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At the LHC, protons in the beam 1 have the four momentum
and protons in the beam 2 have the four momentum
They collide at the collison point at the center of the detectors (ATLAS, CMS, LHCb, ALICE). Since the velocity is very very near to the light velocity 
![0.94\left[\mathrm{GeV/c^{2}}\right]](http://s0.wp.com/latex.php?latex=0.94%5Cleft%5B%5Cmathrm%7BGeV%2Fc%5E%7B2%7D%7D%5Cright%5D+&bg=fafad3&fg=6f5e4e&s=0 "0.94\left[\mathrm{GeV/c^{2}}\right]")
With this approximation,E holds. In the collision center-of-mass frame, the vectorial sum of the three-momenta of the colliding particles is zero
At the LHC, protons of both beams have the same and opposite momentum in the laboratory frame. It may be called asymmetric collider. (A good example of asymmetric collider is KEKB or PEP2 B-factory colliders).
The four momentum (energy and three-momentum) of the colliding two-proton system is
in the laboratory frame, and we call the total energy 2E as the collision energy.
It is often expressed as
by using the Lorentz invariant
Let parton 
the squared collision energy of the colliding 
the rapidity of the
the measure
where
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\left(p^{\prime\mu}+p^{\mu}\right)+2m\gamma^{\mu}\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29%2B2m%5Cgamma%5E%7B%5Cmu%7D%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\left(p^{\prime\mu}+p^{\mu}\right)+2m\gamma^{\mu}\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(2p^{\prime\mu}+2p^{\mu}-4m\gamma^{\mu}\right)\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%282p%5E%7B%5Cprime%5Cmu%7D%2B2p%5E%7B%5Cmu%7D-4m%5Cgamma%5E%7B%5Cmu%7D%5Cright%29%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(2p^{\prime\mu}+2p^{\mu}-4m\gamma^{\mu}\right)\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(2p^{\prime\mu}+2p^{\mu}-p^{\prime}\!\!\!/\gamma^{\mu}-\gamma^{\mu}p\!\!\!/-2m\gamma^{\mu}\right)\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%282p%5E%7B%5Cprime%5Cmu%7D%2B2p%5E%7B%5Cmu%7D-p%5E%7B%5Cprime%7D%5C%21%5C%21%5C%21%2F%5Cgamma%5E%7B%5Cmu%7D-%5Cgamma%5E%7B%5Cmu%7Dp%5C%21%5C%21%5C%21%2F-2m%5Cgamma%5E%7B%5Cmu%7D%5Cright%29%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(2p^{\prime\mu}+2p^{\mu}-p^{\prime}\!\!\!/\gamma^{\mu}-\gamma^{\mu}p\!\!\!/-2m\gamma^{\mu}\right)\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(\gamma^{\mu}p^{\prime}\!\!\!/+p\!\!\!/\gamma^{\mu}-2m\gamma^{\mu}\right)\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%5Cgamma%5E%7B%5Cmu%7Dp%5E%7B%5Cprime%7D%5C%21%5C%21%5C%21%2F%2Bp%5C%21%5C%21%5C%21%2F%5Cgamma%5E%7B%5Cmu%7D-2m%5Cgamma%5E%7B%5Cmu%7D%5Cright%29%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(\gamma^{\mu}p^{\prime}\!\!\!/+p\!\!\!/\gamma^{\mu}-2m\gamma^{\mu}\right)\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left\{ \gamma^{\mu}\left(p^{\prime}\!\!\!/-m\right)+\left(p\!\!\!/-m\right)\gamma^{\mu}\right\} \right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5C%7B+%5Cgamma%5E%7B%5Cmu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5C%21%5C%21%5C%21%2F-m%5Cright%29%2B%5Cleft%28p%5C%21%5C%21%5C%21%2F-m%5Cright%29%5Cgamma%5E%7B%5Cmu%7D%5Cright%5C%7D+%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left\{ \gamma^{\mu}\left(p^{\prime}\!\!\!/-m\right)+\left(p\!\!\!/-m\right)\gamma^{\mu}\right\} \right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left\{ \gamma^{\mu}\left(p^{\prime}\!\!\!/-p\!\!\!/\right)-\left(p^{\prime}\!\!\!/-p\!\!\!/\right)\gamma^{\mu}\right\} \right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5C%7B+%5Cgamma%5E%7B%5Cmu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5C%21%5C%21%5C%21%2F-p%5C%21%5C%21%5C%21%2F%5Cright%29-%5Cleft%28p%5E%7B%5Cprime%7D%5C%21%5C%21%5C%21%2F-p%5C%21%5C%21%5C%21%2F%5Cright%29%5Cgamma%5E%7B%5Cmu%7D%5Cright%5C%7D+%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left\{ \gamma^{\mu}\left(p^{\prime}\!\!\!/-p\!\!\!/\right)-\left(p^{\prime}\!\!\!/-p\!\!\!/\right)\gamma^{\mu}\right\} \right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\left(p_{\nu}^{\prime}-p_{\nu}\right)\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%5Cgamma%5E%7B%5Cmu%7D%5Cgamma%5E%7B%5Cnu%7D-%5Cgamma%5E%7B%5Cnu%7D%5Cgamma%5E%7B%5Cmu%7D%5Cright%29%5Cleft%28p_%7B%5Cnu%7D%5E%7B%5Cprime%7D-p_%7B%5Cnu%7D%5Cright%29%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\left(p_{\nu}^{\prime}-p_{\nu}\right)\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]\left(p_{\nu}^{\prime}-p_{\nu}\right)\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cgamma%5E%7B%5Cmu%7D%2C%5Cgamma%5E%7B%5Cnu%7D%5Cright%5D%5Cleft%28p_%7B%5Cnu%7D%5E%7B%5Cprime%7D-p_%7B%5Cnu%7D%5Cright%29%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]\left(p_{\nu}^{\prime}-p_{\nu}\right)\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]q_{\nu}\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cgamma%5E%7B%5Cmu%7D%2C%5Cgamma%5E%7B%5Cnu%7D%5Cright%5Dq_%7B%5Cnu%7D%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)-\frac{1}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]q_{\nu}\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)+i\sigma^{\mu\nu}q_{\nu}\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cfrac%7B1%7D%7B2m%7D%5Cleft%5B%5Cleft%28p%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%5Cright%29%2Bi%5Csigma%5E%7B%5Cmu%5Cnu%7Dq_%7B%5Cnu%7D%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\frac{1}{2m}\left[\left(p^{\prime\mu}+p^{\mu}\right)+i\sigma^{\mu\nu}q_{\nu}\right]u\left(p\right)")
![=\bar{u}\left(p^{\prime}\right)\left[\frac{p^{\prime\mu}+p^{\mu}}{2m}+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cleft%5B%5Cfrac%7Bp%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%7D%7B2m%7D%2B%5Cfrac%7Bi%5Csigma%5E%7B%5Cmu%5Cnu%7Dq_%7B%5Cnu%7D%7D%7B2m%7D%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "=\bar{u}\left(p^{\prime}\right)\left[\frac{p^{\prime\mu}+p^{\mu}}{2m}+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\right]u\left(p\right)")
![\therefore \ \ \ \bar{u}\left(p^{\prime}\right)\gamma^{\mu}u\left(p\right) =\bar{u}\left(p^{\prime}\right)\left[\frac{p^{\prime\mu}+p^{\mu}}{2m}+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\right]u\left(p\right)](http://s0.wp.com/latex.php?latex=%5Ctherefore+%5C+%5C+%5C+%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cgamma%5E%7B%5Cmu%7Du%5Cleft%28p%5Cright%29+%3D%5Cbar%7Bu%7D%5Cleft%28p%5E%7B%5Cprime%7D%5Cright%29%5Cleft%5B%5Cfrac%7Bp%5E%7B%5Cprime%5Cmu%7D%2Bp%5E%7B%5Cmu%7D%7D%7B2m%7D%2B%5Cfrac%7Bi%5Csigma%5E%7B%5Cmu%5Cnu%7Dq_%7B%5Cnu%7D%7D%7B2m%7D%5Cright%5Du%5Cleft%28p%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "\therefore \ \ \ \bar{u}\left(p^{\prime}\right)\gamma^{\mu}u\left(p\right) =\bar{u}\left(p^{\prime}\right)\left[\frac{p^{\prime\mu}+p^{\mu}}{2m}+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\right]u\left(p\right)")
まで遷移する確率は、次のように書ける。
なら、
ならば、
に関してまず$latex\int dx_{i}$の積分を行い、それから
と和をとったのだが、時間は連続的な量なので、もう$x$は時間の関数な
になっている。ゆえに、
というのは関数の関数になっていて、ある変数
をどうやって決めるか。古典極限に持っていくと、量子力学的とは違って、経路が確実に1つに定まることから、あの汎関数積分を古典経路を、次のようなstationary conditionより把握し、![\left.\frac{\delta}{\delta x\left(t\right)}\biggl(\mathrm{phase}\left[x\left(t\right)\right]\biggr)\right|_{x=x_{\mathrm{cl}}}=0](http://s0.wp.com/latex.php?latex=%5Cleft.%5Cfrac%7B%5Cdelta%7D%7B%5Cdelta+x%5Cleft%28t%5Cright%29%7D%5Cbiggl%28%5Cmathrm%7Bphase%7D%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%5Cbiggr%29%5Cright%7C_%7Bx%3Dx_%7B%5Cmathrm%7Bcl%7D%7D%7D%3D0&bg=fafad3&fg=6f5e4e&s=0 "\left.\frac{\delta}{\delta x\left(t\right)}\biggl(\mathrm{phase}\left[x\left(t\right)\right]\biggr)\right|_{x=x_{\mathrm{cl}}}=0")
![\left.\frac{\delta}{\delta x\left(t\right)}\biggl(S\left[x\left(t\right)\right]\biggr)\right|_{x=x_{\mathrm{cl}}} =\left.\frac{\delta}{\delta x\left(t\right)}\left(\int L\left(x\left(t\right),\dot{x}\left(t\right);t\right)dt\right)\right|_{x=x_{\mathrm{cl}}}=0](http://s0.wp.com/latex.php?latex=%5Cleft.%5Cfrac%7B%5Cdelta%7D%7B%5Cdelta+x%5Cleft%28t%5Cright%29%7D%5Cbiggl%28S%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%5Cbiggr%29%5Cright%7C_%7Bx%3Dx_%7B%5Cmathrm%7Bcl%7D%7D%7D+%3D%5Cleft.%5Cfrac%7B%5Cdelta%7D%7B%5Cdelta+x%5Cleft%28t%5Cright%29%7D%5Cleft%28%5Cint+L%5Cleft%28x%5Cleft%28t%5Cright%29%2C%5Cdot%7Bx%7D%5Cleft%28t%5Cright%29%3Bt%5Cright%29dt%5Cright%29%5Cright%7C_%7Bx%3Dx_%7B%5Cmathrm%7Bcl%7D%7D%7D%3D0&bg=fafad3&fg=6f5e4e&s=0 "\left.\frac{\delta}{\delta x\left(t\right)}\biggl(S\left[x\left(t\right)\right]\biggr)\right|_{x=x_{\mathrm{cl}}} =\left.\frac{\delta}{\delta x\left(t\right)}\left(\int L\left(x\left(t\right),\dot{x}\left(t\right);t\right)dt\right)\right|_{x=x_{\mathrm{cl}}}=0")
をphaseとして使えそうである。でもただし、これはあくまでも
のような古典極限で成り立つものだし、次元の問題もある。ゆえに、phaseとしては
を使うことにしよう。実際この遷移振幅をSchroedinger方程式に入れてみよう。
![i\hbar\frac{\partial}{\partial t}e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]} =\left(-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V\left(x\right)\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}](http://s0.wp.com/latex.php?latex=i%5Chbar%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7De%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D+%3D%5Cleft%28-%5Cfrac%7B%5Chbar%5E%7B2%7D%7D%7B2m%7D%5Cfrac%7Bd%5E%7B2%7D%7D%7Bdx%5E%7B2%7D%7D%2BV%5Cleft%28x%5Cright%29%5Cright%29e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D&bg=fafad3&fg=6f5e4e&s=0 "i\hbar\frac{\partial}{\partial t}e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]} =\left(-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V\left(x\right)\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}")
![-\frac{\partial S}{\partial t}\cdot e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]} =\left\{ -\frac{\hbar^{2}}{2m}\frac{d}{dx}\left(\frac{i}{\hbar}\cdot\frac{dS}{dx}\cdot e^{S\left[x\left(t\right)\right]}\right)\right\} +V\left(x\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}](http://s0.wp.com/latex.php?latex=-%5Cfrac%7B%5Cpartial+S%7D%7B%5Cpartial+t%7D%5Ccdot+e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D+%3D%5Cleft%5C%7B+-%5Cfrac%7B%5Chbar%5E%7B2%7D%7D%7B2m%7D%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28%5Cfrac%7Bi%7D%7B%5Chbar%7D%5Ccdot%5Cfrac%7BdS%7D%7Bdx%7D%5Ccdot+e%5E%7BS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D%5Cright%29%5Cright%5C%7D+%2BV%5Cleft%28x%5Cright%29e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D&bg=fafad3&fg=6f5e4e&s=0 "-\frac{\partial S}{\partial t}\cdot e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]} =\left\{ -\frac{\hbar^{2}}{2m}\frac{d}{dx}\left(\frac{i}{\hbar}\cdot\frac{dS}{dx}\cdot e^{S\left[x\left(t\right)\right]}\right)\right\} +V\left(x\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}")
![=\left\{ -\frac{\hbar^{2}}{2m}\left(\left(\frac{i}{\hbar}\right)^{2}\cdot\left(\frac{dS}{dx}\right)^{2}\cdot e^{S\left[x\left(t\right)\right]}+\frac{i}{\hbar}\cdot\frac{d^{2}S}{dx^{2}}\cdot e^{S\left[x\left(t\right)\right]}\right)\right\} +V\left(x\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}](http://s0.wp.com/latex.php?latex=%3D%5Cleft%5C%7B+-%5Cfrac%7B%5Chbar%5E%7B2%7D%7D%7B2m%7D%5Cleft%28%5Cleft%28%5Cfrac%7Bi%7D%7B%5Chbar%7D%5Cright%29%5E%7B2%7D%5Ccdot%5Cleft%28%5Cfrac%7BdS%7D%7Bdx%7D%5Cright%29%5E%7B2%7D%5Ccdot+e%5E%7BS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D%2B%5Cfrac%7Bi%7D%7B%5Chbar%7D%5Ccdot%5Cfrac%7Bd%5E%7B2%7DS%7D%7Bdx%5E%7B2%7D%7D%5Ccdot+e%5E%7BS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D%5Cright%29%5Cright%5C%7D+%2BV%5Cleft%28x%5Cright%29e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D&bg=fafad3&fg=6f5e4e&s=0 "=\left\{ -\frac{\hbar^{2}}{2m}\left(\left(\frac{i}{\hbar}\right)^{2}\cdot\left(\frac{dS}{dx}\right)^{2}\cdot e^{S\left[x\left(t\right)\right]}+\frac{i}{\hbar}\cdot\frac{d^{2}S}{dx^{2}}\cdot e^{S\left[x\left(t\right)\right]}\right)\right\} +V\left(x\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}")
![=\frac{1}{2m}\left(\frac{dS}{dx}\right)^{2}\cdot e^{S\left[x\left(t\right)\right]}-\frac{i\hbar}{2m}\frac{d^{2}S}{dx^{2}}\cdot e^{S\left[x\left(t\right)\right]}+V\left(x\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}](http://s0.wp.com/latex.php?latex=%3D%5Cfrac%7B1%7D%7B2m%7D%5Cleft%28%5Cfrac%7BdS%7D%7Bdx%7D%5Cright%29%5E%7B2%7D%5Ccdot+e%5E%7BS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D-%5Cfrac%7Bi%5Chbar%7D%7B2m%7D%5Cfrac%7Bd%5E%7B2%7DS%7D%7Bdx%5E%7B2%7D%7D%5Ccdot+e%5E%7BS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D%2BV%5Cleft%28x%5Cright%29e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D&bg=fafad3&fg=6f5e4e&s=0 "=\frac{1}{2m}\left(\frac{dS}{dx}\right)^{2}\cdot e^{S\left[x\left(t\right)\right]}-\frac{i\hbar}{2m}\frac{d^{2}S}{dx^{2}}\cdot e^{S\left[x\left(t\right)\right]}+V\left(x\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}")
としたら、
![\left\langle x_{b}\right|e^{-\frac{i}{\hbar}HT}\left|x_{a}\right\rangle = U\left(x_{a},x_{b};T\right) =\int\mathcal{D}x\left(t\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}](http://s0.wp.com/latex.php?latex=%5Cleft%5Clangle+x_%7Bb%7D%5Cright%7Ce%5E%7B-%5Cfrac%7Bi%7D%7B%5Chbar%7DHT%7D%5Cleft%7Cx_%7Ba%7D%5Cright%5Crangle+%3D+U%5Cleft%28x_%7Ba%7D%2Cx_%7Bb%7D%3BT%5Cright%29+%3D%5Cint%5Cmathcal%7BD%7Dx%5Cleft%28t%5Cright%29e%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7DS%5Cleft%5Bx%5Cleft%28t%5Cright%29%5Cright%5D%7D&bg=fafad3&fg=6f5e4e&s=0 "\left\langle x_{b}\right|e^{-\frac{i}{\hbar}HT}\left|x_{a}\right\rangle = U\left(x_{a},x_{b};T\right) =\int\mathcal{D}x\left(t\right)e^{\frac{i}{\hbar}S\left[x\left(t\right)\right]}")
軸がvelocity。
に近づくほど、rapidityはさらに早いスピードで大きくなる。
は
matrixである。ここに更なる
を作用させても、それもLorentz transformationであろう。つまり
もLorentz変換である。ここで数学の群が浮かぶ。つまり、こういうLorentz transformationたちは群をなしているのである。としたら、群論の言葉で、
ごとに、そのdimensionalityを
とし、その表現行列を
行列である。
![\left[J^{\mu\nu},\ J^{\rho\sigma}\right] =-i\left(g^{\mu\rho}J^{\nu\sigma}-g^{\mu\sigma}J^{\nu\rho}-g^{\nu\rho}J^{\mu\sigma}+g^{\nu\sigma}J^{\mu\rho}\right)](http://s0.wp.com/latex.php?latex=%5Cleft%5BJ%5E%7B%5Cmu%5Cnu%7D%2C%5C+J%5E%7B%5Crho%5Csigma%7D%5Cright%5D+%3D-i%5Cleft%28g%5E%7B%5Cmu%5Crho%7DJ%5E%7B%5Cnu%5Csigma%7D-g%5E%7B%5Cmu%5Csigma%7DJ%5E%7B%5Cnu%5Crho%7D-g%5E%7B%5Cnu%5Crho%7DJ%5E%7B%5Cmu%5Csigma%7D%2Bg%5E%7B%5Cnu%5Csigma%7DJ%5E%7B%5Cmu%5Crho%7D%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "\left[J^{\mu\nu},\ J^{\rho\sigma}\right] =-i\left(g^{\mu\rho}J^{\nu\sigma}-g^{\mu\sigma}J^{\nu\rho}-g^{\nu\rho}J^{\mu\sigma}+g^{\nu\sigma}J^{\mu\rho}\right)")
![{\psi_{L}}_{\alpha} \rightarrow{\left(\exp\left[-i\theta_{k}\frac{\sigma^{k}}{2}-\beta_{k}\frac{\sigma^{k}}{2}\right]\right)_{\alpha}}^{\beta}{\psi_{L}}_{\beta}](http://s0.wp.com/latex.php?latex=%7B%5Cpsi_%7BL%7D%7D_%7B%5Calpha%7D+%5Crightarrow%7B%5Cleft%28%5Cexp%5Cleft%5B-i%5Ctheta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7Bk%7D%7D%7B2%7D-%5Cbeta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7Bk%7D%7D%7B2%7D%5Cright%5D%5Cright%29_%7B%5Calpha%7D%7D%5E%7B%5Cbeta%7D%7B%5Cpsi_%7BL%7D%7D_%7B%5Cbeta%7D&bg=fafad3&fg=6f5e4e&s=0 "{\psi_{L}}_{\alpha} \rightarrow{\left(\exp\left[-i\theta_{k}\frac{\sigma^{k}}{2}-\beta_{k}\frac{\sigma^{k}}{2}\right]\right)_{\alpha}}^{\beta}{\psi_{L}}_{\beta}")
![{\psi_{R}}^{\dot{\alpha}} \rightarrow{\left(\exp\left[-i\theta_{k}\frac{\sigma^{k}}{2}+\beta_{k}\frac{\sigma^{k}}{2}\right]\right)^{\dot{\alpha}}}_{\dot{\beta}}{\psi_{R}}^{\dot{\beta}}](http://s0.wp.com/latex.php?latex=%7B%5Cpsi_%7BR%7D%7D%5E%7B%5Cdot%7B%5Calpha%7D%7D+%5Crightarrow%7B%5Cleft%28%5Cexp%5Cleft%5B-i%5Ctheta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7Bk%7D%7D%7B2%7D%2B%5Cbeta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7Bk%7D%7D%7B2%7D%5Cright%5D%5Cright%29%5E%7B%5Cdot%7B%5Calpha%7D%7D%7D_%7B%5Cdot%7B%5Cbeta%7D%7D%7B%5Cpsi_%7BR%7D%7D%5E%7B%5Cdot%7B%5Cbeta%7D%7D&bg=fafad3&fg=6f5e4e&s=0 "{\psi_{R}}^{\dot{\alpha}} \rightarrow{\left(\exp\left[-i\theta_{k}\frac{\sigma^{k}}{2}+\beta_{k}\frac{\sigma^{k}}{2}\right]\right)^{\dot{\alpha}}}_{\dot{\beta}}{\psi_{R}}^{\dot{\beta}}")
はright-handed spinorのような変換則を持つ。![{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}{\psi_{L}^{*}}_{\dot{\beta}}](http://s0.wp.com/latex.php?latex=%7B%5Cpsi_%7BL%7D%5E%7B%2A%7D%7D_%7B%5Cdot%7B%5Calpha%7D%7D+%5Crightarrow%7B%5Cleft%28%5Cexp%5Cleft%5Bi%5Ctheta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D-%5Cbeta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D%5Cright%5D%5Cright%29_%7B%5Cdot%7B%5Calpha%7D%7D%7D%5E%7B%5Cdot%7B%5Cbeta%7D%7D%7B%5Cpsi_%7BL%7D%5E%7B%2A%7D%7D_%7B%5Cdot%7B%5Cbeta%7D%7D&bg=fafad3&fg=6f5e4e&s=0 "{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}{\psi_{L}^{*}}_{\dot{\beta}}")
![\left(i\sigma^{2}\right)^{\dot{\gamma}\dot{\alpha}}{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow\left(i\sigma^{2}\right)^{\dot{\gamma}\dot{\alpha}}{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}\left(-i\sigma^{2}\right)_{\dot{\beta}\dot{\delta}}\left(i\sigma^{2}\right)^{\dot{\delta}\dot{\kappa}}{\psi_{L}^{*}}_{\dot{\kappa}}](http://s0.wp.com/latex.php?latex=%5Cleft%28i%5Csigma%5E%7B2%7D%5Cright%29%5E%7B%5Cdot%7B%5Cgamma%7D%5Cdot%7B%5Calpha%7D%7D%7B%5Cpsi_%7BL%7D%5E%7B%2A%7D%7D_%7B%5Cdot%7B%5Calpha%7D%7D+%5Crightarrow%5Cleft%28i%5Csigma%5E%7B2%7D%5Cright%29%5E%7B%5Cdot%7B%5Cgamma%7D%5Cdot%7B%5Calpha%7D%7D%7B%5Cleft%28%5Cexp%5Cleft%5Bi%5Ctheta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D-%5Cbeta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D%5Cright%5D%5Cright%29_%7B%5Cdot%7B%5Calpha%7D%7D%7D%5E%7B%5Cdot%7B%5Cbeta%7D%7D%5Cleft%28-i%5Csigma%5E%7B2%7D%5Cright%29_%7B%5Cdot%7B%5Cbeta%7D%5Cdot%7B%5Cdelta%7D%7D%5Cleft%28i%5Csigma%5E%7B2%7D%5Cright%29%5E%7B%5Cdot%7B%5Cdelta%7D%5Cdot%7B%5Ckappa%7D%7D%7B%5Cpsi_%7BL%7D%5E%7B%2A%7D%7D_%7B%5Cdot%7B%5Ckappa%7D%7D&bg=fafad3&fg=6f5e4e&s=0 "\left(i\sigma^{2}\right)^{\dot{\gamma}\dot{\alpha}}{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow\left(i\sigma^{2}\right)^{\dot{\gamma}\dot{\alpha}}{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}\left(-i\sigma^{2}\right)_{\dot{\beta}\dot{\delta}}\left(i\sigma^{2}\right)^{\dot{\delta}\dot{\kappa}}{\psi_{L}^{*}}_{\dot{\kappa}}")
![\varepsilon^{\dot{\gamma}\dot{\alpha}}{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow\varepsilon^{\dot{\gamma}\dot{\alpha}}{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}\varepsilon_{\dot{\beta}\dot{\delta}}\varepsilon^{\dot{\delta}\dot{\kappa}}{\psi_{L}^{*}}_{\dot{\kappa}}](http://s0.wp.com/latex.php?latex=%5Cvarepsilon%5E%7B%5Cdot%7B%5Cgamma%7D%5Cdot%7B%5Calpha%7D%7D%7B%5Cpsi_%7BL%7D%5E%7B%2A%7D%7D_%7B%5Cdot%7B%5Calpha%7D%7D+%5Crightarrow%5Cvarepsilon%5E%7B%5Cdot%7B%5Cgamma%7D%5Cdot%7B%5Calpha%7D%7D%7B%5Cleft%28%5Cexp%5Cleft%5Bi%5Ctheta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D-%5Cbeta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D%5Cright%5D%5Cright%29_%7B%5Cdot%7B%5Calpha%7D%7D%7D%5E%7B%5Cdot%7B%5Cbeta%7D%7D%5Cvarepsilon_%7B%5Cdot%7B%5Cbeta%7D%5Cdot%7B%5Cdelta%7D%7D%5Cvarepsilon%5E%7B%5Cdot%7B%5Cdelta%7D%5Cdot%7B%5Ckappa%7D%7D%7B%5Cpsi_%7BL%7D%5E%7B%2A%7D%7D_%7B%5Cdot%7B%5Ckappa%7D%7D&bg=fafad3&fg=6f5e4e&s=0 "\varepsilon^{\dot{\gamma}\dot{\alpha}}{\psi_{L}^{*}}_{\dot{\alpha}} \rightarrow\varepsilon^{\dot{\gamma}\dot{\alpha}}{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)_{\dot{\alpha}}}^{\dot{\beta}}\varepsilon_{\dot{\beta}\dot{\delta}}\varepsilon^{\dot{\delta}\dot{\kappa}}{\psi_{L}^{*}}_{\dot{\kappa}}")
![\left(i\sigma^{2}\psi_{L}^{*}\right)^{\dot{\gamma}} \rightarrow{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)^{\dot{\gamma}}}_{\dot{\delta}}\left(i\sigma^{2}\psi_{L}^{*}\right)^{\dot{\delta}}](http://s0.wp.com/latex.php?latex=%5Cleft%28i%5Csigma%5E%7B2%7D%5Cpsi_%7BL%7D%5E%7B%2A%7D%5Cright%29%5E%7B%5Cdot%7B%5Cgamma%7D%7D+%5Crightarrow%7B%5Cleft%28%5Cexp%5Cleft%5Bi%5Ctheta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D-%5Cbeta_%7Bk%7D%5Cfrac%7B%5Csigma%5E%7B%2Ak%7D%7D%7B2%7D%5Cright%5D%5Cright%29%5E%7B%5Cdot%7B%5Cgamma%7D%7D%7D_%7B%5Cdot%7B%5Cdelta%7D%7D%5Cleft%28i%5Csigma%5E%7B2%7D%5Cpsi_%7BL%7D%5E%7B%2A%7D%5Cright%29%5E%7B%5Cdot%7B%5Cdelta%7D%7D&bg=fafad3&fg=6f5e4e&s=0 "\left(i\sigma^{2}\psi_{L}^{*}\right)^{\dot{\gamma}} \rightarrow{\left(\exp\left[i\theta_{k}\frac{\sigma^{*k}}{2}-\beta_{k}\frac{\sigma^{*k}}{2}\right]\right)^{\dot{\gamma}}}_{\dot{\delta}}\left(i\sigma^{2}\psi_{L}^{*}\right)^{\dot{\delta}}")
![S^{\mu\nu}=\frac{i}{4}\left[\gamma^{\mu},\gamma^{\nu}\right]](http://s0.wp.com/latex.php?latex=S%5E%7B%5Cmu%5Cnu%7D%3D%5Cfrac%7Bi%7D%7B4%7D%5Cleft%5B%5Cgamma%5E%7B%5Cmu%7D%2C%5Cgamma%5E%7B%5Cnu%7D%5Cright%5D&bg=fafad3&fg=6f5e4e&s=0 "S^{\mu\nu}=\frac{i}{4}\left[\gamma^{\mu},\gamma^{\nu}\right]")
![\Lambda_{\frac{1}{2}}=\exp\left[-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}\right]](http://s0.wp.com/latex.php?latex=%5CLambda_%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Cexp%5Cleft%5B-%5Cfrac%7Bi%7D%7B2%7D%5Comega_%7B%5Cmu%5Cnu%7DS%5E%7B%5Cmu%5Cnu%7D%5Cright%5D&bg=fafad3&fg=6f5e4e&s=0 "\Lambda_{\frac{1}{2}}=\exp\left[-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}\right]")
![=\exp\left[\frac{\omega_{\mu\nu}}{8}\left(\begin{array}{cc} {\left(\sigma^{\mu}\bar{\sigma}^{\nu}-\sigma^{\nu}\bar{\sigma}^{\mu}\right)_{\alpha}}^{\beta} & 0\\ 0 & {\left(\bar{\sigma}^{\mu}\sigma^{\nu}-\bar{\sigma}^{\nu}\sigma^{\mu}\right)^{\dot{\alpha}}}_{\dot{\beta}} \end{array}\right)\right]](http://s0.wp.com/latex.php?latex=%3D%5Cexp%5Cleft%5B%5Cfrac%7B%5Comega_%7B%5Cmu%5Cnu%7D%7D%7B8%7D%5Cleft%28%5Cbegin%7Barray%7D%7Bcc%7D++%7B%5Cleft%28%5Csigma%5E%7B%5Cmu%7D%5Cbar%7B%5Csigma%7D%5E%7B%5Cnu%7D-%5Csigma%5E%7B%5Cnu%7D%5Cbar%7B%5Csigma%7D%5E%7B%5Cmu%7D%5Cright%29_%7B%5Calpha%7D%7D%5E%7B%5Cbeta%7D+%26+0%5C%5C++0+%26+%7B%5Cleft%28%5Cbar%7B%5Csigma%7D%5E%7B%5Cmu%7D%5Csigma%5E%7B%5Cnu%7D-%5Cbar%7B%5Csigma%7D%5E%7B%5Cnu%7D%5Csigma%5E%7B%5Cmu%7D%5Cright%29%5E%7B%5Cdot%7B%5Calpha%7D%7D%7D_%7B%5Cdot%7B%5Cbeta%7D%7D++%5Cend%7Barray%7D%5Cright%29%5Cright%5D&bg=fafad3&fg=6f5e4e&s=0 "=\exp\left[\frac{\omega_{\mu\nu}}{8}\left(\begin{array}{cc} {\left(\sigma^{\mu}\bar{\sigma}^{\nu}-\sigma^{\nu}\bar{\sigma}^{\mu}\right)_{\alpha}}^{\beta} & 0\\ 0 & {\left(\bar{\sigma}^{\mu}\sigma^{\nu}-\bar{\sigma}^{\nu}\sigma^{\mu}\right)^{\dot{\alpha}}}_{\dot{\beta}} \end{array}\right)\right]")
![\left[\left(-\sigma_{\alpha\dot{\gamma}}^{i}\right),\sigma^{j\dot{\gamma}\beta}\right]=-2i\varepsilon^{ijk}{\sigma_{\alpha}^{k}}^{\beta}](http://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cleft%28-%5Csigma_%7B%5Calpha%5Cdot%7B%5Cgamma%7D%7D%5E%7Bi%7D%5Cright%29%2C%5Csigma%5E%7Bj%5Cdot%7B%5Cgamma%7D%5Cbeta%7D%5Cright%5D%3D-2i%5Cvarepsilon%5E%7Bijk%7D%7B%5Csigma_%7B%5Calpha%7D%5E%7Bk%7D%7D%5E%7B%5Cbeta%7D&bg=fafad3&fg=6f5e4e&s=0 "\left[\left(-\sigma_{\alpha\dot{\gamma}}^{i}\right),\sigma^{j\dot{\gamma}\beta}\right]=-2i\varepsilon^{ijk}{\sigma_{\alpha}^{k}}^{\beta}")
, ![\left[\sigma^{i\dot{\alpha}\gamma},\left(-\sigma_{\gamma\dot{\beta}}^{j}\right)\right]=-2i\varepsilon^{ijk}{\sigma^{k\dot{\alpha}}}_{\dot{\beta}}](http://s0.wp.com/latex.php?latex=%5Cleft%5B%5Csigma%5E%7Bi%5Cdot%7B%5Calpha%7D%5Cgamma%7D%2C%5Cleft%28-%5Csigma_%7B%5Cgamma%5Cdot%7B%5Cbeta%7D%7D%5E%7Bj%7D%5Cright%29%5Cright%5D%3D-2i%5Cvarepsilon%5E%7Bijk%7D%7B%5Csigma%5E%7Bk%5Cdot%7B%5Calpha%7D%7D%7D_%7B%5Cdot%7B%5Cbeta%7D%7D&bg=fafad3&fg=6f5e4e&s=0 "\left[\sigma^{i\dot{\alpha}\gamma},\left(-\sigma_{\gamma\dot{\beta}}^{j}\right)\right]=-2i\varepsilon^{ijk}{\sigma^{k\dot{\alpha}}}_{\dot{\beta}}")
![\theta_{k}\equiv\frac{1}{2}\omega_{ij}\varepsilon_{ijk}\](http://s0.wp.com/latex.php?latex=%5Ctheta_%7Bk%7D%5Cequiv%5Cfrac%7B1%7D%7B2%7D%5Comega_%7Bij%7D%5Cvarepsilon_%7Bijk%7D%5C+&bg=fafad3&fg=6f5e4e&s=0 "\theta_{k}\equiv\frac{1}{2}\omega_{ij}\varepsilon_{ijk}\")
, 
![\Lambda_{\frac{1}{2}}=\left(\begin{array}{cc} {\exp\left[-\frac{i}{2}\left(\theta_{k}-i\beta_{k}\right)\sigma^{k}\right]_{\alpha}}^{\beta} & 0\\ 0 & {\exp\left[-\frac{i}{2}\left(\theta_{k}+i\beta_{k}\right)\sigma^{k}\right]^{\dot{\gamma}}}_{\dot{\delta}} \end{array}\right)](http://s0.wp.com/latex.php?latex=%5CLambda_%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bcc%7D++%7B%5Cexp%5Cleft%5B-%5Cfrac%7Bi%7D%7B2%7D%5Cleft%28%5Ctheta_%7Bk%7D-i%5Cbeta_%7Bk%7D%5Cright%29%5Csigma%5E%7Bk%7D%5Cright%5D_%7B%5Calpha%7D%7D%5E%7B%5Cbeta%7D+%26+0%5C%5C++0+%26+%7B%5Cexp%5Cleft%5B-%5Cfrac%7Bi%7D%7B2%7D%5Cleft%28%5Ctheta_%7Bk%7D%2Bi%5Cbeta_%7Bk%7D%5Cright%29%5Csigma%5E%7Bk%7D%5Cright%5D%5E%7B%5Cdot%7B%5Cgamma%7D%7D%7D_%7B%5Cdot%7B%5Cdelta%7D%7D++%5Cend%7Barray%7D%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "\Lambda_{\frac{1}{2}}=\left(\begin{array}{cc} {\exp\left[-\frac{i}{2}\left(\theta_{k}-i\beta_{k}\right)\sigma^{k}\right]_{\alpha}}^{\beta} & 0\\ 0 & {\exp\left[-\frac{i}{2}\left(\theta_{k}+i\beta_{k}\right)\sigma^{k}\right]^{\dot{\gamma}}}_{\dot{\delta}} \end{array}\right)")
はrotationの角度を、
はboostのrapidityを表す。つまり、spatial rotationとLorentz boostの部分に分かれられているのである。
![{\left(\exp\left[\frac{i}{2}\left(\theta_{k}-i\beta_{k}\right)\sigma^{k}\right]\right)_{\alpha}}^{\gamma}\sigma_{\gamma\dot{\gamma}}^{\mu}{\left(\exp\left[-\frac{i}{2}\left(\theta_{k}+i\beta_{k}\right)\sigma^{k}\right]\right)^{\dot{\gamma}}}_{\dot{\alpha}}={\Lambda^{\mu}}_{\nu}\sigma_{\alpha\dot{\alpha}}^{\nu}](http://s0.wp.com/latex.php?latex=%7B%5Cleft%28%5Cexp%5Cleft%5B%5Cfrac%7Bi%7D%7B2%7D%5Cleft%28%5Ctheta_%7Bk%7D-i%5Cbeta_%7Bk%7D%5Cright%29%5Csigma%5E%7Bk%7D%5Cright%5D%5Cright%29_%7B%5Calpha%7D%7D%5E%7B%5Cgamma%7D%5Csigma_%7B%5Cgamma%5Cdot%7B%5Cgamma%7D%7D%5E%7B%5Cmu%7D%7B%5Cleft%28%5Cexp%5Cleft%5B-%5Cfrac%7Bi%7D%7B2%7D%5Cleft%28%5Ctheta_%7Bk%7D%2Bi%5Cbeta_%7Bk%7D%5Cright%29%5Csigma%5E%7Bk%7D%5Cright%5D%5Cright%29%5E%7B%5Cdot%7B%5Cgamma%7D%7D%7D_%7B%5Cdot%7B%5Calpha%7D%7D%3D%7B%5CLambda%5E%7B%5Cmu%7D%7D_%7B%5Cnu%7D%5Csigma_%7B%5Calpha%5Cdot%7B%5Calpha%7D%7D%5E%7B%5Cnu%7D&bg=fafad3&fg=6f5e4e&s=0 "{\left(\exp\left[\frac{i}{2}\left(\theta_{k}-i\beta_{k}\right)\sigma^{k}\right]\right)_{\alpha}}^{\gamma}\sigma_{\gamma\dot{\gamma}}^{\mu}{\left(\exp\left[-\frac{i}{2}\left(\theta_{k}+i\beta_{k}\right)\sigma^{k}\right]\right)^{\dot{\gamma}}}_{\dot{\alpha}}={\Lambda^{\mu}}_{\nu}\sigma_{\alpha\dot{\alpha}}^{\nu}")
![{\left(\exp\left[\frac{i}{2}\left(\theta_{k}+i\beta_{k}\right)\sigma^{k}\right]\right)^{\dot{\beta}}}_{\dot{\delta}}\bar{\sigma}^{\mu\dot{\delta}\delta}{\left(\exp\left[-\frac{i}{2}\left(\theta_{k}-i\beta_{k}\right)\sigma^{k}\right]\right)_{\delta}}^{\beta}={\Lambda^{\mu}}_{\nu}\bar{\sigma}^{\nu\dot{\beta}\beta}](http://s0.wp.com/latex.php?latex=%7B%5Cleft%28%5Cexp%5Cleft%5B%5Cfrac%7Bi%7D%7B2%7D%5Cleft%28%5Ctheta_%7Bk%7D%2Bi%5Cbeta_%7Bk%7D%5Cright%29%5Csigma%5E%7Bk%7D%5Cright%5D%5Cright%29%5E%7B%5Cdot%7B%5Cbeta%7D%7D%7D_%7B%5Cdot%7B%5Cdelta%7D%7D%5Cbar%7B%5Csigma%7D%5E%7B%5Cmu%5Cdot%7B%5Cdelta%7D%5Cdelta%7D%7B%5Cleft%28%5Cexp%5Cleft%5B-%5Cfrac%7Bi%7D%7B2%7D%5Cleft%28%5Ctheta_%7Bk%7D-i%5Cbeta_%7Bk%7D%5Cright%29%5Csigma%5E%7Bk%7D%5Cright%5D%5Cright%29_%7B%5Cdelta%7D%7D%5E%7B%5Cbeta%7D%3D%7B%5CLambda%5E%7B%5Cmu%7D%7D_%7B%5Cnu%7D%5Cbar%7B%5Csigma%7D%5E%7B%5Cnu%5Cdot%7B%5Cbeta%7D%5Cbeta%7D&bg=fafad3&fg=6f5e4e&s=0 "{\left(\exp\left[\frac{i}{2}\left(\theta_{k}+i\beta_{k}\right)\sigma^{k}\right]\right)^{\dot{\beta}}}_{\dot{\delta}}\bar{\sigma}^{\mu\dot{\delta}\delta}{\left(\exp\left[-\frac{i}{2}\left(\theta_{k}-i\beta_{k}\right)\sigma^{k}\right]\right)_{\delta}}^{\beta}={\Lambda^{\mu}}_{\nu}\bar{\sigma}^{\nu\dot{\beta}\beta}")
![H=\int\frac{d^{3}p}{\left(2\pi\right)^{3}}E_{\mathbf{p}}\left(a_{\mathbf{p}}^{\dagger}a_{\mathbf{p}}+b_{\mathbf{p}}^{\dagger}b_{\mathbf{p}}+\frac{1}{2}\left[a_{\mathbf{p}},a_{\mathbf{p}}^{\dagger}\right]+\frac{1}{2}\left[b_{\mathbf{p}},b_{\mathbf{p}}^{\dagger}\right]\right)](http://s0.wp.com/latex.php?latex=H%3D%5Cint%5Cfrac%7Bd%5E%7B3%7Dp%7D%7B%5Cleft%282%5Cpi%5Cright%29%5E%7B3%7D%7DE_%7B%5Cmathbf%7Bp%7D%7D%5Cleft%28a_%7B%5Cmathbf%7Bp%7D%7D%5E%7B%5Cdagger%7Da_%7B%5Cmathbf%7Bp%7D%7D%2Bb_%7B%5Cmathbf%7Bp%7D%7D%5E%7B%5Cdagger%7Db_%7B%5Cmathbf%7Bp%7D%7D%2B%5Cfrac%7B1%7D%7B2%7D%5Cleft%5Ba_%7B%5Cmathbf%7Bp%7D%7D%2Ca_%7B%5Cmathbf%7Bp%7D%7D%5E%7B%5Cdagger%7D%5Cright%5D%2B%5Cfrac%7B1%7D%7B2%7D%5Cleft%5Bb_%7B%5Cmathbf%7Bp%7D%7D%2Cb_%7B%5Cmathbf%7Bp%7D%7D%5E%7B%5Cdagger%7D%5Cright%5D%5Cright%29&bg=fafad3&fg=6f5e4e&s=0 "H=\int\frac{d^{3}p}{\left(2\pi\right)^{3}}E_{\mathbf{p}}\left(a_{\mathbf{p}}^{\dagger}a_{\mathbf{p}}+b_{\mathbf{p}}^{\dagger}b_{\mathbf{p}}+\frac{1}{2}\left[a_{\mathbf{p}},a_{\mathbf{p}}^{\dagger}\right]+\frac{1}{2}\left[b_{\mathbf{p}},b_{\mathbf{p}}^{\dagger}\right]\right)")
は上げ下げにマイナスつくんだけど、
はつかない。
だったら2つ上げ下げだから、マイナスが打ち消しあう。
がダミーなのに着目すると、簡単な計算で次がわかる。
, 
, 
, 
