Hi!
I am studying Dirac's equation and I already have understood the derivation. Following Griffiths, from factoring Einstein's energy relation with the gamma matrices:
##
(\gamma^\mu p_\mu + m)(\gamma^\mu p_\mu - m) = 0
##
You take any of the two factors, apply quantization and you arrive to...
The problem is given in the summary.
My attempt: Assume that ##\psi^\prime (x^\prime)## is a solution of the Dirac equation in the primed frame, given the transformation ##x\mapsto x^\prime :=\Lambda^{-1}x## and ##\psi^\prime (x^\prime)=S\psi(x)##, we have
$$
\begin{align*}
0&=(\gamma^\mu...
The Dirac equation for an electron in the presence of an electromagnetic 4-potential ##A_\mu##, where ##\hbar=c=1##, is given by
$$\gamma^\mu\big(i\partial_\mu-eA_\mu\big)\psi-m_e\psi=0.\tag{1}$$
I assume the Weyl basis so that
$$\psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}\hbox{ and...
Given that the Minkowski metric implies the Lorentz transformations and special relativity, why do the equations of relativistic quantum mechanics, i.e., the Dirac and Klein-Gordon equations, require a mass term to unite quantum mechanics and special relativity? Shouldn't their formulation in...
I derive the quadratic form of Dirac equation as follows
$$\lbrace[i\not \partial-e\not A]^2-m^2\rbrace\psi=\lbrace\left( i\partial-e A\right)^2 + \frac{1}{2i} \sigma^{\mu\nu}F_{\mu \nu}-m^2\rbrace\psi=0$$
And I need to find the form of the spin dependent term to get the final expression
$$g...
Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to
$$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$
My question is how the above mentioned terms can be written as...
When working on the exercise 3.2 of Peskin's QFT, I find one of the calculating steps confused for me. I read the solution, which is showed in the picture. I just don't understand the boxed part.
I know it involved the Dirac equation, and the solution seems to treat the momentum as a operator...
Hello everybody!
I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##.
$$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$
If I consider ##\psi = \begin{pmatrix}...
some notes:
There was actually no proof given why ##u^s(p)## or ##v^s(p)## should solve the Dirac equation, only a statement that one could prove it using the identity
$$(\sigma\cdot p)(\bar\sigma\cdot p)=p^2=m^2.$$
We were using the Wely-representation of the ##\gamma##-matrices, if this should...
I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$
The solution up to first order is
$$ \psi(x) =...
Dirac wanted to fix the problems with the Klein-Gordon equation by seeking a new solution to it.
He wanted a relativistic solution so it makes sense that the solution needed to satisfy Einstein's energy-momentum relation.
But why did it need to be of first order in time- and...
Here I am considering the one particle free Dirac equation. As is known the spin operator does not commute with the Hamiltonian. However, the solutions to the Dirac equation have a constant spinor term and only an overall phase factor which depends on time. So as the solution evolves in time...
Homework Statement
Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and...
I’m attempting to learn QFT on my own and would like to get an idea of just how much I still do not know.
Consider a system consisting only of electrons and for the purpose of this question, pretend that particle creation and annihilation never occur.
QUESTION: Would Dirac’s famous...
I continue to be occupied with the first chapter of Lessons on Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf.
I am on page 24, where they derive equations 1.5.67, which are:
##(\gamma^\mu p_\mu-m)u(p)=0## and...
So I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen, the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf
I am in the discussion of the Dirac equation, on page 21, trying to go from equation 1.5.49 to 1.5.51. And I get stuck.
Equation...
I understand that momentum, rest mass and energy can be put on the sides of a right triangle such that the Pythagorean Theorem suggests E^2=p^2+m^2. I understand that the Dirac equation says E=aypy+axpx+azpz+Bm and that when we square both sides the momentum and mass terms square while the cross...
If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian, see Peskin and Schroeder
$$
H = \int\frac{d^3p}{(2\pi)^3}E_p
\sum_{s=1}^2
\Big(
a^{s\dagger}_\textbf{p}a^s_\textbf{p}...
Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
I am finding references and good books for the conceptual understanding of Breit Interactions. Are there any books which specifically include the topic? The related topics explaining the formulation Breit Hamiltonian and its involvement to the correction to the atomic structure calculation are...
I'm reading the book Quantum Field Theory and the Standard Model by Matthew Schwartz and currently I'm studying the chapter 17 titled "The anomalous magnetic moment" which is devoted to computing the corrections due to QFT to the g factor.
My main issue is in the begining of the chapter, where...
I was reading that one of the successes of the Dirac equation was that it was able to account for the fine structure of some of the differences in the spectrum of the hydrogen atom.
But the Dirac equation is about subatomic particles moving at relativistic velocities. But an electron around the...
Consider the Dirac Lagrangian,
L =\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi,
where \overline{\psi}=\psi^{\dagger}\gamma^{0} , and show that, for \alpha\in\mathbb{R} and in the limit m\rightarrow0 , it is invariant under the chiral transformation...
Graphene's Hamiltonian contains first order derivatives (from the momentum operators) which aren't invariant under simple spatial rotations. So it initially appears to me that it isn't invariant under rotation. From reading around I see that we also have to perform a rotation on the Pauli...
$$i\frac{\partial \phi}{\partial t} = \frac{1}{2m} (\sigma .P)(\sigma .P)\phi + eφ\phi$$
Rewriting the equation by using B = ∇ × A and e = −|e| (electron charge) leads to a Schr¨odinger like equation:
$$i\frac{\partial \phi}{\partial t} =[ \frac{1}{2m} (-i∇ + |e|A)^2 + \frac{|e|}{2m} σ.B - |e|φ...
I just started learning this so I am a bit lost. This is where I am lost http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_7/node1.html .
Why when E>0, we use $$\phi_p=
\begin{pmatrix}
1 \\
0 \\
\end{pmatrix}
$$ or $$
\begin{pmatrix}
0 \\...