What Is a Special Unitary Group?

  • Context: High School 
  • Thread starter Thread starter ChrisisC
  • Start date Start date
  • Tags Tags
    Group
Click For Summary

Discussion Overview

The discussion revolves around the concept of Special Unitary Groups (SU groups), particularly in the context of quantum mechanics. Participants explore definitions and properties of SU groups, as well as the challenges in understanding them.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about the meaning of SU groups and seeks a simple explanation.
  • Another participant provides a definition of SU groups as the group of complex (n x n) matrices that are unitary with a determinant of 1, including a mathematical representation.
  • A further comment suggests that the provided definition may omit important properties and notes that there are alternative ways to define SU groups based on the invariance of the complex inner product.
  • One participant questions whether the original poster (OP) might find the Wikipedia article too complex, implying that the explanation sought may require a more accessible approach.

Areas of Agreement / Disagreement

There is no clear consensus on the best way to explain SU groups, as participants express differing views on the adequacy of existing resources and the complexity of the topic.

Contextual Notes

Some participants note that the definition provided may not encompass all important properties of SU groups, indicating potential limitations in understanding without further context.

ChrisisC
Messages
54
Reaction score
4
I constantly read physics topics that are generally more QM, and i always find descriptions of SU groups. I have no idea what they mean? this is not a discussion topic and i don't mind if it's taken down but i really would like a simple, yet informative answer! Thanks!
 
Physics news on Phys.org
You could have easily found an answer on Wikipedia
https://en.wikipedia.org/wiki/Special_unitary_group
which raises the question, what is it that you didn't find there and hope to find here?
The shortest answer might be: It is the group of complex ##(n \times n)## matrices, which are unitary of determinant ##1##:
##SU(n,\mathbb{C}) = \{ A \in \mathbb{M}(n,\mathbb{C})\,\vert \, A\cdot A^\dagger = 1 \,\wedge \, \det A = 1\}##.
##A^\dagger## here is the matrix mirrored at the main diagonal and taken the complex conjugate entries: ##A^\dagger = \bar{A}^t##.

This definition leaves out a couple of important properties and isn't the only one possible. As a group of linear transformation one can also define it by the properties of these transformations, namely the invariance of the complex inner product: ##\langle Ux,Uy \rangle = \langle x,y \rangle## etc.
 
Since the thread is marked "B", I suspect the OP can't understand the wikipedia article, which is above a B level.
 

Similar threads

  • · Replies 46 ·
2
Replies
46
Views
6K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 7 ·
Replies
7
Views
3K
  • · Replies 1 ·
Replies
1
Views
675
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 52 ·
2
Replies
52
Views
20K
  • · Replies 17 ·
Replies
17
Views
10K
  • · Replies 50 ·
2
Replies
50
Views
5K
  • · Replies 11 ·
Replies
11
Views
3K
  • · Replies 35 ·
2
Replies
35
Views
9K