Symmetry Group Freedom: Choosing How Groups Act on Coordinates

In summary, the question discusses whether there is freedom in choosing how a symmetry group acts on coordinates, or if the group itself imposes the way it acts. The conversation also mentions deriving generators from coordinate transformations, specifically for rotations around the X axis and reflections in the XY plane. The speaker suggests trying an exercise to see the differences between the two cases.
  • #1
kent davidge
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One could argue that this question should be posted on the maths forum, but I see it so frequently in physics that I find it more productive to ask it here.

In a symmetry group, do we have freedom of choice of how the group is going to act in the coordinates? Or is the way the group act on the coordinates imposed on us by the group itself?

An example of what I mean: do I have the freedom to identify a group of order 2 as the responsible for rotations around say, X axis, or this same group I could say that is the group of reflections in say, XY plane?
 
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  • #2
One writes out the transformation between coordinates, and then derives generators from that. E.g., the coordinate transformations corresponding to rotations around the X axis are in terms of an angle ##\theta##. Then by taking a derivative wrt ##\theta## at ##\theta=0## one can derive differential operators which generate the finite transformations. (I can post a bit more detail if you need it).

Try this exercise for both your cases. You'll find that the ordinary rotation group is very different from the group of reflections.
 
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1. What is a symmetry group?

A symmetry group is a mathematical concept that refers to a set of transformations or operations that leave an object or system unchanged. In other words, it is a group of symmetries that preserve the structure or characteristics of an object.

2. How do symmetry groups act on coordinates?

Symmetry groups act on coordinates by transforming the coordinates in a way that preserves the symmetry of the object. This means that the coordinates are rearranged or rotated in a specific manner that maintains the overall shape and structure of the object.

3. What is symmetry group freedom?

Symmetry group freedom refers to the ability to choose how a symmetry group acts on coordinates. In other words, it is the flexibility to determine the specific transformations or operations that will be used to preserve the symmetry of an object.

4. Why is choosing how groups act on coordinates important?

Choosing how groups act on coordinates is important because it allows for more control and customization in mathematical calculations and models. It also allows for a deeper understanding of the symmetries present in a system and how they can be manipulated.

5. How is symmetry group freedom used in scientific research?

Symmetry group freedom is used in scientific research in various fields such as physics, chemistry, and mathematics. It is used to study the symmetries present in natural phenomena and to develop mathematical models and theories that accurately describe and predict these phenomena. It also plays a crucial role in understanding the fundamental laws and principles of the universe.

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