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\begin {equation} T_{\mu \nu \lambda }=\varepsilon _{\mu \nu \lambda }\varphi +\frac 12\left ( \eta _{\nu \lambda }t_\mu -\eta _{\mu \lambda }t_\nu \right ) +\varepsilon _{\mu \nu \alpha }X^\alpha {}_\lambda \mathrm {\ }, \label {3.1} \end {equation}
\begin{equation*} T _ { \mu \nu \lambda } = \varepsilon _ { \mu \nu \lambda } \varphi + \frac { 1 } { 2 } ( \eta _ { \nu \lambda } t _ { \mu } - \eta _ { \mu \lambda } t _ { \nu } ) + \varepsilon _ { \mu \nu \alpha } X ^ { \alpha } { } _ { \lambda } ~ , \end{equation*}
0203158_page007
isolated
\begin {eqnarray}\delta x^\mu =\omega ^\mu {}_\nu x^\nu +\frac {1}{\theta ^2}p_\nu \omega ^{[\nu }{}_\rho \theta ^{\rho \mu ]}, \quad \delta p^\mu =\omega ^\mu {}_\nu p^\nu . \end {eqnarray}
\begin{equation*} \delta x ^ { \mu } = \omega ^ { \mu } { } _ { \nu } x ^ { \nu } + \frac { 1 } { \theta ^ { 2 } } p _ { \nu } \omega ^ { [ \nu } { } _ { \rho } \theta ^ { \rho \mu ] } , \quad \delta p ^ { \mu } = \omega ^ { \mu } { } _ { \nu } p ^ { \nu } . \end{equation*}
0207274_page005
isolated
\begin {equation} M_{abc}=N_{gha}P^{gh}{}_{bc} \end {equation}
\begin{equation*} M _ { a b c } = N _ { g h a } P ^ { g h } { } _ { b c } \end{equation*}
0208218_page004
isolated
\begin {displaymath} p=(\gamma -1)\rho \end {displaymath}
\begin{equation*} p = ( \gamma - 1 ) \rho \end{equation*}
0209015_page002
isolated
\begin {equation} -i\bar {\kappa }_{\alpha }\gamma ^{a}\kappa _{(\beta )}e_{a} =-i\left [\mathcal {D}\mathcal {S}\Gamma _{s}(T^{I})\right ]_{\alpha \beta } R^{L}{}_{I} k_{(L)}\, , \end {equation}
\begin{equation*} - i \bar { \kappa } _ { \alpha } \gamma ^ { a } \kappa _ { ( \beta ) } e _ { a } = - i { [ D S \Gamma _ { s } ( T ^ { I } ) ] } _ { \alpha \beta } R ^ { L } { } _ { I } k _ { ( L ) } \, , \end{equation*}
0209069_page009
isolated
\begin {equation} \{Q_{(\alpha )},Q_{(\beta )}\}= -i\left [\mathcal {D}\mathcal {S}\Gamma _{s}(T^{I})\right ]_{\alpha \beta } (R^{a}{}_{I} P_{a} + R^{M}{}_{I}M)\, . \end {equation}
\begin{equation*} \{ Q _ { ( \alpha ) } , Q _ { ( \beta ) } \} = - i { [ D S \Gamma _ { s } ( T ^ { I } ) ] } _ { \alpha \beta } ( R ^ { a } { } _ { I } P _ { a } + R ^ { M } { } _ { I } M ) \, . \end{equation*}
0209069_page009
isolated
\begin {equation}\label {pary2.1} { L}^{(0)}_1 = \frac {m\dot {x}^2_1}{2} - k \epsilon _{ij} \dot {x}_i \ddot {x}_j\, . \end {equation}
\begin{equation*} L _ { 1 } ^ { ( 0 ) } = \frac { m \dot { x } _ { 1 } ^ { 2 } } { 2 } - k \epsilon _ { i j } \dot { x } _ { i } \ddot { x } _ { j } \, . \end{equation*}
0210112_page001
isolated
\begin {eqnarray} R_{\mu \nu } -\ft 12 R\, g_{\mu \nu }&=& \ft 12(\partial _\mu \phi \, \partial _\nu \phi -\ft 12 (\partial \phi )^2\, g_{\mu \nu }) +\ft 12 e^{-\ft 32\phi }\, (F_{\mu \rho }\, F_\nu {}^\rho -\ft 14F_2^2\, g_{\mu \nu })\\ &&+\ft 1{4} e^\phi \, (F_{\mu \rho \sigma }\, F_\nu {}^{\rho \sigma } -\ft 16 F_3^2\, g_{\mu \nu }) + \ft 1{12} e^{-\ft 12\phi }\, (F_{\mu \rho \sigma \lambda }\, F_\nu {}^{\rho \sigma \lambda } -\ft 18 F_4^2\, g_{\mu \nu })\\ && -9m^2\, (A_\mu \, A_\nu + 4A_\rho \, A^\rho \, g_{\mu \nu }) - 36m^2\, e^{\ft 32\phi }\, g_{\mu \nu } \\ &&-\ft 92 m\, (D_\mu A_\nu +D_\nu A_\mu -2D_\rho A^\rho \, g_{\mu \nu })\\ &&+\ft 34m\, (A_\mu \partial _\nu \phi + A_\nu \partial _\mu \phi -A^\rho \partial _\rho \phi \, g_{\mu \nu })\\ D_\nu (e^{-\ft 32\phi }\, F_\mu {}^\nu ) &=& 12m\partial _\mu \phi +18m^2\, A_\mu + 9m\, e^{-\ft 32\phi }\, A^\nu \, F_{\mu \nu } -\ft 16 e^{-\ft 12\phi }\, F_{\mu \nu \rho \sigma }F^{\nu \rho \sigma }\\ D^{\sigma }(e^{-\ft 12\phi }F_{\mu \nu \rho \sigma })&=& -6m\,e^{\phi }\, F_{\mu \nu \rho }+6m\, e^{-\ft 12\phi }\, A^{\sigma }F_{\mu \nu \rho \sigma } -\ft 1{144}\epsilon _{\mu \nu \rho \sigma _1\dots \sigma _7}\, F^{\sigma _1\dots \sigma _4}\,F^{\sigma _5\sigma _6\sigma _7}\\ D^{\sigma }(e^{\phi }F_{\mu \nu \sigma })&=& 6m\,e^{\phi }\, A^\sigma \, F_{\mu \nu \sigma }+\ft 12\, e^{-\ft 12\phi }\, F_{\mu \nu \sigma \rho }\, F^{\sigma \rho } + \ft 1{1152} \epsilon _{\mu \nu \rho _1\dots \rho _8}\, F^{\rho _1\dots \rho _4} F^{\rho _5\dots \rho _8}\\ \end {eqnarray}
\begin{align*} R _ { \mu \nu } - \frac { 1 } { 2 } R \, g _ { \mu \nu } & = & \frac { 1 } { 2 } ( \partial _ { \mu } \phi \, \partial _ { \nu } \phi - \frac { 1 } { 2 } ( \partial \phi ) ^ { 2 } \, g _ { \mu \nu } ) + \frac { 1 } { 2 } e ^ { - \frac { 3 } { 2 } \phi } \, ( F _ { \mu \rho } \, F _ { \nu } { } ^ { \rho } - \frac { 1 } { 4 } F _ { 2 } ^ { 2 } \, g _ { \mu \nu } ) \\ & & + \frac { 1 } { 4 } e ^ { \phi } \, ( F _ { \mu \rho \sigma } \, F _ { \nu } { } ^ { \rho \sigma } - \frac { 1 } { 6 } F _ { 3 } ^ { 2 } \, g _ { \mu \nu } ) + \frac { 1 } { 1 2 } e ^ { - \frac { 1 } { 2 } \phi } \, ( F _ { \mu \rho \sigma \lambda } \, F _ { \nu } { } ^ { \rho \sigma \lambda } - \frac { 1 } { 8 } F _ { 4 } ^ { 2 } \, g _ { \mu \nu } ) \\ & & - 9 m ^ { 2 } \, ( A _ { \mu } \, A _ { \nu } + 4 A _ { \rho } \, A ^ { \rho } \, g _ { \mu \nu } ) - 3 6 m ^ { 2 } \, e ^ { \frac { 3 } { 2 } \phi } \, g _ { \mu \nu } \\ & & - \frac { 9 } { 2 } m \, ( D _ { \mu } A _ { \nu } + D _ { \nu } A _ { \mu } - 2 D _ { \rho } A ^ { \rho } \, g _ { \mu \nu } ) \\ & & + \frac { 3 } { 4 } m \, ( A _ { \mu } \partial _ { \nu } \phi + A _ { \nu } \partial _ { \mu } \phi - A ^ { \rho } \partial _ { \rho } \phi \, g _ { \mu \nu } ) \\ D _ { \nu } ( e ^ { - \frac { 3 } { 2 } \phi } \, F _ { \mu } { } ^ { \nu } ) & = & 1 2 m \partial _ { \mu } \phi + 1 8 m ^ { 2 } \, A _ { \mu } + 9 m \, e ^ { - \frac { 3 } { 2 } \phi } \, A ^ { \nu } \, F _ { \mu \nu } - \frac { 1 } { 6 } e ^ { - \frac { 1 } { 2 } \phi } \, F _ { \mu \nu \rho \sigma } F ^ { \nu \rho \sigma } \\ D ^ { \sigma } ( e ^ { - \frac { 1 } { 2 } \phi } F _ { \mu \nu \rho \sigma } ) & = & - 6 m \, e ^ { \phi } \, F _ { \mu \nu \rho } + 6 m \, e ^ { - \frac { 1 } { 2 } \phi } \, A ^ { \sigma } F _ { \mu \nu \rho \sigma } - \frac { 1 } { 1 4 4 } \epsilon _ { \mu \nu \rho \sigma _ { 1 } \ldots \sigma _ { 7 } } \, F ^ { \sigma _ { 1 } \ldots \sigma _ { 4 } } \, F ^ { \sigma _ { 5 } \sigma _ { 6 } \sigma _ { 7 } } \\ D ^ { \sigma } ( e ^ { \phi } F _ { \mu \nu \sigma } ) & = & 6 m \, e ^ { \phi } \, A ^ { \sigma } \, F _ { \mu \nu \sigma } + \frac { 1 } { 2 } \, e ^ { - \frac { 1 } { 2 } \phi } \, F _ { \mu \nu \sigma \rho } \, F ^ { \sigma \rho } + \frac { 1 } { 1 1 5 2 } \epsilon _ { \mu \nu \rho _ { 1 } \ldots \rho _ { 8 } } \, F ^ { \rho _ { 1 } \ldots \rho _ { 4 } } F ^ { \rho _ { 5 } \ldots \rho _ { 8 } } \end{align*}
0211192_page010
isolated
\begin {eqnarray} \delta \lambda &=& \ft 12 m e^{\frac {3}{4} \phi } \gamma _{11} \epsilon -\ft 18 e^{-\frac {3}{4} \phi } \slashed {F}_2 \gamma _{11} \epsilon -\ft 13 \slashed {\mathfrak {D}} \phi \epsilon -\ft {1}{144} e^{-\frac {1}{4} \phi } \slashed {F}_4 \epsilon +\ft {1}{18} e^{\frac {1}{2} \phi } \slashed {F}_3 \gamma _{11} \epsilon \\ \delta \psi _a&=&D_a \epsilon +\ft {9}{16}m e^{\frac {3}{4} \phi } \gamma _a \gamma _{11} \epsilon -m\ft {9}{16} \gamma _a \slashed {A}_1\epsilon -\ft {1}{64} e^{-\frac {3}{4} \phi }({}_a \slashed {F}_2-14 \slashed {F}_{(2)a})\gamma _{11} \epsilon \\&&+\ft {1}{48} e^{\frac {1}{2} \phi }(9\slashed {F}_{(3)a}-{}_a \slashed {F}_3) \gamma _{11} \epsilon +\ft {1}{128} e^{-\frac {1}{4} \phi }(\ft {20}{3} \slashed {F}_{(4)a}-{}_a \slashed {F}_4)\epsilon \end {eqnarray}
\begin{align*} \delta \lambda & = & 12me34\phi\gamma11\epsilon-18e-34\phi\slash F 2\gamma11\epsilon-13\slash D \phi\epsilon-1144e-14\phi\slash F 4\epsilon+118e12\phi\slash F 3\gamma11\epsilon\\ \delta \psi _ { a } & = & Da\epsilon+916me34\phi\gammaa\gamma11\epsilon-m916\gammaa\slash A 1\epsilon-164e-34\phi(a\slash F 2-14\slash F (2)a)\gamma11\epsilon\\ & & +148e12\phi(9\slash F (3)a-a\slash F 3)\gamma11\epsilon+1128e-14\phi(203\slash F (4)a-a\slash F 4)\epsilon \end{align*}
0211192_page010
isolated
\begin {equation} \mathfrak {D}_a \phi =\partial _a \phi +\ft 32 m A_a \end {equation}
\begin{equation*} D _ { a } \phi = \partial _ { a } \phi + \frac { 3 } { 2 } m A _ { a } \end{equation*}
0211192_page010
isolated
\begin {equation} \star \left (dx^{\mu _1}\wedge \dots dx^{\mu _n}\right )=\frac {\sqrt {-{\rm det}(g)}}{(10-n)!}G^{\mu _1\nu _1}\dots G^{\mu _n\nu _n}\epsilon _{\rho _1\dots \rho _{10-n}\nu _1\dots \nu _n}\,dx^{\rho _1}\wedge \dots dx^{\rho _{10-n}} \label {hodgecurvo} \end {equation}
\begin{equation*} \star ( d x ^ { \mu _ { 1 } } \wedge \ldots d x ^ { \mu _ { n } } ) = \frac { \sqrt { - d e t ( g ) } } { ( 1 0 - n ) ! } G ^ { \mu _ { 1 } \nu _ { 1 } } \ldots G ^ { \mu _ { n } \nu _ { n } } \epsilon _ { \rho _ { 1 } \ldots \rho _ { 1 0 - n } \nu _ { 1 } \ldots \nu _ { n } } \, d x ^ { \rho _ { 1 } } \wedge \ldots d x ^ { \rho _ { 1 0 - n } } \end{equation*}
0212287_page014
isolated
\begin {equation} K\phi ^{\ell }_{jm}=-\kappa \phi ^{\ell }_{jm}. \end {equation}
\begin{equation*} K \phi _ { j m } ^ { l } = - \kappa \phi _ { j m } ^ { l } . \end{equation*}
0301199_page003
isolated
\begin {equation} H= { {\mathbf \alpha }}\cdot ({\bf p}-i\beta {\bf {\hat r}}(W(r)+ {K\over r})) +\beta M+V(r) \end {equation}
\begin{equation*} H = \alpha \cdot ( p - i \beta \hat { r } ( W ( r ) + \frac { K } { r } ) ) + \beta M + V ( r ) \end{equation*}
0301199_page003
isolated
\begin {equation} \left (-i\rho _2{d\over dr} +W\rho _1-(E-V)+M\rho _3\right )\Phi =0. \end {equation}
\begin{equation*} ( - i \rho _ { 2 } \frac { d } { d r } + W \rho _ { 1 } - ( E - V ) + M \rho _ { 3 } ) \Phi = 0 . \end{equation*}
0301199_page003
isolated
\begin {equation} \Phi =e^{-{i\rho _2 \eta }}\hat \Phi \end {equation}
\begin{equation*} \Phi = e ^ { - i \rho _ { 2 } \eta } \hat { \Phi } \end{equation*}
0301199_page003
isolated
\begin {eqnarray} {}&&\left [-i\rho _2{d\over dr}-(E-V)+\rho _1(W\cos 2\eta -M\sin 2\eta )\right ]\hat \Phi \\ {}&&+ \left [\rho _3 (W\sin 2\eta +M\cos 2\eta )\right ]\hat \Phi =0. \end {eqnarray}
\begin{align*} & & [ - i \rho _ { 2 } \frac { d } { d r } - ( E - V ) + \rho _ { 1 } ( W \operatorname { c o s } 2 \eta - M \operatorname { s i n } 2 \eta ) ] \hat { \Phi } \\ & & + [ \rho _ { 3 } ( W \operatorname { s i n } 2 \eta + M \operatorname { c o s } 2 \eta ) ] \hat { \Phi } = 0 . \end{align*}
0301199_page003
isolated
\begin {eqnarray} W={V\over \sin 2\eta }. \end {eqnarray}
\begin{equation*} W = \frac { V } { \operatorname { s i n } 2 \eta } . \end{equation*}
0301199_page003
isolated
\begin {eqnarray} {}&&\left [-i\rho _2{d\over dr}-E+V(1+\rho _3)+\rho _1({V\over \tan 2\eta }- M\sin 2\eta )\right ]\hat \Phi \\ {}&&+ \left [\rho _3 M\cos 2\eta \right ]\hat \Phi =0 \end {eqnarray}
\begin{align*} & & [ - i \rho _ { 2 } \frac { d } { d r } - E + V ( 1 + \rho _ { 3 } ) + \rho _ { 1 } ( \frac { V } { \operatorname { t a n } 2 \eta } - M \operatorname { s i n } 2 \eta ) ] \hat { \Phi } \\ & & + [ \rho _ { 3 } M \operatorname { c o s } 2 \eta ] \hat { \Phi } = 0 \end{align*}
0301199_page003
isolated
\begin {equation} \label {FG} \hat F_{\ell j}={1\over {E+M\cos 2\eta }}({d\over dr}+{V\over \tan 2\eta }- M\sin 2\eta )\hat G_{\ell j} \end {equation}
\begin{equation*} \hat { F } _ { l j } = \frac { 1 } { E + M c o s 2 \eta } ( \frac { d } { d r } + \frac { V } { \operatorname { t a n } 2 \eta } - M \operatorname { s i n } 2 \eta ) \hat { G } _ { l j } \end{equation*}
0301199_page003
isolated
\begin {equation} \left [-{d^2\over dr^2}+({V\over \tan 2\eta })^2+2EV- {1\over \tan 2\eta }{dV\over dr}- (E^2-M^2)\right ]\hat G_{\ell j}=0. \end {equation}
\begin{equation*} [ - \frac { d ^ { 2 } } { d r ^ { 2 } } + ( \frac { V } { t a n 2 \eta } ) ^ { 2 } + 2 E V - \frac { 1 } { \operatorname { t a n } 2 \eta } \frac { d V } { d r } - ( E ^ { 2 } - M ^ { 2 } ) ] \hat { G } _ { l j } = 0 . \end{equation*}
0301199_page003
isolated
\begin {equation} S = \frac 1{2\kappa ^2}\int d^Dx\sqrt {-g}e^{-2\Phi }(-2V(\phi _0)), \end {equation}
\begin{equation*} S = \frac { 1 } { 2 \kappa ^ { 2 } } \int d ^ { D } x \sqrt { - g } e ^ { - 2 \Phi } ( - 2 V ( \phi _ { 0 } ) ) , \end{equation*}
0302010_page007
isolated
\begin {equation} c = 3\alpha 'V(\Phi _0). \end {equation}
\begin{equation*} c = 3 \alpha ^ { \prime } V ( \Phi _ { 0 } ) . \end{equation*}
0302010_page007
isolated
\begin {eqnarray} D_0^x \tilde U[A](x;x') & = & D_0^x e^{x_0 \, a_0(x)} \, U[A](x;x') \, e^{-x_0' \, a_0(x')} \\ &= & \left [D_0^x , e^{x_0 \, a_0(x)} \right ] \, U[A](x;x') \, e^{-x_0' \, a_0(x')} + e^{x_0 \, a_0(x)} D_0^x \, U[A](x;x') \, e^{-x_0' \, a_0(x')} \\ &= & a_0(x) \, \tilde U[A](x;x') \end {eqnarray}
\begin{align*} D _ { 0 } ^ { x } \widetilde { U } [ A ] ( x ; x ^ { \prime } ) & = & D _ { 0 } ^ { x } e ^ { x _ { 0 } \, a _ { 0 } ( x ) } \, U [ A ] ( x ; x ^ { \prime } ) \, e ^ { - x _ { 0 } ^ { \prime } \, a _ { 0 } ( x ^ { \prime } ) } \\ & = & [ D _ { 0 } ^ { x } , e ^ { x _ { 0 } \, a _ { 0 } ( x ) } ] \, U [ A ] ( x ; x ^ { \prime } ) \, e ^ { - x _ { 0 } ^ { \prime } \, a _ { 0 } ( x ^ { \prime } ) } + e ^ { x _ { 0 } \, a _ { 0 } ( x ) } D _ { 0 } ^ { x } \, U [ A ] ( x ; x ^ { \prime } ) \, e ^ { - x _ { 0 } ^ { \prime } \, a _ { 0 } ( x ^ { \prime } ) } \\ & = & a _ { 0 } ( x ) \, \widetilde { U } [ A ] ( x ; x ^ { \prime } ) \end{align*}
0302142_page012
isolated
\begin {equation}\label {1} m^2 = \left (2\kappa \sinh \left (\frac {P_0}{2\kappa }\right )\right )^2 - \vec {P}\,{}^2\, e^{P_0/\kappa }. \end {equation}
\begin{equation*} m ^ { 2 } = { ( 2 \kappa \operatorname { s i n h } ( \frac { P _ { 0 } } { 2 \kappa } ) ) } ^ { 2 } - \vec { P } \, { } ^ { 2 } \, e ^ { P _ { 0 } \slash \kappa } . \end{equation*}
0302157_page002
isolated
\begin {equation}\label {2} m^2 = \frac {P_{0}^2 - \vec {P}{}^2}{\left (1- \frac {P_0}\kappa \right )^2}, \end {equation}
\begin{equation*} m ^ { 2 } = \frac { P _ { 0 } ^ { 2 } - \vec { P } { } ^ { 2 } } { { ( 1 - \frac { P _ { 0 } } { \kappa } ) } ^ { 2 } } , \end{equation*}
0302157_page002
isolated
\begin {equation} \big \{X_{1}, X_{2}\big \}=\theta , \label {NCpos} \end {equation}
\begin{equation*} \{ X _ { 1 } , X _ { 2 } \} = \theta , \end{equation*}
0303099_page001
isolated
\begin {equation} L= \frac {m\dot {\vx }^2}{2} + \frac {\kappa }{2}\,\dot {\vx }\times \ddot {\vx }. \label {LSZlag} \end {equation}
\begin{equation*} L = \frac { m \dot { \vec { x } } ^ { 2 } } { 2 } + \frac { \kappa } { 2 } \, \dot { \vec { x } } \times \ddot { \vec { x } } . \end{equation*}
0303099_page001
isolated
\[ \beta _\mu ^{(1)}\beta _\nu ^{(1)}=\varepsilon ^{[\alpha \mu ] ,[\alpha \nu ]} + \delta _{\mu \nu }\varepsilon ^{\alpha ,\alpha } -\varepsilon ^{\nu ,\mu } ,\hspace {0.3in} \beta _\mu ^{(\widetilde {1})}\beta _\nu ^{(1)}=e_{\mu \nu \rho \omega }\varepsilon ^{\widetilde {\omega },\rho } , \]
\begin{equation*} \beta _ { \mu } ^ { ( 1 ) } \beta _ { \nu } ^ { ( 1 ) } = \varepsilon ^ { [ \alpha \mu ] , [ \alpha \nu ] } + \delta _ { \mu \nu } \varepsilon ^ { \alpha , \alpha } - \varepsilon ^ { \nu , \mu } , \hspace{21.68pt} \beta _ { \mu } ^ { ( \widetilde { 1 } ) } \beta _ { \nu } ^ { ( 1 ) } = e _ { \mu \nu \rho \omega } \varepsilon ^ { \widetilde { \omega } , \rho } , \end{equation*}
0304091_page022
isolated
\[ \beta _\mu ^{(0)}\beta _\nu ^{(1)}=\varepsilon ^{0,[\mu \nu ]} ,\hspace {0.3in}\beta _\mu ^{(\widetilde {0})}\beta _\nu ^{(\widetilde {0})}=\delta _{\mu \nu }\varepsilon ^{\widetilde {0},\widetilde {0}} +\varepsilon ^{\widetilde {\mu },\widetilde {\nu }} , \]
\begin{equation*} \beta _ { \mu } ^ { ( 0 ) } \beta _ { \nu } ^ { ( 1 ) } = \varepsilon ^ { 0 , [ \mu \nu ] } , \hspace{21.68pt} \beta _ { \mu } ^ { ( \widetilde { 0 } ) } \beta _ { \nu } ^ { ( \widetilde { 0 } ) } = \delta _ { \mu \nu } \varepsilon ^ { \widetilde { 0 } , \widetilde { 0 } } + \varepsilon ^ { \widetilde { \mu } , \widetilde { \nu } } , \end{equation*}
0304091_page022
isolated
\begin {equation} \beta _\mu ^{(\widetilde {1})}\beta _\nu ^{(\widetilde {0})}=\frac {1}{2}e_{\nu \mu \rho \omega }\varepsilon ^{ [\rho \omega ],\widetilde {0}} ,\hspace {0.3in}\beta _\mu ^{(1)}\beta _\nu ^{(\widetilde {1})}=e_{\mu \nu \rho \omega }\varepsilon ^{\omega ,\widetilde {\rho }} ,\label {96} \end {equation}
\begin{equation*} \beta _ { \mu } ^ { ( \widetilde { 1 } ) } \beta _ { \nu } ^ { ( \widetilde { 0 } ) } = \frac { 1 } { 2 } e _ { \nu \mu \rho \omega } \varepsilon ^ { [ \rho \omega ] , \widetilde { 0 } } , \hspace{21.68pt} \beta _ { \mu } ^ { ( 1 ) } \beta _ { \nu } ^ { ( \widetilde { 1 } ) } = e _ { \mu \nu \rho \omega } \varepsilon ^ { \omega , \widetilde { \rho } } , \end{equation*}
0304091_page022
isolated
\[ \beta _\mu ^{(\widetilde {0})}\beta _\nu ^{(\widetilde {1})}=\frac {1}{2}e_{ \mu \nu \rho \omega }\varepsilon ^{\widetilde {0},[\rho \omega ]} ,~~\beta _\mu ^{(\widetilde {1})}\beta _\nu ^{(\widetilde {1})}= \delta _{\mu \nu } \left (\varepsilon ^{\widetilde {\alpha },\widetilde {\alpha }} + \frac {1}{2}\varepsilon ^{[\rho \omega ],[\rho \omega ]} \right )- \varepsilon ^{\widetilde {\nu },\widetilde {\mu }} + \varepsilon ^{[\alpha \nu ],[\mu \alpha ]} , \]
\begin{equation*} \beta _ { \mu } ^ { ( \widetilde { 0 } ) } \beta _ { \nu } ^ { ( \widetilde { 1 } ) } = \frac { 1 } { 2 } e _ { \mu \nu \rho \omega } \varepsilon ^ { \widetilde { 0 } , [ \rho \omega ] } , ~ ~ \beta _ { \mu } ^ { ( \widetilde { 1 } ) } \beta _ { \nu } ^ { ( \widetilde { 1 } ) } = \delta _ { \mu \nu } ( \varepsilon ^ { \widetilde { \alpha } , \widetilde { \alpha } } + \frac { 1 } { 2 } \varepsilon ^ { [ \rho \omega ] , [ \rho \omega ] } ) - \varepsilon ^ { \widetilde { \nu } , \widetilde { \mu } } + \varepsilon ^ { [ \alpha \nu ] , [ \mu \alpha ] } , \end{equation*}
0304091_page022
isolated
\[ \beta _\mu ^{(1)}\beta _\nu ^{(0)}=\varepsilon ^{[\nu \mu ] ,0} ,\hspace {0.3in}\beta _\mu ^{(0)}\beta _\nu ^{(0)}=\delta _{\mu \nu } \varepsilon ^{0,0} + \varepsilon ^{\mu ,\nu } . \]
\begin{equation*} \beta _ { \mu } ^ { ( 1 ) } \beta _ { \nu } ^ { ( 0 ) } = \varepsilon ^ { [ \nu \mu ] , 0 } , \hspace{21.68pt} \beta _ { \mu } ^ { ( 0 ) } \beta _ { \nu } ^ { ( 0 ) } = \delta _ { \mu \nu } \varepsilon ^ { 0 , 0 } + \varepsilon ^ { \mu , \nu } . \end{equation*}
0304091_page022
isolated
\[ \frac {1}{2}\Gamma _{[\mu } \Gamma _{\nu ]}\equiv \frac {1}{2}\left (\Gamma _\mu \Gamma _\nu -\Gamma _\nu \Gamma _\mu \right )=\varepsilon ^{[\alpha \mu ],[\alpha \nu ]}- \varepsilon ^{[\alpha \nu ][ \alpha \mu ]} +\varepsilon ^{\mu ,\nu }-\varepsilon ^{\nu ,\mu } + \varepsilon ^{\widetilde {\mu },\widetilde {\nu }} - \varepsilon ^{\widetilde {\nu },\widetilde {\mu }} \]
\begin{equation*} \frac { 1 } { 2 } \Gamma _ { [ \mu } \Gamma _ { \nu ] } \equiv \frac { 1 } { 2 } ( \Gamma _ { \mu } \Gamma _ { \nu } - \Gamma _ { \nu } \Gamma _ { \mu } ) = \varepsilon ^ { [ \alpha \mu ] , [ \alpha \nu ] } - \varepsilon ^ { [ \alpha \nu ] [ \alpha \mu ] } + \varepsilon ^ { \mu , \nu } - \varepsilon ^ { \nu , \mu } + \varepsilon ^ { \widetilde { \mu } , \widetilde { \nu } } - \varepsilon ^ { \widetilde { \nu } , \widetilde { \mu } } \end{equation*}
0304091_page023
isolated
\[ +e_{\mu \nu \rho \omega }\left (\varepsilon ^{\widetilde {\omega },\rho } - \varepsilon ^{\rho ,\widetilde {\omega }}\right ) + \varepsilon ^{0,[\mu \nu ]}-\varepsilon ^{[\mu \nu ],0} + \frac {1}{2}e_{\mu \nu \rho \omega }\left (\varepsilon ^{\widetilde {0}, [\rho \omega ]} -\varepsilon ^{[\rho \omega ],\widetilde {0}}\right ) . \]
\begin{equation*} + e _ { \mu \nu \rho \omega } ( \varepsilon ^ { \widetilde { \omega } , \rho } - \varepsilon ^ { \rho , \widetilde { \omega } } ) + \varepsilon ^ { 0 , [ \mu \nu ] } - \varepsilon ^ { [ \mu \nu ] , 0 } + \frac { 1 } { 2 } e _ { \mu \nu \rho \omega } ( \varepsilon ^ { \widetilde { 0 } , [ \rho \omega ] } - \varepsilon ^ { [ \rho \omega ] , \widetilde { 0 } } ) . \end{equation*}
0304091_page023
isolated
\[ \frac {1}{2}\overline {\Gamma }_{[\mu } \overline {\Gamma }_{\nu ]}\equiv \frac {1}{2}\left (\overline {\Gamma }_\mu \overline {\Gamma }_\nu -\overline {\Gamma }_\nu \overline {\Gamma }_\mu \right )=\varepsilon ^{[\alpha \mu ],[\alpha \nu ]}- \varepsilon ^{[\alpha \nu ][ \alpha \mu ]} +\varepsilon ^{\mu ,\nu }-\varepsilon ^{\nu ,\mu } + \varepsilon ^{\widetilde {\mu },\widetilde {\nu }} - \varepsilon ^{\widetilde {\nu },\widetilde {\mu }} \]
\begin{equation*} \frac { 1 } { 2 } \overline { \Gamma } _ { [ \mu } \overline { \Gamma } _ { \nu ] } \equiv \frac { 1 } { 2 } ( \overline { \Gamma } _ { \mu } \overline { \Gamma } _ { \nu } - \overline { \Gamma } _ { \nu } \overline { \Gamma } _ { \mu } ) = \varepsilon ^ { [ \alpha \mu ] , [ \alpha \nu ] } - \varepsilon ^ { [ \alpha \nu ] [ \alpha \mu ] } + \varepsilon ^ { \mu , \nu } - \varepsilon ^ { \nu , \mu } + \varepsilon ^ { \widetilde { \mu } , \widetilde { \nu } } - \varepsilon ^ { \widetilde { \nu } , \widetilde { \mu } } \end{equation*}
0304091_page023
isolated
\[ -e_{\mu \nu \rho \omega }\left (\varepsilon ^{\widetilde {\omega },\rho } - \varepsilon ^{\rho ,\widetilde {\omega }}\right ) - \varepsilon ^{0,[\mu \nu ]}+\varepsilon ^{[\mu \nu ],0} - \frac {1}{2}e_{\mu \nu \rho \omega }\left (\varepsilon ^{\widetilde {0}, [\rho \omega ]} -\varepsilon ^{[\rho \omega ],\widetilde {0}}\right ) , \]
\begin{equation*} - e _ { \mu \nu \rho \omega } ( \varepsilon ^ { \widetilde { \omega } , \rho } - \varepsilon ^ { \rho , \widetilde { \omega } } ) - \varepsilon ^ { 0 , [ \mu \nu ] } + \varepsilon ^ { [ \mu \nu ] , 0 } - \frac { 1 } { 2 } e _ { \mu \nu \rho \omega } ( \varepsilon ^ { \widetilde { 0 } , [ \rho \omega ] } - \varepsilon ^ { [ \rho \omega ] , \widetilde { 0 } } ) , \end{equation*}
0304091_page023
isolated
\[ \frac {1}{2}\overline {\Gamma }_{[\mu } \Gamma _{\nu ]}\equiv \frac {1}{2}\left (\overline {\Gamma }_\mu \Gamma _\nu -\overline {\Gamma } _\nu \Gamma _\mu \right )= e_{\mu \nu \rho \omega }\left (\varepsilon ^{\widetilde {\rho },\omega } + \varepsilon ^{\omega ,\widetilde {\rho }}\right ) \]
\begin{equation*} \frac { 1 } { 2 } \overline { \Gamma } _ { [ \mu } \Gamma _ { \nu ] } \equiv \frac { 1 } { 2 } ( \overline { \Gamma } _ { \mu } \Gamma _ { \nu } - \overline { \Gamma } _ { \nu } \Gamma _ { \mu } ) = e _ { \mu \nu \rho \omega } ( \varepsilon ^ { \widetilde { \rho } , \omega } + \varepsilon ^ { \omega , \widetilde { \rho } } ) \end{equation*}
0304091_page023
isolated
\[ + \varepsilon ^{0,[\nu \mu ]}+\varepsilon ^{[\nu \mu ],0} + \frac {1}{2}e_{\mu \nu \rho \omega }\left (\varepsilon ^{\widetilde {0}, [\rho \omega ]} +\varepsilon ^{[\rho \omega ],\widetilde {0}}\right ) . \]
\begin{equation*} + \varepsilon ^ { 0 , [ \nu \mu ] } + \varepsilon ^ { [ \nu \mu ] , 0 } + \frac { 1 } { 2 } e _ { \mu \nu \rho \omega } ( \varepsilon ^ { \widetilde { 0 } , [ \rho \omega ] } + \varepsilon ^ { [ \rho \omega ] , \widetilde { 0 } } ) . \end{equation*}
0304091_page023
isolated
\[ J_{\mu \nu }=\frac 14\left ( \Gamma _{[\mu } \Gamma _{\nu ]} +\overline {\Gamma }_{[\mu } \overline {\Gamma }_{\nu ]}\right ) \]
\begin{equation*} J _ { \mu \nu } = \frac { 1 } { 4 } ( \Gamma _ { [ \mu } \Gamma _ { \nu ] } + \overline { \Gamma } _ { [ \mu } \overline { \Gamma } _ { \nu ] } ) \end{equation*}
0304091_page023
isolated
\[ =\varepsilon ^{[\alpha \mu ],[\alpha \nu ]}- \varepsilon ^{[\alpha \nu ][ \alpha \mu ]} +\varepsilon ^{\mu ,\nu }-\varepsilon ^{\nu ,\mu } + \varepsilon ^{\widetilde {\mu },\widetilde {\nu }} - \varepsilon ^{\widetilde {\nu },\widetilde {\mu }} . \]
\begin{equation*} = \varepsilon ^ { [ \alpha \mu ] , [ \alpha \nu ] } - \varepsilon ^ { [ \alpha \nu ] [ \alpha \mu ] } + \varepsilon ^ { \mu , \nu } - \varepsilon ^ { \nu , \mu } + \varepsilon ^ { \widetilde { \mu } , \widetilde { \nu } } - \varepsilon ^ { \widetilde { \nu } , \widetilde { \mu } } . \end{equation*}
0304091_page023
isolated