SU(3)¶
Defining representation \(\bf{3}\) of su(3) algebra
- class groups.su3.SU3_DEFINING(p=1, q=0, dtype=torch.complex128, device='cpu')[source]¶
- Parameters:
p (int) – (p,q) labels of the highest weight state of su(3) representation. For defining representation
p=1, q=0.q (int) –
dtype (torch.dtype) – data type of matrix representation of operators
device (int) – device on which the torch.tensor objects are stored
Build the defining representation \(\bf{3}\) of su(3) Lie algebra using the Cartan-Weyl (C-W) basis. In terms of the standard Gell-Mann matrices \(\lambda\), the C-W basis is:
\[\begin{split}\begin{align*} T^\pm &= \frac{1}{2} (\lambda_1 \pm i\lambda_2) = (F_1 \pm iF_2)\\ T^z &= \frac{1}{2} \lambda_3 = F_3\\ V^\pm &= \frac{1}{2} (\lambda_4 \pm i\lambda_5) = (F_4 \pm iF_5)\\ U^\pm &= \frac{1}{2} (\lambda_6 \pm i\lambda_7) = (F_6 \pm iF_7)\\ Y &= \frac{1}{\sqrt{3}} \lambda_8 = \frac{2}{\sqrt{3}} F_8 \end{align*}\end{split}\]The quadratic Casimir operator of su(3) can be expressed in terms of the C-W basis, defined as follow.
\[\begin{split}\begin{align*} C_1 = \sum_{k}{F_k F_k} &= \frac{1}{2} (T^+ T^- + T^- T^+ + V^+ V^- + V^- V^+ + U^+ U^- + U^- U^+) \\ &+ T^z T^z + \frac{3}{4} Y Y \end{align*}\end{split}\]- C1()[source]¶
- Returns:
The quadratic Casimir of su(3) as rank-4 for tensor
- Return type:
torch.tensor
- Cartan_Weyl()[source]¶
- Returns:
vector of generators forming Cartan-Weyl basis ordered as [T^+, T^-, T^z, V^+, V^-, U^+, U^-, Y].
- Return type:
torch.tensor
Returns a rank-3 tensor with first index running over generators.
SU(3) with explicit U(1)xU(1) structure¶
Defining representation \(\bf{3}\) of su(3) algebra in block-sparse form leveraging abelian U(1)xU(1) subgroup.