Questions tagged [dirac-matrices]

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1answer
262 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 ...
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3answers
484 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}\\ -\...
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1answer
129 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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0answers
48 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
62 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
119 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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4answers
14k 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 ...
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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^\...
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0answers
85 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
78 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
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 ...
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1answer
91 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 ...
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1answer
325 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 ...
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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
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1answer
116 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
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1answer
125 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 ...
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0answers
189 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^{...
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1answer
123 views

How is $\gamma^{\mu}$ defined in the anti commutation relation $\{{\gamma_{5},\gamma^{\mu}}\}$?

how is $\gamma^{\mu}$ defined in the anti commutation relation $\{\gamma_{5},\gamma^{\mu}\}$? does it make a difference if you write the index ${^\mu}$ lower? what does usually change if the index is ...
1
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1answer
129 views

Signature of trace of Dirac Matrices

I came across this question in my problem set: Let $\gamma^\mu$, $\mu=0,1,2,3$ be the Dirac matrices, satisfying: \begin{eqnarray} \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}I, \:\:\:...
2
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1answer
181 views

The photon propagator term in Peskin & Schroeder Eq. 6.38

In Peskin and Shcroeder, when calculating the one-loop vertex correction, the line above Eq. (6.38) reads $$ \rightarrow \int \frac{d^4 k}{(2\pi)^4} \frac{-ig_{\nu\rho}}{(k-p)^2 + iϵ} \bar{u}(p') (-...
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2answers
818 views

Dirac spinors in 2+1 dimensions

In 3+1 dimensions, Dirac spinors have four complex components. In 2+1 dimensions, the representation of the Clifford algebra by $\sigma^3$ and $-i\sigma^3\sigma^i$, with $i\in\{1,2\}$ is 2-dimensional,...
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0answers
158 views

QED parity invariance and gamma 5 [closed]

Why does the invariance of parity of QED indicate that gamma-5 can not appear in Feynman diagrams?
3
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1answer
345 views

Spin Operator in terms of Gamma-5

The QM spin operator can be expressed in terms of gamma matrices and I am trying to do an exercise where I prove an identity which uses $\gamma^5$ and ${\mathbf{\alpha}}$: $$\mathbf{S}=\frac{1}{2}\...
2
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0answers
245 views

Gamma matrices in higher (even) spacetime dimensions

Suppose we write the gamma matrices in this following representation: \begin{align*} \gamma^{0}=\begin{pmatrix} \,\,0 & \mathbb{1}_{2}\,\,\\ \,\,\mathbb{1}_{2} & 0\,\, \end{...
5
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2answers
499 views

Choice of Dirac gamma matrix representation and definition of adjoint spinor

Is the definition of the adjoint spinor $\bar{\psi}=\psi^\dagger \gamma^0$ forcing a particular choice of representation of the Dirac matrices (or a subset of the possible choices)? More precisely, I ...
5
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2answers
804 views

spinor vs vector indices of Dirac gamma matrices

I am struggling to understand the nature of the components of the Dirac matrices. If we view the four components of a Dirac spinor as $\psi^a$ with $a$ being a 'spinor' index, then if a gamma matrix ...
0
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1answer
191 views

When can the Minkowski metric be treated as a “number”?

I am starting to study QED at the moment. I can not wrap my head around why the metric ($g_{\mu\nu}$) is used as a number sometimes. In this case it is pretty obvious that it has to be the number ...
2
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1answer
41 views

Dirac Equation Dimensionality

In Griffith's Introduction to Elementary Particles, the Dirac equation is given during its derivation as (Equation 7.19): $$ \gamma ^ \mu p_\mu - mc = 0 $$ However, the dimensions don't seem to make ...
2
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0answers
395 views

Spin covariant derivative of gamma matrices?

Where can I find a general expression (on curved manifolds) in local coordinates, for the following: $$\nabla^S_{\mu}\gamma^{\nu} = ?$$ $\nabla^S_{\mu} = \partial_{\mu} + \omega^S_{\mu}$ is the spin ...
1
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1answer
91 views

Weak isospin current

I cannot understand the product of a Dirac gamma matrix and a Pauli matrix in this formula of the weak isospin current: $$J_α^i(x)=\frac12\bar \psi_L(x)\gamma_\alpha\tau^i\psi_L(x),$$ where $γ_α$ is ...
2
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2answers
128 views

What is the meaning of $\not{p}$ in physics?

I am reading Srednicki's QFT book in physics. On page 286, the formula $(45.16)$ has a notation $\not{p}$. What is the meaning of $\not{p}$ in physics? Thank you very much.
1
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1answer
554 views

Some general formula with trace of gamma matrices relating $\gamma^{(d+1)}$

I want to figure out the trace of gamma matrices relating with $\gamma^{(d+1)}$ for even $d$ dimensional case. First define $\gamma^{(d+1)}$ as \begin{align} \gamma^{(d+1)} = \gamma^1 \gamma^2 \...
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1answer
224 views

Trace of $2n$ gamma matrices

To proof $$\mathrm{Tr}(\gamma_{\mu_1}\cdots\gamma_{\mu_{2n}}) =\mathrm{Tr}(\gamma_{\mu_{2n}}\cdots\gamma_{\mu_1}),$$ I use $\gamma_\mu^\dagger=\gamma^0\gamma_\mu\gamma^0$ and get $$\cdots=\mathrm{Tr}...
1
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1answer
155 views

How to prove $\gamma^0=(\gamma^0)^T$?

The Dirac gamma matrix $\gamma^0$ is symmetric in Dirac, Weyl and Majorana representation. Is it in general true that $\gamma^0=(\gamma^0)^T$? Can it be proved that $\gamma^0=(\gamma^0)^T$ in a ...
0
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1answer
298 views

Solution of the Dirac equation by Pauli four-vector

Reading through David Tong lecture notes on QFT. On page 100, he solves the Dirac equation by Pauli four-vector. See below link: QFT notes by Tong, Chapter 4 In (4.107) he gives the solution in ...
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2answers
756 views

Solution of Dirac equation-Positive and Negative energy

For particles defined with positive energy, we use $$\phi= \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ or $$ \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix} $$...
2
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1answer
91 views

Dirac operator identity separating out gamma matrices

I am trying to show that $$(i\gamma^\mu\partial_\mu -e\gamma^\mu A_\mu)^2 = (i\partial_\mu -eA_\mu)^2 -\frac{e}{2} \sigma^{\mu\nu}F_{\mu\nu},$$ where $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\...
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2answers
334 views

Need help with solution of the Dirac equation

$$\left(\vec\sigma \cdot \vec{p} \right)^2=\left(\vec\sigma \cdot \vec{p}\right) \left(\vec\sigma \cdot \vec{p} \right)=\vec{p} \cdot \vec{p}+\mathrm{i}\left(\vec\sigma \cdot \left[ \vec{p} \times \...
2
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1answer
1k views

Charge conjugation operator and gamma matrices

The gamma matrices are defined by their anticommutation relations, which are symmetrical in permutations of $\gamma_1, \gamma_2, \gamma_3$. Given this symmetry, why is the change conjugation operator $...
0
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2answers
37 views

Derivative of an Expression with respect to One Component of Strain

I recently come across a paper in which the notation of some equation confuses me a lot. Let's say, if I have an expression represented by delta $\delta_{jk},\delta_{jl}$, infinitesimal strain tensor $...
3
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1answer
1k views

Transformation between Weyl and Dirac representation of Gamma matrices

I want to find a similarity transformation $T$ between the Weyl representation and the Dirac representation of the gamma matrices: $\gamma_W^\mu=T \gamma_D^\mu T^{-1}$. It turns out that I can look at ...
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0answers
95 views

Dirac Fields and Derivatives (Am I gaining extra minus signs?)

I've given myself a severe headache jumping between East/West Coast sign conventions; I have picked up an extra minus sign and could do with a hand. I am currently using $\eta=\textrm{Diag}[-,+,+,+]$ ...
2
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0answers
321 views

Charge conjugation in chiral representation

I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find ...
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2answers
612 views

Property of Charge Conjugation Operator

In class, we have defined the Charge Conjugation Operator ($C$) such that: \begin{equation} C \left(\gamma^\mu\right)^T C^{-1} = - \gamma ^\mu , \end{equation} \begin{equation} \psi^C \equiv C\,\...
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0answers
108 views

How are the covariant Pauli matrices defined?

When doing calculations with Weyl spinors, terms like $\theta\sigma^\mu\theta^\dagger$ appear. I know that for 3+1 spacetime dimensions, $\sigma^\mu = (\textbf{1}, \sigma^i)$ with $i=1,2,3$ the usual ...
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1answer
395 views

The fifth gamma matrix and fermion fields

I am aware of the various relations with Dirac spinors and chirality but how does the fifth gamma matrix $\gamma^5$ behave with fermion fields, $\psi$? Does the fifth gamma matrix have any particular ...
3
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0answers
1k views

Dirac trace theorem [closed]

I am unable to prove exactly one trace identity that appears in the appendix of Peskin and Schroeder's QFT book. Can someone help me? The theorem [Appendix A.4 eqn (A.28)] says that the order of $\...
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1answer
774 views

Trace of Gamma Matrices [closed]

If I have: $Tr(\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\gamma^{\beta}\gamma^{\rho}\gamma^{\gamma}\gamma^{\sigma}\gamma^{\delta})$ and I want to get it re-ordered like $Tr(\gamma^{\alpha}\gamma^{\...
0
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1answer
281 views

Trace technology with polarisation vectors

Consider $d$-dimensional gamma matrix structures. I have an expression like $$ \sum_{h_2=\pm}\text{Tr}(\not{\xi}_2\not{p}_3\bar{\not{\xi}}_2\not{p}_1), $$ where $\not p=p^\mu \eta_{\mu\nu}\gamma^\nu$ ...
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2answers
441 views

Zee's use of Kronecker Product in “QFT in a Nutshell” to represent Dirac matrices

In his book Quantum Theory in a Nutshell (2nd edition, p. 94), Zee describes the Dirac gamma matrices and lists a representation using Pauli matrices and the identity matrix. For example he writes $$ ...