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.
I'm reading an article (http://prb.aps.org/abstract/PRB/v82/i4/e045122) but I have some problems understanding certain definitions. The authors have introduced a basis of certain bands (four) and then continue to give the transformation matrices of the symmetry operators. One (rotation) of them...
Dear guys,
I know that gamma matrices have some relations, like
\gamma^0{\gamma^\mu}^\dagger\gamma^0 = \gamma^\mu \quad---(*)
And I am wondering if this is representation independent?
Consider,
S\gamma^0S^{-1}S{\gamma^\mu}^\dagger S^{-1}S\gamma^0 S^{-1} = S\gamma^\mu S^{-1}...
Hi all,
I'm interested in proving/demonstrating/understanding why the Dirac gamma matrices, plus the associated tensor and identity, form a complete basis for 4\times4 matrices.
In my basic QFT course, the Dirac matrices were introduced via the Dirac equation, and we proved various...
I'm trying to show that the generators of the spinor representation:
M^{\mu \nu}=\frac{1}{2}\sigma^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu]
obey the Lorentz algebra:
[M^{\mu \nu},M^{\rho \sigma}]=i(\delta^{\mu \rho}M^{\nu \sigma}-\delta^{\nu \rho}M^{\mu \sigma}+\delta^{\nu \sigma}M^{\mu...
Hi
My QFT course assumes the following notation for gamma matrices:
\gamma ^{\mu_1 \mu_2 \mu_3 \mu_4} = {\gamma ^ {[\mu_1}}{\gamma ^ {\mu_2}}{\gamma ^ {\mu_3}}{\gamma ^ {\mu_4 ]}}
what does the thing on the right hand side actually mean? Its seems to be a commutator of some sort.
Dear guys,
I read a derivation of the dimension of gamma matrices in a d dimension space, which I don't quite understand.
First of all, in d dimension, where d is even.
One assumes the dimension of gamma matrices which satisfy
\{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu\nu}...
I'm very confused
By performing a lorentz transformation on a spinor \psi\rightarrow S(\Lambda)\psi(\Lambda x) and imposing covariance on the Dirac equation i\gamma^{\mu}\partial_{\mu}\psi=0 we deduce that the gamma matrices transform as
S(\Lambda)\gamma^{\mu}...
It is well known that at times we do need explicit representations for the Dirac gamma matrices while doing calculations with fermions. Recently I found two different expressions for Majorana representation for the gamma matrices. In one paper, the form used is:
\gamma_{0} = \left(...
Homework Statement
Show that tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) = 0
Homework Equations
(anti-)commutation rules for the gammas, trace is cyclic
The Attempt at a Solution
I can do
tr(\gamma^{\mu}\gamma^{\nu}\gamma^{5}) = -tr(\gamma^{\mu}\gamma^{5}\gamma^{\nu}) = -...
Hello everybody,
I have to calculate the matrix element of the process gg-->ttbar-->lnub lnub (ttbar dileptonic decay) using FORM.
I have three feynman dyagrams for such a process. When I calculate the interference term i have as output thousands of terms with Levi civita tensors inside (but...
Hi everyone,
From the condition:
\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu} = 2g_{\mu\nu}
how does one formally proceed to show that the objects \gamma_{\mu} must be 4x4 matrices? I unfortunately know very little about Clifford algebras, and for this special relativity project...