Questions tagged [dirac-matrices]

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States of spin of a quantum mechanical particle

Assuming a spin is prepared in the positive $x$-direction ($|r\rangle$) and a measuring apparatus is oriented on the $z$ axis, does this equation apply? $$|r\rangle=\frac{1}{\sqrt{2}}|u\rangle+\frac{...
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135 views

Relation between Dirac spinors, quaternions, and bicomplex numbers

Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ...
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1answer
43 views

Question about $\gamma^{0}$ matrix

I see different definitions in different places so here is my question. why is $\gamma^{0}$ sometimes defined as a 2 by 2 matrix and sometimes as a 4 by 4 matrix? shouldn't a definition be something ...
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1answer
111 views

Signature of trace of Dirac Matrices

I came across this question in my problem set: Let $\gamma^\mu$, $\mu=0,1,2,3$ be the Dirac matrices, satisfying: \begin{eqnarray} \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}I, \:\:\:...
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92 views

Dirac Fields and Derivatives (Am I gaining extra minus signs?)

I've given myself a severe headache jumping between East/West Coast sign conventions; I have picked up an extra minus sign and could do with a hand. I am currently using $\eta=\textrm{Diag}[-,+,+,+]$ ...
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151 views

Transpose Noether current from $U(1)$ symmetry of the free Dirac field

When I read through the notes of a particle physics script there is the following identity that I don't understand $$\left(\psi^T {\gamma^\mu}^T \bar{\psi}^T \right)^T = -\bar{\psi}\gamma^\mu \psi.$$ ...
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136 views

Lorentz invariance of the axial current

I want to determine whether $$\bar{\psi}\gamma^\mu \gamma^5 \psi \bar{\psi}\gamma_\mu \gamma^5\psi = j_5^\mu j_{5\mu},$$ where $\bar{\psi} = \psi^\dagger \gamma^0 $ and $\gamma^5 = -i\gamma^0\gamma^1\...
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140 views

Product of Lorentz transformation with gamma matrices

In my QFT course, we wrote the following things : $$ S(\Lambda)^{-1} \gamma^\mu S(\Lambda)=\Lambda^{\mu ~ .}_{. ~ \nu} ~ \gamma^\nu $$ So when we ""apply"" a Lorentz boost to gamma matrices, they ...
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168 views

Dimension of Representation of Majorana Fermions with Euclidean Metric?

It is possible to represent the Dirac matrices in the Majorana basis using $N= 2^{⌊d/2⌋}$-dimensional matrices, as shown here. This source uses a Minkowski metric. It would then be possible to move to ...
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268 views

What is the Hermitian conjugate of the 4-vector momentum in Dirac equation?

I am quite confused about the Hermitian conjugate of the 4-vector momentum $p=(p^0, p^1, p^2, p3)$. The confusion mainly arises when deriving the Dirac adjoint and the charge current probability. (1)....
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210 views

Basic calculus of the adjoint spinor being transformed under parity

In Modern Particle Physics (p.287) Thompson says that under the parity transformation of the adjoint spinor we have $$\bar u=u^\dagger\gamma^0\rightarrow^p (\hat Pu)^\dagger\gamma^0= u^\dagger\gamma^{...
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491 views

Behaviour of Dirac Bilinears

Dirac bilinears transform in the Lorentz indices as, $\bar{\psi}\psi$ scalar $\bar{\psi}\gamma^\mu\psi$ vector $\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor $\bar{\psi}\gamma^{\mu}\...
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300 views

Dirac Spinors as Eigenvalues of Helicity Matrix

I am trying (unsuccessfully) to verify this relation regarding the helicity of Dirac spinors: $$ { \sigma }_{ \vec { p } }u_{ r }\left( \vec { p } \right) =\frac { \vec { \Sigma } \cdot \vec { p } ...
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235 views

What is $\langle G_{\mu\nu}\rangle\langle G_{\mu\nu}\rangle$ for the Dirac gamma matrices?

Given the following 16 matrix multiplications of the Dirac gamma matrices \begin{align} G_{\mu\nu} = \dfrac{1}{2} \begin{pmatrix} I && \gamma_{0} && i\gamma_{123} && i\gamma_{...
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3answers
422 views

Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?

Is spinor $\psi$ actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar? Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of ...
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1k views

What is unitarily similar matrices?

In one of tasks I met the concept of unitarily similar matrices: in particular, I need to prove that sets $\gamma_{\mu}, -\gamma_{\mu}$ (Dirac gamma matrices) are unitarily similar. I don't know what ...
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114 views

Do physical results for spinors depend on the Clifford algebra representation?

As I understand it, the Dirac equation and its solutions depend on the representation of the $\gamma$ matrices one uses. So if I were to use the Dirac representation I would get different mathematical ...
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2answers
329 views

Need help with solution of the Dirac equation

$$\left(\vec\sigma \cdot \vec{p} \right)^2=\left(\vec\sigma \cdot \vec{p}\right) \left(\vec\sigma \cdot \vec{p} \right)=\vec{p} \cdot \vec{p}+\mathrm{i}\left(\vec\sigma \cdot \left[ \vec{p} \times \...
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118 views

Transforming a QFT identity (quantisation of the Dirac field)

When (falsely) quantizing the Dirac-field, Peskin/Schroeder (Introduction to Quantum Field Theory) get with $$\psi(\vec{x})=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_{\vec{p}}}}e^{i\vec{p}\cdot\...
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2k views

Gamma matrices and trace operator

I'm trying to show that the trace of the product of the following three Gamma (Dirac) matrices is zero, i.e. $$\text{tr}(\gamma_{\mu} \gamma_{\nu} \gamma_{5})=0 \text{.}$$ I attempted to use the fact ...
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1answer
94 views

Why is Dirac equation a matrix equation?

According to Wikipedia's Dirac equation article, the Dirac equation can be written in form $$ i\hbar\gamma^{\mu}\partial_{\mu}\psi-mc\psi=0, $$ where $\gamma^{\mu}$ are gamma matrices which are $4 \...
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143 views

QFT, show that $(\vec{p}\cdot \vec{\sigma})^2 = |p|^2$

I am following Peskin. During the derivation of the Dirac field (boosting the solution from the rest frame), we use that $(\vec{p}\cdot \vec{\sigma})^2 = |p|^2$. Where $\vec{p}$ is momentum vector, ...
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58 views

Reality of Dirac kinetic term

The Dirac kinetic term is $$\mathscr{L}_{\text{ferm}}=-i\bar{\psi}\gamma^\mu D_\mu\psi$$ where $\bar{\psi}\equiv \psi^\dagger \gamma^0$. Here I've assumed the mostly plus metric, so $\left(\gamma^0\...
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1answer
75 views

How do I construct a Palantini action within Clifford algebra?

I want to define the following two object with spinor-type indices: $${\hat{e}}^{\alpha\beta}(x)\equiv e_n^\mu(x) \gamma_n^{\alpha\beta}\partial_\mu$$ $$\omega^{\alpha\beta}(x) \equiv e^\mu_p(x)W_\...
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1answer
60 views

Proving identity $\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu} $$ Yet when I try to prove this,...
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2answers
325 views

Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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1answer
87 views

weak interaction are not parity invariant

I'm having some hard time trying to see why the left-handed lagrangian for fermions $\psi$, $$\mathcal{L} := G\overline{\psi}_{1L}\gamma^\mu\psi_{2L}\overline{\psi}_{3L}\gamma_\mu\psi_{4L}$$ is not ...
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457 views

About the fifth gamma matrix [closed]

How can one prove that $$\gamma^5=\frac{i}{4!}\varepsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}$$ from the following: $$\gamma^5:=i\gamma^0\gamma^1\gamma^2\gamma^...
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715 views

Solution of Dirac equation-Positive and Negative energy

For particles defined with positive energy, we use $$\phi= \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ or $$ \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix} $$...
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2answers
125 views

How to calculate the trace below?

I am currently reading Peskin's QFT book on my own. Though it introduces the Trace Technology in Section 5.1, the trace calculations in the following sections are still far from clear to me. Here is ...
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1answer
341 views

Dimensionality of Gamma Matrices

If I express the Dirac equation in the form of $$i\hbar \frac{\partial}{\partial t} \psi_a(x) = \left(-i\hbar c(\alpha^j)_{ab}\partial _j + mc^2(\beta)_{ab}\right)\psi_b(x),$$ with the constraints $...
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196 views

Components of Dirac equation solve the Klein Gordan equation derivation

On page 90 of this set of lecture notes on quantum field theory, http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf a simple derivation is given to show that each component Dirac equation solves the ...
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1answer
458 views

Invertibility of Dirac matrices

Suppose we have Dirac equation in the following form : $i\partial_t\psi = (-i\vec{\alpha}\cdot \nabla +m\beta)\psi$ and assume that the Klein-Gordon equation is satisfied, i.e. $\partial_t^2 \psi =(...
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47 views

Regarding properties of matrices involved in Dirac equation

In this document, after equation 62 on page 9, the author says that we can rewrite $\alpha^i \alpha^j \partial_i \partial_j$ as $\frac{1}{2} (\alpha^i \alpha^j + \alpha^j \alpha^i)\partial_i\...
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128 views

Question about Majorana spinor's property

I am reading the BBS, Exercise 5.1 This exercise is nothing but showing that two Majorana spinors $\Theta_1$ and $\Theta_2$ \begin{align} \bar{\Theta}_1 \Gamma_{\mu} \Theta_2 = -\bar{\Theta}_2 \...
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515 views

Dimension of gamma matrices in higher dimensional Dirac equations

Reading about Dirac's equation in higher dimensional space-times I have read that the gamma matrices are $2^{[D/2]}\times{}2^{[D/2]}$. So, if we have $D=11$, for example, how is this formula supposed ...
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78 views

Microcausality for Dirac's current

I`m supposed to show as an exercises that for the Dirac field's associated current: $$j^\mu=\bar{\Psi}\gamma^\mu\Psi$$ The microcausality relation holds: $$ [j^\mu(x),j^\nu(y)]=0 \text{ for } (x-y)^...
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97 views

A few questions about spinors and gamma matrices

I am following BUSSTEPP Lectures on Supersymmetry and trying to show that the Wess-Zumino action is invariant under SUSY transformations. I encountered the following questions about spinors and gamma ...
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1answer
129 views

Dirac matrices in 1+1 dimensions

Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
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29 views

Evaluating a trace with two factors of $\gamma^5$

In the process of calculating a spin-averaged square amplitude in QFT, I came across the following expression: $$ \text{Tr}\left[\gamma^\mu\gamma^5\gamma^\alpha\gamma^\nu\gamma^5\gamma^\beta\right] $$ ...
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1answer
137 views

C and T Symmetry of Free Dirac Lagrangian

I want to show the $C$ and $T$ symmetry of the free Dirac Lagrangian $$\mathcal{L}=\overline{\psi}\left(i\gamma^\mu\partial_\mu-m\right)\psi.$$ Following the notation of Peskin, Schroeder, we have ...
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1answer
133 views

What is the transpose of Lorentz transformation under spinor representation?

Let $S$ be the Lorentz transfortmation under spinor representation, and from any quantum field theory textbooks, we know that $$ S^\dagger=\gamma^0S^{-1}\gamma^0 \\ S^{-1}=\gamma^0S^\dagger\gamma^0 $$ ...
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125 views

Gamma matrices in curved spacetime

How to raise and lower indices of gamma matrix in curved spacetime? Do we raise and lower the index of gamma matrix with $ g_{\mu \nu} $?
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1answer
210 views

Trace of $2n$ gamma matrices

To proof $$\mathrm{Tr}(\gamma_{\mu_1}\cdots\gamma_{\mu_{2n}}) =\mathrm{Tr}(\gamma_{\mu_{2n}}\cdots\gamma_{\mu_1}),$$ I use $\gamma_\mu^\dagger=\gamma^0\gamma_\mu\gamma^0$ and get $$\cdots=\mathrm{Tr}...
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2answers
522 views

Property of Charge Conjugation Operator

In class, we have defined the Charge Conjugation Operator ($C$) such that: \begin{equation} C \left(\gamma^\mu\right)^T C^{-1} = - \gamma ^\mu , \end{equation} \begin{equation} \psi^C \equiv C\,\...
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1answer
687 views

Trace of Gamma Matrices [closed]

If I have: $Tr(\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\gamma^{\beta}\gamma^{\rho}\gamma^{\gamma}\gamma^{\sigma}\gamma^{\delta})$ and I want to get it re-ordered like $Tr(\gamma^{\alpha}\gamma^{\...
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2answers
334 views

Derivation of generators of Lorentz group for spinor representation

I want to prove $$S^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu].$$ I started from $$[\gamma^\mu,S^{\alpha\beta}]=(J^{\alpha\beta})^\mu_\nu \gamma^\nu$$ Putting the value of $(J^{\alpha\beta})^\mu_\...
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1answer
78 views

Spin states in a finite potential well

i have a question concerning an electron in an attractive potential well. Let's suppose the potential function is defined as $$V = \left\{ \begin{array}{cl}0, & \mbox{for } z < 0\\ ...
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1answer
73 views

What is the implication of the existence of a non-singular matrix $S: \gamma^\lambda S=S \gamma^m u$? [closed]

$$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}I_4$$ Pauli's fundamental theorem states that given two sets of matrices $\gamma^\mu$ and $\gamma^\nu$ which obey the commutation rules (...
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474 views

Magnetic moment of the electron and gamma matrices

What is the relation between the magnetic moment of the electron and $\sigma^{\mu\nu}= \frac{i}{2}[\gamma^\mu,\gamma^\nu]$ ? that I'd like to answer this question: Consider the coupling: $\psi' \...