# 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

C2()[source]
Returns

The cubic Casimir of su(3) as rank-6 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.

I()[source]
Returns

Identity operator of irrep

Return type

torch.tensor

J_Gell_Mann()[source]
Returns

$$\vec{\lambda}$$ vector of Gell-Mann matrices

Return type

torch.tensor

Returns a rank-3 tensor with first index running over generators.

TM()[source]
Returns

$$T^-$$ operator of irrep

Return type

torch.tensor

TP()[source]
Returns

$$T^+$$ operator of irrep

Return type

torch.tensor

TZ()[source]
Returns

$$T^z$$ operator of irrep

Return type

torch.tensor

UM()[source]
Returns

$$U^-$$ operator of irrep

Return type

torch.tensor

UP()[source]
Returns

$$U^+$$ operator of irrep

Return type

torch.tensor

VM()[source]
Returns

$$V^-$$ operator of irrep

Return type

torch.tensor

VP()[source]
Returns

$$V^+$$ operator of irrep

Return type

torch.tensor

Y()[source]
Returns

$$Y$$ operator of irrep

Return type

torch.tensor

# 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.