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\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#4, Rev. B, due Wednesday, 2024-02-07 }
\end{center}

\vspace{.3 in}


{\bf 1.} The {\bf \href{https://en.wikipedia.org/wiki/pauli_matrices}{Pauli matrices}}, defined as $\sigma_{x}\equiv\pmatrix{0&1\cr1&0}$, $\sigma_{y}\equiv\pmatrix{0&\!\!\!\!-i\cr i&0}$, $\sigma_{z}\equiv\pmatrix{1&0\cr0&\!\!\!-1}$, describe quantum mechanical spin $\frac12$ particles in a space with two basis vectors representing spin up and down.

{\bf \quad a)} 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 the basis vectors $\hat x,\hat y,\hat z$, and encode the structure of the dot and cross products just as $1,i$ do in the complex plane.  What are the inverses of $\sigma_{k}$?

[{\it 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 $\sigma^{\mu\nu}$.  They encode the structure of spacetime in special relativity.]

{\bf \quad b)} Show that $(I,\sigma_{x},\sigma_{y},\sigma_{z})$ is a basis of the generalized vector space of complex $2\times 2$ matrices: they are independent and any matrix can be written as the linear combination $A=\pmatrix{a & b \cr c & d} = I A_{0} + \sigma_{x} A_{x} + \sigma_{y} A_{y} + \sigma_{z} A_{z}$ by solving for $A_{0}, A_{x}, A_{y}, A_{z}$.

\qquad What are the original basis matrices $P_{1,2}$ and $\sigma_{\pm}$, such that $A=P_{1} a + \sigma_{+} b + \sigma_{-} c + P_{2} d$?

{\bf \quad c)} Verify that the transformation matrices 
$U_{x}=\frac{1}{\sqrt2}\pmatrix{1 & 1\cr 1 &\!\!\!\! -1}$ and $U_{y}=\frac{1}{\sqrt2}\pmatrix{1 & 1\cr i &\!\!\!\! -i}$ are {\it unitary}, $U_{x}\ad U_{x}=U_{y}\ad U_{y}=I$, so that $U_{x}^{-1}=U_{x}\ad$ and $U_{y}^{-1}=U_{y}\ad$.

{\bf \quad d)} Verify the {\it eigenequations} $\sigma_{x}U_{x}=U_{x}\sigma_{z}$ and $\sigma_{y}U_{y}=U_{y}\sigma_{z}$.  Calculate the {\it similarity transform} $U_{x}\sigma_{k}U_{x}^{-1}$ of each of the Pauli matrices $\sigma_{k}=\sigma_{x},\sigma_{y},\sigma_{z}$.  For $i,j{=}x,y,z$, find the matrices $U_{ij}$ that transform $\sigma_{i}$ into $\sigma_{j}=U_{ij}\sigma_{i}U_{ij}^{-1}$.  by rearranging the eigenequations above. In these equations, ${}_{i,j}$ are not component indices, but label (distinguish) different $U$ matrices.

{\bf \quad e)} The {\it trace} of a matrix is the sum of its diagonal elements: $\mathrm{tr} A=A_{ii}$.  Calculate $\mathrm{tr}(0)$ (the zero matrix) and $\mathrm{tr}(\sigma_i)$.  [{\it bonus}: Show that $\mathrm{tr}(ABC)=\mathrm{tr}(BCA)=\mathrm{tr}(CAB)$ but not necessarily $\mathrm{tr}(ABC)=\mathrm{tr}(CBA)$.]  Use this to show that if $A'=UAU\ad$, then $\mathrm{tr}(A')=\mathrm{tr}(A)$, i.e. the trace, or `perimeter', of $A$ is invariant.

{\bf \quad f)} The {\it determinant} of a matrix is the fully antisymmetric product of one element in each row or column: $\mathrm{det} A=\varepsilon_{ijk}A_{1i}A_{2j}A_{3k}=\varepsilon_{ijk}A_{i1}A_{j2}A_{k3}$, which has the properties $\mathrm{det}(A\ad)=\mathrm{det}(A)^{*}$ and $\mathrm{det}(AB)=\mathrm{det}(A)\mathrm{det}(B)$.  Calculate $\mathrm{det}(I)$ and $\mathrm{det}(\sigma_i)$.  Show that if $U\ad U=I$ then $|\mathrm{det}(U)|=1$, and if $A'=UAU\ad$, then $\mathrm{det}(A')=\mathrm{det}(A)$, i.e. the determinant, or `volume', of $A$ is invariant.  %Show that $\exp(\mathrm{tr}(A))=\mathrm{det}(\exp(A))$, for example, $\exp(\mathrm{tr}(0))=1=\mathrm{det}(\exp(0)=I)$.


{\bf 2. Rotations} of the vector $\bv v=\pmatrix{x\cr y}$  are generated by the matrix $M=\pmatrix{0&\!\!\!\!-1\cr1&0}$, which represents a $90^{\circ}$ CCW rotation---just as complex rotations $e^{i\phi}$ are generated by $i$, where $i^{2}=-1$.

{\bf \quad a)} Show that $M\bs v$ rotates $\bs v$ 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 $M^{T}=-M$ (the generator is asymmetric) and thus $R^{T}R=I$ (verify it), just like $(e^{i\phi})^{*}(e^{i\phi})=1$.  Show that $dR=MRd\phi$, just like $de^{i\phi}=ie^{i\phi}d\phi$ in H02\#1d.

{\bf \quad c)} The cross product $\times$ generates rotation in 3d.  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 matrix for a CCW rotation of angle $v$ about the $\bh v$-axis is $R_{\bs v} = e^{\bs M \cdot \bs v} = I \cos v + \bs M{\cdot} \bh v \sin v + \bh v \bh v^{T} (1-\cos v)$, the \href{https://en.wikipedia.org/wiki/Rodrigues\%27_rotation_formula}{\it Rodrigues' formula}. The third term projects out the axis of rotation $\bh v$.  Verify this formula for the familiar case $\bv v=\bh z\phi$.

{\bf \quad d)} [{\it bonus}: The Pauli matrices also generate rotation. Calculate $e^{-i\sigma_{x}\phi}$, $e^{-i\sigma_{y}\phi}$, $e^{-i\sigma_{z}\phi}$.  Which transformation corresponds to $R$ and the complex rotation $e^{i\phi}$?]


[{\it bonus}: {\bf 3. Minkowski space}.  We will derive the Lorentz transformations using the Lorentz metric, which encodes Einstein's principle of special relativity that the speed of light $c$ is constant in any reference frame.

{\bf \quad a)} Show that the principle of relativity can be written as $\bs x^{2}=\bs x\cdot \bs x=\bs x^{T}g \bs x=0$ for the {\it space-time vector} $\bs x=(ct\; x)^{T}$ with components  representing the distance $ x=c t$ traveled by a photon in time $t$, where the matrix $g=\pmatrix{-1&0\cr\phantom{-}0&1}$ is called the {\it Minkowski metric}.  In three-dimensional space, $\bs x=(ct\;  x\;  y\;  z)^{T}$ is a 4-vector, and the metric becomes a $4\times4$ matrix $g=\rm{diag}(-1,1,1,1)$.

{\bf \quad b)} Momentum and energy can also be combined into the space-time vector $\bs p=(E/c\; p)^{T}$.  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 Einstein's equation $E=mc^{2}$ when $p=0$.

{\bf \quad c)} Normal rotations $R$ keep the length of vector constant by preserving the Euclidean metric $I$: $R^{T}IR=I$.  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 to first order in $d\alpha$.  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 of the features of special relativity.  Verify that $\Lambda^{T}g\Lambda=g$ and plot the rotated basis vectors.  Calculate the relativistic addition rule for velocities $\beta_{1}$ and $\beta_{2}$ from $\Lambda=\Lambda_{2}\Lambda_{1}$.

{\bf \quad e)} The Dirac matrices $\gamma^{\mu}$ generate Lorentz rotations. Calculate $e^{-i\gamma_{\mu}v^{\mu}}$.]


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