# Questions tagged [dirac-matrices]

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### Proving identity $\mathrm{Tr}[\gamma^{\mu}\gamma^{\nu}] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$$ Yet when I try to prove this,...
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### Dirac matrices in 1+1 dimensions

Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
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### Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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### Evaluating a trace with two factors of $\gamma^5$

In the process of calculating a spin-averaged square amplitude in QFT, I came across the following expression: $$\text{Tr}\left[\gamma^\mu\gamma^5\gamma^\alpha\gamma^\nu\gamma^5\gamma^\beta\right]$$ ...
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### Confusion with trace of gamma matrices

Using $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} \mathbf{1}$, it is easy to show that: \begin{align*} \operatorname{tr} \gamma^\mu \gamma^\nu = 4\eta^{\mu\nu} \end{align*} Now, it is also true that ...
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### Gamma matrices in (2+1)

I am sure that is very well-known question and see on this site several similar questions but I would like to specify the answer 1) I know that in $(2+1)$-dimensions one can construct $\gamma$-...
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### What is the relationship between the Lorentz group and the $CL(1,3)$ algebra?

In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ...
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### Vertex with spinorial structure and scattering amplitude

Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$ ...
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### Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?

Is spinor $\psi$ actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar? Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of ...
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### Missing identity element in the Clifford relation

While studying the Dirac equation, $$\left(i\gamma^{\mu} \partial_{\mu} - m\right)\psi = 0.$$ I have been finding difficulty understanding the following summarisation of the algebra that the $\gamma$-...
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### Simplification of an expression with gamma matrices [closed]

I am trying to understand the details of the calculations involved in determining the electron's anomalous magnetic moment to second order: the $\alpha/{2\pi}$ term. There is just one step, where an ...
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### What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
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### Relation between Dirac spinors, quaternions, and bicomplex numbers

Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ...
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### Møller scattering amplitudes

In order to compute the scattering cross section for Møller scattering, one needs the amplitudes for both the $t$- and the $u$-channel. Since the cross section is proportional to $|\mathcal{M}|^2$, ...
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### Gamma matrices in curved spacetime

How to raise and lower indices of gamma matrix in curved spacetime? Do we raise and lower the index of gamma matrix with $g_{\mu \nu}$?
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### How can I prove this relation involving gamma matrices?

Let $$l^{\mu} = l^{\mu}_{\parallel} + l^{\mu}_{\bot}$$ be a D-dimensional vector living in a Minkowskian space; the only non-zero components of $l^{\mu}_{\parallel}$ are the first four, while the only ...
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### Identities of Pauli matrices in two-component spinor formalism

I'm reading the review by H. K. Dreiner, H. E. Haber and S. P. Martin (arXiv:0812.1594) about the two-component spinor formalism. There are some identities and notational conventions which lead to ...
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### Do physical results for spinors depend on the Clifford algebra representation?

As I understand it, the Dirac equation and its solutions depend on the representation of the $\gamma$ matrices one uses. So if I were to use the Dirac representation I would get different mathematical ...