\relax \citation{Se} \@writefile{toc}{\contentsline {chapter}{\numberline {8}Geometry of transformation groups}{175}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \newlabel{ch!lie}{{8}{175}} \citation{Fn} \@writefile{lof}{\contentsline {figure}{\numberline {8.0}{\ignorespaces The geometry of the group of planar rotations}}{176}} \newlabel{fig!rot_and_S1}{{8.0}{176}} \@writefile{toc}{\contentsline {section}{\numberline {8.1}Topology on groups}{176}} \newlabel{sec!top on groups}{{8.1}{176}} \@writefile{toc}{\contentsline {paragraph}{Example 1}{177}} \@writefile{toc}{\contentsline {paragraph}{Example 2}{177}} \@writefile{toc}{\contentsline {paragraph}{Example 3}{177}} \@writefile{toc}{\contentsline {paragraph}{Example 4}{177}} \@writefile{toc}{\contentsline {paragraph}{Example 5}{178}} \@writefile{toc}{\contentsline {section}{\numberline {8.2}Dimension counting}{178}} \newlabel{sec!parameters}{{8.2}{178}} \@writefile{toc}{\contentsline {paragraph}{Proof}{179}} \@writefile{toc}{\contentsline {section}{\numberline {8.3}Compact and noncompact groups}{180}} \newlabel{decomp of GL(n)}{{8.3}{180}} \@writefile{toc}{\contentsline {paragraph}{Proof}{180}} \@writefile{toc}{\contentsline {paragraph}{Example 1}{181}} \@writefile{toc}{\contentsline {paragraph}{Example 2}{181}} \@writefile{toc}{\contentsline {paragraph}{Example 3}{181}} \@writefile{toc}{\contentsline {paragraph}{Discussion}{182}} \@writefile{toc}{\contentsline {paragraph}{Proof}{182}} \@writefile{toc}{\contentsline {section}{\numberline {8.4}Components}{182}} \newlabel{SOn conn}{{8.4}{182}} \@writefile{toc}{\contentsline {paragraph}{Remark}{182}} \@writefile{toc}{\contentsline {paragraph}{Proof}{182}} \@writefile{toc}{\contentsline {section}{\numberline {8.5}The geometry of $\SO (n)$}{183}} \newlabel{sec!topson}{{8.5}{183}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.5.1}Quaternions}{184}} \newlabel{quat sec}{{8.5.1}{184}} \@writefile{toc}{\contentsline {paragraph}{Proof}{185}} \@writefile{toc}{\contentsline {paragraph}{Remark}{185}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.5.2}Quaternions and rotations}{186}} \newlabel{quat rot}{{8.5.2}{186}} \@writefile{toc}{\contentsline {paragraph}{Proof}{186}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.5.3}Spheres and special orthogonal groups}{187}} \newlabel{ssec!sph_and_so}{{8.5.3}{187}} \@writefile{toc}{\contentsline {paragraph}{Proof}{187}} \@writefile{toc}{\contentsline {paragraph}{Remark}{188}} \@writefile{toc}{\contentsline {section}{\numberline {8.6}The group $\SU (2)$}{188}} \newlabel{sec!su2}{{8.6}{188}} \@writefile{toc}{\contentsline {paragraph}{Remark}{188}} \citation{Fn} \newlabel{eq!aabar}{{1}{189}} \@writefile{toc}{\contentsline {paragraph}{Proof}{189}} \@writefile{toc}{\contentsline {section}{\numberline {8.7}The electron spin in quantum mechanics}{189}} \newlabel{sec!spin}{{8.7}{189}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.7.1}The story of the electron spin}{189}} \newlabel{stern}{{8.7.1}{189}} \@writefile{lof}{\contentsline {figure}{\numberline {8.7a}{\ignorespaces The Stern--Gerlach experiment}}{190}} \citation{Fn} \@writefile{toc}{\contentsline {subsection}{\numberline {8.7.2}Measuring spin: the Stern--Gerlach device}{191}} \@writefile{lof}{\contentsline {figure}{\numberline {8.7b}{\ignorespaces The modified Stern--Gerlach device}}{191}} \@writefile{lof}{\contentsline {figure}{\numberline {8.7c}{\ignorespaces Two identical SG devices}}{192}} \@writefile{lof}{\contentsline {figure}{\numberline {8.7d}{\ignorespaces Two different SG devices}}{192}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.7.3}The spin operator}{193}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.7.4}Rotate the device}{194}} \citation{Fn} \citation{Fn} \@writefile{toc}{\contentsline {subsection}{\numberline {8.7.5}The solution}{195}} \newlabel{sec! spin and rot}{{8.7.5}{195}} \@writefile{toc}{\contentsline {section}{\numberline {8.8}Preview of Lie groups}{196}} \newlabel{sec!lie}{{8.8}{196}} \citation{FH} \citation{St} \@writefile{toc}{\contentsline {section}{Exercises to Chapter\nobreakspace {}8}{197}} \newlabel{ex!count_ecul_params}{{2}{197}} \newlabel{soparam}{{3}{198}} \newlabel{GLn ex}{{4}{198}} \newlabel{Opq ex}{{5}{198}} \newlabel{ex!decomp_gs}{{6}{198}} \newlabel{quat mult ex}{{8}{198}} \newlabel{quat conj ex}{{9}{199}} \newlabel{quat frame ex}{{12}{199}} \newlabel{sumult}{{14}{199}} \newlabel{direct ex}{{15}{199}} \citation{ConSmi} \newlabel{ex!su2}{{20}{200}} \newlabel{ex!4dim}{{22}{200}} \newlabel{ex!octonions}{{23}{200}} \newlabel{herm_ex}{{24}{201}} \@setckpt{lie}{ \setcounter{page}{202} \setcounter{equation}{1} \setcounter{enumi}{24} \setcounter{enumii}{2} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{1} \setcounter{mpfootnote}{0} \setcounter{part}{0} \setcounter{chapter}{8} \setcounter{section}{8} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{5} \setcounter{table}{0} \setcounter{parentequation}{0} \setcounter{cor}{9} \setcounter{corn}{2} \setcounter{constr}{1} \setcounter{principle}{2} \setcounter{prop}{34} \setcounter{lem}{4} \setcounter{lemn}{4} \setcounter{cla}{2} \setcounter{thm}{24} }