Questions tagged [dirac-matrices]

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Contracting gamma matrices with explicit indices

So I was calulating the matrix element of an interaction and arrived at the following contraction $$\gamma^\mu_{ab}\gamma_{\mu\,cd}$$ With $a,b,c,d$ spinor indices that are never contracted with ...
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31 views

Spinorial representation of Lorentz group for solution to Dirac equation

In my relativistic quantum mechanics course, we found plane wave solution to the Dirac equation by first studying it the reference frame of the particle. Using a plane wave solution for both positive ...
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38 views

Proving Completeness Relation for Dirac Spinors

So the problem I'm attempting is: And I'm trying to verify the completeness relations (2nd two equations). Attempt So $u_{r}(\mathbf{p})=\sqrt{E+m} \begin{pmatrix}\chi_{r}\\ \frac{\mathbf{\sigma .p}...
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1answer
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Can we get the product of two gamma matrices in Quantum Mechanics to yield the matrix with i on the diagonal and zero elsewhere?

Can we get the product of two gamma matrices in Quantum Mechanics to yield the matrix with i on the diagonal and zero elsewhere? There can be different basis for the gamma matrices. I wonder if a ...
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1answer
82 views

Spinors, Spacetime and Clifford algebra

I'm looking to understand the intrinsic connection that Clifford algebra allows one to make between spin space and spacetime. For a while now I've trying to wrap my head around how the Clifford ...
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1answer
28 views

Electron muon scattering unpolarised cross-section calculation with traces

Following my class on RQM, we wanted to evaluate the unpolarised cross section for the following process $$ e^+ e^- \rightarrow \mu^+ \mu^- $$ In doing so, whenever summing over spinor indices, a ...
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1answer
63 views

Is $\gamma_\mu \gamma^\mu$ a unit operator?

Is the term: $$γ^μ γ_μ$$ An identity matrix? Since,if we start with both the Dirac equation, $$(iγ^μ ∂_μ-m)Ѱ=0$$ We find that, $$iγ^μ ∂_μ=m$$ If we square both sides, we get, $$-γ^μ γ_μ∂^μ ∂_μ=m^{2}$$...
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1answer
84 views

Does the geometric algebra of curved space have a matrix representation?

Suppose the geometric algebra defined by $$ \frac{1}{2}(e_\mu e_\nu +e_\nu e_\mu)=g_{\mu\nu} $$ where $e_\mu,e_\nu$ are generators of the algebra, and where $g_{\mu\nu}$ are elements of the reals. I ...
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3answers
534 views

Physical interpretation of gamma matrices

Just out of plain curiosity, I want to ask: What are/is the physical interpretation(s) of the gamma matrices? If there is none, is it right to assume that it is just a mathematical fudge-factor?
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1answer
178 views

Dirac equation without $i$

In Witten's review paper "Fermion path integrals and topological phases", the Dirac equation (Eq(2.2)) is $$(\gamma^{\mu}D_{\mu}-m)\psi=0$$ which appears very strange to me. Initially I thought this ...
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0answers
34 views

Traces of gamma matrices in $d$ dimensions

For $d=4$, some identities of the traces of gamma matrices are: $tr[\gamma_\mu] = 0$ $tr[\gamma_\mu \gamma_\nu ] = 4g_{\mu\nu}$ $tr[\gamma_\mu\gamma_\alpha\gamma_\nu] = 0$ $tr[\gamma_\mu\gamma_\alpha\...
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27 views

How can you take common factor out from this commutator?

So I need to compute $\left\lbrace \psi (\vec{x}, t), \psi^{\dagger} (\vec{y}, t) \right\rbrace$ where both $\psi$ are Dirac fields $\psi(x) = \int \frac{d^3 k}{(2 \pi)^3} \frac{m}{w_k} \left[ \sum_{\...
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1answer
63 views

Spin of electron using Levi Civita symbol

Problem is probably trivial, but I can't seem to find a fault in the argument. If we write the Dirac equation as $$(c\vec{\alpha}\cdot \mathbf{p} + \beta mc^2) \psi = i\hbar \frac{\partial \psi}{\...
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1answer
82 views

Question about the Dirac adjoint and Feynman slash notation

I was trying to prove the identity $\overline{\displaystyle{\not}{a}\displaystyle{\not}{b}\dots \displaystyle{\not}{p}} = \displaystyle{\not}{p}\dots \displaystyle{\not}{b}\displaystyle{\not}{a}$. On ...
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0answers
68 views

Confusion about gamma matrices in Euclidean spacetime

I have encountered a number of sources with differing definitions of the transition from Minkowski spacetime to Euclidean spacetime. I'd like some clarification as to how to go from Minkowski to ...
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2answers
85 views

What is the correct form of Dirac equation?

Usually the Dirac equation in curved space is written as $$i\Gamma^{\mu}D _{\mu}\Psi-m\Psi=0,$$ where $\Gamma_{\mu}$ are curved space gamma matrices and $D_{\mu}$ is covariante derivative. This ...
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0answers
51 views

Gamma Matrices as nonstandard numbers, and Grassman Numbers

I'm in the process of exploring the Dirac equation and its forms and consequences, and as such have just been initiated into the theory of spinors and their accompanying formalism. One of the things ...
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1answer
55 views

Field equation of motion for this Lagrangian

In my first QFT exam I was supposed to derive the equations of motion for all fields for this Lagrangian: $$\mathcal{L} = \bar{\Psi}(i\gamma^\mu\partial_\mu-M)\Psi+g\bar{\Psi}\gamma^\mu\Psi\bar{\Psi}\...
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1answer
126 views

Almost complex structure $J_i^j$ from covariantly constant spinor $\eta$

In the context of superstring compactification on a 6-manifold which admits a covariantly conserved spinor $\eta$, which we normalize so that $\eta^{\dagger} \eta = 1$, I am trying to show that the ...
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0answers
49 views

Gamma traceless

I read this Under what conditions is a vector-spinor gamma trace free. And also read many papers about higher spin, but no one explains why irreducible spinor is gamma traceless spinor? Can anyone ...
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68 views

Dirac matrices for generalized metric tensors

The Dirac matrices are defined by the relations $$\left [\gamma^{i},\gamma^{j}\right]_{+}=2\eta^{i,j}\mathbb{1}$$ where $[\cdot,\cdot]_{+}$ is the anti-commutator. What happens if I replace $\eta^{i,...
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51 views

Relativistic quantum field theory

Let $\psi(x)$ be solution of Dirac equation $$ (\gamma^\mu\Pi_\mu-mc) \psi(x)=0 $$ where $\Pi_\mu=i\partial_\mu-eA_\mu$ is momentum operator in present electromagnetic field . We consider tow ...
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55 views

Spinor transformations as representations of $\mathrm{SL}(2, \mathbb{C})$

Background In the Weyl representation of the Dirac $\gamma$-matrices, the spinor transformations $S=e^{\frac{1}{2} \omega_{\alpha \beta} \Sigma^{\alpha \beta}} \in \,\mathrm{G}_{\mathrm{L}} \leq \...
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1answer
192 views

Dirac equation derivation

I am working through a set of lecture notes containing a derivation of the Dirac equation following the historical route of Dirac. It states that Dirac postulated a hermitian first-order differential ...
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1answer
58 views

Pauli Matrices multiplication with indicies

I am trying to write product of Pauli matrices in terms of its indicies. I am trying to find a proof of it. $\sigma^{z}_{\mu \nu}\sigma^{z}_{\alpha \beta}=\delta_{\mu \beta}\delta_{\nu \alpha}$ $\...
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44 views

Spin for Dirac fermions proof $\Sigma_{i}=\frac{1}{2} \gamma_{5} \gamma_{0} \gamma_{i}$

The spin for Dirac fermions is defined as: $$\Sigma_{i} \equiv \frac{1}{4}\epsilon_{ijk}\sigma_{jk}$$ Without using an explicit representation for the matrices, I would like to show that the spin ...
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1answer
101 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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Supergravity and Gamma Matrices

On page 101 of Freedman and Van Proeyen's book on Supergravity they find the propagator of the gravitino, however I'm not sure how to work through the steps in (5.30), and hints or answers would be ...
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57 views

Sakurai Quantum Mechanics problems

I just began studying QM on Sakurai's "Modern Quantum Mechanics" and just finished chapter 1. I am now approaching the exercises. On exercise 2 there is a notation I can't understand: "A 2x2 square ...
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1answer
48 views

Commutation of alpha dirac matrix

I want to calculate the commutation of $[\hat{x},\vec{\alpha}\;\vec{p}]$. This boils down to $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}+\left[\hat{x},\hat{\alpha_x}\right]\hat{p_x} +\...
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1answer
150 views

From relativistic equation to find Dirac matrices

Is this possible and then how? $$((\gamma \otimes \mathbf\sigma)\bullet\mathbf p)(\gamma^\prime\otimes\mathbf 1_2) = \gamma\gamma^\prime\otimes\sigma \bullet \mathbf p $$ where $\gamma$ and $\gamma^\...
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2answers
217 views

Can gamma matrices be real in 6 dimensions?

I'm trying to find the really real representation of 6D gamma matrices. The problem is that "do they really exist?" If yes, then how am I supposed to construct them? Thank you!
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35 views

Charge conjugation opearation in odd dimension

In my problem, I use the following set of $\gamma$-matrices (in (2+1) spacetime): $$\gamma^0=\sigma^1;\quad \gamma^{1}=i\sigma^2;\quad \gamma^{2}=i\sigma^{3},$$ where $\sigma^{(i)}$ are usual Pauli ...
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1answer
94 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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110 views

Again on Spin Operator in Dirac Field Theory (Peskin & Schroeder)

Good morning, I've already seen that this topic has been discussed so long, but my doubts remain unchanged. At page 61 of Peskin & Schroeder, An Introduction to QFT, there is the demonstration ...
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1answer
88 views

Maximal anticommuting sets of Dirac matrices

At the end of this webpage, it is said that there exist 6 maximal anticommuting sets each consisting of 5 Dirac $\Gamma$-matrices. I couldn't find anything more in the book cited there, either. I ...
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2answers
173 views

Eigenvalues of $\not{p}$

Let $p_\mu$ be a generic non-null 4-vector, and let $m=\sqrt{p_\mu p^\mu}\neq 0$ (let us choose $\operatorname{Im}m\ge0$). Let also $\gamma^\mu$ be the Dirac gamma matrices. Prove that the ...
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1answer
54 views

Gordon decomposition in Cheng and Li p. 422

In the $\mu \rightarrow e+\gamma$ calculation in Cheng and Li "Gauge theory of elementary particle physics" p.422 they have $$ T=A\bar{u}_e(p-q)(1+\gamma_5)i\sigma_{\lambda\nu}q^\nu\epsilon^\lambda ...
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114 views

Does anyone know how to symmetrize $\gamma$-matrices?

I'm trying to construct the SO(5, 5) $\gamma$-matrices which are real and symmetric. Recently, I have 6 symmetric and 4 antisymmetric $\gamma$-matrices ($6_S + 4_A$ representation). How can I ...
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39 views

Question about Pauli Matrices

I found the following identities about Pauli matrices from the lecture notes of Supersymmetry. $$((\sigma^{\mu})^{\alpha\dot{\alpha}})^{\ast}=(\bar{\sigma}^{\mu})^{\dot{\alpha}\alpha}$$ $$((\sigma_{\...
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1answer
58 views

Square bracket notation of the basis of 16 independent gamma matrices

The question is very simple and I couldn't find an answer. What the notation $\gamma^{ [ \mu} \gamma^{\nu} \gamma^{\rho ]}$ and $\gamma^{ [ \mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma ]}$ means? ...
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1answer
69 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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2answers
89 views

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 ...
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1answer
109 views

Physics Meaning of Trace Technology in QED [closed]

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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107 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
104 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
76 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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35 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
31 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
126 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 ...