| I | (123) | (321) | Basis functions | |
|---|---|---|---|---|
| \(\Gamma^{1}\) | 1 | 1 | 1 | \(v_{1}=s_{1}+s_{2}+s_{3}\) |
| \(\Gamma^{2}\) | 1 | \(\omega\) | \(\omega^{2}\) | \(v_{2}=s_{1}+\omega s_{2}+\omega^{2} s_{3}\) |
| \(\Gamma^{3}\) | 1 | \(\omega^{2}\) | \(\omega\) | \(v_{3}=s_{1}+\omega^{2} s_{2}+\omega s_{3}\) |
columns: | I | (123) | (321) | Basis functions
| \(I\) | (12)(34) | (13)(24) | (14)(23) | Basis functions | |
|---|---|---|---|---|---|
| \(\Gamma^{1}\) | 1 | 1 | 1 | 1 | \(w_{1}=t_{1}+t_{2}+t_{3}+t_{4}\) |
| \(\Gamma^{2}\) | 1 | 1 | -1 | -1 | \(w_{2}=t_{1}+t_{2}-t_{3}-t_{4}\) |
| \(\Gamma^{3}\) | 1 | -1 | 1 | -1 | \(w_{3}=t_{1}-t_{2}+t_{3}-t_{4}\) |
| \(\Gamma^{4}\) | 1 | -1 | -1 | 1 | \(w_{4}=t_{1}-t_{2}-t_{3}+t_{4}\) |
columns: | \(I\) | (12)(34) | (13)(24) | (14)(23) | Basis functions
| Group | Dimension |
|---|---|
| \(U T(p, q)\) | \(p^{2}+q^{2}+p q\) |
| \(H T(p, q)\) | \(p(p+q)\) |
| \(U T(p, q, r)\) | \(p^{2}+q^{2}+r^{2}+p q+p r+q r\) |
| \(\operatorname{Sol}(n)\) | \(n(n+1) / 2\) |
| \(\operatorname{Nil}(n)\) | \(n(n-1) / 2\) |
| \(A(p, q)\) | \(p q\) |
columns: Group | Dimension
| Transformation | \(\mathbf{F}=d \mathbf{p} / d t\) | \(\left(\nabla^{2}-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\right) A_{\mu}=-\frac{4 \pi}{c} j_{\mu}\) |
|---|---|---|
| Galilean | invariant | not invariant |
| Poincaré | not invariant | invariant |
columns: Transformation | \(\mathbf{F}=d \mathbf{p} / d t\) | \(\left(\nabla^{2}-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\right) A_{\mu}=-\frac{4 \pi}{c} j_{\mu}\)
| In group | Relation | In algebra | ||
|---|---|---|---|---|
| \(S^{t}=S^{+1}\) | \(\operatorname{det}(S)=+1\) | \(S=e^{\Sigma}\) | \(\operatorname{Tr} \Sigma=0\) | \(\Sigma^{t}=+\Sigma\) |
| \(O^{t}=O^{-1}\) | \(\operatorname{det}(O)=+1\) | \(O=e^{A}\) | \(\operatorname{Tr} A=0\) | \(A^{t}=-A\) |
columns: In group \(S^{t}=S^{+1}\) | In group \(\operatorname{det}(S)=+1\) | Relation \(S=e^{\Sigma}\) | In algebra \(\operatorname{Tr} \Sigma=0\) | In algebra \(\Sigma^{t}=+\Sigma\)
| \(X_{\eta}\) | \(X_{R}\) | \(X_{L}\) | \(X_{r}\) | \(X_{l}\) | \(X_{\delta}\) | |
|---|---|---|---|---|---|---|
| \(X_{\eta}\) | 0 | \(2 X_{R}\) | \(-2 X_{L}\) | \(X_{r}\) | \(-X_{l}\) | 0 |
| \(X_{R}\) | 0 | \(-4 X_{\eta}\) | 0 | \(-2 X_{r}\) | 0 | |
| \(X_{L}\) | 0 | \(2 X_{l}\) | 0 | 0 | ||
| \(X_{r}\) | 0 | \(-X_{\delta}\) | 0 | |||
| \(X_{l}\) | 0 | 0 | ||||
| \(X_{\delta}\) | 0 |
columns: | \(X_{\eta}\) | \(X_{R}\) | \(X_{L}\) | \(X_{r}\) | \(X_{l}\) | \(X_{\delta}\)
| \(\hat{n}+\frac{1}{2} I\) | \(a^{\dagger} a^{\dagger}\) | \(a a\) | \(a^{\dagger}\) | \(a\) | I | |
|---|---|---|---|---|---|---|
| \(\hat{n}+\frac{1}{2} I\) | 0 | \(2 a^{\dagger} a^{\dagger}\) | \(-2 a a\) | \(a^{\dagger}\) | \(-a\) | 0 |
| \(a^{\dagger} a^{\dagger}\) | 0 | \(-4\left(\hat{n}+\frac{1}{2} I\right)\) | 0 | \(-2 a^{\dagger}\) | 0 | |
| \(a a\) | 0 | \(2 a\) | 0 | 0 | ||
| \(a^{\dagger}\) | 0 | \(-I\) | 0 | |||
| \(a\) | 0 | 0 | ||||
| I | 0 |
columns: | \(\hat{n}+\frac{1}{2} I\) | \(a^{\dagger} a^{\dagger}\) | \(a a\) | \(a^{\dagger}\) | \(a\) | I
| \(\mu\) | \(\sigma\) | Algebra | Singular subspace |
|---|---|---|---|
| +1 | +1 | \(\mathfrak{s} \mathfrak{o}(3,2)\) | |
| -1 | +1 | \(\mathfrak{s} \mathfrak{o}(4,1)\) | |
| -1 | -1 | \(\mathfrak{s} \mathfrak{o}(5)\) | |
| +1 | 0 | Poincare | translations \(t_{\mu}\) |
| 0 | 0 | Galilei | translations \(t_{\mu}\), boosts \(\mathbf{b}\) |
columns: \(\mu\) | \(\sigma\) | Algebra | Singular subspace
| \(\left[\begin{array}{cc|c}0 & \theta & v_{1} \\ -\theta & 0 & v_{2} \\ \hline v_{1} & v_{2} & 0\end{array}\right]\) | \(\left[\begin{array}{cc|cc}0 & \theta & v_{1} & t_{1} \\ -\theta & 0 & v_{2} & t_{2} \\ \hline v_{1} & v_{2} & 0 & t_{3} \\ 0 & 0 & 0 & 0\end{array}\right]\) | \(\left[\begin{array}{cc|cc}0 & \theta & v_{1} & t_{1} \\ -\theta & 0 & v_{2} & t_{2} \\ \hline 0 & 0 & 0 & t_{3} \\ 0 & 0 & 0 & 0\end{array}\right]\) |
|---|---|---|
| Lorentz | Poincare | Galilei |
columns: \(\left[\begin{array}{cc|c}0 & \theta & v_{1} \\ -\theta & 0 & v_{2} \\ \hline v_{1} & v_{2} & 0\end{array}\right]\) | \(\left[\begin{array}{cc|cc}0 & \theta & v_{1} & t_{1} \\ -\theta & 0 & v_{2} & t_{2} \\ \hline v_{1} & v_{2} & 0 & t_{3} \\ 0 & 0 & 0 & 0\end{array}\right]\) | \(\left[\begin{array}{cc|cc}0 & \theta & v_{1} & t_{1} \\ -\theta & 0 & v_{2} & t_{2} \\ \hline 0 & 0 & 0 & t_{3} \\ 0 & 0 & 0 & 0\end{array}\right]\)
| \(\left[H_{1}, H_{2}\right]\) | = | 0 |
|---|---|---|
| \(\left[\mathbf{H}, E_{ \pm 2 \mathbf{e}_{1}}\right]\) | = | \(( \pm 2 / \sqrt{12}, 0) E_{ \pm 2 \mathrm{e}_{1}}\) |
| \(\left[\mathbf{H}, E_{ \pm 2 \mathbf{e}_{2}}\right]\) | = | \((0, \pm 2 / \sqrt{12}) E_{ \pm 2 \mathrm{e}_{2}}\) |
| \(\left[\mathbf{H}, E_{ \pm \mathbf{e}_{1} \pm \mathbf{e}_{2}}\right]\) | = | \(( \pm / \sqrt{12}, \pm / \sqrt{12}) E_{ \pm \mathbf{e}_{1} \pm \mathbf{e}_{2}}\) |
| \(\left[E_{+2 \mathrm{e}_{1}}, E_{-2 \mathrm{e}_{1}}\right]\) | = | \((2 / \sqrt{12}) H_{1}\) |
| \(\left[E_{+2 \mathrm{e}_{2}}, E_{-2 \mathrm{e}_{2}}\right]\) | = | \((2 / \sqrt{12}) H_{2}\) |
| \(\left[E_{ \pm \mathbf{e}_{1} \pm \mathbf{e}_{2}}, E_{-\left( \pm \mathbf{e}_{1} \pm \mathbf{e}_{2}\right)}\right]\) | = | \((1 / \sqrt{12})\left( \pm H_{1} \pm H_{2}\right)\) |
| \(\left[E_{+2 \mathbf{e}_{1}}, E_{-\left(\mathbf{e}_{1}+\mathbf{e}_{2}\right)}\right]\) | = | \(*(1 / \sqrt{6}) E_{\mathbf{e}_{1}-\mathbf{e}_{2}}\) |
| \(\left[E_{-\mathbf{e}_{1}+\mathbf{e}_{2}}, E_{-\mathbf{e}_{1}-\mathbf{e}_{2}}\right]\) | = | \((1 / \sqrt{6}) E_{-2 \mathrm{e}_{1}}\) |
| \(\left[E_{-\mathbf{e}_{1}-\mathbf{e}_{2}}, E_{+2 \mathbf{e}_{1}}\right]\) | = | \((1 / \sqrt{6}) E_{+\mathbf{e}_{1}-\mathbf{e}_{2}}\) |
| \(\left[E_{-2 \mathbf{e}_{1}}, E_{\mathbf{e}_{1}-\mathbf{e}_{2}}\right]\) | = | \((-1 / \sqrt{6}) E_{-\mathbf{e}_{1}-\mathbf{e}_{2}}\) |
| \(\left[E_{+\mathbf{e}_{1}-\mathbf{e}_{2}}, E_{+\mathbf{e}_{1}+\mathbf{e}_{2}}\right]\) | = | \((-1 / \sqrt{6}) E_{+2 \mathrm{e}_{1}}\) |
| \(\left[E_{+\mathbf{e}_{1}+\mathbf{e}_{2}}, E_{-2 \mathbf{e}_{2}}\right]\) | = | \((-1 / \sqrt{6}) E_{+\mathbf{e}_{1}-\mathbf{e}_{2}}\) |
| \(\left[E_{+2 \mathbf{e}_{2}}, E_{-\mathbf{e}_{1}-\mathbf{e}_{2}}\right]\) | = | \(*(1 / \sqrt{6}) E_{-\mathbf{e}_{1}+\mathbf{e}_{2}}\) |
| \(\left[E_{-\mathbf{e}_{1}-\mathbf{e}_{2}}, E_{+\mathbf{e}_{1}-\mathbf{e}_{2}}\right]\) | = | \((1 / \sqrt{6}) E_{-2 \mathrm{e}_{2}}\) |
| \(\left[E_{+\mathbf{e}_{1}-\mathbf{e}_{2}}, E_{+2 \mathbf{e}_{2}}\right]\) | = | \((1 / \sqrt{6}) E_{+\mathbf{e}_{1}+\mathbf{e}_{2}}\) |
| \(\left[E_{-2 \mathbf{e}_{2}}, E_{+\mathbf{e}_{1}+\mathbf{e}_{2}}\right]\) | = | \((-1 / \sqrt{6}) E_{+\mathbf{e}_{1}-\mathbf{e}_{2}}\) |
| \(\left[E_{+\mathbf{e}_{1}+\mathbf{e}_{2}}, E_{-\mathbf{e}_{1}+\mathbf{e}_{2}}\right]\) | = | \((-1 / \sqrt{6}) E_{+2 \mathrm{e}_{2}}\) |
| \(\left[E_{-\mathbf{e}_{1}+\mathbf{e}_{2}}, E_{-2 \mathbf{e}_{2}}\right]\) | = | \((-1 / \sqrt{6}) E_{-\mathbf{e}_{1}-\mathbf{e}_{2}}\) |
columns: \(\left[H_{1}, H_{2}\right]\) | = | 0
| \((i \phi)^{j} f_{j}\) | Representation | |||
|---|---|---|---|---|
| \(\frac{1}{2}\) | 1 | \(\frac{3}{2}\) | 2 | |
| 2 | 3 | 4 | 5 | |
| \(f_{0}\) | \(\cos (\phi / 2)\) | 1 | \(\frac{9}{8} \cos \left(\frac{\phi}{2}\right)-\frac{1}{8} \cos \left(\frac{3 \phi}{2}\right)\) | 1 |
| \((i \phi)^{1} f_{1}\) | \(2 i \sin (\phi / 2)\) | \(i \sin (\phi)\) | \(\frac{9 i}{4} \sin \left(\frac{\phi}{2}\right)-\frac{i}{12} \sin \left(\frac{3 \phi}{2}\right)\) | \(\frac{i}{3} \sin (\phi)-\frac{i}{6} \sin (2 \phi)\) |
| \((i \phi)^{2} f_{2}\) | \(\cos (\phi)-1\) | \(-\frac{1}{2} \cos \left(\frac{\phi}{2}\right)+\frac{1}{2} \cos \left(\frac{3 \phi}{2}\right)\) | \(-\frac{5}{4}+\frac{1}{3} \cos (\phi)-\frac{1}{12} \cos (2 \phi)\) | |
| \((i \phi)^{3} f_{3}\) | \(-i \sin \left(\frac{\phi}{2}\right)+\frac{i}{3} \sin \left(\frac{3 \phi}{2}\right)\) | \(-\frac{i}{3} \sin (\phi)+\frac{i}{6} \sin (2 \phi)\) | ||
| \((i \phi)^{4} f_{4}\) | \(\frac{1}{4}-\frac{1}{3} \cos (\phi)+\frac{1}{12} \cos (2 \phi)\) | |||
columns: \((i \phi)^{j} f_{j}\) | Representation \(\frac{1}{2}\) | Representation 1 | Representation \(\frac{3}{2}\) | Representation 2
| \(\cos ^{2}(\alpha, \beta)\) | \(\theta(\alpha, \beta)\) | \(n=\frac{2 \alpha \cdot \beta}{\alpha \cdot \alpha}\) | \(n^{\prime}=\frac{2 \alpha \cdot \beta}{\beta \cdot \beta}\) | \(\frac{\alpha \cdot \alpha}{\beta \cdot \beta}=\frac{n^{\prime}}{n}\) |
|---|---|---|---|---|
| 1 | \(\frac{\pi}{2} \pm \frac{\pi}{2}\) | \(\pm 2\) | \(\pm 2\) | 1 |
| 3 | \(\frac{\pi}{2} \pm \frac{\pi}{3}\) | \(\pm 3\) | \(\pm 1\) | \(3^{-1}\) |
| \(\pm 1\) | \(\pm 3\) | \(3^{+1}\) | ||
| \(\frac{2}{4}\) | \(\frac{\pi}{2} \pm \frac{\pi}{4}\) | \(\pm 2\) | \(\pm 1\) | \(2^{-1}\) |
| \(\pm 1\) | \(\pm 2\) | \(2^{+1}\) | ||
| \(\frac{1}{4}\) | \(\frac{\pi}{2} \pm \frac{\pi}{6}\) | \(\pm 1\) | \(\pm 1\) | 1 |
| 0 | \(\frac{\pi}{2}\) | 0 | 0 | - |
columns: \(\cos ^{2}(\alpha, \beta)\) | \(\theta(\alpha, \beta)\) | \(n=\frac{2 \alpha \cdot \beta}{\alpha \cdot \alpha}\) | \(n^{\prime}=\frac{2 \alpha \cdot \beta}{\beta \cdot \beta}\) | \(\frac{\alpha \cdot \alpha}{\beta \cdot \beta}=\frac{n^{\prime}}{n}\)
| \(\alpha_{1}\) | \(\alpha_{2}\) | \(\alpha_{3}\) | \(\alpha_{l-1}\) | \(\alpha_{l}\) | ||
|---|---|---|---|---|---|---|
| \(A_{l-1}\) | \(\mathbf{e}_{1}-\mathbf{e}_{2}\) | \(\mathbf{e}_{2}-\mathbf{e}_{3}\) | \(\mathbf{e}_{3}-\mathbf{e}_{4}\) | ... | \(\mathbf{e}_{l-1}-\mathbf{e}_{l}\) | |
| \(D_{l}\) | " | " | " | ... | " | \(\mathbf{e}_{l-1}+\mathbf{e}_{l}\) |
| \(B_{l}\) | " | ,, | " | ... | " | \(\mathbf{e}_{l}\) |
| \(C_{l}\) | " | ,, | " | ... | " | \(2 \mathbf{e}_{l}\) |
columns: | \(\alpha_{1}\) | \(\alpha_{2}\) | \(\alpha_{3}\) | | \(\alpha_{l-1}\) | \(\alpha_{l}\)
| \(p\) | \(q\) | \(r\) | Root space |
|---|---|---|---|
| \(p\) | 2 | 2 | \(D_{p+2}\) |
| 3 | 3 | 2 | \(E_{6}\) |
| 4 | 3 | 2 | \(E_{7}\) |
| 5 | 3 | 2 | \(E_{8}\) |
columns: \(p\) | \(q\) | \(r\) | Root space
| Canonical form | Boson operators | Coordinates and derivatives | Fermion operators |
|---|---|---|---|
| \(H_{i}\) | \(b_{i}^{\dagger} b_{i}\) | \(x^{i} \partial_{i}\) | \(f_{i}^{\dagger} f_{i}\) |
| \(E_{+\mathbf{e}_{i}-\mathbf{e}_{j}}\) | \(b_{i}^{\dagger} b_{j}\) | \(x^{i} \partial_{j}\) | \(f_{i}^{\dagger} f_{j}\) |
columns: Canonical form | Boson operators | Coordinates and derivatives | Fermion operators
| Canonical form | \(C_{l}\) | \(D_{l}\) | ||
|---|---|---|---|---|
| Boson operators | Coordinates and derivatives | Fermion operators | Coordinates and derivatives | |
| \(H_{i}\) | \(b_{i}^{\dagger} b_{i}+\frac{1}{2}\) | \(x^{i} \partial_{i}\) | \(f_{i}^{\dagger} f_{i}+\frac{1}{2}\) | \(x^{i} \partial_{i}+\frac{1}{2}\) |
| \(E_{+\mathbf{e}_{i}-\mathbf{e}_{j}}\) | \(b_{i}^{\dagger} b_{j}\) | \(x^{i} \partial_{j}\) | \(f_{i}^{\dagger} f_{j}\) | \(x^{i} \partial_{j}\) |
| \(E_{+\mathbf{e}_{i}+\mathbf{e}_{j}}\) | \(b_{i}^{\dagger} b_{j}^{\dagger}\) | \(x^{i} x^{j}\) | \(f_{i}^{\dagger} f_{j}^{\dagger}\) | \(x^{i} x^{j}\) |
| \(E_{-\mathbf{e}_{i}-\mathbf{e}_{j}}\) | \(b_{i} b_{j}\) | \(\partial_{i} \partial_{j}\) | \(f_{i} f_{j}\) | \(\partial_{i} \partial_{j}\) |
| \(E_{+2 \mathrm{e}_{i}}\) | \(b_{i}^{\dagger} b_{i}^{\dagger}\) | \(x^{i} x^{i}\) | ||
| \(E_{-2 \mathbf{e}_{i}}\) | \(b_{i} b_{i}\) | \(\partial_{i} \partial_{i}\) | ||
columns: Canonical form | \(C_{l}\) Boson operators | \(C_{l}\) Coordinates and derivatives | \(D_{l}\) Fermion operators | \(D_{l}\) Coordinates and derivatives
| Mapping | Real form | Root space | Condition |
|---|---|---|---|
| Block submatrix | \(\mathfrak{s} \mathfrak{o}(p, q)\) | \(D_{n}\) | \(p+q=2 n\) |
| \(\mathfrak{s} \mathfrak{o}(p, q)\) | \(B_{n}\) | \(p+q=2 n+1\) | |
| \(\mathfrak{s} \mathfrak{u}(p, q)\) | \(A_{n-1}\) | \(p+q=n\) | |
| \(\mathfrak{s} \mathfrak{p}(p, q)\) | \(C_{n}\) | \(p+q=n\) | |
| Subfield restriction | \(\mathfrak{s l}(n ; \mathbb{R})\) | \(A_{n-1}\) | |
| \(\mathfrak{s} \mathfrak{p}(2 n ; \mathbb{R})\) | \(C_{n}\) | ||
| Field embedding | \(\mathfrak{s} \mathfrak{o}^{*}(2 n)\) | \(D_{n}\) | |
| \(\mathfrak{s} \mathfrak{u}^{*}(2 n)\) | \(A_{2 n-1}\) |
columns: Mapping | Real form | Root space | Condition
| \(A_{1}\) | ~ | \(B_{1}\) | ~ | \(C_{1}\) |
|---|---|---|---|---|
| \(\mathfrak{s} \mathfrak{u}(2)\) | ~ | \(\mathfrak{s} \mathfrak{o}(3)\) | ~ | \(\mathfrak{s} \mathfrak{p}(1)=\mathfrak{u} \mathfrak{s} \mathfrak{p}(2)\) |
| \(\mathfrak{s u}(1,1)=\mathfrak{s l}(2 ; \mathbb{R})\) | ~ | \(\mathfrak{s} \mathfrak{o}(2,1)\) | ~ | \(\mathfrak{s} \mathfrak{p}(2 ; \mathbb{R})\) |
| \(D_{2}\) | = | \(A_{1}\) | + | \(A_{1}\) |
| \(\mathfrak{s} \mathfrak{o}(4)\) | = | \(\mathfrak{s} \mathfrak{o}(3)\) | + | \(\mathfrak{s} \mathfrak{o}(3)\) |
| \(\mathfrak{s} \mathfrak{o}^{*}(4)\) | ~ | \(\mathfrak{s} \mathfrak{o}(3)\) | + | \(\mathfrak{s} \mathfrak{o}(2,1)\) |
| \(\mathfrak{s} \mathfrak{o}(3,1)\) | ~ | \(\mathfrak{s} \mathfrak{l}(2 ; \mathbb{C})\) | ||
| \(\mathfrak{s} \mathfrak{o}(2,2)\) | ~ | \(\mathfrak{s} \mathfrak{o}(2,1)\) | + | \(\mathfrak{s} \mathfrak{o}(2,1)\) |
| \(B_{2}\) | = | \(C_{2}\) | ||
| \(\mathfrak{s} \mathfrak{o}(5)\) | ~ | \(\mathfrak{s} \mathfrak{p}(2)=\mathfrak{u} \mathfrak{s} \mathfrak{p}(4)\) | ||
| \(\mathfrak{s} \mathfrak{o}(4,1)\) | ~ | \(\mathfrak{s} \mathfrak{p}(1,1)=\mathfrak{u} \mathfrak{s} \mathfrak{p}(2,2)\) | ||
| \(\mathfrak{s} \mathfrak{o}(3,2)\) | ~ | \(\mathfrak{s} \mathfrak{p}(4 ; R)\) | ||
| \(D_{3}\) | = | \(A_{3}\) | ||
| \(\mathfrak{s} \mathfrak{o}(6)\) | ~ | \(\mathfrak{s} \mathfrak{u}(4)\) | ||
| \(\mathfrak{s} \mathfrak{o}(5,1)\) | ~ | \(\mathfrak{s} \mathfrak{u}^{*}(4)\) | ||
| \(\mathfrak{s} \mathfrak{o}^{*}(6)\) | ~ | \(\mathfrak{s} \mathfrak{u}(3,1)\) | ||
| \(\mathfrak{s} \mathfrak{o}(4,2)\) | ~ | \(\mathfrak{s} \mathfrak{u}(2,2)\) | ||
| \(\mathfrak{s} \mathfrak{o}(3,3)\) | ~ | \(\mathfrak{s} \mathfrak{l}(4 ; \mathbb{R})\) |
columns: \(A_{1}\) | ~ | \(B_{1}\) | ~ | \(C_{1}\)
| Root space | Class \({ }_{\text {rank(character) }}\) | Maximal compact subgroup | |
|---|---|---|---|
| Root space | Dimension | ||
| \(G_{2}\) | \(G_{2(-14)}\) | \(G_{2}\) | 14 |
| \(G_{2(+2)}\) | \(A_{1}+A_{1}\) | 6 | |
| \(F_{4}\) | \(F_{4(-52)}\) | \(F_{4}\) | 52 |
| \(F_{4(-20)}\) | \(B_{4}\) | 36 | |
| \(F_{4(+4)}\) | \(C_{3}+A_{1}\) | 24 | |
| \(E_{6}\) | \(E_{6(-78)}\) | \(E_{6}\) | 78 |
| \(E_{6(-26)}\) | \(F_{4}\) | 52 | |
| \(E_{6(-14)}\) | \(D_{5}+D_{1}\) | 46 | |
| \(E_{6(+2)}\) | \(A_{5}+A_{1}\) | 38 | |
| \(E_{6(+6)}\) | \(C_{4}\) | 36 | |
| \(E_{7}\) | \(E_{7(-133)}\) | \(E_{7}\) | 133 |
| \(E_{7(-25)}\) | \(E_{6}+D_{1}\) | 79 | |
| \(E_{7(-5)}\) | \(D_{6}+A_{1}\) | 69 | |
| \(E_{7(+7)}\) | \(A_{7}\) | 63 | |
| \(E_{8}\) | \(E_{8(-248)}\) | \(E_{8}\) | 248 |
| \(E_{8(-24)}\) | \(E_{7}+A_{1}\) | 136 | |
| \(E_{8(+8)}\) | \(D_{8}\) | 120 | |
columns: Root space | Class \({ }_{\text {rank(character) }}\) | Maximal compact subgroup Root space | Maximal compact subgroup Dimension
| Quadrant | Operator | Edge |
|---|---|---|
| I | \(J_{+}\) | \(n_{2}=0\) |
| I | \(J_{-}\) | \(n_{1}=0\) |
| III | \(J_{+}\) | \(n_{1}=-1\) |
| III | \(J_{-}\) | \(n_{2}=-1\) |
columns: Quadrant | Operator | Edge
| Root space | Quotient \(G^{\prime} / H\) | Dimension \(P\) | Rank \(P\) |
|---|---|---|---|
| \(A_{p+q-1}\) | \(S U(p, q) / S[U(p) \otimes U(q)]\) | \(2 p q\) | \(\min (p, q)\) |
| \(A_{n-1}\) | \(S L(n ; \mathbb{R}) / S O(n)\) | \(\frac{1}{2}(n+2)(n-1)\) | \(n-1\) |
| \(A_{2 n-1}\) | \(S U^{*}(2 n) / U \operatorname{Sp}(2 n)\) | \((2 n+1)(n-1)\) | \(n-1\) |
| \(B_{p+q}\) | \(S O(p, q) / S O(p) \otimes S O(q)\) | \(p q\) | \(\min (p, q)\) |
| \(D_{p+q}\) | \(S O(p, q) / S O(p) \otimes S O(q)\) | \(p q\) | \(\min (p, q)\) |
| \(D_{n}\) | \(S O^{*}(2 n) / U(n)\) | \(n(n-1)\) | [ \(n / 2\) ] |
| \(C_{p+q}\) | \(U S p(2 p, 2 q) / U S p(2 p) \otimes U S p(2 q)\) | \(4 p q\) | \(\min (p, q)\) |
| \(C_{n}\) | \(S p(2 n ; \mathbb{R}) / U(n)\) | \(n(n+1)\) | \(n\) |
columns: Root space | Quotient \(G^{\prime} / H\) | Dimension \(P\) | Rank \(P\)
| Root space | \(G^{\prime} / H\) | \(\operatorname{Dim} G^{\prime}\) | \(\operatorname{Dim} H\) | \(\operatorname{Dim} P\) | Rank \(P\) |
|---|---|---|---|---|---|
| \(G_{2}\) | \(G_{2(+2)} /\left(A_{1} \oplus A_{1}\right)\) | 14 | 6 | 8 | 2 |
| \(F_{4}\) | \(F_{4(-20)} / B_{4}\) | 52 | 36 | 16 | 1 |
| \(F_{4(+4)} /\left(C_{3} \oplus A_{1}\right)\) | 52 | 24 | 28 | 4 | |
| \(E_{6}\) | \(E_{6(-26)} / F_{4}\) | 78 | 52 | 26 | 2 |
| \(E_{6(-14)} /\left(D_{5} \oplus D_{1}\right)\) | 78 | 46 | 32 | 2 | |
| \(E_{6(+2)} /\left(A_{5} \oplus A_{1}\right)\) | 78 | 38 | 40 | 4 | |
| \(E_{6(+6)} / C_{4}\) | 78 | 36 | 42 | 6 | |
| \(E_{7}\) | \(E_{7(-25)} /\left(E_{6} \oplus D_{1}\right)\) | 133 | 79 | 54 | 3 |
| \(E_{7(-5)} /\left(D_{6} \oplus A_{1}\right)\) | 133 | 69 | 64 | 4 | |
| \(E_{7(+7)} / A_{7}\) | 133 | 63 | 70 | 7 | |
| \(E_{8}\) | \(E_{8(-24)} /\left(E_{7} \oplus A_{1}\right)\) | 248 | 136 | 112 | 4 |
| \(E_{8(+8)} / D_{8}\) | 248 | 120 | 128 | 8 |
columns: Root space | \(G^{\prime} / H\) | \(\operatorname{Dim} G^{\prime}\) | \(\operatorname{Dim} H\) | \(\operatorname{Dim} P\) | Rank \(P\)
| Operation | \(i \sigma_{1}\) | \(i \sigma_{2}\) | \(i \sigma_{3}\) |
|---|---|---|---|
| \(T_{1}=\) block matrix decomposition | -1 | -1 | +1 |
| \(T_{2}=\) complex conjugation | -1 | +1 | -1 |
| \(T_{3}=T_{1} T_{2}\) | +1 | -1 | -1 |
columns: Operation | \(i \sigma_{1}\) | \(i \sigma_{2}\) | \(i \sigma_{3}\)
| Bessel function | → | harmonic oscillator eigenfunction |
|---|---|---|
| Bessel equation | → | Schrödinger equation for harmonic oscilator |
columns: Bessel function | → | harmonic oscillator eigenfunction
| Equation | \(A\) | \(B\) | \(C\) |
|---|---|---|---|
| Klein-Gordon | \(-l(l+1)+\left(e^{2} / \hbar c\right)^{2}\) | \(2 E e^{2} /(\hbar c)^{2}\) | \(\left[E^{2}-\left(m c^{2}\right)^{2}\right] /(\hbar c)^{2}\) |
| Schrödinger | \(-l(l+1)\) | \(2 m e^{2} / \hbar^{2}\) | \(2 \mathrm{~m} \mathrm{~W} / \hbar^{2}\) |
columns: Equation | \(A\) | \(B\) | \(C\)
| Procedure | Result | |
|---|---|---|
| 1 | Locate singularities | \(0, \infty\) |
| 2 | Determine analytic behavior at singular points | \[ \begin{aligned} & r \rightarrow 0: R \simeq r^{\gamma}, \gamma(\gamma-1)+A=0 \\ & r \rightarrow \infty: R \simeq e^{\lambda r}, \quad \lambda^{2}+C=0 \end{aligned} \] |
| 3 | Keep only \(\mathcal{L}^{2}\) solutions | \(\gamma=\frac{1}{2}+\sqrt{\left(\frac{1}{2}\right)^{2}-A}, \lambda=-\sqrt{-C}\) |
| 4 | Look for solutions with proper asymptotic behavior | \(R=r^{\gamma} e^{\lambda r} f(r)\) |
| 5 | Construct differential equation for \(f(r)\) | \(\left[\left(r D^{2}+2 \gamma D\right)+(2 \lambda \gamma+B+2 \lambda r D)\right] f(r)=0\) |
| 6 | Construct recursion relation | \(f_{j+1}=-\frac{2 \lambda(j+\gamma)+B}{j(j+1)+2 \gamma(j+1)} f_{j}\) |
| 7 | Look at asymptotic behavior | \[ \begin{aligned} & f \simeq e^{-2 \lambda r} \text { if series does not terminate } \\ & \simeq e^{+1 \lambda r} \text { if series does terminate }(\lambda<0) \end{aligned} \] |
| 8 | Construct quantization condition | \[ \begin{aligned} & 2 \lambda(n+\gamma)+B=0 \text { or } \\ & n+\frac{1}{2}+\sqrt{\left(\frac{1}{2}\right)^{2}-A}=\frac{B}{2 \sqrt{-C}} \end{aligned} \] |
| 9 | Construct explicit solutions | \[ \begin{aligned} & E=\frac{m c^{2}}{\sqrt{1+\left(\alpha / N^{\prime}\right)^{2}}}, W=-\frac{1}{2} m c^{2} \alpha^{2} \frac{1}{N^{2}} \\ & N^{\prime}=n+\frac{1}{2}+\sqrt{\left(l+\frac{1}{2}\right)^{2}-\alpha^{2}}, \quad N=n+l+1 \end{aligned} \] |
columns: | Procedure | Result
| \(m\) | \(l=0\) | \(l=1\) | \(l=2\) | \(l=3\) |
|---|---|---|---|---|
| 0 | \(\sqrt{\frac{1}{4 \pi}}\) | \(\sqrt{\frac{3}{4 \pi}} \cos \theta\) | \(\sqrt{\frac{5}{16 \pi}}\left(3 \cos ^{2} \theta-1\right)\) | \(\sqrt{\frac{7}{16 \pi}}\left(5 \cos ^{3} \theta-3 \cos \theta\right)\) |
| \(\pm 1\) | \(\mp \sqrt{\frac{3}{8 \pi}} \sin \theta e^{ \pm i \phi}\) | \(\mp \sqrt{\frac{15}{8 \pi}} \cos \theta \sin \theta e^{ \pm i \phi}\) | \(\mp \sqrt{\frac{21}{64 \pi}} \sin \theta\left(5 \cos ^{2} \theta-1\right) e^{ \pm i \phi}\) | |
| \(\pm 2\) | \(\sqrt{\frac{15}{32 \pi}} \sin ^{2} \theta e^{ \pm 2 i \phi}\) | \(\sqrt{\frac{105}{32 \pi}} \sin ^{2} \theta \cos \theta e^{ \pm 2 i \phi}\) | ||
| \(\pm 3\) | \(\mp \sqrt{\frac{35}{64 \pi}} \sin ^{3} \theta e^{ \pm 3 i \phi}\) |
columns: \(m\) | \(l=0\) | \(l=1\) | \(l=2\) | \(l=3\)
| H | \(\frac{p^{2}}{2 m}-\frac{\alpha}{r}\) | \(\sqrt{p^{2}+m^{2}}-\frac{\alpha}{r}\) |
|---|---|---|
| \(A_{4}\) | \(\mathbf{r} \cdot \mathbf{p}-i\) | \(\mathbf{r} \cdot \mathbf{p}-i\) |
| \(\Gamma_{4}\) | \(\frac{1}{2}(r \mathbf{p} \cdot \mathbf{p}-r)\) | \(\frac{1}{2}\left(r \mathbf{p} \cdot \mathbf{p}-r-\frac{\alpha^{2}}{r}\right)\) |
| \(\Gamma_{5}\) | \(\frac{1}{2}(r \mathbf{p} \cdot \mathbf{p}+r)\) | \(\frac{1}{2}\left(r \mathbf{p} \cdot \mathbf{p}+r-\frac{\alpha^{2}}{r}\right)\) |
| \(\Theta\) | \(r\left(H_{S}-W\right)\) | \(r\left\{\left(H_{K G}+\frac{\alpha}{r}\right)^{2}-\left(E+\frac{\alpha}{r}\right)^{2}\right\}\) |
| \(A\left(\Gamma_{5}+\Gamma_{4}\right)+B\left(\Gamma_{5}-\Gamma_{4}\right)+C\) | \(A\left(\Gamma_{5}+\Gamma_{4}\right)+B\left(\Gamma_{5}-\Gamma_{4}\right)+C\) | |
| \(A\) | \(1 / 2 m\) | 1 |
| B | \(-W\) | \(m^{2}-E^{2}\) |
| C | \(-\alpha\) | \(-2 \alpha E\) |
columns: H | \(\frac{p^{2}}{2 m}-\frac{\alpha}{r}\) | \(\sqrt{p^{2}+m^{2}}-\frac{\alpha}{r}\)
| Particle | Rest energy (MeV) |
|---|---|
| electron \(e^{ \pm}\) | 0.511 |
| mu meson \(\mu^{ \pm}\) | 105.7 |
| tau meson \(\tau^{ \pm}\) | 1784.0 |
| proton, antiproton \(p^{ \pm}\) | 938.26 |
| deuteron \(d^{+}\) | 1875.6 |
| tritium \(t^{+}\) | 2809.4 |
| pi meson \(\pi^{ \pm}\) | 139.6 |
| sigma meson \(\Sigma^{ \pm}\) | 1385.0 |
| cascade meson \(\Xi^{-}\) | 1533.0 |
| omega \(\Omega^{-}\) | 1672.0 |
columns: Particle | Rest energy (MeV)
| Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune |
|---|---|---|---|---|---|---|---|
| 0.39 | 0.72 | 1.00 | 1.52 | 5.20 | 9.54 | 19.18 | 30.06 |
columns: Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune
| Time period | Approach | Strengths | Weaknesses |
|---|---|---|---|
| Nineteenth century | Manifestly covariant | Fields have elegant transformation properties | Many fields represent nonphysical states |
| Twentieth century | Hilbert space | All linear superpositions represent physical states | Transformation properties are complicated |
columns: Time period | Approach | Strengths | Weaknesses
| \(|k\rangle\) | \(\left.\begin{array}{cc}j & j^{\prime} \\ \mu & \mu^{\prime}\end{array}\right\}\) | |
|---|---|---|
| \(\{I, a\}\) | \(|k\rangle e^{i k \cdot a}\) | \(\left.\begin{array}{cc}j & j^{\prime} \\ v & v^{\prime}\end{array}\right\rangle_{v v^{\prime} ; \mu \mu^{\prime}}\) |
| \{ \(\Lambda\), 0\} | \(|\Lambda k\rangle\) | \(\left.\begin{array}{cc}j & j^{\prime} \\ v & v^{\prime}\end{array}\right\rangle D_{v v^{\prime} ; \mu \mu^{\prime}}^{j j^{\prime}}(\Lambda)\) |
columns: | \(|k\rangle\) | \(\left.\begin{array}{cc}j & j^{\prime} \\ \mu & \mu^{\prime}\end{array}\right\}\)
| Electric | Magnetic | |
|---|---|---|
| Charge density | \(\rho_{e}(\mathbf{x}, t)=\sum_{j} e_{j} \mathbf{x}_{j}(t)\) | \(\rho_{m}(\mathbf{x}, t)=\sum_{j} m_{j} \mathbf{x}_{j}(t)\) |
| Current density | \(\mathbf{J}_{e}(\mathbf{x}, t)=\sum_{j} e_{j} \frac{d \mathbf{x}_{j}(t)}{d t}\) | \(\mathbf{J}_{m}(\mathbf{x}, t)=\sum_{j} m_{j} \frac{d \mathbf{x}_{j}(t)}{d t}\) |
| Conservation law | \(\nabla \cdot \mathbf{J}_{e}(\mathbf{x}, t)+\frac{\partial \rho_{e}(\mathbf{x}, t)}{\partial t}=0\) | \(\nabla \cdot \mathbf{J}_{m}(\mathbf{x}, t)+\frac{\partial \rho_{m}(\mathbf{x}, t)}{\partial t}=0\) |
columns: | Electric | Magnetic
| Infinitesimal generators | Canonical coordinates | Constraint | ||||
|---|---|---|---|---|---|---|
| \(\xi(x, y)\) | \(\eta(x, y)\) | \(\zeta(x, y, p)\) | \(R(x, y)\) | \(S(x, y)\) | \(T(x, y, p)\) | \(d S / d R\) |
| 0 | 1 | 0 | \(x\) | \(y\) | \(p\) | \(T\) |
| 1 | 0 | 0 | \(y\) | \(x\) | \(p\) | \(1 / T\) |
| \(1 / a\) | \(-1 / b\) | 0 | \(a x+b y\) | \(b x-a y\) | \(p\) | \((b-a T) /(a+b T)\) |
| \(x\) | 0 | \(-p\) | \(y\) | \(\ln x\) | \(x p\) | \(1 / T\) |
| 0 | \(y\) | \(p\) | \(x\) | \(\ln y\) | \(p / y\) | \(T\) |
| \(x / a\) | \(y / b\) | \((1 / b-1 / a) p\) | \(y^{b} / x^{a}\) | \(b \ln y\) | \(p / x^{(a / b-1)}\) | \((b T / R) /\left(b T-a R^{(1 / b)}\right)\) |
| \(x\) | \(y\) | 0 | \(y / x\) | \(\ln y\) | \(p\) | \((T / R) /(T-R)\) |
| \(x\) | \(-y\) | \(-2 p\) | \(x y\) | \(\ln y\) | \(x^{2} p\) | \((T / R) /(T+R)\) |
| \(2 x\) | \(y\) | \(-p\) | \(y^{2} / x\) | \(\ln y\) | \(y p\) | \((T / R) /(2 T-R)\) |
| \(x\) | \(2 y\) | \(p\) | \(y / x^{2}\) | \(\ln y\) | \(p / x\) | \((T / R) /(T-2 R)\) |
| \(y\) | 0 | \(-p^{2}\) | \(y\) | \(x / y\) | \(x-y / p\) | \(-T / R^{2}\) |
| 0 | \(x\) | 1 | \(x\) | \(y / x\) | \(x p-y\) | \(T / R^{2}\) |
| \(-y\) | \(x\) | \(1+p^{2}\) | \(\sqrt{x^{2}+y^{2}}\) | \(\tan ^{-1}(y / x)\) | \((y-x p) /(x+y p)\) | \(-T / R\) |
| 1 | \(y / x\) | \((p x-y) / x^{2}\) | \(y / x\) | \(x\) | \((x p-y) / x^{2}\) | \(1 / T\) |
| \(a\) | \(x\) | 1 | \(x^{2}-2 a y\) | \(x / a\) | \(x-a p\) | \(1 /(2 a T)\) |
| \(a\) | \(y\) | \(p\) | \(x-a \ln y\) | \(x / a\) | \(p / y\) | \((1 / a) /(1-a T)\) |
| \(x\) | \(b\) | \(-p\) | \(e^{y} / x^{b}\) | \(y / b\) | \(e^{y} * p^{b}\) | \((b R)^{-1} /\) |
| \(\left[1-b(R / T)^{(1 / b)}\right]\) | ||||||
| \(y\) | \(b\) | \(-p^{2}\) | \(y^{2}-2 b x\) | \(y / b\) | \(y-b / p\) | \(1 /(2 b T)\) |
| 0 | \(e^{f(x)}\) | \(f^{\prime} e^{f}\) | \(x\) | \(y / e^{f}\) | \(p-y f^{\prime}\) | \(T / e^{f(R)}\) |
| \(x^{2}\) | \(x y\) | \(y-x p\) | \(y / x\) | \(1 / x\) | \(x p-y\) | \(1 / T\) |
| \(x y\) | \(y^{2}\) | \(y p-x p^{2}\) | \(y / x\) | \(1 / y\) | \(y / p-x\) | \(1 /\left(T R^{2}\right)\) |
| \(x y\) | 0 | \(-y p-x p^{2}\) | \(y\) | \((\ln x) / y\) | \(y /(x p)-\ln x\) | \(T / R^{2}\) |
| 0 | \(x y\) | \(y+x p\) | \(x\) | \((\ln y) / x\) | \(x p / y-\ln y\) | \(T / R^{2}\) |
| \(g(y)\) | 0 | \(-g^{\prime} p^{2}\) | \(y\) | \(x / g\) | \(1 / p-x g^{\prime} / g\) | \(T / g(R)\) |
| 0 | \(f(x)\) | \(f^{\prime}\) | \(x\) | \(y / f\) | \(f p-f^{\prime} y\) | \(T / f^{2}(R)\) |
| \(f(x)\) | 0 | \(-f^{\prime} p\) | \(y\) | \(F\left(F^{\prime} f=1\right)\) | \(p f\) | \(1 / T\) |
| 0 | \(g(y)\) | \(g^{\prime} p\) | \(x\) | \(G\left(G^{\prime} g=1\right)\) | \(p / g\) | \(T\) |
| \(x^{k+1}\) | \(k x^{k} y\) | \(x^{k}\left(k^{2} y / x-p\right)\) | \(y / x^{k}\) | \(1 / x^{k}\) | \(x p-k y\) | \(-k / T\) |
| \(k x y^{k}\) | \(y^{k+1}\) | \(y^{k}\left(p-k^{2} x p^{2} / y\right)\) | \(x / y^{k}\) | \(1 / y^{k}\) | \(y / p-k x\) | \(-k / T\) |
columns: Infinitesimal generators \(\xi(x, y)\) | Infinitesimal generators \(\eta(x, y)\) | Infinitesimal generators \(\zeta(x, y, p)\) | Canonical coordinates \(R(x, y)\) | Canonical coordinates \(S(x, y)\) | Canonical coordinates \(T(x, y, p)\) | Constraint \(d S / d R\)
| 1830 | Galois | solve algebraic equations |
|---|---|---|
| 1874 | Lie | solve differential equations |
| 1969 | Risch | integrate in closed form |
| ? | - | solve determining equations for \(\xi, \eta\) |
| ? | - | solve determining equations for \(R, S, T\) |
| ? | - | solve \(F(R,-, T)=0\) for \(R\). |
columns: 1830 | Galois | solve algebraic equations
| \(k\) | Potential type | Transformation |
|---|---|---|
| -1 | Coulomb | \(\gamma^{2}=\beta^{3}\) |
| 0 | no force | \(\gamma^{2}=\beta^{2}\) |
| +1 | local gravitational potential | \(\gamma^{2}=\beta^{1}\) |
| +2 | harmonic oscillator | \(\gamma^{2}=\beta^{0}\) |
columns: \(k\) | Potential type | Transformation
| \(v_{i}\) | Infinitesial | \(e^{\epsilon v_{i}} f(x, t)=\) |
|---|---|---|
| \(v_{1}\) | \(\partial_{x}\) | \(f(x-\epsilon, t)\) |
| \(v_{2}\) | \(\partial_{t}\) | \(f(x, t-\epsilon)\) |
| \(v_{3}\) | \(u \partial_{u}\) | \(e^{\epsilon} f(x, t)\) |
| \(v_{4}\) | \(x \partial_{x}+2 t \partial_{t}\) | \(f\left(e^{-\epsilon} x, e^{-2 \epsilon} t\right)\) |
| \(v_{5}\) | \(2 t \partial_{x}-x u \partial_{u}\) | \(e^{-\epsilon x+\epsilon^{2} t} f(x-2 \epsilon t, t)\) |
| \(v_{6}\) | \(4 x t \partial_{x}+4 t^{2} \partial_{t}-\left(x^{2}+2 t\right) u \partial_{u}\) | \(\lambda e^{-\epsilon \lambda^{2} x^{2}} f\left(\lambda^{2} x, \lambda^{2} t\right)\) where \(\lambda^{2}=1 /(1+4 \epsilon t)\) |
columns: \(v_{i}\) | Infinitesial | \(e^{\epsilon v_{i}} f(x, t)=\)
| Symmetry | Conserved quantity |
|---|---|
| Space displacements | momentum |
| Time displacements | energy |
| Space-time displacements | four-momentum |
| Rotations | angular momentum |
columns: Symmetry | Conserved quantity