# Tag Info

Accepted

### Relation between the Dirac Algebra and the Lorentz group

It's pretty annoying that P&S just give you $$S^{\mu \nu} = \frac{i}{4} [\gamma^{\mu},\gamma^{\nu}]$$ from thin air, here is a way to derive it similar to Bjorken-Drell's derivation (who start ...
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### Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

While this is an old question, I imagine many encounter geometric algebra through David Hestenes's writing on zitterbewegung, so I'd like to offer a different perspective than the existing answers. ...
• 763

### Relation between the Dirac Algebra and the Lorentz group

I want to know if we take any arbitrary metric $g_{\mu\nu}$ on some space $V$, will the generators defined as $S^{\mu\nu}$ generate a Lie group whose elements are transformations on $V$ that conserve ...
• 15.3k
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### Understanding Wikipedia's definition of a spinor

The spin group has a multiple irreducible representations of dimension 4. Two of them are the left- and right-handed spin-3/2 representations. The other one is called the vector representation. "...
• 4,335
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### Is there an elegant proof of the existence of Majorana spinors?

To answer the confusion between the three sources you list: Using the signature convention of Figueroa O'Farrill, we have Majorana pinor representations for $p - q \pmod 8 = 0,6,7$ and Majorana spinor ...
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### How are Clifford algebras related to Dirac Equation

The vector space is Minkowski spacetime. The quadratic form is the spacetime interval. The relationship to the Dirac equation is that when you start with Minkowski spacetime, the vector represent ...
• 25.6k

### Understanding Wikipedia's definition of a spinor

Here's a purely mathematical answer. The Lie algebra $\mathfrak{so}(1,3)$ of the Lorentz group is a simple 6-dimensional real Lie algebra which is isomorphic to $\mathfrak{sl}_2(\Bbb{C})$ viewed as a ...
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### Does $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbb1$ determine the hermiticity of the gamma matrices?

I don't think so. If $\gamma^\mu$ is a set of gamma matrices obeying the clifford algebra, then so is $S^{-1}\gamma^\mu S$ for any invertible $S$. But unless $S$ is unitary, $\gamma^\mu$ being ...
• 53.7k

### Dimension of Dirac $\gamma$ matrices

Thats a good question. To answer this lets start with Clifford algebra generated by $\gamma$ matrices. $$\gamma_{\mu}\gamma_{\nu}+ \gamma_{\mu}\gamma_{\nu}=2\eta_{\mu\nu}$$ ...
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• 63.3k
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### Does the geometric algebra of curved space have a matrix representation?

In curved space the coordinate basis will not be orthonormal, but an orthonormal basis still exists, and can be used to construct an arbitrary basis. So... Start with $\gamma_\mu$ as a matrix ...
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### Is $\gamma_\mu \gamma^\mu$ a unit operator?
You are using $\mu$ as an index too many times. When you square $\gamma^\mu \partial_\mu$, you have to use a different index for the two factors. Here is a correct derivation: Starting with \$(i \...