https://haifei.pro/en/tags/relativity/Recent content in Relativity on Haifei's HomeHugoenhfwang132@gmail.com (hfwang132)hfwang132@gmail.com (hfwang132)This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.Wed, 09 Jul 2025 19:23:54 +0800https://haifei.pro/en/post_20250709_%E8%87%AA%E6%97%8B%E4%B8%BA%E4%BB%80%E4%B9%88%E4%B8%8D%E6%98%AF%E7%9B%B8%E5%AF%B9%E8%AE%BA%E6%95%88%E5%BA%94/Wed, 09 Jul 2025 19:23:54 +0800hfwang132@gmail.com (hfwang132)https://haifei.pro/en/post_20250709_%E8%87%AA%E6%97%8B%E4%B8%BA%E4%BB%80%E4%B9%88%E4%B8%8D%E6%98%AF%E7%9B%B8%E5%AF%B9%E8%AE%BA%E6%95%88%E5%BA%94/<p>Conclusion: integer spin is a classical effect, whereas half-integer spin is a quantum effect and has little to do with relativity.</p> <p>The reason is simple: spin-1 is the smallest faithful representation of SO(3), while spin-1/2 is the smallest faithful representation of SU(2).</p> <p>So why can quantum mechanics lift SO(3) to SU(2)? Because quantum states are rays and are equivalent up to a global phase. In other words, quantum mechanics requires projective representations of SO(3). And projective representations of SO(3) are in one-to-one correspondence with representations of SU(2) (Bargmann&rsquo;s theorem). This is why quantum mechanics considers SU(2) rather than SO(3).</p>https://haifei.pro/en/post_20210226_%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8F%98%E6%8D%A2%E7%9A%84%E5%87%A0%E4%BD%95%E7%9B%B4%E8%A7%89/Fri, 26 Feb 2021 23:05:45 +0800hfwang132@gmail.com (hfwang132)https://haifei.pro/en/post_20210226_%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8F%98%E6%8D%A2%E7%9A%84%E5%87%A0%E4%BD%95%E7%9B%B4%E8%A7%89/<p>You do not need to know any formulas to gain an intuitive understanding of various phenomena in special relativity (length contraction, time dilation, the twin paradox, etc.).</p> <hr> <h2 id="a-small-modification-to-the-galilean-transformation">A Small Modification to the Galilean Transformation</h2> <p>Let us first look at the classical Galilean transformation. In a t-x diagram, the Galilean transformation appears as a shear transformation. Points slide along lines parallel to the x-axis.</p> <figure class="post-figure" style="--post-figure-width: 80%;"><img src="https://haifei.pro/post_20210226_%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8F%98%E6%8D%A2%E7%9A%84%E5%87%A0%E4%BD%95%E7%9B%B4%E8%A7%89/images/v2-1434b48b249a720bee168ee9630156e7_r.jpg" loading="lazy"/></figure> <p>Galilean transformation, viewed from Little Red&rsquo;s and Little Green&rsquo;s perspectives</p>