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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Derivation of covariant derivative of Spinor

I am following this derivation for the covariant derivative of spinors. I have some questions about this derivation: On page 3 they use the fact, that \begin{align*} V^a(x) = \bar{\Psi}(x)\gamma^a\...
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Dirac spinor manipulation [closed]

I'm trying to simplify the following relationship as much as possible: $$\partial_\mu\bar\psi[\gamma^\mu,\gamma^\nu]\psi$$ where $\psi$ satisfies the dirac equation. Using the fact that $[\gamma^\mu,\...
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Peskin and Schroeder's QFT eq. (7.88)

On Peskin and Schroeder's QFT book, page 251, the book discussed how things will be changed in $d$ dimensions. For example $g_{\mu \nu}g^{\mu \nu}=d$. In eq. (7.88), the book gave how Dirac matrices ...
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Connection between Pauli matrices and Quaternions? [duplicate]

reading this sentence from wikipedia: The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 ...
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Density Matrices in Quantum Mechanics

I have a question about the physical meanings of various matrices expressed in Dirac bra-kets. I take it that $\frac{1}{2}|A\rangle\langle A| + \frac{1}{2}|B\rangle\langle B|$ can be interpreted as a ...
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Hermitian conjugates of Dirac equation

I am given the following Dirac equations: $$ (\gamma^{\mu}p_{\mu} - m)u_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(1) $$ $$ (\gamma^{\mu}p_{\mu} + m)v_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(2) $$ where u are the ...
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Lorentz invariance and Dirac Slash Notation

I have the following factor inside a scattering amplitude $$\Big[\bar{u}(p')\not{l}\gamma^{\mu}\not{l}u(p)\Big] \ \Big[\bar{u}(q')\gamma_{\mu}u(q)\Big]$$ where $p$ and $p'$ are the initial and final ...
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Prove that Majorana mass term is Lorentz invariance

I have a homework to prove that using one kind of chirality, let's say left-handed, we can construct a mass term. The argument is to show that this term $\psi^TCP_L\psi$ is satisfy the dimensionality ...
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Pauli matrices from anticommutator

I want construct the Pauli matrices starting from $$\sigma_i=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$$ by using only the anti-commutation relation $$ \sigma^i\sigma^h+\sigma^h\sigma^i=\{\...
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Degrees of freedom in a spinor in $d$ dimensions (following Polchinski & Lounesto)

I am working through several texts on spinors and trying to deepen my understanding of this fascinating concept. In many ways I have found Polchinski's great Appendix B of String Theory, volume 2 to ...
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How to dimensionally reduce the 3+1 D Dirac equation into the 1+1D Dirac equation?

In 3+1D the Dirac equation looks like $$i\partial_\mu \gamma^\mu \Psi -m\Psi=0.$$ If we only consider $x$-direction, then it should reduce to $$i\partial_t\gamma^0\Psi =(-i\partial_x\gamma^1+m)\Psi.$$ ...
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Charge Conjugation

For my calculations I need to know how charge conjugation acts on the Spin 3/2 propagator. The charge conjugation operator $C$ is calculated as $C= i\gamma^2\gamma^0$. The Spin 3/2 propagator is $$ S_{...
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Peskin & Schroeder's QFT book page 161

I am trouble with below derivation in Peskin & Schroeder's QFT book on page 161. At the bottom of this page, $$ \begin{aligned} \operatorname{tr}\left[\not p^{\prime} \gamma^{\mu} \not k \gamma^{\...
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Lorentz boost property of gamma matrices

I was watching this video where he boosted the Dirac equation. He reached this equation: $$S^{-1}(\Lambda)\gamma^\mu S(\Lambda)=\Lambda^\mu{}_\nu \gamma^\nu$$ My question is since $\gamma^\mu$ is a ...
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How charge conjugation operator flips chirality

I am trying to understand the charge conjugation operator by reading several references online. Until I come to a point which mention that using the anticommutation properties of the Dirac-$\gamma$ ...
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Why is it that the Quantised Dirac field spin has non-half integral eigenvalues?

The z-spin operator is: $$\sum_{\vec{p}, r} \frac{m}{E} u_r^{\dagger} (\vec{p}) \Sigma_zu_r(\vec{p})N_r(\vec{p})$$ For a state like $|\vec{p}=p_3 \vec{k}, s_z=\frac{1}{2}\rangle$, this operator ...
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Why does Tong uses Euclidean Gamma matrices in this step of deriving the Chiral Anomaly?

In David Tong's GT notes on page 137, he uses the trace identity for Euclidean gamma matrices given by $$\text{Tr}(\gamma^5\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=4\epsilon^{\mu\nu\rho\...
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Do I have to take the trace over gamma matrices in Yukawa vertex correction?

Given the Yukawa coupling $\mathcal{L}_{\text{int}}=g\phi\bar{\psi}\psi$, if I want to compute the correction to one loop to the vertex, I would write something like this $$\Lambda\sim g\int\frac{d^...
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Connection between Dirac-Matrices and Clifford Algebra

I have to give a quick talk at our University about the Dirac Equation and the Clifford Algebra. Since I am still at the beginning of my studies, my knowledge in this regard is still very limited. ...
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Dirac's gammas in scalar Lagrangian

Can one write a Lagrangian for a scalar field using Dirac's gamma matrices? If so, is it useful? This question came to my mind when trying to write a general (nonrenormalizable) Lagrangian for a ...
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Noether current of Dirac field

Under $U(1)$ symmetry of the Dirac Lagrangian I get, up to a constant, a current $$j^\mu=\bar\psi\gamma^\mu\psi.$$ How can I express it using creation and annihilation operators? For example it's easy ...
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Gamma matrices in the 4-momentum operator

I'm trying to compute the operator $\hat P^\mu$ from $$\hat P^i=\int d^3x\hat\tau^{0i}$$ where $\hat\tau^{\mu\nu}$ is the stress energy tensor for the Dirac Lagrangian. In my case I have $$\hat\tau^{...
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Charge conjugation of the Dirac action

I'm following David Tong's convention on charge conjugation, $$ \psi^c = C \psi^* \ , \qquad C^\dagger C = 1, \qquad C^{-1}\gamma^\mu C = - (\gamma^\mu)^* \ , $$ where $\gamma^0$ is hermitian, while $\...
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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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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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3 votes
2 answers
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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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1 vote
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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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