光円錐座標系

${\displaystyle dx^{2}=\eta _{\mu \nu }\,dx^{\mu }dx^{\nu }=(dx^{1})^{2}+(dx^{2})^{2}+\cdots +(dx^{d-1})^{2}+(dx^{d})^{2}-(dx^{d+1})^{2}}$

${\displaystyle {\begin{cases}x^{+}=(x^{d+1}+x^{d})/{\sqrt {2}}\\x^{-}=(x^{d+1}-x^{d})/{\sqrt {2}}\\\end{cases}}}$

${\displaystyle dx^{2}=\eta _{\mu \nu }\,dx^{\mu }dx^{\nu }=(dx^{1})^{2}+(dx^{2})^{2}+\cdots +(dx^{d-1})^{2}-2dx^{+}dx^{-}}$

${\displaystyle \eta ={\begin{pmatrix}1&0&\cdots &0&0&0\\0&1&&0&0&0\\\vdots &&\ddots &&\vdots &\vdots \\0&0&&1&0&0\\0&0&\cdots &0&0&-1\\0&0&\cdots &0&-1&0\\\end{pmatrix}}}$

${\displaystyle x_{+}=-x^{-},~x_{-}=-x^{+}}$

ローレンツ変換

${\displaystyle {\begin{cases}x'^{d}=x^{d}\cosh \beta -x^{d+1}\sinh \beta \\x'^{d+1}=x^{d+1}\cosh \beta -x^{d}\sinh \beta \\\end{cases}}}$

${\displaystyle {\begin{cases}x'^{+}=\mathrm {e} ^{-\beta }x^{+}\\x'^{-}=\mathrm {e} ^{+\beta }x^{-}\\\end{cases}}}$

粒子の運動

${\displaystyle t_{\text{lc}}={\frac {1}{c}}x^{+}}$

${\displaystyle V^{\mu }={\frac {dx^{\mu }}{dt_{\text{lc}}}}}$

${\displaystyle V^{+}=c}$

${\displaystyle S[x]=-mc\int ds=-mc\int {\sqrt {-\eta _{\mu \nu }V^{\mu }V^{\nu }}}\,dt_{\text{lc}}}$

${\displaystyle p_{\mu }={\frac {\partial L}{\partial V^{\mu }}}={\frac {mcV_{\mu }}{\sqrt {-V^{2}}}}}$

${\displaystyle H=p_{-}V^{-}+\sum _{i=1}^{d-1}p_{i}V^{i}-L=-V^{+}p_{+}=cp^{-}}$

${\displaystyle E_{\text{lc}}=cp^{-}}$

${\displaystyle -m^{2}c^{2}=p^{2}=\sum _{i=1}^{d-1}(p^{i})^{2}-2p^{+}p^{-}}$

${\displaystyle E_{\text{lc}}={\frac {c}{2p^{+}}}\left(m^{2}c^{2}+\sum _{i=1}^{d-1}(p^{i})^{2}\right)}$

関連項目

参考文献