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?
$\{γ^5,γ^m\} = 0$
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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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Time reversal symmetry on bilinears
I wanted to check the electroweak standard model Lagrangian is invariant under CPT transformation by first checking how the bilinears of $\bar{\psi} \psi$, $\bar{\psi}\gamma_5 \psi$ $\bar{\psi}\gamma^{...
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Inverting the Propagator [closed]
I would like to know how I can invert the following expression:
$$
S^{\mu\nu}=(\!\!\not{p} -m)\eta^{\mu\nu}+\gamma^{\mu}\!\!\!\not{p}\gamma^{\nu}+m\gamma^{\mu}\gamma^{\nu} \ ,
$$
to get $(S^{−1})^{\mu\...
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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?
user353451
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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 ...
- 309
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votes
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Possible error in Hatfield's "Quantum Field Theory of Point Particles and Strings"?
In problem 8.1 of Hatfield's Quantum Field Theory of Point Particles and Strings, the reader is tasked with calculating the trace $$\operatorname {tr}(\gamma_0\frac{(\not p_2 + m)}{2m}\gamma_0\frac{(\...
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Dirac notation and spin confusion [closed]
In my book I have $$\chi = a \chi_+ + b \chi_- = \begin{bmatrix}a \\ b \end{bmatrix} \tag{1}$$
Also, $$| 1/2, 1/2\rangle = \chi_+$$
The way I see that is that $a \chi_+ =|a/2 , a/2\rangle$, but ...
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Square of invariant matrix element involving the Levi-Civita symbol
Consider the invariant matrix element
$$\mathcal M=2ige\frac{\epsilon^{\mu \nu \rho \sigma}k_{1 \mu}\epsilon_\nu(k_1,s_1)\bar{u}(p_2,r_2)\gamma_\sigma u(p_1,r_1)q_\rho}{q^2}, \quad q=k_2-k_1$$
...
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Why is this form of writing the six antisymmetric gamma matrices correct?
I encountered the following expression in Ashok Das' QFT Lectures:
$$\sigma_{\mu \nu } =\frac{i}{2}[\gamma ^\mu,\gamma^\nu]=i(\eta ^{\mu \nu}-\gamma^\nu \gamma^\mu)=-i(\eta ^{\mu \nu}-\gamma^\mu \...
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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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Dirac spinor field anti-commutation
I am thinking the anti-commutation property of Dirac field! First, note that the equal time anti-commutation relation (from P&S's QFT):
$$\{ \psi_a(\mathbf{x}),\psi_b^{\dagger}(\mathbf{y}) \}=\...
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Commutator with generator of Lorentz group shows transformation property
The generators of the Lorentz group in spinorial representation are
\begin{align}
M^{\mu\nu} = \frac{i}{4}[\gamma^\mu,\gamma^\nu]
\end{align}
where $[\gamma^\mu,\gamma^\nu]$ is the commutator of two ...
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Why a $4\times 4$ double degenerate Hamiltonian can be obtained with Clifford algebra matrices?
in the book - Topological Insulators and Topological Superconductors, B. Andrei Bernevig and Taylor L. Hughes, Princeton University Press (2013), pag 148 - the authors say that in order to have a $4\...
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Trace of product of six Pauli matrices
Using the standard definition of the Pauli matrices with the zeroth included, i.e.
$$ \sigma^{\mu} = (I, \sigma^i) $$
$$ \bar \sigma^{\mu} = (I, -\sigma^i) $$
it's a standard result that
$$ Tr[\sigma^...
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Field shift in Generating functional for the Dirac field
On P&S's QFT page 302, eq.(9.73) defined the generating functional for the Dirac field.
$$Z[\bar{\eta}, \eta]=\int \mathcal{D} \bar{\psi} \mathcal{D} \psi \exp \left[i \int d^4 x[\bar{\psi}(i \not ...
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Plane Wave Relations for Dirac Spinors
I am trying to show the following relationships: $\bar{u}_{\pm p\sigma}\gamma^\mu u_{\pm p\sigma'} = 2p^\mu \delta_{\sigma\sigma'}$, $\bar{u}_{\pm p\sigma} u_{\pm p\sigma'} = \pm 2m\delta_{\sigma\...
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How to understand the completeness of the Dirac spinor and why?
I'm searching around to see why $$\sum u^s\bar{u}^s=(\gamma^\mu p_\mu+mc)$$ and $$\sum v^s\bar{v}^s=(\gamma^\mu p_\mu - mc)$$ is called the completeness relation.
Also wondering the same question for ...
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Euclidean Dirac Lagrangian is NOT Hermitian?
In $\mathbb{R}^4$ with the Euclidean metric, there are two entirely independent Dirac spinors, which I write as $\psi$ and $\overline{\psi}$.
This is because the universal double cover of the ...
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A representation-neutral derivation of $g = 2$ from the Dirac equation
This question is largely based on Thomson's Modern Particle Physics. In the appendix of this book, Thomson uses the Dirac-Pauli representation of the Dirac equation to derive the effect of an external ...
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Dirac propagator in Peskin & Schroeder's Book
I read Quantum Field Theory Book written by Peskin & Schroeder, and when a commutator about Dirac field is compile,
he compile a general commutator:
$$ [\psi_a(x),\overline{\psi}_b(x)] $$ having ...
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Massless QED Vs Massive QED Spin operator
In massless QED we know that the left-handed and the right-handed Weyl spinors are eigenvectors of the Spin operator and obey, hence, an equation of the form
$$\vec{\Sigma}\cdot \hat{p} u_{\pm}(p)=hu_{...
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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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Dirac spinor manipulation [closed]
I'm trying to simplify the following relationship as much as possible:
$$\partial_\mu\bar\psi[\gamma^\mu,\gamma^\nu]\psi$$
where $\psi$ satisfies the dirac equation.
Using the fact that $[\gamma^\mu,\...
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Peskin and Schroeder's QFT eq. (7.88)
On Peskin and Schroeder's QFT book, page 251, the book discussed how things will be changed in $d$ dimensions. For example $g_{\mu \nu}g^{\mu \nu}=d$.
In eq. (7.88), the book gave how Dirac matrices ...
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Connection between Pauli matrices and Quaternions? [duplicate]
reading this sentence from wikipedia:
The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 ...
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Density Matrices in Quantum Mechanics
I have a question about the physical meanings of various matrices expressed in Dirac bra-kets.
I take it that $\frac{1}{2}|A\rangle\langle A| + \frac{1}{2}|B\rangle\langle B|$ can be interpreted as a ...
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Hermitian conjugates of Dirac equation
I am given the following Dirac equations:
$$
(\gamma^{\mu}p_{\mu} - m)u_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(1)
$$
$$
(\gamma^{\mu}p_{\mu} + m)v_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(2)
$$
where u are the ...
- 11
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vote
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answers
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Lorentz invariance and Dirac Slash Notation
I have the following factor inside a scattering amplitude
$$\Big[\bar{u}(p')\not{l}\gamma^{\mu}\not{l}u(p)\Big] \ \Big[\bar{u}(q')\gamma_{\mu}u(q)\Big]$$
where $p$ and $p'$ are the initial and final ...
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Prove that Majorana mass term is Lorentz invariance
I have a homework to prove that using one kind of chirality, let's say left-handed, we can construct a mass term. The argument is to show that this term $\psi^TCP_L\psi$ is satisfy the dimensionality ...
- 11
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votes
0
answers
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Pauli matrices from anticommutator
I want construct the Pauli matrices starting from $$\sigma_i=\begin{bmatrix}
a & b\\
c& d
\end{bmatrix}$$
by using only the anti-commutation relation
$$
\sigma^i\sigma^h+\sigma^h\sigma^i=\{\...
- 205
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votes
0
answers
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Degrees of freedom in a spinor in $d$ dimensions (following Polchinski & Lounesto)
I am working through several texts on spinors and trying to deepen my understanding of this fascinating concept. In many ways I have found Polchinski's great Appendix B of String Theory, volume 2 to ...
- 161
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votes
1
answer
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How to dimensionally reduce the 3+1 D Dirac equation into the 1+1D Dirac equation?
In 3+1D the Dirac equation looks like
$$i\partial_\mu \gamma^\mu \Psi -m\Psi=0.$$
If we only consider $x$-direction, then it should reduce to
$$i\partial_t\gamma^0\Psi =(-i\partial_x\gamma^1+m)\Psi.$$
...
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