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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Chiral Transformation and Dirac Bilinear

I need to compute the following Dirac bilinears: $$\overline{\psi} \psi \quad \text{and} \quad \overline{\psi} \gamma_\mu \psi$$ Under the following Chiral transformation: $$\psi \rightarrow \psi' = \...
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Question about the parity violation of weak interaction Lagrangian

In the textbook of A. Zee, Quantum Field Theory in a Nutshell, the author states that the following Lagrangian: $$ \mathcal{L} = G (\overline{\psi}_{1L} \gamma^\mu \psi_{2L})(\overline{\psi}_{3L} \...
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What is the correct way of looking at the Dirac field?

All quantum fields are operators in QFT. However, the Dirac field operator $\hat{\psi}$ has the following difference with the scalar field operator $\hat{\phi}$: For the $\hat{\psi}$, it makes sense ...
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How to simplify this integral using Wick rotation?

The integral is $$\sum_{n=1}^{\infty}\int \frac{d^4k}{(2\pi)^4}\frac{1}{2n}Tr\left[\left(\frac{ia\!\not\! k\,\mathcal{C}}{k^2+i\epsilon}\right)^{2n}\right]$$ I have to simplify it using Wick rotation, ...
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Is there a fast way to simplify gamma matrix tensors?

I’m doing QFT homework, and our professor said Fermions require a lot of algebra. And I’m finding he’s right. I want to either get a mastery of these operations or find useful shortcuts. Since it’s ...
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Srednicki eq. (1.27): $\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}$

Srednicki, QFT, p. 8 writes $$\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}\tag{1.27}.$$ What does exactly $ab$ here denote? Assume I have a matrix X [0 1] [2 3] and does a ...
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Witten on the hermitian of the Dirac operator

I happened to read Witten SU(2) anomaly paper (1982), and come back to digest again what I said the hermitian of the Dirac operator. According to Prahar https://physics.stackexchange.com/a/701287/...
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Is the Dirac operator $i \not D$ or $\gamma^0 i \not D$ and nonabelian gauge field $A_\mu = A_\mu^\alpha T^\alpha$ hermitian?

In quantum mechanics, it is common to write the momentum operator $$P = i \partial_x.$$ It turns out that $p$ is hermitian although $i^\dagger = -i$ we also have $\partial_x ^ \dagger=-\partial_x$. It ...
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About Einstein's sum rule and Dirac equation

I am studying the Dirac equation and I'm having some trouble about something that I think should be trivial. I'm working in a (1+1)-dimensional Minkowski spacetime with signature $(+, -)$, i.e., $ds^2=...
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A problem with QED

I have a small problem with the understanding of QED. The equations of motion in QED are $\square A^\mu=e\bar{\psi}\gamma^\mu\psi$ $\left(i\gamma^\mu\partial_\mu-m\right)\psi=e\gamma^\mu A_\mu\psi$ If ...
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Symmetry arguments and derivation for product of gamma matrices and derivatives

I am trying to work with the Dirac equation and the solution for the Klein-Gordon equation for some derivation and I stomped on the following problem in my derivation. $\gamma^{\mu} \gamma^{\nu} \...
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What is the deep connection between the two ways to get Pauli matrices?

The first way is the one in which we start with the commutator relations $[J_x, J_y]=ihJ_z$, etc. We consider the simultanelus eigenbasis of $J^2$ and $J_z$ : $|j,m\rangle$. We then obtain their ...
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Transformation of Chern-Simons type action under large $SO(4)$ gauge transformation

For the Chern-Simons action $$S = \kappa \epsilon^{\mu\nu\delta}tr(A_\mu\partial_\nu A_\delta + \frac{2}{3}A_\mu A_\nu A_\delta)$$ under a large gauge transformation $$A_\mu \rightarrow g^{-1}A_\mu g +...
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Parity of object made up of Dirac spinors and gamma matrices

I am reading Introduction to Elementary Particles by Griffiths, specifically the chapter on Dirac equation. Griffiths states without proof, that the expression $\bar{\psi}\gamma^\mu\gamma^5\psi$ is a ...
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Self or complex Weyl representation in Polchinski

In p.433 of Polchinski String theory volume 2, he said that: In $d$ spacetime dimension, For $d = 2 \mod 4$, each Weyl representation is its own conjugate. ($B \Gamma B^{-1} =-\Gamma $) For $d = 0 \...
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The relation of gamma matrix between field operator change and chirality in Peskin and Schroeder (page 165)

I'm reading Compton scattering in Peskin's book (page 165) and there is a sentence I can't understand. The third sentence in the above paragraph says that three $\gamma$-matrix between field operator ...
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Decomposing product of spinor representation of orthogonal group

I am reading A. Zee's group theory book. In the chapter of spinor representation, page 416, he was trying to describe how to decompose of the product of spinor representation into irreducible ...
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Lowering the spacetime index of a Dirac matrix

$\gamma_\mu\partial^\mu$=$\gamma^\nu\partial_\nu$ Does the above equation hold for Gamma matrices? If so, why?
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Is Dirac equation valid only for spin-$\frac{1}{2}$ particles?

It is usually said that Dirac got his equation by looking for the square root of the 4-momentum norm (see Dirac’s coop here). The relativistic 4-momentum norm is $$(E)^2-(\mathbf{p}c)^2=(mc^2)^2 \tag{...
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How to calculate the trace of six gamma matrices multiplied to $\gamma_5$?

I read from Weinberg that, the gamma matrices have the following property: \begin{equation} \text{Tr}\{\gamma_5 \gamma_\mu \gamma_\nu \gamma_\rho \gamma_\sigma\}=4i\epsilon_{\mu \nu \rho \sigma} \end{...
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QFT $\sigma$-model - Commutator of Pauli and Gamma matrices

To give some contest, I'm working on a $\sigma$-model given by the Lagrangian $$L = \bar\psi i \partial_\mu\gamma^\mu \psi - g\bar\psi (\sigma + i\vec\pi \cdot \vec\tau \gamma^5)\psi + \frac{1}2(\...
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How to show that $\sigma^\mu_{\alpha \dot{\alpha}} \bar\sigma_\mu^{\dot{\beta} \beta} = 2\delta_\alpha ^\beta \delta_\dot{\alpha}^\dot\beta$?

Consider the usual definitions $\sigma^\mu = (1, \sigma^i)$ and $\bar\sigma^\mu = (1, -\sigma^i)$, is it possible to show $$\sigma^\mu_{\alpha \dot{\alpha}} \bar\sigma_\mu^{\dot{\beta} \beta} = 2\...
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Trouble deriving the symmetrised canonical energy-momentum tensor for the Dirac Lagrangian

I've recently learned of a method which can symmetrise the canonical energy-momentum tensor, $$T_{\mu\nu} = \eta_{\mu\nu} \mathcal{L} - \sum_{a} \frac{\partial \mathcal{L}}{\partial (\partial^\mu \...
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Space-time split in Geometric Algebra

I was performing some calculations with Geometric Algebra today and I've found myself stuck with a simple operation that I don't know how to answer. Considering that you have the usual vector ...
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Lagrangian density transformation

In a calculation of a Lagrangian density $$ \mathcal{L}=\bar{\psi}\left(i \gamma^{\mu} \partial_{\mu}-m+i \gamma_{5} m^{\prime}\right) \psi. $$ In order to see if it is invariant or not with the ...
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To construct a Lorentz scalar we use $\psi^{\dagger}\gamma^{0}\psi$. Could we use $\gamma^{5}$ instead of $\gamma^{0}$ seen as both are Hermitian?

Both $\gamma^{0}$ and $\gamma^{5}$ are Hermitian, so could we replace $\gamma^{0}$ with $\gamma^{5}$ to construct a Lorentz scalar with the same properties as $\bar{\psi}\psi$?
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How to show that $\bar{\psi}\gamma^{\mu}\psi$ goes to 0 if $\psi$ is are not both left/right handed?

How do you show, using the Dirac matrices, that the above expression is $0$? I have tried substituting $\psi^{\dagger}\gamma^{0}$ in for $\bar{\psi}$ but I cannot find any identities linking $\gamma^{...
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Dirac equation: four-derivative spinor

I am confused where the four-derivative acts on. $$ (i\gamma^\mu \partial^\mu-m)\psi=0 $$ I thought that the derivative will act on the spinor. But is that wrong? What is the relation of it with the ...
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Gamma matrix, four derivative exchange [closed]

I'm trying to study about the dirac equation, and the adjoint of it. I have few questions about that. **After finishing writing, I realized that all $\partial ^\mu$ should be $\partial_\mu$. 1. $$ \...
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How do I project on the Dirac and Pauli structures?

I have a term which is $A_1\gamma^{\mu}+A_2 [\gamma^\mu,\gamma^\nu]p_\nu$ where $A_1$ and $A_2$ are constants and $p_\nu$ is a momentum where $p^2$ is much greater than zero. How can I project on the ...
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What do the gamma matrices mean in QED’s Feynman diagrams?

From page 245 of the PDF of Matthew Schwartz's book (page 226 of the textbook), he shows a Feynmann diagram in QED like this: We can see the spinors/propagator for the electrons and the photons ...
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How to convert a singlet current in electroweak theory to a doublet current?

I am referring to an aspect of Glashow-Salam-Weinberg theory of leptons. More specifically a step in the book 'Gauge Theory of Weak Interactions' by Walter Greiner page 147. We can write the ...
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Why is working infinitesimally allowed here?

In page 42 of Peskin and Schroeder, for $$\left( \mathcal{J}^{\mu\nu} \right)_{\alpha\beta} = i \left( {\delta^\mu}_{\alpha} {\delta^\nu}_{\beta} - {\delta^\mu}_{\beta} {\delta^\nu}_{\alpha} \right)$$ ...
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Pauli matrices: lower index vs upper index

I have read some identities about the Pauli matrices in 4-vector notation and I am a little confused. as $$\sigma ^\mu=(I,\sigma ^i);\qquad \overline{\sigma}^\mu=(I,-\sigma ^i).$$ But what exactly is $...
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How is spin derived in different numbers of dimensions? [closed]

I understand that spin is derived from combining special relativity and quantum mechanics, but I don't know the details of how. I know that the Dirac Equation takes spin into account, but only works ...
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Why must $\gamma^\mu$ matrices in Dirac equation be linearly independent?

In Ashok Das' Lecture on Quantum Field Theory, pg 20, it was said that the Dirac equation is derived by considering the Einstein relation $$p^\mu p_\mu=m$$ as a matrix relation, namely an $n\times n$ ...
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Parity transformation on the adjoint spinor

In eq. (3.128) of page 66 of "An Introduction to QFT", by Peskin & Schroeder, a step involves: $$P\,\overline{\psi}\left(t,\textbf{x}\right)P^{-1}=P\,\psi^{\dagger}\left(t,\textbf{x}\...
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Lorentz Transformation - Representation for Rotations, the Scalar Product and Kronecker Delta

I'm currently reading this article about the Dirac Fermion by Yehonatan Viernik. In section 3.3 (looking at Lorentz transformations), we define matrices $S ^{ij}$ to be representations for rotations. ...
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Pauli matrices in general relativity

Just as in a tetrad formalism one brings the gamma matrices from the local Lorentz frame to the manifold through: $$\gamma^\mu = \gamma^a e^\mu_a.$$ Can one do that for the Pauli spin matrices?
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Amount of SUSY preseved by branes

Consider 10d superstring theory, a D5 brane extended in the 012789 directions, and an NS5 brane extended in the 012345 directions. I have read that this configuration preserves 1/4 of the ...
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Four-current and identity

I am studying QFT and I have found the following identity, which I don't know how to prove. $$\bar{u}(p_{1})\not p_{1} \gamma_{\mu} \not p_{2}u(p_{2})=\bar{u}(p_{1})((q^{2}+m^{2})\gamma_{\mu}-2im\...
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Why three gamma matrix between initial and final electron lead to the same helicity?

Quote Peskin page 165 Suppose that the initial electron is right-handed... Since this term contains three $\gamma$-matrix in (5.97) between $\bar u$ and $u$ the final electron must also be right-...
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Representation of the Gamma matrices and the metric

Suppose I have the Gamma matrices given by $\gamma ^\mu$. Under some unitary transformation $U$ I can consider $\tilde{\gamma^{\mu}} = U\gamma ^\mu U^\dagger$. Since I have: $$ \{ \gamma^\mu, \gamma^\...
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Sign discrepancy in CPT theorem

I'm reading Fundamentals of Neutrino Physics and Astrophysics by Carlo Giunti and Chung W. Kim (2007). In chapter 2 section 11.5, they talk about the CPT theorem and how each fermion bilinear ...
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Fierz identity and its application

I know there are very good explanations for Fierz identity in this platform. But I have a question about its application. Consider a matrix element $M$ and we can write $M$ in the basis of 16 Dirac ...
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Do the Dirac matrices form a proper four-vector?

I'm seeing many conflicting statements about Dirac's gamma matrices. Some say it's a four-vector, some say it isn't, some say it's an invariant four vector. I know the Dirac matrices satisfy the ...
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Symmetries of two Massless Fermion

I'm considering the following free Lagrangian (density): $$\mathcal{L}= \bar{\Psi} \left(i \displaystyle{\not}{\partial}\mathbb{1}_{2}\right)\Psi$$ Where $\Psi$ is a doublet of two fermion fields $\...
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Charge conjugation matrix in terms of rank-2 antisymmetric tensors

I am having trouble with the sign when computing the charge conjugation matrix in the Weyl representation, namely I am yielding an additional minus sign. Let me start with some of the conventions I ...
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One-loop diagram calculation: Wick rotation and gamma matrices

I have a question about a calculation which is performed in David Tong's lecture notes on Gauge Theory (page 400, chapter 8). At the bottom (see below for a screenshot), we want to calculate the one-...
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Anti-commutation relation for gamma matrices; when/why did the definition change?

The anti-commutation relation defining gamma matrices is presently given by $$\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}$$ where appears the matric tensor (see for instance the ...
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