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\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#3, Rev. B, due Monday, 2022-02-07 }
\end{center}

\vspace{.3 in}

{\bf 0.} The {\bf \href{https://en.wikipedia.org/wiki/pauli_matrices}{Pauli matrices}}, defined as $\sigma_{x}\equiv\pmatrix{0&1\cr1&0}$, $\sigma_{y}\equiv\pmatrix{0&\!\!\!\!-i\cr i&0}$, $\sigma_{z}\equiv\pmatrix{1&0\cr0&\!\!\!-1}$, are used to describe quantum mechanical spin-1/2 particles.  Show that $\sigma_{j}\sigma_{k}=\delta_{jk}I+i\varepsilon_{jk\ell}\sigma_{\ell}$.  What is the {\it triple product} $\sigma_{x}\sigma_{y}\sigma_{z}$?  Thus $\sigma_{j}$ acts like a basis vector, the product of the Pauli matrices has the structure of the dot, cross, and triple products.

[bonus: Show that product of {\it \href{https://en.wikipedia.org/wiki/gamma_matrices}{Dirac matrices}} is $\gamma^{\mu}\gamma^{\nu}=g^{\mu\nu}I+i\sigma^{\mu\nu}$ and calculate the components $g^{\mu\nu}$ and the matrices $\sigma^{\mu\nu}$.]

{\bf 1. Vector rotations} can be generated by the matrix $M=\pmatrix{0&\!\!\!\!-1\cr1&0}$ in the same way complex rotations $e^{i\phi}$ are generated by $i$ with the property $i^{2}=-1$, representing a $90^{\circ}$ rotation.

{\bf \quad a)} Show that $M\bs v$ also rotates the vector $\bs v=\pmatrix{x&y}^{T}$ by $90^{\circ}$ CCW, and that  $M^{2}=-I$.

{\bf \quad b)} Show that $R=e^{M\phi}=I\cos\phi + M\sin\phi$ and calculate the components of $R$.  Show that $R^{T}R=I$ and that $dR=MRd\phi$, similar to $(e^{i\phi})^{*}(e^{i\phi})=1$ and $de^{i\phi}=ie^{i\phi}d\phi$.  %Note that matrix multiplication is non-commutativ e.

{\bf \quad c)} The cross product $\times$ also generates rotation.  Calculate the matrices $\bs M=(M_{x},M_{y},M_{z})$, where $\bh x\times\bs v=M_{x}\bs v$, etc.  Show that the components of $M_{\ell}$ are $(M_{\ell})_{jk}=\varepsilon_{kj\ell}$ and that $M_{\ell}^{2}=-P_{\perp\ell}$, where $P_{\perp\ell}$ projects perpendicular to $\bh e_{\ell}$.  Thus the general rotation matrix $R_{\bs v}$ for a CCW rotation by an angle $v=|\bs v|$ about the $\bh v$-axis is given by the \href{https://en.wikipedia.org/wiki/Rodrigues\%27_rotation_formula}{\it Rodrigues' formula} $R_{\bs v} = e^{\bs M \cdot \bs v} = I \cos v + \bs M{\cdot} \bh v \sin v + \bh v \bh v^{T} (1-\cos v)$. The third term projects out the axis of rotation $\bh v$.
% \bs m in 404h02

{\bf 2. Minkowski space}: We will derive the Lorentz transformations using matrices from Einstein's principle of special relativity that the speed of light $c$ is constant in any reference frame.

{\bf \quad a)} Let $\Delta x=c\Delta t$ for the distance $\Delta x$ travelled by a photon in time $\Delta t$.  Combining them into a {\it space-time vector} $\bs x=\pmatrix{c\Delta t &\Delta x}^{T}$, show that this is equivalent to $\bs x^{2}=\bs x\cdot \bs x=\bs x^{T}g \bs x=0$, where the matrix $g=\pmatrix{-1&0\cr0&1}$ is called the {\it Minkowski metric}.  In three dimensional space the metric is a $4\times4$ matrix and $x^{2}$ becomes $r^{2}=x^{2}+y^{2}+z^{2}$.

{\bf \quad b)} Momentum and energy can also be combined into a space-time vector $\bs p=\pmatrix{E/c&p}^{T}$.  Show that invariance of the dot product $\bs p^{T}g\bs p=-(mc)^{2}$ leads to the formula $E^{2}=(pc)^{2}+(mc^{2})^{2}$, which reduces to $E=mc^{2}$ when $p=0$.

{\bf \quad c)} Normal rotations $R$ keep the length of vector constant---they preserve Euclidian metric $I$, or $R^{T}IR=I$.  Likewise, Lorentz transformations $\Lambda$ preserve the Minkowski metric: $\Lambda^{T}g\Lambda=g$.  For a small `rotation' $\Lambda=I+Gd\alpha$ generated by $G$, show that $G^{T}g+gG=0$. %or the adjoint $\tilde G\equiv g^{-1}G^{T}g = -G$, that is G is antisymmetric
Show that $G=\pmatrix{0&1\cr1&0}$ satisfies this relation.  Note the difference between $G$ and $M$!  

{\bf \quad d)} Show that $G^{2}=I$ and therefore $\Lambda=e^{G\alpha}=I\cosh\alpha+G\sinh\alpha = \gamma\pmatrix{1&\beta\cr\beta&1}$, where $\beta=\tanh\alpha=v/c$ and $\gamma=\cosh\alpha=(1-\beta^{2})^{-1/2}$.  Thus the Lorentz transformations $\bs x'=\Lambda \bs x$ are $t'=\gamma(t+vx/c^{2})$ and $x'=\gamma(x+vt)$, which encode all features of special relativity.

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