F Sketch
Author:Peir-Ru Wang
Update
(S20)2025.04.23: Lie Derivative and Spin (Lie algebra)
(S18)2025.04.21: Lie Derivative and Spin
SO group satisfied $$ \eta^{\alpha\beta} = \Lambda^\alpha_{\ \gamma} \Lambda^\beta_{\ \epsilon} \eta^{\gamma\epsilon} \qquad \Lambda^\alpha_{\ \gamma} = \delta^\alpha_\gamma + \omega^\alpha_{\ \gamma} $$ where \( \eta^{\alpha\beta}\) is metric tensor.
Check SO condition:
| Upper index $\eta^{\alpha\beta}$ | Lower index $\eta_{\alpha\beta}$ |
|---|---|
| $$ \begin{aligned} \eta^{\alpha\beta} &= (\delta^\alpha_\gamma + \omega^\alpha_{\ \gamma}) (\delta^\beta_\epsilon + \omega^\beta_{\ \epsilon}) \eta^{\gamma\epsilon} \\ 0 &= \delta^\alpha_\gamma \omega^\beta_{\ \epsilon} \eta^{\gamma\epsilon} + \omega^\alpha_{\ \gamma} \delta^\beta_\epsilon \eta^{\gamma\epsilon} \\ 0 &= \omega^\beta_{\ \epsilon} \eta^{\alpha\epsilon} + \omega^\alpha_{\ \gamma} \eta^{\gamma\beta} \\ 0 &= \omega^{\alpha\beta} + \omega^{\beta\alpha} \end{aligned} $$ | $$ \begin{aligned} \eta_{\alpha\beta} &= (\delta^\gamma_\alpha + \omega^\gamma_{\ \alpha}) (\delta^\epsilon_\beta + \omega^\epsilon_{\ \beta}) \eta_{\gamma\epsilon} \\ 0 &= \delta^\gamma_\alpha \omega^\epsilon_{\ \beta} \eta_{\gamma\epsilon} + \omega^\gamma_{\ \alpha} \delta^\epsilon_\beta \eta_{\gamma\epsilon} \\ 0 &= \omega^\epsilon_{\ \beta} \eta_{\alpha\epsilon} + \omega^\gamma_{\ \alpha} \eta_{\gamma\beta} \\ 0 &= \omega_{\alpha\beta} + \omega_{\beta\alpha} \end{aligned} $$ |
Let：
$$ \delta X = \delta X^\gamma\, \partial_\gamma = (\omega^\gamma_{\ \epsilon} x^\epsilon) \partial_\gamma, \qquad \delta X^\gamma = \omega^\gamma_{\ \epsilon} x^\epsilon $$$$ \delta X^\gamma = \omega^\gamma_{\ \epsilon} x^\epsilon $$
| Vector field | 1-form field |
|---|---|
| $V = V^\alpha(p)\, \partial_\alpha$ | $W = W_\alpha(p)\, dx^\alpha$ |
| $$ \begin{aligned} (\hat{\mathcal{L}}_{\delta X} V)^\alpha &= {V^\alpha}_{\ ,\gamma} \delta X^\gamma - V^\gamma \delta X^\alpha_{\ ,\gamma} \\ &={V^\alpha}_{\ ,\gamma} {\omega^\gamma}_{\epsilon} x^\epsilon - V^\gamma \left({\omega^\alpha}_{\epsilon} x^\epsilon \right)_{,\gamma} \end{aligned} $$ | $$ \begin{aligned} (\hat{\mathcal{L}}_{\delta X} W)_\alpha &= W_{\alpha,\gamma} \delta X^\gamma + W_\gamma \delta X^\gamma_{\ ,\alpha} \\ &= W_{\alpha,\gamma} \omega^\gamma_{\ \epsilon} x^\epsilon + W_\gamma \left({\omega^\gamma}_{\epsilon} x^\epsilon \right)_{,\alpha} \end{aligned} $$ |
| Let $(\omega^\alpha_{\epsilon})_{,\gamma} = 0$ (global transformation) | |
| $$ \begin{aligned} (\hat{\mathcal{L}}_{\delta X} V)^\alpha &= {V^\alpha}_{,\gamma} {\omega^\gamma}_{\ \epsilon} x^\epsilon - V^\gamma {\omega^\alpha}_{\epsilon} {x^\epsilon}_{,\gamma} \\ &= {\omega^\gamma}_{\ \epsilon} x^\epsilon \partial_\gamma V^\alpha - V^\gamma {\omega^\alpha}_{\epsilon} {\delta^\epsilon}_{\gamma} \\ &= \omega^{\gamma\epsilon} x_\epsilon \partial_\gamma V^\alpha - {\omega^\alpha}_{\gamma} V^\gamma \\ &= \omega^{\gamma\epsilon} \frac{1}{2} \underbrace{(x_\epsilon \partial_\gamma - x_\gamma \partial_\epsilon)}_{\text{Angular}} V^\alpha - \underbrace{{\omega^\alpha}_{\gamma}}_{\text{Spin}} V^\gamma \end{aligned} $$ | $$ \begin{aligned} (\hat{\mathcal{L}}_{\delta X} W)_\alpha &= W_{\alpha,\gamma}{\omega^\gamma}_{\epsilon} x^\epsilon + W_\gamma {\omega^\gamma}_{\epsilon} {x^\epsilon}_{,\alpha} \\ &= {\omega^\gamma}_{\epsilon} x^\epsilon \partial_\gamma W_\alpha + W_\gamma {\omega^\gamma}_{\epsilon} {\delta^\epsilon}_{\alpha} \\ &= {\omega}^{\gamma\epsilon} x_\epsilon \partial_\gamma W_\alpha + {\omega^\gamma}_{\alpha} W_\gamma \\ &= \omega^{\gamma\epsilon} \frac{1}{2} \underbrace{(x_\epsilon \partial_\gamma - x_\gamma \partial_\epsilon)}_{\text{Angular}} W_\alpha + \underbrace{{\omega^\gamma}_{\alpha}}_{\text{Spin}} W_\gamma \end{aligned} $$ |
Let \(G\) is a Lie group acting on a manifold \(M\), and \(\mathfrak{g}\) is the Lie algebra of \(G\). Let \( \mathfrak{X}(M) := \Gamma(TM) \) is the set of all (smooth) section on tengent bundle of \(M\).
The induced vector field on \(M\) by \(G\) (Fundamental vector field):
Since \( G \) acts on \( M \), the infinitesimal action induces a Lie algebra homomorphism \( \mathfrak{g} \to \mathfrak{X}(M) \), sometimes called the fundamental vector fielda. More precisely, it is the induced vector field \(\widetilde{X} \in \mathfrak{X}(M) \) on \( M \) induced by the action of \( G \). If \(G=O\left(p,q\right)\) is an orthoganal group, then we called \(\widetilde{X}\) is a Killing vector field.
For example, let \(G=SO(3)\) and \(\sigma_x,\sigma_y,\sigma_z \in \mathfrak{so} (3) \) are the rotation generator along \(x,y,z\)-axis, respectively. The corresponds indueced vector fields are $$ \sigma_x \to \widetilde{X}_x= z\partial_y-y\partial_z =\frac{i}{\hbar} \hat{L}_x $$ $$ \sigma_y \to \widetilde{X}_y= z\partial_x-x\partial_z =\frac{i}{\hbar} \hat{L}_y $$ $$ \sigma_z \to \widetilde{X}_z= x\partial_y-y\partial_x =\frac{i}{\hbar} \hat{L}_z $$ , respectively. Note that \(\hat{L}_x,\hat{L}_y,\hat{L}_z \) are angular momentum operartors in quantum physics.
Furthermore, the Lie derivative \(\hat{\mathcal{L}}_{\widetilde{X}}\) with respect to \(\widetilde{X} \) is a infinite-dimensional Lie algebra representation of \(\mathfrak{g}\).\(^a\)Principal Fiber Bundles, Michael Kunzinger, https://www.mat.univie.ac.at/~mike/teaching/ss20/pfb.pdf, (Backup)
The adjoint representation of \(\mathfrak{g}\) :
Let \(\hat{T}_a \) is the generator of \(\mathfrak{g}\) and \(f^c_{ab}\) is the structure constant of \(\mathfrak{g}\) defined by \(\left[\hat{T}_a ,\hat{T}_b \right]=f^c_{ab}\hat{T}_c \). For \(X,Y\in \mathfrak{g} \), the adjoint action of \(X\) on \(Y\) is \(ad_X Y =\left[X,Y\right]\).
The adjoint representation \(ad\) of \(\hat{T}_a \) is $$\left(ad_{\hat{T}_a}\right)^b_c=f^b_{ac} $$.
Lie Derivative of vector field on \(M \) with respect to Killing vector field induced by \(G\) group :
We denote the induced vector field \(\widetilde{X}_\alpha \in \mathfrak{X}(M) \) coresspond to \(\hat{T}_\alpha \in \mathfrak{g}\) of \(G\).
Let \(V=V^\alpha \partial_\alpha \in \mathfrak{X}(M) \), and \(\widetilde{X}_\theta=\theta^\beta \widetilde{X}_\beta \) is the induced vector field induced by the action of \( G \) acting on \(M\) \( (\theta=\theta^\beta\hat{T}_\beta \in \mathfrak{g} )\). Note that \(\theta^\beta\) is a constant functon on \(M\). Then the Lie derivative of \(V\) with respect to \(\widetilde{X}_\theta\) is $$ \hat{\mathcal{L}}_{\widetilde{X}_\theta} V= \hat{\mathcal{L}}_{\theta^\beta \widetilde{X}_\beta} \left(V^\alpha \partial_\alpha\right)=\theta^\beta\left[ \left(\hat{\mathcal{L}}_{\widetilde{X}_\beta} V^\alpha\right) \partial_\alpha+V^\alpha \left(\hat{\mathcal{L}}_{\widetilde{X}_\beta} \partial_\alpha\right) \right]=\theta^\beta\left[ \left(\widetilde{X}_\beta V^\alpha\right) \partial_\alpha+V^\alpha \left[\widetilde{X}_\beta,\partial_\alpha\right] \right] $$ Using previous result, we have: $$ \hat{\mathcal{L}}_{\widetilde{X}_\theta} V= \theta^\beta\left[ \left(\frac{i}{\hbar}\hat{L}_\beta V^\alpha\right) \partial_\alpha+V^\alpha ad_{\widetilde{X}_\beta}\partial_\alpha \right] = \theta^\beta\left[ \underbrace{\frac{i}{\hbar}\hat{L}_\beta }_{Angular}V^\alpha +V^\alpha \underbrace{ad_{\widetilde{X}_\beta}}_{Spin} \right] \partial_\alpha $$
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