幾何代数によるYang-Mills項 - うべの時空代数

$\def\ou#1#2#3{\overset{#2}{\underset{#3}{#1}}}\def\os#1#2{\overset{#2}{#1}}\def\diff{\mathrm{d}}\def\biff{\mathrm{b}}\def\D{\mathrm{D}}\def\p{\mathrm{p}}\def\pu#1{\underset{#1}{\mathrm{p}}}\def\p{\mathrm{p}}\def\qu#1{\underset{#1}{\mathrm{q}}}\def\G#1#2{\overset{#1}{\underset{#2}{\Gamma}}}\def\abs#1{|#1|}$
Lie代数 $A$ と幾何代数 $B$ の積をとりあえず
$A\lhd B=B\rhd A$
と定義する．これはKronecker積とみることができる．とりあえず平坦なMinkowski空間を考え，その幾何代数の基底を $\ou{\gamma}{}{\mu}$ とし，ゲージ場に関するLie代数の生成子を $T_a$ とする．ここでゲージ場を
$A:=\ou{A}{}{\mu}\lhd\ou{\gamma}{\mu}{}=\left\{\ou{A}{}{\mu}{}_aT_a\right\}\lhd\ou{\gamma}{\mu}{}$
と定義し，非可換ゲージ理論におけるDirac 作用素を
$\mathcal{D}:=\ou{\mathcal{D}}{}{\mu}\lhd\ou{\gamma}{\mu}{}=\left\{\pu{\mu}I_N+ig\ou{A}{}{\mu}\right\}\lhd\ou{\gamma}{\mu}{}=I_N\lhd\D+igA$
と定義する． $I_N$ はLie代数のサイズの単位行列であり， $\D$ は通常のDirac 作用素である．
$A\mathcal{D}$ は重力を無視して幾何代数の基底の微分を0とするとKronecker積における混合積性質と同様に計算できて
$A\mathcal{D}=A\left\{\pu{\nu}I_N\lhd\ou{\gamma}{\nu}{}\right\}+igA^2\\ =\left\{\ou{A}{}{\mu}\lhd\ou{\gamma}{\mu}{}\right\}\left\{\pu{\nu}I_N\lhd\ou{\gamma}{\nu}{}\right\}+ig\left\{\ou{A}{}{\mu}\lhd\ou{\gamma}{\mu}{}\right\}\left\{\ou{A}{}{\nu}\lhd\ou{\gamma}{\nu}{}\right\}\\ =\left\{\ou{A}{}{\mu}\pu{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\ou{\gamma}{\nu}{}\right\}+ig\left\{\ou{A}{}{\mu}\ou{A}{}{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\ou{\gamma}{\nu}{}\right\}\\ =\left\{\ou{A}{}{\mu}\pu{\nu}+ig\ou{A}{}{\mu}\ou{A}{}{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\vee\ou{\gamma}{\nu}{}+\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}\\ =\left\{\ou{A}{\mu}{}\pu{\mu}+ig\ou{A}{}{\mu}\ou{A}{\mu}{}\right\}+\left\{\ou{A}{}{\mu}\pu{\nu}+ig\ou{A}{}{\mu}\ou{A}{}{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}$
となる．行列積 $AB$ の定義できる行列 $A,\ B$ と幾何代数 $a,\ b$ に対して
$\left\{A\lhd a\right\}\wedge\left\{B\lhd b\right\}:=\left\{AB\right\}\lhd\left\{a\wedge b\right\}\\ \left\{A\lhd a\right\}\vee\left\{B\lhd b\right\}:=\left\{A\lhd a\right\}\left\{B\lhd b\right\}-\left\{A\lhd a\right\}\wedge\left\{B\lhd b\right\}=\left\{AB\right\}\lhd\left\{a\vee b\right\}$
とwedge積とvee積を拡張する．
$F:=-A\wedge\mathcal{D}=-A\wedge\left\{\pu{\nu}I_N\lhd\ou{\gamma}{\nu}{}\right\}-igA\wedge A\\ =-\left\{\ou{A}{}{\mu}\lhd\ou{\gamma}{\mu}{}\right\}\wedge\left\{\pu{\nu}I_N\lhd\ou{\gamma}{\nu}{}\right\}-ig\left\{\ou{A}{}{\mu}\lhd\ou{\gamma}{\mu}{}\right\}\wedge\left\{\ou{A}{}{\nu}\lhd\ou{\gamma}{\nu}{}\right\}\\ =-\left\{\ou{A}{}{\mu}\pu{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}-ig\left\{\ou{A}{}{\mu}\ou{A}{}{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}\\ =-\left\{\ou{A}{}{\mu}\pu{\nu}+ig\ou{A}{}{\mu}\ou{A}{}{\nu}\right\}\lhd\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}$
成分を計算する．添字の丸括弧は総和を取らないことを意味する．
$-\left\{\ou{A}{}{(\mu)}\pu{(\nu)}+ig\ou{A}{}{(\mu)}\ou{A}{}{(\nu)}\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}-\left\{\ou{A}{}{(\nu)}\pu{(\mu)}+ig\ou{A}{}{(\nu)}\ou{A}{}{(\mu)}\right\}\lhd\left\{\ou{\gamma}{(\nu)}{}\wedge\ou{\gamma}{(\mu)}{}\right\}\\ =-\left\{\ou{A}{}{(\mu)}\pu{(\nu)}+ig\ou{A}{}{(\mu)}\ou{A}{}{(\nu)}\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}+\left\{\ou{A}{}{(\nu)}\pu{(\mu)}+ig\ou{A}{}{(\nu)}\ou{A}{}{(\mu)}\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}\\ =\left\{\ou{A}{}{(\nu)}\pu{(\mu)}-\ou{A}{}{(\mu)}\pu{(\nu)}-ig\ou{A}{}{(\mu)}\ou{A}{}{(\nu)}+ig\ou{A}{}{(\nu)}\ou{A}{}{(\mu)}\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}\\ =\left\{\ou{A}{}{(\nu)}\pu{(\mu)}-\ou{A}{}{(\mu)}\pu{(\nu)}+ig\left\{-\ou{A}{}{(\mu)}{}_a\ou{A}{}{(\nu)}{}_bT_aT_b+\ou{A}{}{(\nu)}{}_b\ou{A}{}{(\mu)}{}_aT_bT_a\right\}\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}\\ =\left\{\ou{A}{}{(\nu)}\pu{(\mu)}-\ou{A}{}{(\mu)}\pu{(\nu)}-ig\ou{A}{}{(\mu)}{}_a\ou{A}{}{(\nu)}{}_b\left\{T_aT_b-T_bT_a\right\}\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}\\ =\left\{\ou{A}{}{(\nu)}\pu{(\mu)}-\ou{A}{}{(\mu)}\pu{(\nu)}-ig\ou{A}{}{(\mu)}{}_a\ou{A}{}{(\nu)}{}_bif_{abc}T_c\right\}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}\\ =\left\{\ou{A}{}{(\nu)}{}_c\pu{(\mu)}-\ou{A}{}{(\mu)}{}_c\pu{(\nu)}+gf_{abc}\ou{A}{}{(\mu)}{}_a\ou{A}{}{(\nu)}{}_b\right\}T_c\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}\\ =\ou{F}{}{(\mu)(\nu)}{}_cT_c\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}=\ou{F}{}{(\mu)(\nu)}\lhd\left\{\ou{\gamma}{(\mu)}{}\wedge\ou{\gamma}{(\nu)}{}\right\}$
つまり
$F=-A\wedge\mathcal{D}=\frac{1}{2}\ou{F}{}{\mu\nu}{}_aT_a\lhd\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}=\frac{1}{2}\ou{F}{}{\mu\nu}\lhd\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}$
が成り立つ．また自身とのvee積は
$F\vee F=\frac{1}{4}\ou{F}{}{\mu\nu}\ou{F}{}{\rho\sigma}\lhd\left\{\left\{\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\right\}\vee\left\{\ou{\gamma}{\rho}{}\wedge\ou{\gamma}{\sigma}{}\right\}\right\}\\ =\frac{1}{4}\ou{F}{}{\mu\nu}\ou{F}{}{\rho\sigma}\lhd\left\{\left\{\ou{\gamma}{\nu}{}\vee\ou{\gamma}{\rho}{}\right\}\left\{\ou{\gamma}{\mu}{}\vee\ou{\gamma}{\sigma}{}\right\}-\left\{\ou{\gamma}{\mu}{}\vee\ou{\gamma}{\rho}{}\right\}\left\{\ou{\gamma}{\nu}{}\vee\ou{\gamma}{\sigma}{}\right\}\right\}\\ =-\frac{1}{2}\ou{F}{}{\mu\nu}\ou{F}{\mu\nu}{}$
であるためYang-Mills項は
$\mathcal{L}_{\mathrm{YM}}=\mathrm{Tr}(F\vee F)$
となる．これは一般の幾何代数でも成り立つ．