Questions tagged [dirac-matrices]

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24 views

Traces in 't Hooft-Veltman scheme

I'm currently looking at the 't Hooft-Veltman regularization scheme and I'm a bit confused on how exactly one calculates traces in this scheme. As far as I understand one has to divide the $D$-...
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1answer
26 views

Four-momentum matrix dot product sigma matrix

In the QFT book of Peskin and Schroeder, they introduce the notation: \begin{align} \sigma^{\mu}=(I,\sigma^{i})\\ \bar{\sigma^{\mu}}=(I,-\sigma^{i}). \end{align} On page 46 (Eq.(3.50)), They take the ...
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49 views

Clarifying Griffiths syntax when solving Feynman Diagrams

In Griffiths textbook "Introduction to Elementary Particles", he details how to use Feynman diagrams and walks through some examples of solving some essential particle interactions. The ...
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1answer
55 views

Deriving the Generalized Fierz Transformation from Schroeder's Textbook

I am self studying QFT from the textbook An Introduction of Quantum Field Theory and the corresponding solutions from Zhong-Zhi Xianyu. The generalized Fierz Transformation is derived in problem 3.6. ...
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70 views

(Non-)Hermiticity of Dirac operator

I have a Dirac operator given by \begin{equation} D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}), \end{equation} where $A_{\mu}$ ...
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36 views

Respresentation Dirac group and Lie algebra

I am reading Peskin and Schroeder's book on QFT and have some difficulties with representation groups. Let's start with the Lorentz group since it is easier. let $\Lambda$ be a Lorentz transformation, ...
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1answer
56 views

Gamma identities computations

I got my BsC in Physics many years ago (18yrs) but at those times QFT was not covered during the career, only in Ph.D. courses. Now I'm studying it at my own during my free time using several books on ...
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1answer
34 views

Are the $\alpha_i$ and $\beta$ matrices in Dirac equation unique in the Dirac representation?

Related question is in Are the $\alpha$ and $\beta$ matrices of the Dirac equation unique? . However, it does not solve my problem. My professor asked us to prove that in the Dirac representation, $$\...
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1answer
44 views

Proof of gamma matrix trace identity from Griffiths Introduction to Particle Physics [closed]

Griffiths states that the product of eq (9.8): $ 8[p_1^\mu p_3^\nu + p_1^\nu p_3^\mu -g^{\mu\nu}(p_1 \cdot p_3) -i\varepsilon^{\mu\nu\lambda\sigma}p_{1\lambda}p_{3\sigma}]$ and eq (9.9) $ 8[p_{2\mu} ...
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1answer
85 views

Trace of Dirac matrices

I was calculating the trace of two Dirac matricies and I used their anti-commutation relations: $$ Tr(\gamma^{\mu} \gamma^{\nu}) = -Tr(\gamma^{\nu} \gamma^{\mu}) - Tr(2\eta^{\mu\nu}) $$ $$ = - Tr(\...
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1answer
52 views

Trace of two Dirac matrices in 4 dimensions

I want to show that tr($\gamma^\mu \gamma^\nu$) = 4$\eta^{\mu \nu}$. I know that {$\gamma^\mu , \gamma^\nu$} = 2$\eta^{\mu\nu}I_4$ and tr($\gamma^\mu \gamma^\nu$) = tr($\gamma^\nu \gamma^\mu$), so tr(...
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51 views

Accounts on the solutions of the Dirac equation

Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. As it is well known, there are different representations for the matrices $\gamma^{\mu}$, $\mu = 0,1,2,3$, the most famous ones ...
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1answer
54 views

Does $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbb1$ determine the hermiticity of the gamma matrices?

If I remember correctly, the derivation of the Dirac equation requires that $\gamma^0$ is Hermitian while $\gamma^i$ for $i=1,2,3$ is anti-Hermitian. This is clearly true for the standard Dirac ...
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33 views

Plane waves solutions for Dirac equation in terms of eigenstates of helicity

Suppose $\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices. Given a momentum ${\bf{p}}$, we define the helicity operator: $$ h = \frac{1}{2}\begin{pmatrix} {\bf{\sigma}}\cdot {\bf{\hat{p}}...
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40 views

Equivalence of Dirac matrix representations

I'm currently having fun with the Dirac matrices $\gamma^\mu$ which appear in the relativistic Fermion field equation (aka the Dirac equation). They need to satisfy the anticommutation relation $$ \{\...
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38 views

Current divergence

How do I calculate the current divergence $$\partial_\mu A_{ud}^\mu = \partial_\mu\left (\overline{u} \gamma^\mu \gamma^5 d(x)\right )?$$ I don't understand exactly how to manipulate gamma matrices.
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41 views

Expression of $\not{p}$ in Dirac equation

In scattering amplitudes, page 9, equation (2.6), (2.7), $\not{p}$ (in the Dirac equation (2.4)) is as follows: \begin{align} \not{p} = \left( \begin{matrix} 0 & p_{a\dot{b}} \\ p^{\dot{a}b} & ...
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182 views

Proof that $\gamma_{5}^2=1$

I need to prove that $\gamma_{5}^2=1$, and in order to do this I wrote: \begin{equation} (\gamma_{5})^2=\gamma^{5}\gamma_{5}=\left(-\frac{i}{4!}\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\...
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1answer
45 views

Sign error when deriving Weyl spinor transformation laws (3.37) in Peskin Schroesder

I am having some trouble deriving the transormation laws for the weyl spinors, equation (3.37) in the Peskin Schroesder book on quantum field theory. Beginning with the relation $\psi\to(1-\frac{i}{2}\...
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1answer
63 views

A question involving chiral transformations and gamma matrices

I'm looking at a calculation that involves an infinitesimal transformation of a Dirac fermion field: $$\Psi \rightarrow e^{i \beta \gamma^5} \Psi.$$ Then the conjugate field $\bar{\Psi} = \Psi^{\...
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1answer
53 views

Trace evaluation in inverse muon decay

Applying Casimir´s trick when averaging over the the initial and summating over the final spin states in the inverse muon decay yields (Griffiths, example 10.1) among others the following trace $$ \...
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64 views

Solution to Dirac equation

We take the solution of Dirac equation as 4 component wave function (Dirac Spinor). But how do we know that it can't be a square or rectangular matrix like 4x2 or 4x4 matrix?
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38 views

Another form of the solutions of the Dirac equation

Consider the Dirac equation $[i\gamma^\mu\partial_\mu-m]\psi=0$ and let me focus in particular on the positive-energy solutions by the ansatz $$ \psi(x)=e^{-ipx}u(\mathbf p,r). $$ Making this ...
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1answer
55 views

Missing sign in Dirac equation

This is very trivial, but it's really bugging me. The ansatz for the Dirac equation in terms of $\boldsymbol\alpha$ and $\beta$ matrices is $$ [i(\partial_t+\boldsymbol\alpha\cdot\boldsymbol\nabla)-\...
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1answer
66 views

Dirac spinor in the chiral basis

In the chiral basis, the gamma matrices take the form $$ \gamma^0=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \gamma^j=\begin{bmatrix}0 & -\sigma^j \\ \sigma^j & 0\end{bmatrix} $$...
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1answer
29 views

What is the correct relation between Dirac matrices and Charge conjugation?

Setup Let $C$ be the charge conjugation operator for spinors and $\gamma$ a Dirac matrix. From this post we conclude that the critical relation between the operator and the Dirac matrices is $$-C(\...
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29 views

Dirac spinors completeness relation for $d>4$

In $d=4$ and in the rest mass frame, the four Dirac spinors $u_s(0)$ and $v_s(0)$ satisfy the completeness relation $$ \sum_{s=1}^{2} u_s(0) \overline{u}_s(0) - \sum_{s=1}^{2} v_s(0) \overline{v}_s(0)...
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1answer
40 views

How to take trace over group and Dirac indices?

I'm currently reading Pokorski's book "Gauge Field Theories" and in Chapter 13 he discusses, among other things, Fujikawa's method of deriving the chiral current (see page 488 and the ...
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71 views

How to invert this matrix?

Is there a smarter method for finding the determinant and inverse of the following matrix, without using the brute force procedure? (When I say brute force, it is to write the matrix with each term ...
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1answer
42 views

The tensor product in the Hamiltonian of graphene

I have the Hamiltonian of pristine graphene \begin{equation} H=v_{F}.\boldsymbol{\gamma}.\boldsymbol{p} \end{equation} with $\boldsymbol{p}=(p_{x},p_{y})$ is the momentum operator, $v_{F}$ is the ...
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1answer
101 views

Proof of $\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=0$

Using $\gamma^{5}\gamma^\mu=-\gamma^\mu\gamma^{5}$ and $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ I obtain \begin{equation}\tag{1} T_{\mu\nu}:=\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=-\mathrm{tr}(\gamma^\...
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44 views

Property of Dirac operator in spinor frame bundle

The Dirac operator is defined by $$d=g(e_\alpha,e_\beta) \gamma(e_\beta)\nabla_{e_\alpha} \tag1$$ Here $\nabla$ is the spin covariante derivative $e_\alpha$ basis of the tangent space and $\gamma$ the ...
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50 views

Reducibility of Lorentz group generators, contrasted with irreducible gamma-matrix representation

After introducing the gamma matrices as $$\gamma^0=-i \pmatrix{\begin{matrix} 0 & \Bbb I_{2x2} \\ \Bbb I_{2x2} & 0 \\ \end{matrix}} , \qquad \gamma^i=-i\pmatrix{\begin{matrix} ...
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58 views

How do you find the projector appearing in $\kappa$-symmetry for Green-Schwarz formulation of superstring?

I am trying to study the Green-Schwarz formulation of the superstring and I am following the string theory book by Becker, Becker, Schwarz. In there, they consider the action $$ S = S_1 + S_2 = -\frac{...
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2answers
80 views

Dirac equation identity

I have the homework problem: Starting from the Dirac equation, derive the identity $$\bar{u}(p') \gamma^{\mu} u(p) = \frac{1}{2m} \bar{u}(p')(p+p')^{\mu} u(p) + \frac{i}{m}\bar{u}(p') \Sigma^{\mu \...
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95 views

Explicit example of dimensional regularization involving $\gamma^5$

I'm currently reading Collins' book Renormalization, Chapter 4. In section 4.5 he introduces $\gamma$-matrices and the trace operation in an arbitrary dimension $d$. In section 4.6 he then talks about ...
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1answer
120 views

Factoring the Laplace operator $\Delta$ in dimensions $D \geq 3$

Consider the Laplace operator in 2 dimensions \begin{equation} \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = \partial^2_x + \partial^2_y \end{equation} By defining the ...
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1answer
99 views

Trace of 4 gamma matrices

Can anyone help me understand how do i execute this trace? $$ Tr(\gamma_\mu(\gamma^\rho P_{1_\rho})\gamma_\nu (\gamma^\sigma K_{1_\sigma})) $$ I know the rule when we have 4 gammas inside the trace, ...
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59 views

Dirac Equation and Notation

I need to ask something regarding the Dirac equation (for a charged particle in an electromagnetic field) with the slash notation, which i fail to understand. We have the Dirac equation with the slash ...
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2answers
50 views

Dirac differential operator and anti-commutation relations

I was studying Dirac's equation and my book says the following. Let $D = i \gamma^{\mu}\partial_{\mu}$, where $\partial_{0} = \partial/\partial t$ and $\partial_{\mu} = \partial / \partial x_{\mu}$ if ...
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1answer
89 views

Standard model notation on doublets

$\require{cancel}$ I have been introduced to electroweak theory in lectures and I wanted to check I understand the notation for the doublets, triplets etc. Take the first generation lepton left handed ...
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1answer
82 views

$\gamma^5$ rotation of chiral fermion in (1) Peskin&Schroeder, (2) Weinberg, or (3) Srednicki

The theta angle due to the chiral gamma^5 rotation of chiral fermion results in the phase alpha(x) that has different + or - sign for (1) Peskin&Schroeder, (2) Weinberg or (3) Srednicki. Here ...
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1answer
46 views

Commutation relation involving $\gamma^5$. Spot the error

I'm trying to prove a relation that is useful when studying general properties of Dirac spinors, namely, that $\left[\gamma_5,\sigma^{\mu\nu}\right]=0$ where $\sigma^{\mu\nu}\sim i\gamma^\mu\gamma^\nu$...
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29 views

Structure function in DIS calculation step

There is one step in the calculation that I am not understanting, but there are many things to present so one can understand and help me: i)$k^2$ and $k_T^2$ are small, so can be neglected; ii)$k^\mu=\...
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1answer
91 views

Gaussian integrals with gamma matrices in their exponents

I should evaluate Gaussian integrals in the 1+1 Minkowski space, which read $$ I_{1}= \int d^{2}k \, {\rm Tr}\big[ \gamma^{5} \gamma^{\eta} \gamma^{\kappa} e^{\alpha k^{\mu}k_{\mu} + \beta \gamma^{\...
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1answer
37 views

Particles physics - Projection matrices

I'm struggling with a question were I'm asked to simply the following expression such that my result only involves one projection matrix. $$P_L \not p \not k P_L $$ Where $$P_L = (\frac{1-\gamma_5}{2})...
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37 views

Fierz Identity calculation

While reading an article, it's said that to simplify the following Dirac structure $$\left(P_Lv_j^d\bar{v}^s_kP_R\right)_{\alpha\beta}\tag{1}\label{1}$$ where $j,k$ are color indices and $d,s$ ...
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1answer
117 views

Derivation of Dirac matrices in spherical polar coordinates

How to derive Dirac $\gamma$ matrices in spherical polar coordinates?
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1answer
81 views

How these two approaches to spinors in curved spacetimes relate?

Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a ...
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1answer
95 views

Why are the 4-vector and bispinor representation of the Lorentz algebra in particular so related?

When learning about the Dirac equation, there are several indications that the fundamental (4-vector) representation and the bispinor representations are connected in some way. To give an example, the ...

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