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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Covariant derivative property

I am trying to demonstrate this propertie $$ \not{D}^2= \mathcal{D}^\mu \mathcal{D}_\mu-\frac{i}{4}\left[\gamma^\mu, \gamma^\nu\right] F_{\mu \nu} $$ where $\not{}~$ is the Feynmann slash, and $D_\mu ...
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Composite system formalism [closed]

I have a doubt about the writing of a linear operator belonging to the composite Hilbert space AB, which of these two is the correct one? $\sum_{i,j,a,b} K_{ijab} \vert i \rangle \langle j \vert \...
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Feynman slash identity

How would one simplify the expression $\gamma^0 {\not}p \;\gamma^0$ ? My guess would be that its $$\begin{align} \gamma^0 {\not}p \;\gamma^0&=\gamma^0 \gamma^{\mu} p_{\mu} \;\gamma^0\\ &=\...
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Can $\gamma^5$ matrices be ignored in $q\bar{q}\to ZZ$ processes?

In the $q\bar{q}\to ZZ$ process, the following Feynman diagram in LO appears: This means for each vertex, the Feynman amplitude contains a term proportional to $(g_V-g_A\gamma^5)$, which makes $D$-...
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Product of Dirac $\gamma^0$ and $\gamma^\mu$ generate a representation of some algebra?

I need your help with an issue about Dirac gamma matrices. Precisely, I need to know if $\gamma^0\gamma^\mu$ generates an irreducible representation of some algebra. This problem has come out in the ...
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Spin operators in relativistic calculations (Dirac theory)

In non-relativistic quantum mechanics, the spin operators associated with a particle of spin 1/2 are proportional to the $2\times 2$ Pauli matrices $$ \widehat{\sigma}_x = \begin{pmatrix} 0 & 1 \\ ...
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Does $\alpha$ and $p$ commute where $\alpha$ is a Dirac matrix and $p$ is a momentum operator?

Can I write $\langle\psi\vert\alpha.p\vert\psi\rangle$ as $\langle\psi\vert p.\alpha\vert\psi\rangle$ ?
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Time reversal operator and Dirac gamma matrices

How would you prove that $T^{-1}\gamma_\mu T=\gamma_\mu$? Being $T$ the time reversal operator defined as $T=\gamma_1\gamma_3 K$ with $K$ the complex conjugate operator and $\gamma$ the Dirac gamma ...
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Positive definiteness of Dirac hamiltonian?

In David Tong's notes, he says that the Hamiltonian (4.92) is positive definite ( see page 112 of chapter five ). Here is equation (4.92) from chapter four. $$ E = \int d^3 x \ T^{00} = \int d^3 x \ \...
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From Klein-Gordon equation to Dirac equation: a wrong "derivation" [closed]

So let us start with the Klein-Gordon equation $$\tag{KG} (-p^\mu p_\mu + m^2)\phi = 0 $$ The idea is to "factorize" the operator $-p^\mu p_\mu + m^2$. \begin{equation}\tag{1} -p^\mu p_\mu + ...
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How exactly is this tensor symmetric?

I am trying to solve P&S Problem 3.3: [...] Let $k^\mu_0, k^\mu_1$ be fixed 4-vectors satisfying $k^2_0 = 0, k^2_1 > = -1, k_0 \cdot k_1 = 0$. Define basic spinors in the following way: Let $...
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On the derivation of Dirac matrices

Is it possible to retrieve the matrix elements of the $\gamma$s by simply knowing their anti-commutation relation: $$ \{\gamma^\mu, \gamma^\nu\}=2\,g^{\mu\nu}\,\mathbb{I}_{4} $$ I'm just trying to ...
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Dirac Equation and the Klein-Gordon Equation

I am trying to solve an exercise in Halzen and Martin's Quarks and Leptons book and got stuck on doing some math. The Dirac equation reads $$i \gamma^{\mu} \partial_{\mu} \psi - m\psi = 0.$$ Now, I ...
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Why $\gamma^5$ pseudoscalar transforms with a determinant?

In the Dirac matrices we define $\gamma^5$ matrix, which transforms like $\bar{\psi}' \gamma^5 \psi'=\bar{\psi} S^{-1} \gamma^5 S \psi=det(a) \bar \psi \gamma^5 \psi$ I try the following: $\bar{\psi}' ...
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4-momentum commutative with gamma matrix?

I was checking the adjoint Dirac equation derivation. Starting from: \begin{equation} (\gamma^\mu p_\mu - m) u = 0. \end{equation} We take the hermitian conjugate on both sides \begin{equation} 0 = u^\...
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I'm trying to prove the chiral invariance of $\bar\psi\gamma_{\mu}\partial^{\mu}\psi$

Considering the chiral transformation $\psi' = \gamma_{5}\psi$ and since $\psi$ is the Dirac spinor, the adjoint spinor can be written as $\bar\psi = \psi^{\dagger}\gamma^{0}$, then: $ (\bar\psi\...
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Proof of normalising the Dirac spinors

I was reading through my particle physics textbook and saw a property of the Dirac spinors that I did not understand. The spinor is defined by $u^s(p)=\sqrt{\frac{E+m}{2m}} \begin{bmatrix}\phi^s \\ \...
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Pairwise product of Pauli Matrices using Levi-Civita

Just a quick question. The professor used $$ \sigma_i \sigma_j = i \sum_k \epsilon_{ijk} \sigma_k$$ to define products of Pauli matrices. This works fine to get $$\sigma_x \sigma_y = i \sigma_z$$ and ...
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How to prove $-i\gamma_2u_{s}^*(p)=v_{s}(p)$ for Dirac spinors?

It should be true and it's obvious for $p^{\mu}=(m,0,0,0)$, but I'm having trouble with the gamma matrices Algebra and prove it for general momentum. I'm using Weyl representation: $$u_{\uparrow}=\...
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Is the solution of the Dirac equation supposed to be a $4\times 4$ matrix?

Rearranging the Dirac Equation, we find that $$\sum_\mu\gamma^\mu \partial_\mu \psi = -imc\frac{2\pi}{h} \psi.$$ It appears that the wavefunction times the constant is equal to the matrix result of ...
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Is there an intuitive way to understand the Lagrangian for magnetic and electric dipole moment?

From the textbook I learned that the electric dipole moment (EDM) and magnetic dipole moment (MDM) has the following Lagrangian: $$\mathcal{L}_{EDM}=F_{\nu\mu}\bar{\psi}\gamma^{5}\left[ \gamma^{\nu},\...
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Explain this step (related to gamma matrices and parity operator)

I am having hard time reproducing a step from the textbook "Lecture Notes on Quantum Field Theory", by Ashok Das. On page 429 ( above equation 11.72), the author is talking about the parity ...
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Imaginary part of the Dirac Lagrangian density as a total derivative

Show that the imaginary part of the Dirac Lagrangian [density] is a total derivative. My attempt: The Dirac Lagrangian density is given by, $$ \mathcal{L} \ = \ -\bar{\psi}(\gamma^{\mu}\partial_{\mu} ...
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How to show that Helicity projectors are Lorentz covariant?

Consider the Helicity projectors $$\Pi_{\pm}:=\frac{\mathbb{1}\pm\gamma^{5}\gamma_{\nu}n^{\nu}}{2}$$ where I define the versor $n^{\mu}\equiv(\frac{|\stackrel{\rightarrow}{p}|}{m},\frac{ω_{p}}{m}\frac{...
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Commutator of gamma matrices with scalar product of 4-vectors

I know about the commutator relationship given in this question: $$\left[\gamma^{\mu},\gamma^{\nu}\right]=2\gamma^{\mu}\gamma^{\nu}-2\eta^{\mu\nu}$$ then, my question is if there exist a similar ...
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How to solve the Dirac Equation and find its eigenfunctions?

Imagine you want to solve the Dirac equation: $$ (i\gamma^\mu \partial_\mu - m )\psi=0 \\ $$ Where $\gamma^\mu$ are the $4 \times 4$ gamma matrices, and $\psi$ is a 4 component spinor. $$ \psi = \...
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Do $p_\mu$ and $\gamma^\mu$ commute?

So I am trying to derive the relation $\bar{u}_{(s)} (\displaystyle{\not}{p} -m)=0$ from the conjugate dirac equation $(i \partial_\mu\bar{\psi}\gamma^\mu+m\bar\psi) = 0$ but I am running into issue. ...
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Uniqueness of Dirac matrices

I am trying to understand the motivation behind the Dirac equation for a free particle $$ i\gamma^\mu\partial_\mu \psi-m\psi=0 \tag{1} $$ I am wondering how to get the concrete form of the matrices $\...
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Square root of the wave operator

How are the Dirac matrices the square root of the wave operator? I keep seeing it mentioned as such but never explained.
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Proof that different representations of gamma matrices are connected by a unitary transformation

The different representations of gamma matrices are related by a similarity transformation \begin{equation} \gamma^{\mu'}=S\gamma^{\mu}S^{-1} \end{equation} for some non-singular $S$. I have to prove ...
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Do matrices with this property appear in physics?

First I should mention that my background is in Mathematics, but I am looking for a motivating example in physics. I apologize in advance if my question does not meet the standards of this site. ...
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How to compute the amplitude of a Feynman diagram with a loop containing a fermion and a scalar?

I know that when we have a Feynman diagram with a fermion loop, we must take the trace and, by doing so, we get rid of the $\gamma$ matrices. What if we have a diagram like the one in the picture ...
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Simplifying some momentum and gamma matrices algebra in invariant amplitude calculation

I'm calculating some `$B_0 \rightarrow K^* \nu_R \nu_L$ decay and right now I'm stuck at invariant amplitude $\mathcal{M}$, which leads from the tensor part of Lagrangian. I'm having trouble ...
Miha Medvesek's user avatar
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Proving multiplication with Dirac adjoint spinor is Lorentz scalar

I'm studying QFT with the book "Quantum Field Theory" by Dr. David Tong, and I have difficulties with understanding some used transformations. On the page 88 the author calculates Hermitian ...
Volodymyr Savin's user avatar
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Spinor Lorentz generators in curved spacetime

The Dirac matrices in curved spacetime are written as $\gamma_{\mu}=e^{a}_{\mu}\gamma_{a}$ where $e^{a}_{\mu}$ are the vielbein fields and $\gamma_{a}$ are the constant Dirac matrices. Given that the ...
physics_2015's user avatar
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Determining the Hermitian operator in a Foldy-Wouthuysen transformation

I am following the mathematical steps of a paper and at some point the authors consider a transformed Hamiltonian of the form $$ \mathcal{H}' = e^{iS} \mathcal{H} e^{-iS}. $$ They then follow a ...
Ron Stean's user avatar
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Why does Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ is frequently written with a factor of $i$?

The tensor Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ has the matrix tensor $\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^\mu,\gamma^\nu\right]$. I can understand that the factor of $\frac{1}{2}$ is a ...
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Projection onto Form factors in the on-shell case

I need to calculate a vertex $$\Gamma^\mu=F_1 \gamma^\mu + \frac{i}{2m}F_2 \sigma^{\mu\nu} k_\nu$$ The vertex contains propagators, which I have taken on-shell with Cutkosky rules. The problem when I ...
Nik's user avatar
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Properties of tensor $\gamma^\mu\gamma^\nu\gamma^\lambda$

As the title suggests, I am trying to derive some properties of the tensor $\Gamma^{\mu\nu\lambda}=\gamma^\mu\gamma^\nu\gamma^\lambda$. My motivation is that $\bar{\psi}\Gamma^{\mu\nu\lambda}\psi$ ...
JavaGamesJAR's user avatar
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Is there a more elegant way to compute the determinant of this Dirac-space matrix?

Consider the Dirac-space matrix \begin{equation} \begin{aligned} D =A+\gamma^0\gamma^jB_j+\gamma^0C &= \begin{pmatrix} (A+C)I_2 & \sigma^jB_j \\ \sigma^jB_j & (A-C)I_2 \end{pmatrix}, \end{...
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Spinors and Gamma matrices [duplicate]

Do gamma matrices and Dirac spinors commute? I mean, if I have the expression $$\gamma_{\mu} \psi,$$ can i say that it´s equal to $$\psi \gamma_{\mu}~?$$
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$\gamma_5$ in non-integer dimension $D$

I need to calculate a trace containing a single $\gamma_5$ in $D$ dimensions. The trace is given by $$\displaylines{ \text{Tr}\bigg[ \gamma^{\alpha\tau\eta} (p_1-p_2)_\tau (p_1)_\eta \left(\gamma\cdot ...
Nik's user avatar
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Why can we choose a gamma matrix to be diagonal?

I am reading the textbook "Lectures on Quantum Field Theory", second edition by Ashok Das. On page 21, in equation (1.83), he writes the properties of gamma matrices: $$ ( \gamma^0)^2 = {...
baba26's user avatar
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2-dimensional spin 1/2 representation of Lorentz Lie Algebra

In Peskin & Schroeder section $3.2$ they begin by telling us that they want to construct spin $1/2$ representations of the Lorentz Lie algebra. One way to do that, they say, is to first find a ...
Leonid's user avatar
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Vague argument on alpha, beta matrices in Bjorken Drell sec 1.3 on Dirac Equation

My question concerns a very vague argument given in section 1.3 (specifically, page 8, right before equation 1.17) in the Bjorken and Drell "Relativistic Quantum Mechanics" book. The ...
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What isomorphisms does $SO(3,1)$ have?

The double cover of $SO(4)$ is isomorphic to $SU(2)\times SU(2)$, which might be related to $SO(3,1)$. Also, I have seen that $so(3,1)=Cl^2(3,1)$. However, the notation confused me a bit, does the ...
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Choosing a basis for the Dirac equation?

During the course of going to the covariant formulation of the Dirac equation, we have the following: $$ i\hbar \partial_{t} \psi = [ c(\mathbf{\alpha}\cdot\mathbf{\hat{p}}) + \beta m c^{2}]\psi $$ At ...
ShKol's user avatar
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1 answer
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Are the eigenvalues of the Dirac operator real?

In 3+1 dimensions, define the Dirac operator as $$\tag{1} \not D = \gamma^\mu (\partial_\mu -i A_\mu )$$ where $A_\mu$ is a $U(1)$ gauge field. Define the following inner product between spinor fields:...
nodumbquestions's user avatar
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A question about some detail in Peskin and Schroeder's book

I have a question in Peskin and Schroeder's book (Quantum field theory) page 46. Why $$\big[ \sqrt{E+p^3}(\frac{1-\sigma^3}{2})+\sqrt{E-p^3}(\frac{1+\sigma^3}{2})\Big]=\sqrt{p.\sigma}~?$$ and why $$(p\...
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The transformation of Gamma matrices under charge cojugation

Consider the Dirac field theory. In this theory the transformation of charge conjugation on a Dirac field is given by, $U(C) \psi(x) U(C)^{-1} = \eta_c C \psi^*(x)$, please note the $C$ give on RHS is ...
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