%  max3.tex    Maxwell bis Feynman , SS 2002 
%             Formelsammlung (mit Baker-Campbell-Hausdorff)  

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\def\aaa{\hbox{\sans A}}
\def\kgll{\hbox{\sc\bf ~:$=\,$}}
\def\kglr{$\,=$\hbox{\sc\bf :~}}
\let\sc=\scriptsize
\def\det{\hspace{.14mm}\rm det\hspace{.1mm}}
\def\any{\hbox{\tiny $\ddagger$}}
\def\hori{\vskip 5mm \hrule \vskip 5mm}

\begin{document}
\rl{M--F~, ~30.~April 2002}

\vskip 3mm
\hspace*{2mm}
{\boldmath$ (\vc a \cdot \vc \s ) (\vc b \cdot \vc \s ) 
 = \vc a \cdot \vc b + \;\,i\, (\vc a \times \vc b ) \cdot \vc \s $}

\hori

{\sc $ B B^{-1}=1 \quad \folg $} {\boldmath$\quad \6_\mu B^{-1} 
              = - B^{-1} (\6_\mu B) B^{-1} $}

\hori

\hspace*{2mm}
{\boldmath$\dis\6_\mu e^{M(x)} = \int_0^1 \!\!\! ds \,\; e^{s M(x)} 
   \lk \6_\mu M(x)\rk e^{(1-s) M(x)} $} \\[2mm]
{\sc $(\!(\,X(s) \kgll (\6_\mu e^{sM}) e^{-sM} \;\; , \;\; 
    \6_s X(s) = (\6_\mu e^{sM} M) e^{-sM} - (\6_\mu e^{sM}) 
    M e^{-sM} = e^{sM} (\6_\mu M) e^{-sM} \;\; , \;\; 
    \int_0^1\! ds \ldots\;\; {\rm und}\; \parallel \cdot e^M\,  )\!)$}  

\hori

\hspace*{2mm}
{\boldmath$  e^A B e^{-A}\; = \;e^{\lk A , \;\; \rk} B \;\; 
   = \; B + \lk A , B \rk  + {1\02} \lk A , 
     \lk A,B \rk \rk + \; \ldots \; $}

{\sc $(\!(\,X(s) \kgll e^{sA} B e^{-sA} \; , \;
     \6_s X(s) = e^{sA} \lk A,B \rk e^{-sA} = \lk A, e^{sA} B e^{-sA}\rk
     = \lk A, X(s) \rk \; , \; X(0)=B \; \folg  \;
        X(s)=e^{s \lk A, \;\,\rk} B \; )\!) $}

\hori

{\ft Baker--Campbell--Hausdorff  ~formula~:}

{\boldmath$\dis e^A e^B = e^{A+B+X} \; \hbox{\sc mit} \; 
X = \sum_{n=1}^\infty  \int_0^1 \!\!\! ds \,\; 
{ \(1 - e^{-s \lk B , \;\;\rk} e^{- \lk A , \;\;\rk} \)^n
  \0 1+n} \, B$} {\ft $\; = \;{1\02} \lk A, B \rk 
  + \cl O ( \hbox{\sc kubisch} ) $}

{\sc $(\!(\; e^A e^{sB} \kglr e^{C(s)}\; , \; C(0)=A \; , \;
   A+B+X = C(1) = A + \int_0^1 \! ds\; C' \; , \; 
   C' = \6_s C(s) \; , \; e^C B = \6_s e^C = \int_0^1\! dt\; 
  e^{tC} C' e^{(1-t)C} \; , \;\; {\rm d.h.} \\[1mm]
  e^C B e^{-C} = \int_0^1\! dt \; e^{tC} C' e^{-tC} 
   \;\; , \;\; e^{\lk C, \;\;\rk} B = \int_0^1\! dt \; 
   e^{\lk tC , \;\; \rk} C'  \;\; , \;\;
  \lk C , \;\;\rk e^{\lk C, \;\;\rk} B = (e^{\lk C, \;\;\rk} -1)\, C' 
   \;\; , \;\; C'= g( e^{-\lk C, \;\;\rk})\, B \;\; {\rm mit} \\[1mm]
  g(z) = {- {\rm ln}(z) \0 1-z } = \sum_{n=0}^\infty \!{(1-z)^n \0 1+n}
  = 1 + \sum_{n=1}^\infty \!{(1-z)^n \0 1+n} \, . \;\, {\rm Und \; nun\;}
  e^{-\lk C, \;\rk} \any = e^{-C} \any e^C 
  = e^{-sB} e^{-A} \any e^A e^{sB} = e^{-s\lk B, \;\rk}
     e^{-\lk A, \;\rk}\any   \;)\!)$}

\hori

\hspace*{2mm}
{\boldmath$ \det (e^A) = e^{{\rm Sp} (A )}$} \quad . \qquad
\hbox{\ft Ist $A= \ln(U)$ m\"oglich, so \ \ } 
  {\boldmath$\det (U) = e^{{\rm Sp} \lk {\rm ln} (U) \rk} $}
\vskip -4.5mm {\sc
\bean  \hspace*{-1mm}
   (\!( \;\6_s \,\det ( e^{s A} ) 
   \glo{4} \e_{j_1 \ldots j_N } \lb 
   ( A e^{sA} )_{1 j_1} ( e^{sA} )_{2 j_2} ( e^{sA} )_{3 j_3} 
      \cdot \ldots  \;\; + \;\; 
    ( e^{sA} )_{1 j_1} (A e^{sA} )_{2 j_2} ( e^{sA} )_{3 j_3} 
   \cdot \ldots \;\; + \;\; \ldots \rb  
   \gluo{5}{9}
   A_{1\ell}\, \e_{j_1 \ldots j_N}
     ( e^{sA})_{\ell j_1} ( e^{sA})_{2j_2} ( e^{sA} )_{3 j_3} 
        \cdot \ldots  \;\; + \;\; 
    A_{2\ell} \,\e_{j_1 \ldots j_N}
     ( e^{sA})_{1 j_1} ( e^{sA})_{\ell j_2} ( e^{sA} )_{3 j_3} 
     \;\; + \;\; \ldots 
    \nonu \\[-7mm] & & 
    \hbox{\sc  ist $\ell \neq$ 1 im 1.~Term (oder $\neq 2$
         im 2.~Term etc.), so kommt es auch an einem}
    \nonu \\[-.4mm] & & 
    \hbox{\sc  anderen Faktor vor, und 
          $(\;)_{\ell j_1} (\;)_{\ell j_n}$ 
          verschwindet wegen $\e$--Antisymmetrie. Ergo} 
    \gluo{8}{4} 
    A_{11} \,\e_{j_1 \ldots j_N} ( e^{sA})_{1j_1} 
   ( e^{sA})_{2j_2} ( e^{sA})_{3j_3} \;\; + \;\; 
    A_{22} \,\e_{j_1 \ldots j_N} ( e^{sA})_{1j_1} 
   ( e^{sA})_{2j_2} ( e^{sA})_{3j_3} \;\; + \;\; \ldots
    \glu{5}   
   {\rm Sp} (A) \;\det ( e^{sA}) \;\;\; \folg \;\;\; 
     \det  ( e^{sA}) = C\, e^{s {\rm Sp} (A)} \;\;\; {\rm und}
     \;\; C=1 \;\;\,{\rm wegen}\;\, \det(e^0)=1 \;\,{\rm bei}
     \;\, s=0\;\; )\!) \nonu \hspace*{1.6cm}
\eea } \vskip -8mm

\hori

{\ft $A$, $B$, $C$, $D$ seien $N\!\times\!N$--Matrizen. 
$\;\underline{\det}$ und $\underline{\Sp}$ verarbeiten 
$2N\!\times\!2N$--Matrizen~:}

\hspace*{2mm}
{\boldmath$ \underline{\det} \( \matrix{ A & B \cr C & D \cr} \) 
 \; = \; \det \( D \) \, \det \( A - B D^{-1} C \)  
 \; = \; \det \( A \) \, \det \( D - C A^{-1} B \) $}

{\sc $(\!(\,\; \(\matrix{A\! & B \cr C\! & D \cr }\) =
        \(\matrix{A\! & 0 \cr 0\! & D \cr }\) \lk
   1 + \(\matrix{A^{-1}\!\!\!\!\!\! & 0 \cr 0\!\!\!\!\!\! & D^{-1} \cr }\) 
             \(\matrix{0\! & B \cr C\! &0 \cr }\) \rk
  \, , \; \underline{\det}(\ldots) = \det(A)\,\det(D)\,
      \underline{\det}\lk 1 + \kaef \rk \, {\rm mit} \,
  \kaef = \( \matrix{ 0 & R \cr S & 0 \cr} \) \; {\rm und}  \\[1.2mm]
  R=A^{-1} B \; , \; S = D^{-1} C\; . \quad
  \ln\(\underline{\det}\lk 1 + \kaef \rk\) = \underline{{\rm Sp}}\({\rm \ln}
  \lk 1 + \kaef \rk \) = \underline{{\rm Sp}}\( \kaef - {1\02} \kaef^2
         + {1\03} \kaef^3 - {1\04} \kaef^4 + \ldots \) \; , \;
   \underline{{\rm Sp}}\( \kaef^{\rm ung.} \) =0 \\[1mm]
    \underline{\det}\lk 1 + \kaef \rk  = e^{\underline{{\rm Sp}}} =
    e^{- {\rm Sp}(RS) - {1\02} {\rm Sp}(RSRS) 
       -{1\03}{\rm Sp}(RSRSRS) - \ldots} \; 
    = e^{{\rm Sp}\lk {\rm ln}(1-RS) \rk} = \det (1-RS) 
    = \det (1-A^{-1}B D^{-1} C) \;\; )\!)$ }

\hori

$ -U_{\prime\mu}U^{-1}= U (U^{-1})_{\prime \mu} \; , \;
-ig\aaa_\mu^U = U (U^{-1})_{\prime \mu} + U (-ig\aaa_\mu) U^{-1} 
 \; , \; D_\mu^U \gll \6_\mu - ig\aaa_\mu^U = U D_\mu U^{-1}\; ,
$ \\[2mm] $
\lk D_\mu\, , D_\nu \rki^U = \lk U D_\mu U^{-1} ,\, U D_\nu U^{-1} \rk
   = U D_\mu D_\nu U^{-1} - U D_\nu D_\mu U^{-1}
   = U \lk D_\mu\, , D_\nu \rk U^{-1} \;\; .  
$ \\[2mm] $
\lk \6_\mu\, , f \rk =  f_{\,\prime \mu} \;\folg \;  
F_{\mu\nu} \gll {i\0g} \lk D_\mu\, , D_\nu \rk 
    = \aaa_{\nu\,\prime\mu} - \aaa_{\mu\,\prime\nu}
     - ig \lk \aaa_\mu\, , \aaa_\nu \rk  
     \; \hbox{\ft und} \; F_{\mu\nu}^U = U F_{\mu\nu} U^{-1}\;$. 


\vskip 3mm\hspace*{-8mm}\tiny (hschulz@itp.uni-hannover.de)
\end{document}




