# Group Theory

## Basics

- Representation: p: G -> GL(V), a map from the group the set of linear maps from V to V (ambigous if Representation means the map p or the resulting set of linear maps)
- Lie Algebra: Vector space + inner product (vector product) that is bilinear, follows the jacobi identity, and has x*x = 0

## Wigner D/d-matrix (sheet 07) -> Dive into angular momentum maths

- Generator operators U(R) = exp(-ihJ), J: operator / infinitesimal rotation
- SO(3) has exactly one irrep of dim 1,3,5,... (=2j+1) -> How does this relate to number of eigenstates?
- SU(2) allows even dim as well (j = half-integer) because SU(2) is double cover of SO(3) -> research
- Wigner D-Matrix gives transition probabilities between eigenstates
- Eigenstates form basis when discrete
- When j= 3, m=-3,-2,-1,0,1,2,3 -> states |3m> are allowed -> real states are superpositions of these basis states / eigenstates -> 7-dim space needed to describe them (even though angular momentum needs only 3d classically) -> remember in general that degrees of freedom explode as 2^n because superpositions are allowed
- j fixes total angular momentum, m fixes z component
- angular momentum operator algebra is inherent to so(3) lie algebra

## Questions / open things

- Where does the inherent need for SO(3) come from? Why are the possible state spaces the irreps of SO(3)?
  -> The operators that rotate states (in physical space) can be called U(R(theta)), where R in SO(3). There is a homomorphism between {U(R)} and SO(3), since U(R1R2)=U(R1)U(R2). Thus U(R) is a representation. 
     Enforcing finite-dim and irrep, this fixes dim = 2j+1.
- What about that L/J algebra? See "Lie Algebra" german wikipedia article -> Beispiele -> Aus der Physik -> Seems as though the relation does not only hold for those three "basis matrices", but all matrices, as long as your in the basis of the L_i (wtf even is a basis of matrices)
- Is there an intuitive notion to see how angular momentum emerges from the shapes of the wave functions?
- How does the emergence of azimuthal dependence in superpositions of (not azimuth-constraining) eigenstates play out mathematically?
- Why is SU(2) the double cover of SO(3)? What does that mean?
- What role do Pauli matrices play (generators of SO(2)? idk)
- How does one come up with the idea that operators and eigenvalues/vectors are the appropriate mathematical tools to describe QM? How does it arise? BC this alone imposes a lot of the quantization maths, since eigenvalues are discrete
- relation between Lie algebras and Lie groups not exactly clear
- Whats the algebra of exp of matrices -> https://de.wikipedia.org/wiki/Matrixexponential
- Is the generator exp(-ihJ) for SO(3) just the most convinient choice or the only choice?

## Next topics

- Character tables and relation between conjugacy classes and irreps
- Further sheets
- Further lectures, explicitly everything after lecture 09
- Wigner-Eckart Theorem (in lecture, sheet and indip research)
  - Clebsch Gordan coeff as seperate research?