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aabbtree77 committed Sep 28, 2024
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2 changes: 1 addition & 1 deletion index.html
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Expand Up @@ -165,7 +165,7 @@ <h2>The Spectrum of a Tensor Field</h2>

<p>The last case, spin <span class="math inline">\(\frac{3}{2})\)</span>, splits into a spinor and <span class="math inline">\((1,\frac{1}{2}) \oplus (\frac{1}{2},1)\)</span>, clf. Weinberg&rsquo;s QFT, Vol. 1, page 232. The latter brings subspaces <span class="math inline">\(\frac{1}{2}\)</span> and <span class="math inline">\(\frac{3}{2}\)</span> with multiplicities 2 and 4, along with their &ldquo;antisubspaces&rdquo;. All of this combined perfectly matches the result of Shankland.</p>

<p>Note that the construction of algebras is skipped, but it is not trivial. For spin <span class="math inline">\(\frac{3}{2})\)</span> Shankland had to spot that <span class="math inline">\(\gamma_{\mu}p^{\mu}\)</span> acts as a primitive which is independent of <span class="math inline">\(p\)</span>, <span class="math inline">\(\gamma\)</span>, and <span class="math inline">\(g\)</span>. This has effectively doubled the basis dimension of the vector-spinor algebra from 5 to 10.</p>
<p>Note that the construction of algebras is skipped, but it is not trivial. For spin <span class="math inline">\(\frac{3}{2}\)</span>, Shankland had to spot that the combination <span class="math inline">\(\gamma_{\mu}p^{\mu}\)</span> acted independently of <span class="math inline">\(p\)</span>, <span class="math inline">\(\gamma\)</span>, and <span class="math inline">\(g\)</span>. This has effectively doubled the basis dimension of the vector-spinor algebra from 5 to 10.</p>

<h2>Other Relevant Algebras</h2>

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2 changes: 1 addition & 1 deletion index.md
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Expand Up @@ -161,7 +161,7 @@ According to group theory, combining indices means taking "tensor products $(m,n

The last case, spin $\frac{3}{2})$, splits into a spinor and $(1,\frac{1}{2}) \oplus (\frac{1}{2},1)$, clf. Weinberg's QFT, Vol. 1, page 232. The latter brings subspaces $\frac{1}{2}$ and $\frac{3}{2}$ with multiplicities 2 and 4, along with their "antisubspaces". All of this combined perfectly matches the result of Shankland.

Note that the construction of algebras is skipped, but it is not trivial. For spin $\frac{3}{2})$ Shankland had to spot that $\gamma_{\mu}p^{\mu}$ acts as a primitive which is independent of $p$, $\gamma$, and $g$. This has effectively doubled the basis dimension of the vector-spinor algebra from 5 to 10.
Note that the construction of algebras is skipped, but it is not trivial. For spin $\frac{3}{2}$, Shankland had to spot that the combination $\gamma_{\mu}p^{\mu}$ acted independently of $p$, $\gamma$, and $g$. This has effectively doubled the basis dimension of the vector-spinor algebra from 5 to 10.

## Other Relevant Algebras

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