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Questions tagged [dirac-matrices]

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Proving an identity relating the gamma matrices

I'm looking to prove the following identity: $$k_a \gamma^a \gamma^\nu K_b \gamma^b p_c \gamma^c \gamma_\nu P_d \gamma^d = 4(p\cdot K)(P\cdot k)$$ I tried this many times but always seem to be stuck ...
1
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1answer
59 views

Physics Meaning of Trace Technology in QED

As it pointed out on page 133 of Peskin and Schroeder, any QED amplitude involving external fermions, when squared and summed or averaged over spins, can be converted to traces of products of Dirac ...
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0answers
28 views

My Struggle with Fierz Identity

I am following BUSSTEPP Lectures on Supersymmetry to learn SUSY. The Lagrangian of a interacting Wess-Zumino model in 4D is given by $$\mathcal{L}=-\frac{1}{2}(\partial_{\mu}S)(\partial^{\mu}S)-\...
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1answer
72 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
61 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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11 views

density of state function for semi Dirac materials

Respected members I have a problem in finding density of state for semi Dirac system (linear dispersion relation in one direction and parabolic in other direction). $$E(k)=\pm\sqrt{{(\hslash^2k_x^...
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1answer
30 views

Can I separate out these Dirac matrix terms?

If I define an object: $$S = \gamma^a A_a + \frac{i}{4}\gamma^a[\gamma^{b},\gamma^{c}] B_{abc}.$$ Is there a way (using dirac matrix formula) to seperate out the terms to either get an expression ...
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1answer
53 views

What do the matrices $\alpha_k$ represent in the Dirac equation?

I have been scouring the internet for an answer. All I have managed to find are the matrices for $k=1,2,3,4,5$. However, I still have no idea they represent, within the equation. Am I correct in ...
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1answer
41 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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1answer
57 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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2answers
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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
22 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
32 views

Confusion with trace of gamma matrices

Using $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} \mathbf{1}$, it is easy to show that: \begin{align*} \operatorname{tr} \gamma^\mu \gamma^\nu = 4\eta^{\mu\nu} \end{align*} Now, it is also true that ...
2
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1answer
71 views

Gamma matrices in (2+1)

I am sure that is very well-known question and see on this site several similar questions but I would like to specify the answer 1) I know that in $(2+1)$-dimensions one can construct $\gamma$-...
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1answer
42 views

Showing hermiticity properties of Dirac matrices using hamiltonians

I want to show $${\gamma^0}^\dagger=\gamma^0\\ {\gamma^i}^\dagger=-\gamma^i.$$ To do this I consider the Dirac equation $$ (i\gamma^\mu\partial_\mu-m)\psi=0$$ and I write it as $$ i\partial_t \psi=(...
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0answers
42 views

How to relate $\gamma^5$ to the spacetime volume form? In regards to axial current anomaly

In playing with gamma matrices of the $\mathcal{\mathscr{C}}l_{1,3}(R)$ variety, it's not uncommon to hear allusions to $\gamma^5$ being related to the volume 4-form. To illustrate the similarities: $...
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1answer
39 views

Geometrized Algebra and Einstein's Equations

The algebraic properties of the pseudoscalar $i$ follows the ordinary rules for imaginary numbers: So its algebraic properties are ~ $i^2 = -1$ the amazing geometric property is that it is an ...
0
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1answer
45 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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0answers
56 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
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1answer
75 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
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Zero momentum in Non relativistic Quantum Mechanics and about Dirac matrices

In relativistic quantum mechanics, we can solve the Dirac's equation with an added condition that the momentum of the particle is $0$. However, such independence isn't provided by the Schrodinger's ...
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0answers
59 views

Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
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1answer
93 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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47 views

Operators, gamma matrices and Lorentz invariance

In class, we have define the following operator: $$\Pi_{\pm} = \frac{1 \pm \gamma^0}{2} \tag1$$ Where, $\gamma^0$ is the usual first gamma matrix in Weyl representation. Applying it to a 4-...
2
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1answer
66 views

Dirac matrices in dimensional regularization, get correct order epsilon

Let us work in dimension $D = 4-2\epsilon$. In 4-dimension, we can write $\text{Tr}[A B]$, where $A$ and $B$ are string of gamma matrices, as $\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$, ...
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1answer
25 views

Weyl basis gamma matrix identity

In finding the scattering amplitude matrix $|\mathcal{M}|^2$, I see the solutions get a way nicer calculation by using that (using Peskin & Schroeder notation): $$(\bar v \gamma^\mu u)^*= \bar u\...
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1answer
53 views

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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1answer
60 views

Derivation of the adjoint of Dirac equation

My goal is to deduce the adjoint of Dirac equation: $$ \overline \psi (i\gamma^\mu \partial_\mu+m)=0 \tag{1} $$ My process: I started with Dirac equation $(i\gamma^\mu \partial_\mu-m)\psi=0$. ...
2
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1answer
100 views

Could there be a pseudovector kinetic term for fermions?

Could there be a kinetic term of the form $\bar{\Psi} \gamma_5 \gamma^\mu \partial_\mu \Psi $ in addition to the usual one? Or is this forbidden by a symmetry?
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0answers
60 views

Can one find Dirac matrices for any spacetime metric?

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{γ_μ,γ_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*...
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2answers
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What is the relationship between the Lorentz group and the $CL(1,3)$ algebra?

In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ...
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0answers
23 views

Vertex with spinorial structure and scattering amplitude

Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$ ...
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3answers
364 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 ...
4
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3answers
108 views

Missing identity element in the Clifford relation

While studying the Dirac equation, $$\left(i\gamma^{\mu} \partial_{\mu} - m\right)\psi = 0.$$ I have been finding difficulty understanding the following summarisation of the algebra that the $\gamma$-...
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2answers
158 views

Simplification of an expression with gamma matrices [closed]

I am trying to understand the details of the calculations involved in determining the electron's anomalous magnetic moment to second order: the $\alpha/{2\pi}$ term. There is just one step, where an ...
3
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1answer
101 views

What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
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0answers
71 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
162 views

Møller scattering amplitudes

In order to compute the scattering cross section for Møller scattering, one needs the amplitudes for both the $t$- and the $u$-channel. Since the cross section is proportional to $|\mathcal{M}|^2$, ...
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0answers
23 views

Gamma matrices field in a using coframe field

I want to see if I got some things straight: Proper frame field and co frame field for an explicit metric (I take Schwarzschild metric in Schwarzschild coordinates) Uplift (cast) the gamma matrices (...
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1answer
82 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
134 views

How can I prove this relation involving gamma matrices?

Let $$l^{\mu} = l^{\mu}_{\parallel} + l^{\mu}_{\bot}$$ be a D-dimensional vector living in a Minkowskian space; the only non-zero components of $l^{\mu}_{\parallel}$ are the first four, while the only ...
3
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1answer
221 views

Identities of Pauli matrices in two-component spinor formalism

I'm reading the review by H. K. Dreiner, H. E. Haber and S. P. Martin (arXiv:0812.1594) about the two-component spinor formalism. There are some identities and notational conventions which lead to ...
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0answers
41 views

Doubt about a Weyl relation

I'm trying to show: $\overline{\sigma}^\mu\sigma^\nu\overline{\sigma}^\rho = \eta^{\mu\nu}\overline{\sigma}^\rho+\eta^{\nu\rho}\overline{\sigma}^\mu-\eta^{\mu\rho}\overline{\sigma}^\nu+i\epsilon^{\mu\...
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1answer
51 views

Squares difference with gamma matrix

$\newcommand{\fsl}[1]{#1\kern-0.4em\raise0.22ex\hbox{/}}$How can I simplify the difference of squares $p^2 - m^2$ in order to obtain $$\frac{p^2 - m^2}{\fsl{p} + m} = \fsl{p} - m~?$$ (where $\fsl{p}=\...
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1answer
76 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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0answers
41 views

Please explain the 2 equalities

why $p^{I}$ can be factored out? what things can be factored out in commutation relations more generally? also in calculating commutation relations does order matter? multiplying from left or right or ...
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0answers
37 views

What's the result of the following multiplication

The question is simple, what is the result of the following multiplication of traces by the metric? $Tr\left[\gamma^\alpha\gamma^\nu\gamma^\beta\gamma^\tau\gamma^5\right]Tr\left[\gamma^\delta\gamma^\...
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0answers
54 views

How do I compute QFT trace of matrices at the end of a Feynman diagram?

At the end of a Feynman diagram I have (p and p' are two incoming momenta): $$Tr[\not p' \gamma^\mu \not p \gamma^\nu(\frac{1+\gamma^5}{2})]=Tr[\gamma^\mu p_\mu' \gamma^\mu \gamma^\nu p_\nu \gamma^\nu(...
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1answer
58 views

A question about the decoupling of Dirac equation in 1+1 dimension

It is said that in 1+1 dimension, if we take $\gamma^0=i\sigma^2$ and $\gamma^1=\sigma^1$, then the two components of dirac spinor $\psi_L$(upper component) and $\psi_R$(lower component) decouple in ...
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0answers
43 views

Mapping from spinor to tetrad

I am reading the journel by Patrick l. Nash: mapping from tetrad to Dirac spinor. While reading this ,I came across the term concrete real 4*4 irreducible representation of SO(3,3). I know SO(3) is ...