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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Generalized Clifford algebra
I wish to know if generalized versions of the Clifford algebra exists such that
$$\{\gamma^{\dagger}_{\mu}, \gamma_{\nu}\} = 2\eta_{\mu\nu}$$
In physics, we usually work with the simplification $\...
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Conventions for Wick rotations of the Dirac Lagrangian
I have some questions about Wick rotations of the Dirac Lagrangian in different signatures. I have seen similar questions, but none of them explain things in the way I need. In fact, I was trying to ...
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Dimensionality of spinor representation for group ${\rm Spin}(n)$
Rather simple question that I'm sure has been asked before, but I haven't been able to figure out a way to phrase it that the search engine understands.
As I understand, the group ${\rm Spin}(n)$ is ...
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Action of time-reversal on Dirac bilinears
I am sorry if this may sound like a duplicate but I couldn't find any satisfying answers anywhere online or on the forum. In Peskin and Schroeder, it is said that $$T\bar\psi \gamma^\mu \psi T^{-1} = \...
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Doubt on Feynman slash notation to prove the LSZ reduction formula for fermions
I'm trying to prove the LSZ reduction formula for fermions, but I'm having some issues with signs.
In particular, I found this reference that seems to use the following identities at the end of page 2:...
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Matrix-index ordering for $\sigma^\mu$ vs $\bar\sigma^\mu$
In $d=4$ we can write a theory of $N$ free weyl fermions
$$
\mathcal{L} = i \bar\psi_{a}(x) \bar\sigma^{\mu} \partial_\mu\,\psi^{a}(x)
$$
with
$$
(\sigma^\mu)_{\alpha\dot\alpha} = (\mathbf{1},\sigma^i)...
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Constructing a Lorentz Scalar from Bilinear Covariants
I'm probably just overthinking this, but here we go:
Let $\gamma^\mu$ be the standard Dirac matrices defined by $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$, where $\eta^{\mu\nu}$ is the Minkowski ...
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Generalizing Pauli 4-vectors $\sigma_\mu$ to other dimensions ($\mathbb R^{p,q}$)
As we know, the Pauli 4-vectors has this property:
$$Lv^\mu\sigma_\mu L^\dagger=\rho(L)^\mu{}_\nu v^\nu\sigma_\mu,$$
where $\rho:\mathrm{SL}(2,\mathbb C)\to\mathrm{SO^+}(1,3)$ is the covering map of ...
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Chiral transformation and a factor of two
In Freedman and Van Proeyen's Supergravity book in section 5.3.2 the expression $$e^{-i\beta \gamma_*} = \cos 2\beta -i \gamma_* \sin 2\beta$$ is used, with $\gamma_*$ the highest rank element in the ...
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"Naturalness" of the 't Hooft-Veltman-Breitenlohner-Maison (HVBM) scheme in Dimensional Regularization
Disclaimer: This question is fairly subjective based on what one considers a natural construction. With that in mind, let's continue.
I'm trying to understand more complicated examples of dimensional ...
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Some questions about Clifford algebras and Dirac $\gamma$ matrices
I have some questions about the Dirac $\gamma$ matrices and their relation with the Clifford algebra $\mathrm{Cl}(1,3)$. Consider the Minkovski metric given by $\eta=\mathrm{diag}(1,-1,-1,-1)$. $\eta$ ...
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Physical interpretation of Pauli matrices acting on a Weyl spinor
Letโs say $\psi$ is a two-component Weyl spinor. What is the physical interpretation of a Pauli matrix acting on this Weyl spinor? In other words, what does $\sigma_{i}\psi$ represent (where $\sigma_{...
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How to explain the Dirac matrices in physical-meaning terms?
I was explaining the Dirac equation, $\left(i \hbar \gamma^\mu \partial_\mu-mc \right) \psi = 0$, to a friend who understands some of the mathematics but not all of the physics. We were just talking ...
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Is the definition of $\gamma_5$ unique?
This is a slightly subjective question: Is there a "natural" way to justify the definition of $\gamma_5$?
I have something like this in mind: Apart from Clifford algebra obeyed by the $\...
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Relation between Lorentz and spinor representations of the Lorentz group generators
In Peskin and Schroeder's 'An Introduction to Quantum Field Theory', the representation of te Lorentz transformation generator in the space of Dirac matrices is given in Equation (3.23) as
$$
S^{\mu\...
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