Questions tagged [dirac-matrices]

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1answer
87 views

Lie algebra/group/basis of the four gamma matrices along with the identity?

Do the four gamma matrices along with the identity element constitute a lie algebra? With real coefficients we have $$ \mathbf{v}_{\mathbb{R}}=aI+t\gamma_0+x\gamma_1+y\gamma_2+z\gamma_3 \tag{real ...
5
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3answers
748 views

Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) $$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...
3
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1answer
146 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 ...
3
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1answer
202 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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3answers
316 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 $|\mathcal{M}|^2$, ...
1
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2answers
58 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 ...
0
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1answer
38 views

Dirac equation plane with solution

Using the Dirac equation with or $p=$ zero and the $\gamma^0$ matrix defined as $$\gamma^0=\begin{pmatrix}0 & \sigma_0 \\ \sigma_0 & 0\end{pmatrix} = \begin{pmatrix}0 & \bf{I} \\ \bf{I} &...
3
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1answer
127 views

On the Euclidean action for QCD

The Euclidean action for QCD reads, (see e.g., Eq. (45) in "ABC of instantons" by Novikov, Shifman, Vainshtein, and Zakharov) $$S_E=\int d^4 x\left[\frac{1}{4}G^a_{\mu\nu}G^a_{\mu\nu}+\psi^\dagger(-i\...
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1answer
188 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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1answer
41 views

Fierz identities and anticommutation relations

Let us consider the following term $$\bar\psi(p_1)M\psi(p_2) \bar\psi(p_3)N\psi(p_4)$$ According to Ortin's book (equation (D.65)), from the Fierz identity we would have something like $$\bar\psi(...
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2answers
60 views

Doubt about identity on the Wikipedia page Feynman slash notation

I have the following doubt about the identity in this page: $$ {a\!\!\!/}{a\!\!\!/} \equiv a^\mu a_\mu \cdot I_4 = a^2 \cdot I_4 $$ however: $${a\!\!\!/}{a\!\!\!/}=a^{\mu}\gamma_{\mu}a_{\mu}\gamma^{...
5
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1answer
100 views

Time reversal for Dirac particles in 2+1 or 6+1 (mod 8) spacetime dimensions

It is my understanding that time reversal invariance for Dirac fermions is usually (in 3+1 dimensions at least) implemented by an antiunitary operator ${\mathfrak T}$ that acts on the Dirac field ...
2
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2answers
95 views

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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1answer
36 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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0answers
58 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
66 views

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 ...
2
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1answer
96 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 ...
2
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1answer
92 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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1answer
69 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, $$-γ^μ γ_μ∂^μ ∂_μ=m^{2}$$...
3
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1answer
232 views

Question about the Dirac equation

Energy and momentum of a particle can be expressed by equation $$E^2=p_1^2c^2+p_2^2c^2+p_3^2c^2+m^2c^4\hspace{40pt}(1)$$ Equation (1) can be divided into $E$ on both sides. We obtain $$E=\frac{v_1}{c}...
4
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3answers
555 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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3answers
5k views

Lorentz transformation of Gamma matrices $\gamma^{\mu}$

From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as \begin{equation} \gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = \Lambda^{\mu}_{\nu}\gamma^{\...
4
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1answer
199 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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1answer
185 views

Definition of probablity current in dirac space not including spatial dimension?

I'm currently reviewing (basic) relativistic quantum mechanics and stumbled upon the probability current in "dirac space", defined as $j^μ = (j^0,\vec j)^\mathrm T$ with $j^0 = c\,ρ = c\,ψ^+ψ$ and $\...
3
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1answer
420 views

Gamma matrices invariant under Lorentz transformation

I know this has been asked before but I just can't seem to get my head around it based on the answers I've read. So the idea is that we have the gamma matrices $\gamma^{\mu}$. Now from my ...
1
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1answer
64 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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0answers
44 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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0answers
31 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_{\...
0
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1answer
100 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
79 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 ...
4
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1answer
139 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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2answers
94 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
52 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
56 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}\...
1
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2answers
66 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
54 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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0answers
75 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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1answer
155 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
427 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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0answers
52 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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1answer
73 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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0answers
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 \...
2
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1answer
272 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
64 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. $\sigma^{z}_{\mu \nu}\sigma^{z}_{\alpha \beta}=\delta_{\mu \beta}\delta_{\nu \alpha}$ $\...
2
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2answers
147 views

$\mu \rightarrow e\gamma$ in the $R_\xi$ gauge: trouble with momenta and Dirac matrices

My first ever question on stackexchange! Sorry if it is clumsy... I am trying to follow the computation of $\mu \rightarrow e \gamma$ in Cheng and Li and am confused about the second and third ligns ...
4
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1answer
4k views

How can pseudospin be a vector? (Graphene)

In graphene science, I don't understand how one interprets pseudospin as a vector. I thought 'pseudospin' was the vector of Pauli matrices. So how can it be a vector that one can plot for example in ...
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0answers
47 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 ...
0
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1answer
106 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 \...
2
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0answers
87 views

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 ...
1
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0answers
69 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 ...