Questions tagged [dirac-matrices]

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Dirac conjugation of a 3x3 matrix

This question might be stupid, but when I compute $\bar{B}$ in the Lagrangian, I have to multiply 3x3 $B$ matrix with 4x4 $\gamma_0$ matrix (Dirac's conjugation) which are incompatible in size. What ...
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1answer
38 views

Sign of pair of Dirac spinor bilinear

I don't understand the following statement: Any pair of Dirac spinors verifies $(\bar{\Psi}_1\Psi_2)^\dagger=\bar{\Psi}_2\Psi_1$ and it is valid for both commuting and anti-commuting (Grassmann-valued)...
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2answers
91 views

Product of an odd number of Dirac $\gamma$ matrices [closed]

Suppose $n$ is an odd number. Why can we write $a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n$ as $$a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n = V_\mu \gamma^\mu + A_\mu \gamma^\mu \gamma_5$$ for some $V_\mu, ...
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1answer
39 views

(Anti)commutation relations for higher-dimensional anti-symmetrized Gamma matrices

Suppose $a,b,c...=0,....,D-1$ are Lorentz indices of $SO(1,D-1)$ tangent space and consider $D$-dimensional Clifford algebra defined by the usual anticommutation relation $$\{\Gamma^a,\Gamma^b\}=2\eta^...
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How does following relation generates Lorentz generator?

In Schwartz book sec 10.3, Schwartz says following: The Lorentz generators when acting on Dirac spinors can be written as $$S^{\mu \nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$$ But what I am able to ...
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1answer
97 views

Matrix “dimensional analysis” of Lagrangians in QFT

Since the important things in the QFT Lagrangian are vectors and matrices, I wanted to do a "matrix dimensional analysis" of each term. The electromagnetic Lagrangian (ignoring all constants ...
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1answer
61 views

Proving an equation built up out of Dirac-$\gamma$ matrices

Given the following Feynman Amplitude: $$\mathscr{M}=\bar{u_s} (\vec p') \Gamma u_r (\vec p) \tag 1$$ Where: $\bar u_s, u_r$ are Dirac spinors ($1\times 4$ and $4 \times 1$ matrices respectively) $\...
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49 views

Dirac matrices in 2 dimensions and $\gamma_5$

The gamma matrices obeying the Clifford algebra can be represented by Pauli matrices in (1+1) and in (1+2) dimensions. But is it possible to define a $\gamma_5$ in these dimensions? In the sense of an ...
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123 views

Why does the trace show up in such expressions?

I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a purely-mathematical common step I do not understand: the showing-up of the trace. ...
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127 views

Normalization for a free Dirac plane wave

I've recently come about the free plane wave solutions to the Dirac equation, and i'm having a hard time proving that the normalization factor $n$ is $$n=\frac{1}{\sqrt{2m(m+\omega)}}$$ Where $\omega$ ...
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1answer
55 views

$\gamma_5 \gamma^\sigma$ expressed with Levi-Civita tensor

We have that $\gamma_5 = -\frac{i}{4!} \epsilon^{\mu \nu \rho \sigma} \gamma_\mu \gamma_\nu \gamma_\rho \gamma_\sigma$. Using this, what approach would be suggested in showing that $\gamma_5 \gamma^\...
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1answer
71 views

Is this a typo in Peskin's QFT?

In ''An intro to QFT (2018)'' chapter 3, Peskin does the following: Let me introduce some notation first, let $v^s_k=\begin{pmatrix}\;\;\,\sqrt{k\cdot\sigma}\,\xi^{-s}\\-\sqrt{k\cdot\bar{\sigma}}\,\...
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1answer
70 views

Calculating traces for triangle diagrams with massless fermions

I am following Schwarz Quantum Field Theory textbook. In particular, I am looking at triangle diagrams with massless fermions. On pg. 623 - 624 Schwarz attempts to calculate $q_\mu^1 M_{5}^{\alpha\mu\...
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1answer
39 views

Expansion of trace in photon self-energy

I am studying through the photon self-energy $$ i\Pi_{\mu\nu}(q) = \int\frac{d^4 k}{(2\pi)^4}Tr\left[(-ie\gamma_\mu)\frac{i(\require{cancel}\cancel k+m)}{k^2-m^2+i\epsilon}(-ie\gamma_\nu)\frac{i(\...
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Is it right to think of Parity as a change of basis in Dirac's Lagrangian?

I'm trying to understand CPT symmetries in the Dirac Lagrangian but, so far, I've had more questions than answers. My naive view of CPT transformations is the following (please don't doubt to correct ...
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How do we know there are only 16 Dirac bilinears?

We know there are 5 types of bilinears in 4 dimensions, all of them add up to contribute with 16 independent DoF (degrees of freedom). Namely, these bilinears are known as: scalar (1DoF), pseudoscalar(...
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Spinor transformation matrix derivation

The spinor in the Dirac equation should transform via a 4x4 matrix $S$ that depends on the specific Lorentz boost/rotation: $\psi '(x')=S(\Lambda )\psi(x)\tag1$ Where S satisfies: $S^{-1}\gamma ^{\...
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An $SL(2,C)$ representation and Dirac Spinor

In PCT, spin and statistics, and all that book, the following example is given: Let $S(A)$ be a representation of $SL(2,C)$ given as : $$S(A)=\frac{1}{2}\left(a^{0} \mathbf{1}+\mathbf{a} \cdot \...
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Is there a bosonic representation of Clifford algebra in (1,3) spacetime?

By a suitable combination of Dirac's $\gamma_\mu$ matrices one can define creation and destruction operators satisfying fermionic anticommutators. Is there a similar result for bosons in the context ...
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How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless?

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless? where I represents the identity matrix I know that $$\{\gamma^\mu , \gamma^\nu\}=\gamma^\mu\gamma^\nu + \...
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Equivalence of the Dirac representation of the Lorentz algebra and its conjugate in even dimensions - Polchinski

In Polchinski's String Theory, Appendix B.1 we look at the smallest irreps of the Clifford algebra in even-dimensional spacetimes $d=2k+2$. In (B.1.16) he defines two operators from the gamma matrices ...
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1answer
67 views

Hermitian Adjoint of Dirac Equation vs Dirac Lagrangian

I have a question about the self-adjointness of the gradient in spinor space. In the derivation of the Dirac adjoint equation, as in Hermitian adjoint of 4-gradient in Dirac equation , it has been ...
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57 views

Why did Dirac choose a linear equation in momentum for formulating a relativistic wavefunction?

The Dirac equation arose out of the need for a quantum mechanical wave equation that treats position and time on an equal basis, just as relativity expects them to be. Now the Schrodinger equation is ...
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116 views

Understanding the Dirac equation

I am reading a physics book where the Dirac equation is being introduced in the form: $$\left[c \boldsymbol{\alpha} \cdot\left(\boldsymbol{p}+\frac{e \boldsymbol{A}}{c}\right)-e \phi+\beta m c^{2}\...
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2answers
61 views

Dirac equation: covariant form and original form

In my lecture book, the Dirac equation is derived and given as the equation: $$i \hbar \gamma^{\mu} \partial_{\mu} \psi-m c \psi=0 \tag{1}$$ Where: $$\gamma^{0}=\left(\begin{array}{rr} 1 & 0 \\ 0 ...
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1answer
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Gamma matrices - Dirac [closed]

I tried to ask this question: Prove that $\{\gamma_\mu , \gamma_\nu\} = 0$, but I was unable to resolve it. Can someone help me?
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Gamma matrices and the metric

There is a step in a Dirac spinor manipulation problem I'm working on that requires me to make the following remark, $$ 2 \gamma^\sigma \eta^{\rho\mu}\mathbb{1}_4 \gamma^\nu = 2 \eta^{\rho\mu}\mathbb{...
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Is this an alternative Dirac Equation in curved space?

The usual covariant derivative for the Dirac equation in curved space is: $$D_\mu \psi = (\partial_\mu - {i \over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$ However, I think I found another ...
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2answers
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Using the infinitesimal-angle form of the rotation operator to perform infinitesimal rotaions on spin-1/2 states

Im stuck on a homework problem where I must use the rotation operator $$\hat{R}_{e_z,d\phi}=\hat{I}-i\frac{\hat{S}_z}{\hbar}d\phi,$$ to act on $|\psi_{\theta,\phi}\rangle=\cos(\theta/2)|\uparrow_z\...
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What is the old (50's) convention on Dirac gamma matrices?

What were the standard relations for gamma matrices in the mid 50's, when 4-vectors where represented by $(x_1, x_2, x_3, ict)$? In particular the values of $\gamma^\mu\gamma^\nu$ , the definition of $...
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1answer
65 views

Partial derivatives in the derivation of a Dirac Spinor

As per JF132's answer to Conservation of the axial current using Dirac equations of motion, "since the gamma matrices $\gamma^\mu$ are $4\times 4$ matrices, and the conjugate Dirac spinors $\bar{\...
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1answer
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Dirac adjoint of a gamma matrix product

I was wondering if, because for generic bounded operators, anti-distributivity applies, i.e. $$(AB)^{\dagger} = B^{\dagger}A^{\dagger},$$ the same is true of gamma matrices. I was asked to prove $$\...
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Going from the Dirac Lagrangian to the adjoint Dirac equation

I am comfortable doing the following calculation, Derivation of the adjoint of Dirac equation, notably — going from the standard Dirac equation to the adjoint Dirac equation via using Dirac ...
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Generalization of Dirac matrices

Is it possible to have a set of 16-dimensional matrices $\gamma_{\mu}^{a}$ such that $$\{\gamma_{\mu}^{a},\gamma_{\nu}^{b}\} = 2\delta^{ab}\eta_{\mu\nu}$$ where $\eta_{\mu\nu}$ is the Minkowski metric ...
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0answers
65 views

Dirac matrix algebra from bosonic creation/anihilation operators?

Using some generic fermionic creation/anihilation operators $a_i$ and $a_i^{\dagger}$ ($i, j = 1, 2, 3, \dots, N$) such that \begin{align} \{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[...
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1answer
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What transformation gives a Weyl-like representation by flipping $\gamma^0$ and $\gamma^5$?

The usual Weyl representation of the Dirac matrices is defined like this: $$\tag{1}\gamma_W^a = T_W \, \gamma^a \, T_W^{-1},$$ where \begin{align}\tag{2} T_W &= \frac{1}{\sqrt{2}} (1 + \gamma^5 \, ...
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Fierz Identity problem in P&S

I have a few questions about the Fierz identity. First of all in general form it has terms as $(\bar u_1\Gamma^Au_2)(\bar u_3\Gamma^Bu_4)$ which is a product of Dirac field bilinears. The question is ...
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1answer
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Quick question on Deriving Klein–Gordon equation from Dirac equation

On page 172 of Schwatz’s QFT book, he derives the Klein–Gordon equation from Dirac equation as following: $$(i \not\partial +m) (i \not\partial -m)\psi=(-\frac{1}{2} \partial_\mu \partial_\nu {\gamma^...
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Do Lorentz rotations transform the Gamma matrices $\gamma_a$?

Do local Lorentz rotations (see below definition) actually transform the Dirac Gamma matrices? If so, how can they collude with coordinate transformations to make the Gamma matrices $\gamma_a$ ...
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1answer
66 views

What is the complex conjugate of two fermionic fields coupled? $(\bar{\psi} \chi)^{\ast} =$ ____?

Suppose $\psi$ and $\chi$ are fermionic fields, and suppose I want to calculate the hermitian conjugate of the operator $\bar{\psi} \chi$ $$ h.c.\ \mathrm{of}\ \bar{\psi} \chi = (\bar{\psi} \chi)^{\...
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For two majorana field $\psi$ and $\chi$, is it true that $\bar{\psi} \chi = - \bar{\chi} \psi$?

For two majorana field $\psi$ and $\chi$, which satisfy $\psi_{c}=\psi$ and $\chi_{c} = \chi$, where the charge-conjugation operation we define as $$ \Psi_{c} = C \Psi^{\ast} $$ where we work in the ...
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1answer
62 views

Spinor and vector representation matrices commutation relation

To show Lorentz invariance of Dirac equation P&S section 3.2 swap $\Lambda$ and $S(\Lambda)$ as both matrices commute. But why is it true? For example taking $${\cal J}^{01}=\left(\begin{matrix} ...
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3answers
122 views

Proving identity $tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5})=-4i\epsilon^{\mu\nu\rho\sigma}$

Im trying to proof the following identity: $tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5})=-4i\epsilon^{\mu\nu\rho\sigma}$ when $\gamma^{\mu},\gamma^{\nu},\gamma^{\rho},\gamma^{\...
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0answers
48 views

Index manipulation of Dirac matrices

In several places I see that Dirac matrix indexes are treated as usual 4-vector indexes that can be changed with the metric tensor, for example $$\gamma_\mu=g_{\mu\nu} \gamma^\nu. $$ Why is it true?
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152 views

Lorentz transformation of Dirac spinor

I'm wondering again what I'm missing in my understanding. In Peskin and Schroeder, as well as in other sources, the spinor representation of Lorentz transformation is given by $$\Lambda_\frac{1}{2}=...
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2answers
140 views

With weak isospin doublets $ \bar{\Psi}^{L}=\Psi^{\dagger L} \gamma^0$ no more true

I'm considering the doublet: $$\Psi_1^L \equiv \begin{pmatrix} \psi_{v_l}^L\\ \psi_{l}^L \end{pmatrix}$$ I know that under $SU(2)$ transformation: $$\Psi^{'L} = e^{\frac{i}{2}\vec{a} \...
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1answer
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Lie algebra/group/basis of the four gamma matrices along with the identity?

Do the four gamma matrices along with the identity element constitute a lie algebra? With real coefficients we have $$ \mathbf{v}_{\mathbb{R}}=aI+t\gamma_0+x\gamma_1+y\gamma_2+z\gamma_3 \tag{real ...
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1answer
47 views

Dirac equation plane with solution

Using the Dirac equation with or $p=$ zero and the $\gamma^0$ matrix defined as $$\gamma^0=\begin{pmatrix}0 & \sigma_0 \\ \sigma_0 & 0\end{pmatrix} = \begin{pmatrix}0 & \bf{I} \\ \bf{I} &...
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1answer
77 views

Fierz identities and anticommutation relations

Let us consider the following term $$\bar\psi(p_1)M\psi(p_2) \bar\psi(p_3)N\psi(p_4)$$ According to Ortin's book (equation (D.65)), from the Fierz identity we would have something like $$\bar\psi(...
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2answers
127 views

Doubt about identity on the Wikipedia page Feynman slash notation

I have the following doubt about the identity in this page: $$ {a\!\!\!/}{a\!\!\!/} \equiv a^\mu a_\mu \cdot I_4 = a^2 \cdot I_4 $$ however: $${a\!\!\!/}{a\!\!\!/}=a^{\mu}\gamma_{\mu}a_{\mu}\gamma^{...

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