What is Gamma matrices: Definition and 62 Discussions
In mathematical physics, the gamma matrices,
{
γ
0
,
γ
1
,
γ
2
,
γ
3
}
{\displaystyle \left\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\right\}}
, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles.
In Dirac representation, the four contravariant gamma matrices are
{\displaystyle \gamma ^{0}}
is the time-like, hermitian matrix. The other three are space-like, antihermitian matrices. More compactly,
γ
0
=
σ
3
⊗
I
{\displaystyle \gamma ^{0}=\sigma ^{3}\otimes I}
, and
γ
j
=
i
σ
2
⊗
σ
j
{\displaystyle \gamma ^{j}=i\sigma ^{2}\otimes \sigma ^{j}}
, where
⊗
{\displaystyle \otimes }
denotes the Kronecker product and the
σ
j
{\displaystyle \sigma ^{j}}
(for j = 1, 2, 3) denote the Pauli matrices.
The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3, 0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma-matrix to be presented below generate the Clifford algebra.
As there was quite rightly some criticism earlier about not following proper theory, I will first demonstrate what I have understood of the gamma matrices of SU(3).
There are 8 gamma matrices that together generate the SU(3) group used in QCD. Gell-Mann used only 2, ##\gamma_3## and...
Hi!
Is it possible to construct gamma matrices satisfying the Clifford algebra ##\{\gamma^\mu, \gamma^\nu \} = \eta^{\mu \nu}## that are *real*, for ##\eta = diag(-1,1,1)##?
I know that I can construct them in principle from sigma matrices, but I don't know how to construct real gamma...
I'm currently working out quantities that include the vector and axialvector currents ##j^\mu_B(x)=\bar{\psi}(x)\Gamma^\mu_{B,0}\psi(x)## where B stands for V (vector) or A (axialvector). The gamma in the middle is a product of gamma matrices and the psi's are dirac spinors. Therefore on the...
Hi all, I am working on a project at the moment, and I have to evaluate the trace by using the Casimir's trick.
The trace form is
$$Tr[(\displaystyle{\not} P +M_{0})\gamma^{\mu}(\displaystyle{\not} P^{'} +M^{'}_{0})(\displaystyle{\not} p^{'}_{1} +m^{'}_{1})\gamma^{\nu}(\displaystyle{\not} p_{1}...
Homework Statement
I define the gamma matrices in this following representation:
\begin{align*}
\gamma^{0}=\begin{pmatrix}
\,\,0 & \mathbb{1}_{2}\,\,\\
\,\,\mathbb{1}_{2} & 0\,\,
\end{pmatrix},\qquad \gamma^{i}=\begin{pmatrix}
\,\,0 &\sigma^{i}\,\,\\
\,\,-\sigma^{i}...
I would like to know where one may operate with tensor quantities in quantum field theory: Minkowski tensors, spinors, effective lagrangians (for example sigma models or models with four quark interaction), gamma matrices, Grassmann algebra, Lie algebra, fermion determinants and et cetera.
I...
Here it is a simple problem which is giving me an headache,Recall from class that in order to build an invariant out of spinors we had to introduce a somewhat
unexpected form for the dual spinor, i.e. ߰ψ = ψ†⋅γ0
Then showing that ߰ is invariant depends on the result that (ei/4⋅σμν⋅ωμν)† ⋅γ0 =...
Hi,
I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra).
1. Homework Statement
Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...
Covariant gamma matrices are defined by
$$\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\{\gamma^{0},-\gamma^{1},-\gamma^{2},-\gamma^{3}\}.$$
-----------------------------------------------------------------------------------------------------------------------------------------------------------...
The gamma matrices ##\gamma^{\mu}## are defined by
$$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$
---
There exist representations of the gamma matrices such as the Dirac basis and the Weyl basis.
---
Is it possible to prove the relation...
Hello,
i have here some identities for gamma matrices in n dimensions to prove and don't know how to do this. My problem is that I am not very familiar with the ⊗ in the equations. I think it should be the Kronecker-product. If someone could give me a explanation of how to work with this stuff...
Hi,
I've been studying Dirac's theory of fermions. A classic topic therein is the proof that the equation is covariant. Invariably authors state that the gamma-matrices have to be considered constants: they do not change under a Lorentz-transformation. I am looking for the reason behind this. It...
Consider a Majorana spinor
$$
\Phi=\left(\begin{array}{c}\phi\\\phi^\dagger\end{array}\right)
$$
and an pseudoscalar current ##\bar\Phi\gamma^5\Phi##. This term is invariant under hermitian conjugation:
$$
\bar\Phi\gamma^5\Phi\to\bar\Phi\gamma^5\Phi
$$
but if I exploit the two component...
Trace of six gamma matrices
I need to calculate this expression:
$$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$
I know that I can express this as:
$$...
Hi all,
I make some exercises in particle physics but I'm stuck in two problems related to Gamma matrices identities,
First: the Fermion propagator ## \frac {i } { /\!\!\!p - m} = i \frac { /\!\!\!p + m } { p^2 - m^2} ## So how ##/ \!\!\!\!p ^2 = p^2 ## ? Where ## /\!\!\!p = \gamma_\mu p^\mu...
I'm reading through some lecture notes and there is a proof that the gamma matrices are traceless that I've never seen before (I've seen the "identity 0" on wikipedia proof) and I can't work out some of the steps:
\begin{align*}
2\eta_{\mu\nu}Tr(\gamma_\lambda) &=...
This is a pretty basic question, but I haven't seen it dealt with in the texts that I have used. In the proof where it is shown that the product of a spinor and its Dirac conjugate is Lorentz invariant, it is assumed that the gamma matrix \gamma^0 is invariant under a Lorentz transformation. I...
Hi,
In QFT we define the projection operators:
\begin{equation}
P_{\pm} = \frac{1}{2} ( 1 \pm \gamma^5)
\end{equation}
and define the left- and right-handed parts of the Dirac spinor as:
\begin{align}
\psi_R & = P_+ \psi \\
\psi_L & = P_- \psi
\end{align}
I was wondering if the left- and...
Dear PhysicsForum,
We have just treated the Dirac equation and its lagrangian during our QFT course, but we have only gone in depth in 3+1 dimensions.
My question is about what happens to spin in 2+1 dimensions. In 3+1 dimensions we have to use 4 by 4 gamma matrices, but in 2+1 dimensions we...
Homework Statement
A proof of equality between two traces of products of gamma matrices.
Tr(\gamma^\mu (1_4-\gamma^5) A (1_4-\gamma^5) \gamma^\nu) = 2Tr(\gamma^\mu A (1_4-\gamma^5) \gamma^\nu)
Where no special property of A is given, so we must assume it is just a random 4x4 matrix.
1_4...
In almost all the books on field theory I've seen, the authors list out the different types of quantities you can construct from the Dirac spinors and the gamma matrices, but I'm confused by how these work. For instance, if $$\overline\psi\gamma^5\psi$$ is a pseudoscalar, how can...
Generally, Gamma matrices with one lower and one upper indices could be constructed based on the Clifford algebra.
\begin{equation}
\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij},
\end{equation}
My question is how to generally construct gamma matrices with two lower indices. There...
Homework Statement
Solve the equation. What is it's trace?Homework Equations
k γμ γ5 o γ\nu γ5
The Attempt at a Solution
I don't think this is reduced enough.
γμkμγ5γ\nuo\nuγ\nuγ5
trace: just got rid of gamma5 with anticommutation.
-Tr[γμkμγ\nuo\nuγ\nu]
\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0
How do I expand
i\hbar \gamma^0
the matrix in this term, I am a bit lost. All the help would be appreciated!
Homework Statement
I need help to expand some matrices
Homework Equations
\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0
The Attempt at a Solution
How do I expand
i\hbar \gamma^0
the matrix in this term, I am a bit lost. All the help would be...
Can anyone help me in proving the following identity:
(\gamma ^{\mu} )^T = \gamma ^0 \gamma ^{\mu} \gamma ^0
I understand that one can proceed by proving it say in standard representation and then proving that it's invariant under unitary transformations. this last thing is the one...
Hi I'm trying to figure out the proof of why the gamma matrices are traceless. I found a proof at wikipedia under 'trace identities' here
http://en.wikipedia.org/wiki/Gamma_matrices
(it's the 0'th identity)
and from the clifford algebra relation
\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu \nu}...
Hi! I'm reading David Tong's notes on QFT and I'm now reading on the chapter on the dirac equation
http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf
and I stumbled across a statement where he claims that
(\gamma^0)^2 = 1 \ \ \Rightarrow \text{real eigenvalues}
while
(\gamma^i)^2 = -1 \...
Homework Statement
So my question is related somehow to the Fierz Identities.
I'm taking a course on QFT. My teacher explained in class that instead of using the traces method one could use another, more intuitive, method. He said that we could use the fact that if we garante that we have the...
I would like to have a general formula, and I am quite sure it must exist, for: \gamma^{\mu}_{ab}\gamma_{\mu \,\alpha\beta} but I didn't succeed at deriving it, or intuiting it, I am troubled by the fact that it must mix dotted and undotted indices.
Hi!
I can define
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
I know that the four gamma matrices \gamma^i\:\:,\;i=0...3 are invariant under a Lorentz transformation. So I can say that also \gamma ^5 is invariant, because it is a product of invariant matrices.
But this equality holds:
\gamma...
Homework Statement
I'm trying to find P_L \displaystyle{\not}p P_L for a left-handed particle.
(I think the answer is zero...)
Homework Equations
P_L = \frac{1}{2} (1-\gamma_5) (the left-handed projection operator)
\displaystyle{\not} p = \gamma_\mu p^\mu (pμ is the 4-momentum)
(γμ, γ5 are...
I've just started a masters in physics after a 4-year break and am having some real trouble getting back into the swing of things! We have been asked to prove some properties of gamma matrices, namely:
1. \gamma^{\mu+}=\gamma^{0}\gamma^{\mu}\gamma^{0}
2. that the matrices have eigenvalues...
I was wondering if there is a representation of gamma matrices unitarily equivalent to the standard representation for which Dirac Spinors with positiv energy and generic momentum have only the first two component different prom zero. Anyone can help me?
Hi,
The typical representation of the Dirac gamma matrices are designed for the +--- metric. For example
/gamma^0 = [1 & 0 \\ 0 & -1] , /gamma^i = [0 & /sigma^i \\ - /sigma^i & 0]
this corresponds to the metric +---
Does anyone know a representation of the gamma matrices for -+++...
Hi,
After having solved some problems I encountered by using Google and often being linked to threads here, I finally decided to register, especially because I sometimes have problems for which I don't find solutions here and now want to ask them by myself :)
Like the following: I am...
Homework Statement
Given that \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1 where 1 is the identity matrix and the \gamma are the gamma matrices from the Dirac equation, prove that:
\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}*1 Homework Equations...
Hi
I've just read the statement that a matrix that commutes with all four gamma matrices / Dirac matrices has to be a multiple of the identity. I don't see that; can anyone tell me why this is true?
Thanks in advance
Typically I understand that projection operators are defined as
P_-=\frac{1}{2}(1-\gamma^5)
P_+=\frac{1}{2}(1+\gamma^5)
where typically also the fifth gamma matrices are defined as
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
and.. as we choose different representations the projection...
Typically I understand that projection operators are defined as
P_-=\frac{1}{2}(1-\gamma^5)
P_+=\frac{1}{2}(1+\gamma^5)
where typically also the fifth gamma matrices are defined as
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
and.. as we choose different representations the projection...
In most of the physics textbooks I read they only give one or two representations of gamma matrices, but none gives a proof, so how can I prove it from the Clifford algebra?
So since I learning QFT a while ago, I've always struggled to understand fermions. I can do computations, but I feel at some level, something fundamental is missing in my understanding. The spinors encountered in QFT develop a lot from "objects that transform under the fundamental representation...
I need help proving the identity
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}=\gamma^{\mu}g^{\nu\rho}+\gamma^{\rho}g^{\mu\nu}-\gamma^{\nu}g^{\mu\rho}+i\epsilon^{\sigma\mu\nu\rho}\gamma_{\sigma}\gamma^{5}