Life with Physics

In mathematics, the first homotopy group of a pathwise-connected topological space $\cal{M}$, denoted by $\pi_1(\cal{M})$, is about the classification of all the loops in $\cal{M}$: The relations between loops that can be continuously deformed into one another is an equivalence relation. Then all loops in $\cal{M}$ can be partitioned into disjoint equivalent classes. Finally, all equivalent classes of loops form a group.  Note: Let $\alpha$ and $\beta$ be two loops in $\cal{M}$, the group multiplication between the equivalent classes $[\alpha]$ and $[\beta]$ is defined as $[\alpha]\cdot[\beta]=[\alpha * \beta]$, where $\alpha * \beta$ is a concatenated loop that we first traverse through the loop $\alpha$ and then through the loop $\beta$. In general, the n-th homotopy group, $\pi_n(\cal{M})$ is about the classification of the n-dimensional sphere $S^n$ in $\cal{M}$. Examples: $\pi_1\left(S^1\right)=\mathbb{Z}$ suggests that we can label each equivalent class of loops in $S^1...

What is a particle? S. Weinberg described a particle simply as "a physical system that has no continuous degrees of freedom except for its total momentum".  Recall that the spacetime admits the so-called Poincare symmetry \begin{equation} {x'}^{\mu} = \Lambda^{\mu}_{\,\,\nu}\,x^{\nu} + a^{\mu}\,.\end{equation} By Noether's theorem , each continuous symmetry indicates a conservation. Thus, the total momentum $p^{\mu}$ is a good quantum because of the spacetime translation invariance. As we shall see below, there are other discrete degrees of freedom, denoted by $\sigma$, that are associated with the Lorentz invariance. So the quantum state of a particle is described as $|p, \sigma\rangle$.  Under the Lorentz transformation $\Lambda$, the one-particle state $|p, \sigma\rangle$ changes to a new state $U(\Lambda)|p,\sigma\rangle$ by a unitary operator $U(\Lambda)$. Note that the Lorentz transformation only changes the reference frames, elementary particles should remain...

Problem:  Show that the Lorentz transformation $W$ that keeps $k^{\mu}\equiv[1, 0,  0,  1]^T$ unchanged forms a two-dimensional Euclidean group SE(2). Solution: As shown in this post , the general solution of the Lorentz transformation $W$ that satisfies $W^{\mu}_{\,\,\,\,\nu}\,k^{\nu}=k^{\mu}$ can be parametrized in three parameters $\alpha,\beta,\theta$ as \begin{equation}W(\alpha, \beta, \theta)=\left[\begin{matrix} 1+(\alpha^2+\beta^2)/2  & -\alpha & -\beta & -(\alpha^2+\beta^2)/2  \\  -\alpha & 1 & 0 & \alpha \\ -\beta & 0 &  1 & \beta \\ (\alpha^2+\beta^2)/2 & -\alpha & -\beta & 1-(\alpha^2+\beta^2)/2 \end{matrix}\right]\left[\begin{matrix} 1 &  &  & \\  & \cos\theta & -\sin\theta & \\ & \sin\theta & \cos\theta  &   \\ & &  & 1\end{matrix}\right]\,.\tag{1}\end{equation} To work out the Lie algebra, we expand $W(\...

Problem:  Let $p^{\mu}\equiv[|\mathbf{p}|, p^1, p^2, p^3]^T$ be the  four-momentum  of a massless particle, assuming $c=1$. In some inertial frame, such four-momentum becomes to $k^{\mu}\equiv[\kappa, 0,  0,  \kappa]^T$.  The problem is to find a Lorentz transformation (matrix) $L^{\mu}_{\,\,\nu}(p)$ such that $p^{\mu} =L^{\mu}_{\,\,\nu}(p)\,k^{\nu}$. (This problem is about the proof of Eq. (2.5.44) in Weinberg's QFT book, Volume I) Solution: To transform $k^{\mu}$ to $p^{\mu}$, we can first boost $k^{\mu}$ to $[|\mathbf{p}|, 0, 0, |\mathbf{p}|]^T$ followed by a spatial rotation to $p^{\mu}$.  Recall the Lorentz boost on the energy and momentum, \begin{equation}\begin{split}E'&=&\gamma\,(E-v\,p^3)\,,\\{p'}^3&=&\gamma\,(p^3-v\,E)\,,\end{split} \end{equation} with $\gamma \equiv 1/\sqrt{1-v^2}$. In the matrix form, \begin{equation}\left[\begin{matrix}E' \\ 0 \\ 0 \\ {p'}^3\end{matrix}\right]\equiv B(v)\, \left[\begin{matrix}E \\ 0 \\...

  Problem:  Let $k^{\mu}\equiv[1, 1,  0,  0]^T$, find a Lorentz transformation (matrix) $W^{\mu}_{\,\,\,\,\nu}$  such that $ W^{\mu}_{\,\,\,\,\nu}\,k^{\nu}=k^{\mu}$. (This problem is about the proof of Eq. (2.5.28) in Weinberg's QFT book, Volume I) Solution: Let $t^{\mu}\equiv[1, 0,  0,  0]^T$, the relations \begin{equation}\begin{split}(Wt)^{\mu}(Wt)_{\mu}&=&t^{\mu}t_{\mu}=-1\,,\\ (Wt)^{\mu}k_{\mu}&=&t^{\mu}k_{\mu}=-1\,, \end{split}\end{equation} suggest that $(Wt)^{\mu}$ is of the form \begin{equation} (Wt)^{\mu}= [1+\zeta\,,\zeta\,, \alpha\,, \beta]^T\,,\end{equation} with \begin{equation}\zeta = (\alpha^2+\beta^2)/2\,.\tag{1}\end{equation} As a result, the Lorentz matrix must take the form of \begin{equation}W = \begin{pmatrix} \begin{matrix} 1+\zeta & -\zeta \\ \zeta & 1-\zeta \end{matrix}  & \mathbf{X} \\  \begin{matrix} \alpha & \quad -\alpha \\ \beta & \quad -\beta \end{matrix}  & \m...

Problem: Let $p^{\mu}\equiv[E, p^1, p^2, p^3]^T$ be the  four-momentum of a particle with the rest mass $M$, assuming $c=1$. In the rest frame , such four-momentum becomes to $k^{\mu}\equiv[M, 0,  0,  0]^T$.  The problem is to find a Lorentz transformation (matrix) $L^{\mu}_{\,\,\nu}(p)$ such that $p^{\mu} =L^{\mu}_{\,\,\nu}(p)\,k^{\nu}$. (This problem is about the proof of Eq. (2.5.24) in Weinberg's QFT book, Volume I) Solution: Recall that the vector form of Lorentz boost of four-momentum in two inertial frames with relative velocity $\mathbf{v}$: \begin{equation}\begin{split} E'&=&\gamma\left(E - v\,\mathbf{n}\cdot\mathbf{p}\right)\,,\\ \mathbf{p}'&=&\mathbf{p} + (\gamma-1)\left(\mathbf{n}\cdot\mathbf{p}\right)\mathbf{n}-\gamma\,E\, v\,\mathbf{n}\,,\end{split}\tag{1}\end{equation} where $v$ is the magnitude of $\mathbf{v}$ and $\mathbf{n}\equiv \mathbf{v}/v$ is the unit vector of  $\mathbf{v}$. $\gamma\equiv \frac{1}{\sqrt{1-v^2}}$ is the L...

To illustrate the concept of relativity of simultaneity in special relativity, Einstein proposed a thought experiment where lightning strikes at the head and tail of a running train (points H and T, in short). There are two observers, Alice on the ground and Bob on the train exactly at the middle between H and T. The two events that lightning strikes H and T occur simultaneously to Alice, but not to Bob. In the analysis of this thought experiment, it is important to distinguish the time that the observers receive the lights ( received time ) and the time that the light sent from both ends of the train ( sent time ). Both Newton and Einstein agree that Bob should receive the light from H earlier than the light from T since Bob is moving toward H and faraway from T. The disagreement is the sent time. Suppose in Alice's frame (ground),  the train's length is $2L$; the train's speed is $v$ and the speed of light is $c$; Bob receives the light sent from H (T) at the time $t_H$ (...