Skip to main content

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.

Filter by
Sorted by
Tagged with
2 votes
1 answer
48 views

$CP$-transformation for fermionic bilinears

I am trying to derive the transformation of the fermionic bilinear $\bar{\psi}\psi$ under $CP$ transformation. I know that $P$ acts as: $$\psi(t, \vec{x}) \xrightarrow{P} \gamma^0 \psi(t, -\vec{x})$$ ...
Damiano Scevola's user avatar
0 votes
1 answer
73 views

Deriving the properties of the Dirac matrices

I am working on the properties of the Dirac matrices, but I cannot figure out the derivations. For example, on proving $\gamma^\mu {\not}{a} \gamma_\mu = -2{\not}{a}$, we first prove that $\gamma^\mu \...
user174967's user avatar
0 votes
1 answer
86 views

Do gamma matrices commute with 4-vectors?

One of my exercises was to prove the identity $$\gamma^\mu\displaystyle{\not}a\gamma_\mu=-2\displaystyle{\not}a.$$ Which is trivial if $\gamma^\mu a_\nu=a_\nu \gamma^\mu$, as follows $$\gamma^\mu\...
agaminon's user avatar
  • 1,645
0 votes
0 answers
46 views

Fierz Identity in 2+1d

Wikipedia states an example of Fierz Identity, under the assumption of commuting spinors, the $\mathrm{V} \times \mathrm{V}$ product that can be expanded as, $$ \left(\bar{\chi} \gamma^\mu \psi\right)\...
Everlin Martins's user avatar
2 votes
0 answers
98 views

How to motivate spinors from the Dirac equation? [closed]

I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator). Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
Integral fan's user avatar
0 votes
1 answer
48 views

Fermi tetrad field: Fermi-Walker tetrad formalism?

These days I'm reading Dirac Eq in GR, and I'm confused about "Fermi tetrad field". Is it Fermi-Walker tetrad formalism?
Lou TY's user avatar
  • 1
1 vote
0 answers
79 views

Is there a $\gamma^{5}$ in $d$-dimensional Clifford Algebra?

I was trying to compute the EW vacuum polarization $$i\Pi_{LL}^{\mu\nu}=(-1)\mu^{\frac{\epsilon}{2}}e^{2}∫\frac{d^{d}k}{(2\pi)^{d}}\frac{Tr\bigl(i\gamma^{\mu}(iP_{L})i(\not{k}+m_{i})(i\gamma^{\nu})P_{...
Filippo's user avatar
  • 477
2 votes
2 answers
312 views

The dimension of the Clifford algebra for the Dirac equation

The Dirac algebra contains sixteen linearly independent elements. In general, a Clifford algebra $\mathcal{C}\!\ell(V,Q)$ generated from a vector space $V$ equipped with a quadratic form $Q$ has ...
Nada Band's user avatar
1 vote
1 answer
55 views

Obtaining the 16 elements of the Clifford algebra from the $\gamma^\mu$ generators

In my study of the Dirac equation, I have fully understood the "linearization" of the relativistic energy to obtain a matrix-valued equation that reduces to the Klein-Gordon equation if the ...
Nada Band's user avatar
0 votes
1 answer
35 views

Question about meaning of "bar"-ing in the context of Dirac fields

Following chapter 38 of Srednicki, "bar"-ing means (based on eq. 38.15) $$\bar{A} = \beta A^\dagger\beta$$ One can show for instance that $$\bar{\gamma^\mu} = \gamma^\mu$$ My question is, ...
JohnA.'s user avatar
  • 1,713
0 votes
2 answers
69 views

Understanding derivation of Klein-Gordon equation from Dirac equation

Above is Tong's notes which shows how the Klein-Gordon equation is derived from Dirac equation. But I don't get why: $\gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} = \frac{1}{2} \{\gamma^{\mu},\...
Stallmp's user avatar
  • 665
0 votes
0 answers
74 views

Dirac field charge conjugation

I struggled a bit to understand the proof of the relation $C\overline\psi\psi C=\overline\psi\psi$ in Peskin's and Schroeder's book An Introduction to Quantum Field Theory (page 70, formula 3.147): $$...
Andrea's user avatar
  • 521
0 votes
1 answer
39 views

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 ...
Gorga's user avatar
  • 161
1 vote
1 answer
110 views

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\\ &=\...
MVPlanet's user avatar
0 votes
0 answers
45 views

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 $D$-...
Ozzy's user avatar
  • 172
1 vote
0 answers
81 views

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 ...
dallla's user avatar
  • 59
3 votes
0 answers
68 views

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 \\ ...
Jacob's user avatar
  • 55
1 vote
2 answers
99 views

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 $\langle\psi\vert p.\alpha\vert\psi\rangle$ ?
Wajahat's user avatar
  • 11
0 votes
1 answer
184 views

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 ...
Salmon's user avatar
  • 941
2 votes
1 answer
82 views

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 \ \...
baba26's user avatar
  • 513
-3 votes
1 answer
116 views

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 + ...
ric.san's user avatar
  • 1,644
0 votes
0 answers
36 views

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 $...
Hrach's user avatar
  • 280
2 votes
1 answer
202 views

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 ...
ric.san's user avatar
  • 1,644
0 votes
1 answer
106 views

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 ...
Newbie's user avatar
  • 3
0 votes
1 answer
191 views

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}' ...
Przemysław Borys's user avatar
0 votes
1 answer
210 views

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^\...
arpg's user avatar
  • 169
0 votes
1 answer
56 views

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\...
Carlos Eduardo Staudt's user avatar
0 votes
1 answer
116 views

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 \\ \...
Chris G's user avatar
  • 51
0 votes
1 answer
49 views

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 ...
Devey Rathore's user avatar
1 vote
2 answers
87 views

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}=\...
Bababeluma's user avatar
0 votes
2 answers
137 views

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 ...
Alexandra's user avatar
  • 111
1 vote
0 answers
112 views

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},\...
Bababeluma's user avatar
0 votes
0 answers
67 views

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 ...
baba26's user avatar
  • 513
2 votes
1 answer
153 views

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} ...
ShKol's user avatar
  • 322
0 votes
1 answer
43 views

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{...
Filippo's user avatar
  • 477
0 votes
1 answer
83 views

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 ...
Lluis Gerardo's user avatar
0 votes
1 answer
136 views

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 = \...
Álvaro Rodrigo's user avatar
0 votes
1 answer
151 views

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. ...
realanswers's user avatar
1 vote
4 answers
277 views

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 $\...
MKO's user avatar
  • 2,200
-1 votes
2 answers
145 views

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.
Amin Khan's user avatar
0 votes
0 answers
122 views

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 ...
Anindita Sarkar's user avatar
3 votes
1 answer
133 views

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. ...
Bumblebee's user avatar
  • 139
0 votes
1 answer
171 views

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 ...
jmaguire's user avatar
  • 313
0 votes
0 answers
41 views

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
1 vote
1 answer
68 views

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
2 votes
0 answers
91 views

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
1 vote
0 answers
57 views

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
1 vote
1 answer
68 views

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 ...
JavaGamesJAR's user avatar
1 vote
0 answers
42 views

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
  • 55
2 votes
1 answer
75 views

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

1
2 3 4 5
12