Questions tagged [dirac-matrices]
Dirac matrices, or gamma matrices, are a set of matrices with specific anticommutation relations that generate a matrix representation of the Clifford algebra which acts on spinors, fundamental to the Dirac equation describing spin-1/2 charged particles.
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$CP$-transformation for fermionic bilinears
I am trying to derive the transformation of the fermionic bilinear $\bar{\psi}\psi$ under $CP$ transformation.
I know that $P$ acts as:
$$\psi(t, \vec{x}) \xrightarrow{P} \gamma^0 \psi(t, -\vec{x})$$
...
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Deriving the properties of the Dirac matrices
I am working on the properties of the Dirac matrices, but I cannot figure out the derivations.
For example, on proving $\gamma^\mu {\not}{a} \gamma_\mu = -2{\not}{a}$, we first prove that $\gamma^\mu \...
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Do gamma matrices commute with 4-vectors?
One of my exercises was to prove the identity
$$\gamma^\mu\displaystyle{\not}a\gamma_\mu=-2\displaystyle{\not}a.$$
Which is trivial if $\gamma^\mu a_\nu=a_\nu \gamma^\mu$, as follows
$$\gamma^\mu\...
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Fierz Identity in 2+1d
Wikipedia states an example of Fierz Identity, under the assumption of commuting spinors, the $\mathrm{V} \times \mathrm{V}$ product that can be expanded as,
$$
\left(\bar{\chi} \gamma^\mu \psi\right)\...
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How to motivate spinors from the Dirac equation? [closed]
I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator).
Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
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Fermi tetrad field: Fermi-Walker tetrad formalism?
These days I'm reading Dirac Eq in GR, and I'm confused about "Fermi tetrad field". Is it Fermi-Walker tetrad formalism?
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Is there a $\gamma^{5}$ in $d$-dimensional Clifford Algebra?
I was trying to compute the EW vacuum polarization
$$i\Pi_{LL}^{\mu\nu}=(-1)\mu^{\frac{\epsilon}{2}}e^{2}∫\frac{d^{d}k}{(2\pi)^{d}}\frac{Tr\bigl(i\gamma^{\mu}(iP_{L})i(\not{k}+m_{i})(i\gamma^{\nu})P_{...
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The dimension of the Clifford algebra for the Dirac equation
The Dirac algebra contains sixteen linearly independent elements. In general, a Clifford algebra $\mathcal{C}\!\ell(V,Q)$ generated from a vector space $V$ equipped with a quadratic form $Q$ has ...
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Obtaining the 16 elements of the Clifford algebra from the $\gamma^\mu$ generators
In my study of the Dirac equation, I have fully understood the "linearization" of the relativistic energy to obtain a matrix-valued equation that reduces to the Klein-Gordon equation if the ...
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Question about meaning of "bar"-ing in the context of Dirac fields
Following chapter 38 of Srednicki, "bar"-ing means (based on eq. 38.15)
$$\bar{A} = \beta A^\dagger\beta$$
One can show for instance that $$\bar{\gamma^\mu} = \gamma^\mu$$ My question is, ...
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Understanding derivation of Klein-Gordon equation from Dirac equation
Above is Tong's notes which shows how the Klein-Gordon equation is derived from Dirac equation. But I don't get why:
$\gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} = \frac{1}{2} \{\gamma^{\mu},\...
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Dirac field charge conjugation
I struggled a bit to understand the proof of the relation $C\overline\psi\psi C=\overline\psi\psi$ in Peskin's and Schroeder's book An Introduction to Quantum Field Theory (page 70, formula 3.147):
$$...
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Covariant derivative property
I am trying to demonstrate this propertie
$$
\not{D}^2= \mathcal{D}^\mu \mathcal{D}_\mu-\frac{i}{4}\left[\gamma^\mu, \gamma^\nu\right] F_{\mu \nu}
$$
where $\not{}~$ is the Feynmann slash, and $D_\mu ...
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Feynman slash identity
How would one simplify the expression
$\gamma^0 {\not}p \;\gamma^0$
?
My guess would be that its
$$\begin{align}
\gamma^0 {\not}p \;\gamma^0&=\gamma^0 \gamma^{\mu} p_{\mu} \;\gamma^0\\
&=\...
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Can $\gamma^5$ matrices be ignored in $q\bar{q}\to ZZ$ processes?
In the $q\bar{q}\to ZZ$ process, the following Feynman diagram in LO appears:
This means for each vertex, the Feynman amplitude contains a term proportional to $(g_V-g_A\gamma^5)$, which makes $D$-...
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