Life with Physics

Inspired by Einstein's quote, "If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker," I recently purchased the book "Euclid's Elements" for my son. With this esteemed work on our bookshelf, I hope it brings an opportunity for my children to one day open its pages and delve into a new world of wonder. This post is to share two interesting discoveries after I read the first three pages of the book. Euclid's fourth postulate At the beginning of the book, Euclid proposes five postulates for plane geometry, which I quote directly from this web : A straight line segment may be drawn from any given point to any other.  A straight line may be extended to any finite length.  A circle may be described with any given point as its center and any distance as its radius.  All right angles are congruent.  If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it toget...

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 $t_H$ (...

Considering special relativity, what is the world line of a moving particle with constant proper acceleration $a$ observed in the still world frame? The proper acceleration is defined as the acceleration measured by the inertial observer to which the particle looks rest during the measurement. Lorentz transformation of acceleration The problem is about the Lorentz transformation of acceleration. For this purpose, we start by the Lorentz transformations in special relativity between a still frame $(t, x)$ and a moving frame $(t', x')$ with constant speed $v$ relative to the still frame: \begin{eqnarray} x'&=&\frac{x-vt}{\sqrt{1-{v^2}/{c^2}}}\,,\tag{1}\\ t'&=& \frac{t-vx/c^2}{\sqrt{1-{v^2}/{c^2}}}\,.\tag{2}\end{eqnarray} A motion with speeds $u_x\equiv \frac{dx}{dt}$ in the still frame and $u'_x\equiv \frac{dx'}{dt'}$ in the moving frame has the transformation: \begin{eqnarray} u'_x = \frac{u_x - v}{1-{vu_x}/{c^2}}\,,\tag{3} \end{eqnarray} ...