17/04/2012
Theoretical Physics Chiral perturbation theory, Physics, Quantum Field Theory Leave a comment
最近はLyXを使わずに、ほとんどiPadのNoteshelfやGoodNotesを活用しているところですから、なかなかTeXが面倒くさくなってきました。
今日はfinite-temperature QCDの1993年までの歴史と、Chiral symmetryに関して調べてみました。
Chiral symmetryの復習は、あれですよ。staggered fermionをこれから使うもんなんでね。
明日はG&Lのstaggered fermionのとこをまた読んでみようと思います。chiral symmetryのSSBのメカニズムも、セミナーでちゃんと説明できるようにまとめときます。
それから、明日やりたいのは、renormalization group methodの復習です。せっかくなので牟田さんの本を使おうと思います。
16/04/2012
Theoretical Physics Leave a comment
宇川先生の講義録の最初あたりは、staggered fermionが結構出没してます。
ってことで、Chiral symmetryを。
マインツ大学の方で、Stefan Schererさんが2学期にわたって授業をやったスライドをもらいました。
14/04/2012
Theoretical Physics Leave a comment
現在、筑波大学の宇川彰副学長の講義録を読んでいます。
これは1993年6月21日から7月2日まで、アメリカのワシントン大学のキャンパスで開催されたUehling夏の学校で、「Lattice QCD at Finite Temperature」という名で行われた講義です。採録はStanley Myintさんにやっていただきました。
12/01/2012
Theoretical Physics ATLAS, ATLAS experiment, Compact Muon Solenoid, Large Hadron Collider, Large Ion Collider Experiment, LHC, LHCb, Proton Leave a comment
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
12/08/2011
Theoretical Physics Gordon Identity, Quantum Field Theory Leave a comment
12/08/2011
Theoretical Physics Path Integral, Stationary phase approximation Leave a comment
1. 粒子がある点
その経路の間に2重スリットを置いてみよう。
superposition principleにより、この場合の遷移振幅は、
のようである。もし
となり、constructive interferenceが起こる。けれど
となり、destructive interferenceが起こる。これより、二重以上のスリットがある場合の一般的な場合は、
と書けるだろう。ここで、位置$x$ が数列であれば、それぞれの
さて、
あれからstationary phase approximationを計算できるかも知れない。でも、古典経路は解析力学の最小作用定理を満たすものだった。
それで、このaction 
ここで、古典極限として
となるが、これはまさに古典力学のHamilton-Jacobi方程式である。ゆえに古典極限がうまく回っている。ゆえに遷移振幅は、最終的に
のようにかける。
09/08/2011
Theoretical Physics Mathematics, rapidity, relativistic, Theory of Relativity Leave a comment
velocity-rapidity graph: 
rapidity-velocity graph: x軸がrapidity。rapidityが増加すればするほど、光速に到達するスピードは遅くなる。
09/08/2011
Theoretical Physics Four-vector, Lorentz covariance Leave a comment
マックスウェル方程式の場合のように、Tensor notationで様々なLorentz変換を見つけることができる。
けれどこのnotation以外にもLorentz transformationは山ほどあるという。
そのすべてのLorentz transformationを一括して眺めたい。としたらすべてのLorentz変換たちが共有している性質を見つけるのが必要である。
任意のn-component multipletのLorentz transformationを考えてみよう。議論の簡単さのために線形変換に限って考える。としても、どんな複雑な変形の仕方だって、そのinfinitesimal transformationは必ず線形だから、大げさに言うまでもない。Lorentz transformationとともにfield argumentも変換するが、こいつは無視して眺めると、
のような変換をするだろう。ここで
さて、Lorentz群の有限次元行列をどうやって表せるか。
Lorentz transformationは「4次元時空での回転」である。なので、量子力学でも容易に記述できる3次元回転群から考えてみよう。
これはスピン量子数
どんな連続群も、ほとんど恒等変換に近いくらいのinfinitesimalな変換は、あるベクトル空間を作る。それはその群のLie algebraという。そのベクトル空間の基底ベクトルはLie algebraの生成子という。
3次元回転群の場合、その演算子は角運動量演算子であり、有名なcommutation relationを満たす。有限な回転演算はgeneratorを指数関数の肩の上に置いたような形になる。演算子の掛け算の規則はgeneratorたちの交換関係によって定まる。
ここでのように、その群のgeneratorを求めて、そいつを指数関数の肩の上に置けば、continuous groupの行列表現を表すことができる。
今はローレンツ群の生成子の交換関係を求めなければいけない。スピン角運動量だけではなく、軌道角運動量も交換関係は満たす。
これを4次元に拡張するとしたら、考えるまででもなくこれだろう。
計算してみれば、わかるが、こいつなら次のような交換関係を満たす。
これがLorentz groupのcommutation relationである。
一方、次のような表現も、Lorentz groupのgeneratorになりえるのがわかる。
それから、以前ポストしたようにSpinor表現もある。
09/08/2011
Theoretical Physics left-handed, right-handed, Spinor, Weyl Spinor Leave a comment
09/08/2011
Theoretical Physics Gamma matrices, Lorentz, Spinor Leave a comment
Gamma matricesを用いて、Lorentz groupをいわゆるSpinor representationで表せる。
Peskinを読んでも、九後さんをわ読んでもわかるが、そのgeneratorはブロック対角化されている。これは、この表現がreducibleであることを意味する。対角成分は2個であって、それぞれは2次元でパウリ行列なので、それならirreducible。
だからDirac spinor fieldも4つの成分を持っているが、その上の2成分と下の2成分がその2つのブロックたちと独立的に作用するということがわかる。その上2成分をleft-handed spinor、下2成分をright-handed spinorという。
Peskinがちょっとうざいのはこういうとこ。Primed & Unprimed Spinorを明記してくれなかった。備忘録の形式で片づけてみよう。Peskinでいう43ページと44ページのことである。
備忘録
Lorentz群のspinor表現のgerenratorを、spinorの足まで全部書いて表せば次のようである。
これをローレンツ変換行列に代入。
それから、次を参照し、
![\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}")
次を代入すると、
![\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}\")
Lorentz変換行列は次のように、もうちょっとわかりやすい形になる。
ここで
spinorのかかわったLorentz変換ではgamma matricesが大事になるが、Gamma matricesはvectoｒの足とspinorの足両方を持っていて、こいつに対するspinor足のLorentz変換と、vector足のLorentz変換は同時に行われる。2つが同時に行われたらgamma matricesは不変。一言でいうと次式のようである。
これをPauli matricesを用いて表すと、次のようである。
![=\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)")
![\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")
![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]}")
![\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]}")
![\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[\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}}")
![\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)")
![{\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}")
