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\begin{document}
\noindent {\bf Physics 251} \hfill {\bf Problem Set 3} \hfill {\bf
Spring 2019}
\myrule
\vskip 0.2in
\centerline{\sl DUE: \ THURSDAY, May 23, 2019}
\vskip 0.3in
\noindent
1.~~(a) A homomorphism from the vector space ${\mathbb{R}^3}$
to the set of traceless Hermitian $2\times 2$ matrices is defined by
${\boldsymbol{\vec{x}}}\rightarrow{\boldsymbol{\vec{x}\cdot\vec{\sigma}}}$, where $\boldsymbol{\vec\sigma}=(\sigma_1\,,\,\sigma_2\,,\,\sigma_3)$ are the Pauli matrices.
First, show that
$\det({\boldsymbol{\vec{x}\cdot\vec{\sigma}}})=-|{\boldsymbol{\vec{x}}}|^2$.
Second, prove the identity:
\begin{equation} \label{eq1}
x_i=\half\Tr\,({\boldsymbol{\vec{x}\cdot\vec{\sigma}}}\,\sigma_i)\,.
\nonumber
\end{equation}
This identity provides the inverse transformation from 
the set of traceless $2\times 2$ Hermitian matrices to 
the vector space $\mathbb{R}^3$.
\vskip 0.1in


(b) Let $U\in {\rm SU(2)}$.  
Show that $U\,{\boldsymbol{\vec{x}\cdot\vec{\sigma}}}
\,U^{-1}={\boldsymbol{\vec{y}\cdot\vec{\sigma}}}$ for some vector
${\boldsymbol{\vec{y}}}$.  Using the results of part (a),
prove that an element of the rotation group exists such that
${\boldsymbol{\vec{y}}}=R{\boldsymbol{\vec{x}}}$
and determine an explicit form for $R\in {\rm SO(3)}$.  Display a
homomorphism from SU(2) onto SO(3) and prove that SO(3)$\iso
{\rm SU(2)}/\mathbb{Z}_2$.
\vskip 0.1in

(c) The Lie group SU(1\,,\,1) is defined as the group of $2\times 2$
matrices $V$ that satisfy $V\sigma_3 V^\dagger=\sigma_3$ and
det~$V=1$.  (Note that $V$ is {\it not} a unitary matrix.)  The Lie
group SO(2\,,\,1) is the group of transformations on vectors
${\boldsymbol{\vec{x}}} \in {\mathbb{R}^3}$ (with determinant equal to one)
that preserves $x_1^2+x_2^2-x_3^2$.  Display
the homomorphism from SU(1\,,\,1) onto SO(2\,,\,1) and compare with part (b).



\vskip 0.3in

\noindent
2. The M\"obius group is defined as the set of linear fractional
transformations:
$$M=\left\{m(z)={az+b\over cz+d}\,,\quad ad-bc=1\right\}\,,$$
where $a$, $b$, $c$, $d$ and $z$ are complex numbers.
\vskip 0.1in

(a) Show that the mapping $f:{\rm SL}(2,\mathbb{C})\rightarrow M$
defined by:
$$f:\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto m(z)$$
is a group homomorphism.  [{\sl HINT:} the multiplication law
on $M$ is defined by the composition of functions.]
\vskip 0.1in

(b) Prove that $M$ is not simply connected and identify its universal
covering group.

\vskip 0.3in
\noindent
3. In class, we showed that the invariant measure on a Lie group manifold is given by
\begin{equation} \label{measure}
d\mu(g)=|{\rm det}~\!c(\boldsymbol{\vec \xi})|\,d\xi_1 d\xi_2\cdots d\xi_n\,,
\end{equation}
where the the matrix elements $c_{jk}(\boldsymbol{\vec\xi})$ are the coefficients
of the Lie algebra element $A^{-1}\partial A/\partial \xi_k$ with respect to some basis,
and $A(\boldsymbol{\vec\xi})$ are elements of the corresponding Lie group that is
parameterized by the coordinates $\boldsymbol{\vec\xi}$.  That is, given an $n$-dimensional
Lie group $G$, the corresponding real Lie algebra $\mathfrak{g}$ consists of
real linear combinations of basis vectors $\mathcal{A}_j\in\mathfrak{g}$.  Since
$A^{-1}\partial A/\partial \xi_k\in\mathfrak{g}$ for any $A\in G$, one can therefore write,
\begin{equation} \label{AA}
A^{-1}\frac{\partial A}{\partial \xi_k}=\sum_{i=1}^n c_{jk}(\boldsymbol{\vec\xi})\mathcal{A}_j\,,
\end{equation}
which defines the coefficients $c_{jk}(\boldsymbol{\vec\xi})$ needed in the determination of
the invariant measure.
\vskip 0.1in

(a)  An element of SO(3) can be parameterized by $\boldsymbol{\vec{\xi}}=(\alpha, \beta,\gamma)$, where $\alpha$, $\beta$ and $\gamma$ are the three Euler angles defined in Appendix E of the class handout entitled \textit{Properties of Proper and Improper Rotation Matrices}.
Using the Euler angle parameterization of the SO(3) group manifold, compute the invariant integration measure $d\mu(g)$ for SO(3).
\vskip 0.1in

(b) The SO(3) group manifold can be also be described as a ball of radius $\pi$ with antipodal
points identified.  A point in the SO(3) group manifold is
specified by a vector ${\boldsymbol{\vec{\xi}}}$ with
$|{\boldsymbol{\vec{\xi}}}|\leq\pi$.  Thus, the SO(3) manifold
is parameterized by ${\boldsymbol{\vec{\xi}}}=(\xi,\theta,\phi)$,
where $(\theta,\phi)$ are the spherical angles (such that
$0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$)
and $\xi$ is the
magnitude of the vector ${\boldsymbol{\vec{\xi}}}$.
[{\sl NOTE:} This is equivalent to the
angle-and-axis parameterization where the rotation angle is $\xi$
and the rotation axis, ${\boldsymbol{\hat\xi}}$, is specified by a polar
angle
$\theta$ and an azimuthal angle $\phi$.]

Show that the the matrix elements of $c(\boldsymbol{\vec\xi})$ defined in eq.~(\ref{AA}) are given by,
\begin{equation} \label{c}
c(\boldsymbol{\vec\xi})_{nk}=\half\epsilon_{\ell nj}R^{-1}_{\ell i}{dR_{ij}\over
d\xi_k}\,,
\end{equation}
and $R_{ij}\equiv R_{ij}({\boldsymbol{\vec\xi}})$ is the SO(3) matrix
given in problem 7(b) of problem set 2.
\vskip 0.1in

(c) [{\sl EXTRA CREDIT}] Using eqs.~(\ref{measure}) and (\ref{c}), evaluate the invariant integration measure $d\mu(g)$ for
the angle-and-axis parameterization of SO(3) and show that
$$d\mu({\boldsymbol{\vec{\xi}}})=2(1-\cos\xi)\sin\theta \,d\theta\, d\phi\,
d\xi\,.$$

\vskip 0.1in
\noindent
{\sl HINT:} First evaluate $d\mu({\boldsymbol{\vec{\xi}}})$ in terms of
Cartesian coordinates $\xi_1$,
$\xi_2$ and $\xi_3$.  Convert to spherical coordinate $(\xi,\theta,\phi)$
at the very end of the calculation.
%\vskip 0.1in
%(c) Compute the total volume of SO(3).  Compare this with the total
%volume of SU(2).
\vskip 0.3in


\noindent
4. Consider a Lie group of transformations $G$ acting on a manifold
$M$. That is, for every $g\in G$, we have $gx=y$ for some $x,y\in M$.
\vskip 0.1in

(a) Let $H$ be the set of all transformations in $G$ that map a given
point $x\in M$ into itself.  Show that $H$ is a subgroup.  $H$ has at
least three names in the mathematical literature: the little group,
the isotropy group, or the stability group of the point $x$.
\vskip 0.1in

(b) Consider the submanifold of $M$ defined by $\{gx\,|\,g\in G\}$, for
fixed $x\in M$.
This is called the {\it orbit} through $x$ with respect to $G$.
Show that there is a one-to-one correspondence between the points of the
orbit and the set of left cosets of $H$.  Explain why we may conclude
that $\{gx\,|\,g\in G\}=G/H$.
Show that the coset space $G/H$ is homogeneous.
\vskip 0.1in

(c) Prove that $S^{n-1}=$\,SO($n$)/SO($n-1$) by considering the action of
the rotation group on the point $(1,0,0,\ldots,0)\in{\mathbb{R}^n}$.
\vskip 0.1in

(d) Prove that $S^{2n-1}=$\,U($n$)/U($n-1$) by considering the action of
the U($n$) matrices on the point $(1,0,0,\ldots,0)\in{\mathbb{C}^n}$.
\vskip 0.1in

(e) Complex projective space $\mathbb{CP}^n$ is defined as the space of
complex lines in ${\mathbb{C}^{n+1}}$ through the origin.
That is, ${\mathbb{CP}^n}$ consists of the set of vectors in
$\mathbb{C}^{n+1}$ (omitting the zero vector) where we identify
$(z_0,z_1,\ldots,z_n)\sim \lambda (z_0,z_1,\ldots,z_n)\,,$
for any nonzero complex number $\lambda$.
Without loss of generality, we can restrict our considerations to the
vectors $\boldsymbol{\vec v}\in \mathbb{C}^{n+1}$ such that $\boldsymbol{\vec v}\cdot\boldsymbol{\vec v}\lsup{\,*}=1$.
Show that ${\rm U}(1)\otimes {\rm U}(n)$
is the little group of the point $z=(1,0,0,\ldots,
0)\in {\mathbb{CP}^n}$, and that ${\mathbb{CP}^n}$ is the orbit through
$z$ with respect to ${\rm U}(n+1)$.  Conclude that
${\mathbb{CP}^n}={\rm U}(n+1)/{\rm U}(1)\otimes {\rm U}(n)$.
\vskip 0.1in

(f) Real projective space $\mathbb{RP}^n$ can be defined analogously
to $\mathbb{CP}^n$ of part (e) by replacing the field of complex numbers
with the field of real numbers.  What coset space can be identified
with $\mathbb{RP}^n$?
\vskip 0.1in

(g) In parts (c)--(f), check that
dim$(G/H)={\rm dim\,}G-{\rm dim\,}H$.
\vskip 0.1in


(h) [{\sl EXTRA CREDIT}]  ${\mathbb{CP}^n}$ is a manifold of $n$
complex (or $2n$ real) dimensions.
${\mathbb{CP}^1}$ is homeomorphic to which well-known two-dimensional
real manifold?


\vskip 0.3in
\noindent
5.  Let $A$ be an even-dimensional
complex antisymmetric $2n\times 2n$ matrix, where $n$
is a positive integer.  We define the \textit{pfaffian} of $A$, denoted by
pf~\!$A$, by:
\begin{equation} \label{pdef}
{\rm pf}~\!A=\frac{1}{2^{n} n!}\sum_{p\in S_{2n}} (-1)^p\,
A_{i_1 i_2}A_{i_3 i_4}\cdots A_{i_{2n-1} i_{2n}}\,,
\end{equation}
where the sum is taken over all permutations
$$
p=\begin{pmatrix} 1 & 2 & \cdots & 2n\\ i_1 & i_2 & \cdots & i_{2n}
\end{pmatrix}
$$
and $(-1)^p$ is the sign of the permutation $p\in S_{2n}$.
%\vskip 0.1in
%
If $A$ is an odd-dimensional complex antisymmetric matrix, the
corresponding pfaffian is defined to be zero.
\vskip 0.1in

(a) By explicit calculation, show that\footnote{In fact, eq.~(\ref{e1}) holds for
all complex antisymmetric $2n\times 2n$ matrices, where $n$ is any positive number.
A general proof will be provided in a class handout.}
\begin{equation} \label{e1}
\det A=({\rm pf}~\!A)^2\,,
\end{equation}
for any $2\times 2$ and $4\times 4$
complex antisymmetric matrix $A$.
\vskip 0.1in

(b) Prove that the determinant of any odd-dimensional complex
antisymmetric matrix
vanishes.  As a result, the definition of the pfaffian in the
odd-dimensional case is consistent
with the result of eq.~(\ref{e1}).
\vskip 0.1in

(c) Given an arbitrary $2n\times 2n$ complex matrix $B$ and complex
antisymmetric $2n\times 2n$ matrix $A$, use the definition of the pfaffian given
in eq.~(\ref{pdef}) to prove
the following identity:
$${\rm pf}~\!(BAB^T)={\rm pf}~\!A\,\det B\,.
$$
\vskip 0.1in

(d) A complex $2n\times 2n$ matrix $S$ is called \textit{symplectic}
if $S^{\T}JS=J$, where $S^{\T}$ is the transpose of $S$ and
$$J\equiv\left(\begin{array}{cc} \phantom{-}\mathds{O} &\,\,\, \mathds{1} \\
-\mathds{1} & \,\,\,\mathds{O}\end{array}\right)\,,
$$
where $\mathds{1}$ is the $n\times n$ identity matrix and $\mathds{O}$
is the $n\times n$ zero matrix.  Prove that the set of $2n\times 2n$
complex symplectic matrices, denoted by ${\rm Sp}(n,\mathbb{C})$,
is a matrix Lie group\footnote{Warning: many authors denote the group of $2n\times 2n$
complex symplectic matrices by ${\rm
Sp}(2n,\mathbb{C})$.}
[\textit{i.e.}, it is a topologically closed subgroup of
${\rm GL}(2n,\mathbb{C})$].
\vskip 0.1in

(e) Prove that if $S$ is a symplectic matrix, then $\det{S}=1$.
\vskip 0.1in

\noindent
{\rm HINT:} It is very easy to prove that $\det{S}=\pm 1$ by taking the
determinant of the equation $S^{\T}JS=J$. To prove that
there are no symplectic matrices with $\det{S}=-1$, use the result of
part (c).
\vskip 0.1in

(f) Using the results of parts (d) and (e), prove that the matrix Lie groups
Sp$(1,\mathbb{C})$ and SL$(2,\mathbb{C})$ are isomorphic.
\vskip 0.3in

\noindent
6. The two-dimensional Poincar\'e group P(2) is the group consisting
of two-dimensional Lorentz transformations [i.e., transformations
on 2-vectors $ct\choose x$ that preserve
$x^2-c^2 t^2$] and translations in time and space.  P(2) can be
represented by $3\times 3$ matrices acting homogeneously on the column vector,
$
\left(\begin{smallmatrix} ct \\ x \\ 1\end{smallmatrix}\right)
$,
in analogy with the two-dimensional
Euclidean group, E(2), worked out in class.
\vskip 0.1in

(a) Find the infinitesimal
generators (i.e., differential operators) of the
corresponding Lie algebra, $\mathfrak{p}(2)$.  Work out the commutation relations
of $\mathfrak{p}(2)$.
\vskip 0.1in

(b) Compute the Cartan-Killing form.  Show that P(2) is noncompact
and non-semisimple.
\vskip 0.1in

(c) Express the Lie algebra $\mathfrak{p}(2)$ as a semidirect sum of two abelian
subalgebras.
\vskip 0.4in


\end{document}
