Linked Questions

16
votes
3answers
2k views

How are anyons possible?

If $|\psi\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator $P$ which switches the states of the two particles. Since the two particles are ...
10
votes
2answers
2k views

What is the significance of Lie groups $SO(3)$ and $SU(2)$ to particle physics?

I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics? The application of Lie groups and their ...
9
votes
3answers
3k views

Applications of analytic continuation to physics

I posted this on math.SE, but didn't get much response. It might fit better on this site. Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the ...
11
votes
3answers
1k views

Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$

I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example: Suppose I have a spin-1/2 particle, lets say an electron, ...
4
votes
1answer
2k views

representation of SU(2)

The question is regarding SU(2) group and SU(2) algebra. The SU(2) group can be generated by exponentiating the generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$ with $t_a$ being three parameters....
1
vote
1answer
5k views

Why are rotation matrices always unitary operators?

Can someone explain why the rotation matrix is a unitary, specifically orthogonal, operator?
3
votes
3answers
2k views

Showing a mapping between $SU(2)$ and $SO(3)$

I know this has been done on this site in a different manner but I'm wondering if it's possible to show the 2:1 Lie group homomorphism between $SU(2)$ and $SO(3)$ using exponentials of the generators ...
5
votes
2answers
1k views

Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
7
votes
1answer
484 views

A Puzzle about $SO(3)$

Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$. For $SO(3)$ case, is the representation $T^a_{ij}=-i\epsilon^{aij}$ fundamental or adjoint? The fundamental representation is defined as ...
4
votes
1answer
1k views

Lie Algebra Generators

We know that rotations are performed via real and orthogonal matrices, $O^{T}=O^{-1}$. We can write $O$ as, (The proper rotations have unit determinant) $$O = \exp(A),$$ where $A^{T}=-A$. In three ...
3
votes
1answer
1k views

$t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$

What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & ...