# Questions tagged [dirac-matrices]

Dirac matrices, or gamma matrices, are a set of matrices with specific anticommutation relations that generate a matrix representation of the Clifford algebra which acts on spinors, fundamental to the Dirac equation describing spin-1/2 charged particles.

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### Is the sign of the mass in the Dirac action irrelevant? [duplicate]

In even dimensions all the representations of the gamma matrices are equivalent, in particular $\gamma^\mu$ and $-\gamma^\mu$ are equivalent. Usually the Dirac Lagrangian is \begin{equation} \psi^\...
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### How is this relationship obtained? [closed]

How do you get in the following relationship that is related to Dirac theory? $\{γ^5,γ^m\} = 0$
1 vote
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### What's the difference between Lorentz transformation properties of Hermitian and Dirac adjoint lepton doublets?

If the lepton doublet transforms like a left-handed Weyl spinor under Lorentz transformations, $$L \longrightarrow exp\left[\frac{1}{2}(i\theta_j\sigma^j - \beta_j\sigma^j)\right]L = \Lambda_{sL}L,$$...
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### Show that Majorana equation implies Klein-Gordon equation

Show that the Majorana equation $i \bar{\sigma}\cdot\partial\chi -im\sigma^2\chi^* = 0$ for 2-component spinors $\chi$ implies the Klein-Gordon equation $(\partial^2+m^2)\chi$. This is part of an ...
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### Are Dirac spinors always eigenvectors of the helicity?

From my lecture notes I found that the helicity projectors are defined as $$\Pi_{\pm}:=\frac{\mathbb{1}\pm\gamma^{5}\not{n}}{2}$$ and it's written that for instance $\Pi_{+}u_{1}=u_{1}$ where $n$ is ...
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### Is Dirac theory just a real Clifford algebra?

The gamma matrices $\gamma^\mu$ appearing in the Dirac equation span the Clifford algebra ${\cal Cl}_{1,3}$ over real numbers. They are generators of Clifford algebra in that sense that their products:...
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### Bilinear covariants of Dirac field

In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
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### Lorentz invariance of the Dirac equation and implicability of the Klein-Gordon equation from the Dirac equation [closed]

I am reading the Peskin & Schroeder's Introduction to quantum field theory, p.42~43 and don't understand some points. In their book p.42 they say that "To show that it (the Dirac equation) is ...
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### Gauge transformation of an adjoint left-handed Weyl spinor in $\rm SU(2)$ fundamental representation

I have a left-handed Weyl spinor field $\Psi_L$ in the fundamental representation of the $\rm SU(2)$ gauge group, which transforms $\Psi_{L,i} \rightarrow \Psi_{L,i} + i\theta^at_{ij}^a\Psi_{L,j}$. ...
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### Why is $( \alpha_i r_i) (\alpha_j r_j ) = \frac{1}{2} \{ \alpha_i , \alpha_j\}r_i r_j$?

Where $\alpha_i= \left( \begin{matrix} 0 & \sigma_i \\ \sigma_i & 0 \end{matrix} \right)$. To me it should just be $( \alpha_i r_i) (\alpha_j r_j ) = \alpha_i \alpha_j r_i r_j$, but it is not. ...
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### Sanity check for a simple calculation involving Dirac spinors and matricies

I've been doing some research related to the Dirac equation and its solutions. To help make sure I understand what's going on, I've done some simple calculations involving planewave solutions. I'm ...
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### How do I keep track of what to differentiate in a Dirac Hamiltonian/Lagrangian?

Suppose we have the dirac Hamiltonian: $$H = \int d^3y\bar\psi(y)_b(-i\gamma^k\partial_k+m)_{bc}\psi(y)_c.$$ My question is should I think the derivative operator $\partial_k$ is acting on the ...
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### Understanding conclusion of Proof that a matrix that commutes with all gamma matrices is proportional to the identity matrix

I am trying to understand why a matrix $M$ that commutes with all gamma matrices $\gamma^\mu$ is proportional to the Identity matrix. I am following example 3.18 in Voja Radovanovic's book on solved ...
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### How to define the inverse of Dirac Gamma Matrices in QFT?

The Dirac gamma matrices are a set defined by the 16 following matrices: $$\Gamma^{(a)}=\{I_{4x4},\gamma^\mu,\sigma^{\mu\nu},\gamma_5\gamma^\mu,\gamma_5\}.\tag{2.122}$$ Now, I wish to determine the ...
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### How does $\sum_s\left[u_a^s(p)\bar u_b^s(p) + v_a^s(-p)\bar v_b^s(-p)\right] = 2E_p\gamma^0_{ab}$?

For Dirac spinors, we have the spin sums for particles, $$\sum_{s=1,2}u_a^s(p)\bar u_b^s(p) = (\not\!p+m)_{ab},$$ and for antiparticles: $$\sum_{s=1,2}v_a^s(p)\bar v_b^s(p) = (\not\!p-m)_{ab}.$$ ...
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### Dirac Lagrangian not Parity Invariant?

I am following the book 'Quantum Field Theory and the Standard Model(2014)' by Matthew D. Schwartz, and I have some incertitude with regards to some subtleties that I have come across while reading ...
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### Parity invariance of Dirac action

The Dirac action is $$S=\int d^4 x \mathcal{L}(x)$$ where the $\mathcal{L}(x)$ is the Lagrangian density given by $$\mathcal{L}(x)=\bar{\psi}(x)(i\gamma^\mu\partial_\mu-m)\psi(x).$$ In proving the ...
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### Difference between position of indexes in Kronecker delta symbol [duplicate]

I am studying the Dirac gamma matrices and have encountered the Kronecker delta $\delta_{ij}$ That I am accustomed to. However, I have also come across a different form, $\delta_{\mu}^{\nu}$. This ...
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### Where can I find a real representation for 8 dimensional gamma matrices?

I understand that gamma matrices can be real in $d = 8$ and with Euclidean signature, with minimal dimension $16\times16$. Does anybody know where I can find such a representation explicitly written?
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### What are Dirac spinors and why did relativistic quantum mechanics need them?

I have a good grasp of the Schrödinger equation and the basics of special relativity But the Dirac equation is alien to me. What are Dirac spinors and why did Dirac use them? 116 views

### What is the relation between Dirac spinors and qubit spin? [closed]

Dirac Spinors is a 4 element vector, and a qubit state vector is two element vector. Two spinors are positive(1 and 4) and negative values (3 and 2), being the first value the spin up and the second ...
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### Trace manipulation in finding the matrix element squared

Here's the $s$-channel of the scalar-fermion scattering given that $L_{int} = -g\phi\bar\psi\psi$: And I found the matrix element of this channel is Then to find $\sum_{s,s'}|\tilde M_1|^2$: I was ...
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### Derivation of covariant derivative of Spinor

I am following this derivation for the covariant derivative of spinors. I have some questions about this derivation: On page 3 they use the fact, that \begin{align*} V^a(x) = \bar{\Psi}(x)\gamma^a\...
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