Questions tagged [dirac-matrices]

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6
votes
2answers
318 views

Diffeomorphisms and the Dirac action

I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach. I heard that there is a special way of ...
0
votes
1answer
130 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 $\gamma^\mu$?
0
votes
1answer
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: $$ \text{Tr}\left[\gamma^\mu\gamma^5\gamma^\alpha\gamma^\nu\gamma^5\gamma^\beta\right] $$ ...
3
votes
2answers
2k views

Fierz identity with Weyl spinors

The following Fierz relation does not seem so obvious to me: \begin{equation} \bar{\psi}_1 \gamma^\mu (1+\gamma_5)\psi_2 \bar{\psi}_3 \gamma_\mu (1-\gamma_5) \psi_4 = -2 \bar{\psi}_1 (1-\gamma_5) \...
3
votes
1answer
180 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 $\gamma$-...
1
vote
1answer
50 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
votes
1answer
2k views

Confusion about slash notation

I am confused about the slash notation and especially taking the square of a slashed operator. Defining $\displaystyle{\not} a \, = \, \gamma^\mu a_\mu$ we have $\,\,$ $\displaystyle{\not} a \...
2
votes
0answers
58 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: $...
1
vote
1answer
66 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
votes
1answer
138 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 ...
2
votes
0answers
66 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, ...
0
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0answers
24 views

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 ...
3
votes
0answers
117 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 $d$ ...
0
votes
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 $$ S^\dagger=\gamma^0S^{-1}\gamma^0 \\ S^{-1}=\gamma^0S^\dagger\gamma^0 $$ ...
3
votes
0answers
181 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}...
0
votes
0answers
62 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-...
3
votes
2answers
666 views

Dirac group representation

I am currently taking a representation theory class (from a physicist), and I am very confused about the Dirac groups' irreducible representations. First of all, all the Dirac matrices in the ...
0
votes
1answer
55 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\...
1
vote
1answer
55 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{...
1
vote
1answer
253 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
votes
1answer
106 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?
4
votes
2answers
218 views

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, ...
0
votes
3answers
423 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 ...
2
votes
0answers
29 views

Vertex with spinorial structure and scattering amplitude

Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$ ...
3
votes
1answer
866 views

A more general completeness relation for Dirac spinors

Assume that we have two 1/2-spin particles with four-momenta $p$ and $p'$. Particle Dirac spinors satisfy the completeness relation $$ \sum_{s=1}^2u_s(p)\overline{u}_s(p)=\not p+m $$ My goal now is to ...
1
vote
2answers
647 views

Why is $\gamma^5$ used to define the projection operator?

The properties of the projection operators are defined as: $$P_+ = \frac{1}{2}(1+\gamma^5)$$ $$P_- = \frac{1}{2}(1-\gamma^5)$$ where $\gamma^5 = -i\gamma^0\gamma^1\gamma^2\gamma^3$ and their key ...
3
votes
4answers
2k views

Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation

My textbook on QFT says that the Dirac equation can be used to show the following relation: $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$$ I have searched around and unable to find how to prove this ...
4
votes
2answers
445 views

Has the relative sign in the Dirac equation any meaning?

I know there are many questions about this topic and also various answers, but it's never stated explicitly, why there is a certain sign before the mass term in the Dirac Lagrangian. I'm also confused,...
2
votes
1answer
71 views

Normalisation of the $\gamma$-matrices

I'm having a little difficulty with understanding the normalisation process of the $\gamma$-matrices. In Thomson Modern Particle Physics 2013, the normalisation of the $\gamma$-matrices are quoted as:...
4
votes
3answers
114 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 $\gamma$-...
-2
votes
2answers
190 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 ...
1
vote
0answers
136 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 ...
0
votes
2answers
1k views

Evaluating the trace of an expression with gamma matrices

I am currently reading Srednicki's Quantum field theory Book and am having some troubles with evaluating the trace of some gamma matrix expressions. For instance in Equation 59.19 Srednicki defines ...
1
vote
1answer
211 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 ...
1
vote
3answers
459 views

How did Dirac come up with the idea of using Pauli matrices?

Dirac equation in natural units is: $$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi=0$$ where $\gamma^{0}=\pmatrix{I_{2} & 0\\ 0 & -I_{2}}$ and $\gamma^{n}=\pmatrix{0 & \sigma_{n}\\ -\...
0
votes
1answer
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} $?
0
votes
0answers
47 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\...
0
votes
1answer
58 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}=\...
0
votes
1answer
115 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 ...
8
votes
4answers
13k views

Commutator of Dirac gamma matrices

Quick question...For some reason I'm having trouble finding an identity or discussion for the commutator of the gamma matrices at the moment...i.e $$\gamma^u\gamma^v-\gamma^v \gamma^u$$ but I am not ...
0
votes
0answers
42 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^\...
0
votes
0answers
76 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(...
2
votes
1answer
73 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 ...
3
votes
0answers
48 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 ...
0
votes
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 ...
1
vote
1answer
298 views

How to prove $\{\gamma^{\mu}, \gamma^{\nu}\}$ ? (notation problem)

I want to prove that $ \{\gamma^{\mu}, \gamma^{\nu} \}=2g^{\mu \nu} $ what are the indices $ \mu$ and $ \nu$ here? because I know the gamma matrices from 0 to 5 and I need to verify the anti ...
1
vote
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 ...
2
votes
1answer
110 views

About the central charge of 4D extended supersymmetry algebra

The 4D SUSY algebra can be written as $$\{ Q_{\alpha}^{A} , Q_{\beta}^{B \dagger} \} = 2 m \delta^{AB} \delta_{\alpha \beta} + 2 i Z^{AB} \Gamma^0_{\alpha \beta}, \tag{B.2.37} $$ in a particular ...
3
votes
1answer
122 views

Is $\displaystyle{\not} p$ a Lorentz Scalar?

I am a bit confused about something. $\gamma^\mu$ is a (Lorentz) vector (c.f. Pesking & Schroeder chapter 3), and so is $p^\mu$, therefore I’d expect their product $\displaystyle{\not}p \triangleq ...
0
votes
0answers
144 views

Proving a General Result for a Trace of $n$ Gamma Matrices

I am attempting to prove a set of results for the products of gamma matrices and traces of products of gamma matrices, but got stuck on this particular one. $$Tr(\gamma^{\mu_1}...\gamma^{\mu_n})=g^{...