TR-034

Wigner \(D\) and Pauli matrices

This report shows the symbolic representation of the Wigner-\(d\) and -\(D\) matrix elements and shows how \(D^\tfrac{1}{2}\) is related to the Pauli matrices.

Wigner \(D\) and Pauli matrices

Import Python libraries Hide code cell content

import sympy as sp
from ampform.io import aslatex
from IPython.display import Math
from sympy import pi
from sympy.physics.matrices import msigma
from sympy.physics.quantum.spin import Rotation as Wigner

j, m, mp = sp.symbols(R"j m m^{\prime}", nonnegative=True, rational=True)
alpha, beta, gamma = sp.symbols("alpha beta gamma", real=True)
half = sp.Rational(0.5)

Big Wigner \(D\) matrices

Helper functions Hide code cell source

def render_wigner_big_d(j: sp.Rational) -> Math:
    symbol = Wigner.D(j, m, mp, alpha, beta, gamma)
    expr = get_wigner_big_d(j)
    return Math(aslatex({symbol: sp.latex(expr)}))


def get_wigner_big_d(j: sp.Rational) -> sp.Matrix:
    dim = int(2 * j + 1)
    return sp.Matrix([
        [
            Wigner.D(j, x - j, y - j, alpha, beta, gamma).doit().simplify()
            for x in range(dim)
        ]
        for y in range(dim)
    ])

render_wigner_big_d(j=half)

\[\begin{split}\displaystyle \begin{array}{rcl} D^{\frac{1}{2}}_{m,m^{\prime}}\left(\alpha,\beta,\gamma\right) &=& \left[\begin{matrix}e^{\frac{i \left(\alpha + \gamma\right)}{2}} \cos{\left(\frac{\beta}{2} \right)} & - e^{- \frac{i \left(\alpha - \gamma\right)}{2}} \sin{\left(\frac{\beta}{2} \right)}\\e^{\frac{i \left(\alpha - \gamma\right)}{2}} \sin{\left(\frac{\beta}{2} \right)} & e^{- \frac{i \left(\alpha + \gamma\right)}{2}} \cos{\left(\frac{\beta}{2} \right)}\end{matrix}\right] \\ \end{array}\end{split}\]

render_wigner_big_d(j=1)

\[\begin{split}\displaystyle \begin{array}{rcl} D^{1}_{m,m^{\prime}}\left(\alpha,\beta,\gamma\right) &=& \left[\begin{matrix}\frac{\left(\cos{\left(\beta \right)} + 1\right) e^{i \left(\alpha + \gamma\right)}}{2} & - \frac{\sqrt{2} e^{i \gamma} \sin{\left(\beta \right)}}{2} & \frac{\left(1 - \cos{\left(\beta \right)}\right) e^{i \left(- \alpha + \gamma\right)}}{2}\\\frac{\sqrt{2} e^{i \alpha} \sin{\left(\beta \right)}}{2} & \cos{\left(\beta \right)} & - \frac{\sqrt{2} e^{- i \alpha} \sin{\left(\beta \right)}}{2}\\\frac{\left(1 - \cos{\left(\beta \right)}\right) e^{i \left(\alpha - \gamma\right)}}{2} & \frac{\sqrt{2} e^{- i \gamma} \sin{\left(\beta \right)}}{2} & \frac{\left(\cos{\left(\beta \right)} + 1\right) e^{- i \left(\alpha + \gamma\right)}}{2}\end{matrix}\right] \\ \end{array}\end{split}\]

Small Wigner \(d\) matrices

Helper functions Hide code cell source

def render_wigner_small_d(j: sp.Rational) -> Math:
    symbol = Wigner.d(j, m, mp, beta)
    expr = get_wigner_small_d(j)
    return Math(aslatex({symbol: sp.latex(expr)}))


def get_wigner_small_d(j: sp.Rational) -> sp.Matrix:
    dim = int(2 * j + 1)
    return sp.Matrix([
        [Wigner.d(j, x - j, y - j, beta).doit().simplify() for x in range(dim)]
        for y in range(dim)
    ])

render_wigner_small_d(j=half)

\[\begin{split}\displaystyle \begin{array}{rcl} d^{\frac{1}{2}}_{m,m^{\prime}}\left(\beta\right) &=& \left[\begin{matrix}\cos{\left(\frac{\beta}{2} \right)} & - \sin{\left(\frac{\beta}{2} \right)}\\\sin{\left(\frac{\beta}{2} \right)} & \cos{\left(\frac{\beta}{2} \right)}\end{matrix}\right] \\ \end{array}\end{split}\]

render_wigner_small_d(j=1)

\[\begin{split}\displaystyle \begin{array}{rcl} d^{1}_{m,m^{\prime}}\left(\beta\right) &=& \left[\begin{matrix}\frac{\cos{\left(\beta \right)}}{2} + \frac{1}{2} & - \frac{\sqrt{2} \sin{\left(\beta \right)}}{2} & \frac{1}{2} - \frac{\cos{\left(\beta \right)}}{2}\\\frac{\sqrt{2} \sin{\left(\beta \right)}}{2} & \cos{\left(\beta \right)} & - \frac{\sqrt{2} \sin{\left(\beta \right)}}{2}\\\frac{1}{2} - \frac{\cos{\left(\beta \right)}}{2} & \frac{\sqrt{2} \sin{\left(\beta \right)}}{2} & \frac{\cos{\left(\beta \right)}}{2} + \frac{1}{2}\end{matrix}\right] \\ \end{array}\end{split}\]

render_wigner_small_d(j=3 * half)

\[\begin{split}\displaystyle \begin{array}{rcl} d^{\frac{3}{2}}_{m,m^{\prime}}\left(\beta\right) &=& \left[\begin{matrix}\frac{3 \cos{\left(\frac{\beta}{2} \right)}}{4} + \frac{\cos{\left(\frac{3 \beta}{2} \right)}}{4} & - \frac{\sqrt{3} \left(\sin{\left(\frac{\beta}{2} \right)} + \sin{\left(\frac{3 \beta}{2} \right)}\right)}{4} & \frac{\sqrt{3} \left(\cos{\left(\frac{\beta}{2} \right)} - \cos{\left(\frac{3 \beta}{2} \right)}\right)}{4} & - \frac{3 \sin{\left(\frac{\beta}{2} \right)}}{4} + \frac{\sin{\left(\frac{3 \beta}{2} \right)}}{4}\\\frac{\sqrt{3} \left(\sin{\left(\frac{\beta}{2} \right)} + \sin{\left(\frac{3 \beta}{2} \right)}\right)}{4} & \frac{\cos{\left(\frac{\beta}{2} \right)}}{4} + \frac{3 \cos{\left(\frac{3 \beta}{2} \right)}}{4} & \frac{\sin{\left(\frac{\beta}{2} \right)}}{4} - \frac{3 \sin{\left(\frac{3 \beta}{2} \right)}}{4} & \frac{\sqrt{3} \left(\cos{\left(\frac{\beta}{2} \right)} - \cos{\left(\frac{3 \beta}{2} \right)}\right)}{4}\\\frac{\sqrt{3} \left(\cos{\left(\frac{\beta}{2} \right)} - \cos{\left(\frac{3 \beta}{2} \right)}\right)}{4} & - \frac{\sin{\left(\frac{\beta}{2} \right)}}{4} + \frac{3 \sin{\left(\frac{3 \beta}{2} \right)}}{4} & \frac{\cos{\left(\frac{\beta}{2} \right)}}{4} + \frac{3 \cos{\left(\frac{3 \beta}{2} \right)}}{4} & - \frac{\sqrt{3} \left(\sin{\left(\frac{\beta}{2} \right)} + \sin{\left(\frac{3 \beta}{2} \right)}\right)}{4}\\\frac{3 \sin{\left(\frac{\beta}{2} \right)}}{4} - \frac{\sin{\left(\frac{3 \beta}{2} \right)}}{4} & \frac{\sqrt{3} \left(\cos{\left(\frac{\beta}{2} \right)} - \cos{\left(\frac{3 \beta}{2} \right)}\right)}{4} & \frac{\sqrt{3} \left(\sin{\left(\frac{\beta}{2} \right)} + \sin{\left(\frac{3 \beta}{2} \right)}\right)}{4} & \frac{3 \cos{\left(\frac{\beta}{2} \right)}}{4} + \frac{\cos{\left(\frac{3 \beta}{2} \right)}}{4}\end{matrix}\right] \\ \end{array}\end{split}\]

Relation to Pauli matrices

Note how the Pauli matrices are related to the rotation around each axis in 3D space, which lives in \(SO(3)\), and each Wigner-\(D\) matrix, which lives in \(SU(2)\):

\[\begin{split} \begin{array}{rcl} \sigma_x &=& i \, D^\tfrac{1}{2}(-\pi,0,0) \\ \sigma_y &=& i \, D^\tfrac{1}{2}(0,+\pi,0) \\ \sigma_z &=& i \, D^\tfrac{1}{2}(0,0,-\pi) \,. \end{array} \end{split}\]

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Math(aslatex({Rf"\sigma_{x}": sp.latex(msigma(i)) for i, x in enumerate("xyz", 1)}))

\[\begin{split}\displaystyle \begin{array}{rcl} \sigma_x &=& \left[\begin{matrix}0 & 1\\1 & 0\end{matrix}\right] \\ \sigma_y &=& \left[\begin{matrix}0 & - i\\i & 0\end{matrix}\right] \\ \sigma_z &=& \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right] \\ \end{array}\end{split}\]

sp.I * get_wigner_big_d(j=half).subs({
    alpha: -pi,
    beta: 0,
    gamma: 0,
})

\[\begin{split}\displaystyle \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]\end{split}\]

sp.I * get_wigner_big_d(j=half).subs({
    alpha: 0,
    beta: pi,
    gamma: 0,
})

\[\begin{split}\displaystyle \left[\begin{matrix}0 & - i\\i & 0\end{matrix}\right]\end{split}\]

sp.I * get_wigner_big_d(j=half).subs({
    alpha: 0,
    beta: 0,
    gamma: -pi,
})

\[\begin{split}\displaystyle \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]\end{split}\]