameters εk to the Cartesian parameters εij. In other words, they tell us how the non-Cartesian parameters are related to the original Cartesian parameters. To find these constants, we can use the given transformation and equate the coefficients of the non-Cartesian parameters on both sides. For example, equating the coefficients of ε12 on both sides, we get:

ε12 = ε1f12(x) + ε2f21(x) + ε3f31(x)

Using the general form of the fki(x) functions, we can substitute in and solve for c1212. This process can be repeated for all the other structure constants.

I hope this helps to clarify the problem for you. Let me know if you have any further questions or if you need any additional help. Good luck!

 

[M98Ranger]

Thank you for providing the context for your question. From my understanding, the problem is asking you to reparametrize the rotation group using the new infinitesimal parameters given, and then calculate the structure constants for these new parameters.

To clarify, the εij mentioned in the problem are not the original Cartesian parameters, but rather the new ones that are given as ε1 = ε23, ε2 = ε31, and ε3 = ε12. These new parameters correspond to rotations around the x, y, and z axes, respectively.

To find the structure constants, you can use the formula you mentioned in (3). However, the functions fki(x) are not determined by the given parameters, but rather they are arbitrary functions that satisfy the condition in (2). This means that there are infinite possibilities for the fki(x) functions, and there is no specific way to find them.

To solve the problem, you can simply use the given parameters to reparametrize the rotation group and then use the formula in (3) to calculate the structure constants. You do not need to find the fki(x) functions explicitly.

I hope this helps clarify the problem. If you are still unsure, I recommend reaching out to your professor or classmates for further assistance. Good luck!