Questions tagged [dirac-equation]

A fully relativistic (Lorentz covariant) description, first put forward by Paul Dirac in 1928, of the first quantized, spin one half fermion with nonzero mass. Physical notions to do with this equation include the Dirac sea, Dirac hole theory, the Klein Paradox and the fine structure of the Hydrogen spectrum.

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Product of an odd number of Dirac $\gamma$ matrices

Suppose $n$ is an odd number. Why can we write $a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n$ as $$a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n = V_\mu \gamma^\mu + A_\mu \gamma^\mu \gamma_5$$ for some $V_\mu, ...
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Canonical commutation relations for Majorana spinor fields

For a Dirac spinor field $\psi$ described by the Lagrangain $ \mathcal{L} = \bar{\psi}( i \gamma^\mu \partial_\mu - m )\psi $, where $\bar{\psi} = \psi^\dagger \gamma^0$, the canonical momenta is ...
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Electrons and holes vs. Electrons and positrons

Drawing parallels between electrons and holes in semiconductors, and electrons and positrons in Dirac equation is certainly useful in the context of learning/teaching the quantum field theory methods, ...
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Is there a “non-linear limit” of the Dirac equation?

I'm just going through old protocols of oral exams students wrote up. One student writes that he was supposed to derive the "non-linear limit" of the Dirac equation during the exam. Is there ...
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Electron neutrinos propagating in the Sun dispersion relation $ E \simeq|\boldsymbol{p}| \pm \sqrt{2} G_{F} n_{e} $

Given the following effective lagrangian describing electron neutrinos propagating in the Sun and scattering via charged current processes with a background of non-relativistic electrons $$ \left[i \...
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Eigenspinor of helicity of electrons

I am reading the chapter in Griffth's introduction to elementary particle. By solving the momentum space Dirac equation and requiring the solution of the spinor to be the eigenspinor of the helicity ...
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Positron in the vicinity of a proton

I was searching for a plane-wave-like decomposition of the electron-positron field operator in QFT around a proton, and I came with this decomposition: \begin{equation} \Psi=\sum_\eta \left( a_\eta F_\...
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Are the $\alpha$ and $\beta$ matrices of the Dirac equation unique?

Assume we are dealing wth three spatial dimensions $d=3$ which requires 3 $\alpha$ matrices. Furthermore assume that we are looking for them in the space of 4-dimensional matrices, not in higher ...
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Why is the concept of state kets not used in relativistic quantum mechanics?

In Sakurai's Quantum Mechanics the concept of a Hilbert space underlying classical quantum wave mechanics (Schrödinger equation) is extensively developed. But when dealing with the Dirac equation this ...
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Why do covariantly constant spinors on the circle require periodic boundary conditions?

I am asking in the context of a paper of Witten on instability of Kaluza-Klein spacetimes (https://www.sciencedirect.com/science/article/pii/0550321382900074). The discussion involves applying Witten'...
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Explicit expansion of the term $\overline{\psi}(i \gamma_\mu \partial^\mu-m) \psi$

Explicit expansion of the term $\overline{\psi}(i \gamma_\mu \partial^\mu-m) \psi$ In QED, one finds the first part of the Lagrangian density to be $\mathcal{L}=\overline{\psi}(i \gamma_\mu \partial^\...
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How do I obtain the Dirac equation from the Euler-Lagrange equation?

Knowing that the free Dirac Lagrangian is : $$\tag{1} \mathcal{L}= \bar{\psi} (i \gamma^\mu \partial_\mu -m ) \psi$$ and that the Euler-Lagrange equation is: $$\tag{2} \frac{\partial \mathcal{L}}{\...
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Normalization for a free Dirac plane wave

I've recently come about the free plane wave solutions to the Dirac equation, and i'm having a hard time proving that the normalization factor $n$ is $$n=\frac{1}{\sqrt{2m(m+\omega)}}$$ Where $\omega$ ...
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Minimal coupling in Dirac equation

In the framework of relativistic quantum mechanics (not QFT) the Dirac equation in presence of external electromagnetic field is obtained by means of the minimal coupling, i.e. the substitution: $$p_{...
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Form of Dirac equation in curved spacetime and its derivation

In this paper and another, the Dirac equation for an uncharged fermion is written as $$i\gamma^{\mu}D_{\mu}\psi + \frac{m}{\hbar}\psi=0 $$ where $D_{\mu} = \partial_{\mu} + \Omega_{\mu}$, $\quad \...
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Dirac equation solution - four-component spinors - left-/right-handed in ultrarelativistic limit

I am confused about the solution to the Dirac equation and how it corresponds to left-/right-handed Weyl spinors. In Srednicki, page 242, it is stated that taking the ultrarelativistic limit ($|\bar{p}...
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Why is $\overline{\Lambda_s \psi} = \bar{\psi} \Lambda_s^{-1}$?

In my text Schwartz writes: since $\bar{\psi} \gamma^\mu \psi$ transforms like a 4-vector, we can deduce that $$\Lambda_s^{-1} \gamma^\mu \Lambda_s = (\Lambda_V)^{\mu \nu} \gamma^\nu, \tag{10.78}$$ ...
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Is this a typo in Peskin's QFT?

In ''An intro to QFT (2018)'' chapter 3, Peskin does the following: Let me introduce some notation first, let $v^s_k=\begin{pmatrix}\;\;\,\sqrt{k\cdot\sigma}\,\xi^{-s}\\-\sqrt{k\cdot\bar{\sigma}}\,\...
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Commutator of momentum and velocity operator in relativistic quantum mechanics

I am currently reading up on Dirac's 'The Principles of Quantum Mechanics'. It is stated there, that for the motion of a free particle with Hamiltonian \begin{equation}H=c(\boldsymbol{\alpha},\...
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Space-reflected solution of the Dirac equation

In Bjorken & Drell vol. 1 "Relativistic quantum mechanics", the parity operator acting on solutions of the Dirac equation is represented as $P=e^{i\phi}\gamma^{0}$, where $\phi$ is a phase. Let's ...
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Solutions to Dirac equation with Gauge field coupling

Looking at the Dirac equation of the form $$\Big(i\gamma^{\mu}(\partial_{\mu}-iA_{\mu})-m\Big)\psi=0$$ There is a simple solution to this equation, which is $$\psi=\exp\Big(i\int^xA_{\mu}dx^{\mu}\...
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Klein-gordon, Dirac equation and the spin of the particles [closed]

This question might seem basic, but how does one conclude that the Klein-Gordon equation describes spin zero particles but Dirac equation describes spin half particles. Thanks. EDIT: Adding more ...
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How do we know there are only 16 Dirac bilinears?

We know there are 5 types of bilinears in 4 dimensions, all of them add up to contribute with 16 independent DoF (degrees of freedom). Namely, these bilinears are known as: scalar (1DoF), pseudoscalar(...
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Spinor transformation matrix derivation

The spinor in the Dirac equation should transform via a 4x4 matrix $S$ that depends on the specific Lorentz boost/rotation: $\psi '(x')=S(\Lambda )\psi(x)\tag1$ Where S satisfies: $S^{-1}\gamma ^{\...
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Evaluating $\sigma^{\mu\nu}F_{\mu\nu}=i\alpha \cdot E+\Sigma\cdot B$ matrix, spin dependent term in quadratic Dirac equation

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\...
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Observables of Dirac equation

So I learned about the Dirac equation which describes a relativistic free particle with spin $\frac{1}{2}$. I get the mathematics but what i can't find nowhere: What are the observables of this ...
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Plane Wave Spinor solutions in 2+1 dimensions

I have read that if we use 2+1 dimensional space instead of the standard 3+1d space used in quantum mechanics, we can write the Dirac equation as $$i \hbar \frac{\partial}{\partial t} \psi(r, t)=\...
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Is the $U(1)_A$ axial vector current even under charge conjugation?

The axial current of a Dirac spinor is given by $j_A^\mu = \bar{\psi} \gamma^5 \gamma^\mu \psi$. In this book, in the paragraph under equation (2.18) it is stated that the current is even under charge ...
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Does the $U(1)$ vector current flip under charge conjugation?

The conserved $U(1)$ current of the Dirac Lagrangian is given by $j^\mu = \bar{\psi} \gamma^\mu \psi$, where $\bar{\psi} = \psi^\dagger \gamma^0$. As this is interpreted as electric current I would ...
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Spinor expansion [closed]

does anyone know how you get this expansion when expanding terms of 4-momentum to linear order, type of expansion is it? Many thanks.
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Hermitian Adjoint of Dirac Equation vs Dirac Lagrangian

I have a question about the self-adjointness of the gradient in spinor space. In the derivation of the Dirac adjoint equation, as in Hermitian adjoint of 4-gradient in Dirac equation , it has been ...
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Dirac action momenta conjugate to conjugate field

Consider the Dirac action $S=\int d^4x\bar{\psi}(x)(i\not\partial-m)\psi(x)$. Since there are no time derivative of $\bar{\psi}$, we get the constraint that its canonical momenta vanishes. This ...
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Why did Dirac choose a linear equation in momentum for formulating a relativistic wavefunction?

The Dirac equation arose out of the need for a quantum mechanical wave equation that treats position and time on an equal basis, just as relativity expects them to be. Now the Schrodinger equation is ...
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Hydrogen relativistic energy levels: Why is this operator equivalent to this matrix?

I am trying to understand the derivation of the relativistic energy levels for the Hydrogen following the annex 2.2 of this notes. Until the equation below I could follow the reasoning: $$\left [-\...
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Spinor method in massless limit

I have this problem where I'm asked to derive an explicit solution for the Dirac equation of massless fermion $p_\mu \gamma^\mu u(p)=0$. I'm instructed to do so by writing $p_\mu \gamma^\mu$ in the ...
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What is the deep difference Between a hole in Dirac sea and an empty orbit around nucleus?

Dirac sea is proposed to reasoning stability of electrons in positive energies. holes that could be occupied by electrons or not (but almost all of them are occupied). Then Dirac propose such a hole ...
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Understanding the Dirac equation

I am reading a physics book where the Dirac equation is being introduced in the form: $$\left[c \boldsymbol{\alpha} \cdot\left(\boldsymbol{p}+\frac{e \boldsymbol{A}}{c}\right)-e \phi+\beta m c^{2}\...
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Dirac equation: covariant form and original form

In my lecture book, the Dirac equation is derived and given as the equation: $$i \hbar \gamma^{\mu} \partial_{\mu} \psi-m c \psi=0 \tag{1}$$ Where: $$\gamma^{0}=\left(\begin{array}{rr} 1 & 0 \\ 0 ...
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Writing the Klein-Gordon in terms of the gamma matrices

Hello I have a quick question regarding the Dirac gamma matrices or Dirac equation and the Klein-Gordon equation. Recall that the Klein-Gordon is given by the following \begin{equation} \left(\Box + ...
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Functional Analytic Square Root of Hamiltonian Alternative to Dirac

I was thinking about the history of the Dirac equation and asked myself, what happens if one simply considers the Schrödinger equation $i\hbar\frac{\partial\phi}{\partial t}=\sqrt{-c^2\hbar^2\Delta+m^...
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Dirac sea interpretation VS the Feynman-Stueckelberg interpretation for antiparticles

I am trying to understand the difference between the Dirac sea interpretation and the Feynman–Stueckelberg interpretation of the negative energy solutions of the Dirac equation. To do so, I would like ...
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Is this an alternative Dirac Equation in curved space?

The usual covariant derivative for the Dirac equation in curved space is: $$D_\mu \psi = (\partial_\mu - {i \over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$ However, I think I found another ...
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Why the imaginary constant on the spin-connection?

I have been studying the spin-connection for the Dirac equation. The covariant derivative is defined as: $$D_\mu \psi = (\partial_\mu - {i \over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$ where $\...
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This pseuso vector equation leads to strange results

Consider this Dirac equation with a pseudo vector field: $$(\gamma^\mu (i\partial_\mu + \gamma^5 B_\mu(x) )+m)\psi(x)=0$$ Now $i\partial_\mu$ can be replaced with the momentum $k_\mu$. But now, ...
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Covariance of Dirac Equation [closed]

In Bjorken and Drell "Relativistic Quantum Mechanics", problem 3 ch.2: Given a free-particle spinor $u(p)$, construct $u(p+q)$ for $q \rightarrow 0$, with $p\cdot q \rightarrow 0$, in terms of $u(p)$...
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What is the old (50's) convention on Dirac gamma matrices?

What were the standard relations for gamma matrices in the mid 50's, when 4-vectors where represented by $(x_1, x_2, x_3, ict)$? In particular the values of $\gamma^\mu\gamma^\nu$ , the definition of $...
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Spectrum of Dirac Hamiltonian

The Dirac Hamiltonian is given by, \begin{aligned} H &=\sum_s\int \frac{d^{3} p}{(2 \pi)^{3}} E_{\vec{p}}\left[b_{\vec{p}}^{s \dagger} b_{\vec{p}}^{s}+c_{\vec{p}}^{s \dagger} c_{\vec{p}}^{s}\right]...
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Partial derivatives in the derivation of a Dirac Spinor

As per JF132's answer to Conservation of the axial current using Dirac equations of motion, "since the gamma matrices $\gamma^\mu$ are $4\times 4$ matrices, and the conjugate Dirac spinors $\bar{\...
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Does a rotating black hole change a left-handed neutrino into a right-handed neutrino?

The Dirac equation in a gravitational field contains a torsion term $\overline{\psi}\gamma^\mu\omega^{nm}_\mu \sigma_{nm}\psi$. In the Weyl represnetation the spinor $\psi$ can be written as $\psi=(\...
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Dirac's equation, boosts and rotations

If we consider Dirac's equation in two different frames of reference $$\left(i\gamma^{\mu} \partial_{\mu}-m c\right) \psi(x)=0,$$ $$\left(i\gamma^{\mu} \partial_{\mu}^{\prime}-m c\right) \psi^{\prime}\...

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