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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Solutions of 1D-Dirac equation for free particle only have positive energy solutions?

Im studying the (stationary) free 1D-Dirac equation $H\Psi(x) =[mc^2\sigma_1-i\hbar c\sigma_3\frac{\partial}{\partial x}]\Psi(x) = E\Psi(x)$. Where $\sigma_1$ and $\sigma_3$ are the pauli matrices. I ...
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Angular-momentum of the Dirac spinor theory

The standard Dirac action $$ S = \int d^4 x \bar \psi (i \gamma^\mu \partial_\mu - m) \psi $$ is invariant under Lorentz transformation. In David Tong's lecture note, eq (4.96) lists that the ...
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Second-order Dirac equation

I'm wondering if one of you could tell me about the following equation: $$\partial_t \Psi = i \sigma_z m - \sigma_y k \partial_x \Psi + i \sigma_y k' \partial_{xx}\Psi$$ where $m, k,k'$ are real ...
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How to find the fine structure constant? [closed]

Is given in the scalar field below the lagrangin. How to find the finite structure constant?
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Normalization of solution of Dirac equation

I know that the solution to the dirac equation are of the form: $\psi(x)=u(\vec{p})e^{ip\cdot x}$ and the spinor can be normalized as $u^\dagger u =E$. I was reading "Lectures on Quantum Field ...
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Can we say that the following equation has substantially the same physical meaning as the Dirac equation?

Dirac's equation is given by $$\left[\beta mc^2-i\hbar\frac{\partial}{\partial t}+c \sum ^3_{n=1} \alpha_n p_n\right]\psi(\vec{x},t)=0\tag1$$ Here, $\alpha_n$ in Eq. (1) is an operator having a ...
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Why is the anticommutation relation for the Dirac field between fields? [duplicate]

The commutation relation for neutral Klein Gordan field is $$[\phi(x,t),\pi(x',t)]=i\delta^3(x-x')$$ with all other commutators zero; The commutation relation for charged Klein Gordan field is $$[\phi(...
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Why doesn't the Hydrogen atom, as described by the Dirac equation, collapse?

In Griffiths quantum mechanics, it's noted that the exact energies for the Dirac equation, involving fine structure, are $$E_{nj} = mc^2 \left\{ \left[1 + \left(\frac{\alpha}{n-(j+1/2)+\sqrt{(j+1/2)^2 ...
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Problem with derivation of the Dirac Hamiltonian

I'm having son trouble when obtaining the Dirac equation. I am working in (1+1)-dimensional Minkowski spacetime with signature $(-, +)$ in coordinates $(t, x)\equiv(1, 2)$. I can think of two ways to ...
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What is the correct way of looking at the Dirac field?

All quantum fields are operators in QFT. However, the Dirac field operator $\hat{\psi}$ has the following difference with the scalar field operator $\hat{\phi}$: For the $\hat{\psi}$, it makes sense ...
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How to calculate surface states in Weyl semimetals?

I'm reading an article https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235127. Fig. 2 in this article shows band structures calculated from Eq. (9), (13), (14), (15), and (16). For example, ...
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Energy-momentum relation for Dirac spinor in curved spacetime

Consider the Dirac equation in curved spacetime \begin{equation} (i\gamma^\mu\nabla_\mu+m)\psi=0 \end{equation} where $\gamma^\mu=e^\mu_a\gamma^a$ is the curved spacetime Gamma matrices, and the ...
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Properties of rest-frame spinors

The four rest-frame spinors ${\displaystyle u^{(s)}\left({\vec {0}}\right),}$ ${\displaystyle \;v^{(s)}\left({\vec {0}}\right)}$ satisfy $${\displaystyle ({p\!\!\!/}-m)u^{(s)}\left({\vec {p}}\right)=0}...
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Why do scalars and fermions have a different result in a Lagrangian?

Consider the Lagrangian for Yukawa theory: $$ \mathcal{L} =i\bar{\psi}\not{\partial}\psi- \bar{\psi}m_F \psi +\frac{1}{2} \partial_\mu \phi \partial^{\mu} \phi - \frac{1}{2}m_s^2 \phi^2 + \mathcal{L}_{...
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How could the neutrino be a Dirac particle

I'm reading up about neutrinos now and I understand the possibility of neutrinos being Majorana particles and further theories can be thought of from that (like the seesaw mechanism). I'm still very ...
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Srednicki eq. (1.27): $\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}$

Srednicki, QFT, p. 8 writes $$\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}\tag{1.27}.$$ What does exactly $ab$ here denote? Assume I have a matrix X [0 1] [2 3] and does a ...
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Does the Schrödinger equation apply to spinors?

I was reading about Larmor precession of the electron in a magnetic field in Griffiths QM when I came across the equation $$ i\hbar \frac{\partial \mathbf \chi}{\partial t} = \mathbf H \mathbf \chi, $$...
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Does the classical electron field exist or not? [duplicate]

On one hand, Dirac equation is supposed to be the equation of a classical field that we quantise. On the other hand, I saw many stackexchange posts saying that the classical limit of the electron ...
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Witten on the hermitian of the Dirac operator

I happened to read Witten SU(2) anomaly paper (1982), and come back to digest again what I said the hermitian of the Dirac operator. According to Prahar https://physics.stackexchange.com/a/701287/...
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Time-reversal transformation acts on the Dirac lagrangian with nonabelian gauge field

Time-reversal transformation acts on the Dirac lagrangian with (non)abelian gauge field Since earlier in Time-reversal transformation acts on the Weyl lagrangian with nonabelian gauge field, we ...
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Time-reversal transformation acts on the Weyl lagrangian with nonabelian gauge field

I would like to show time-reversal transformation acts on the Weyl lagrangian in the familiar 4 dimensional space-time. My notation follows the same as Peskin QFT book, such as that of chapter 3. I ...
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Is the Dirac operator $i \not D$ or $\gamma^0 i \not D$ and nonabelian gauge field $A_\mu = A_\mu^\alpha T^\alpha$ hermitian?

In quantum mechanics, it is common to write the momentum operator $$P = i \partial_x.$$ It turns out that $p$ is hermitian although $i^\dagger = -i$ we also have $\partial_x ^ \dagger=-\partial_x$. It ...
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Got stuck on Dirac's equation related by Lorentz transformation

Currently I'm reading the book Quark & Leptons : An introductory course in modern particle physics by F. Halzen & Alan D. Martin. And have been on Chapter 5. Dirac's Equation. In section 5.6 ...
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Is $\bar{\psi} \psi$ its own complex (*)? transpose ($T$)? hermitian conjugate ($\dagger$)?

Related to this earlier A common standard model Lagrangian mistake? Here I am treating Dirac equation of Dirac field as QFT. You may want to consider the quantized version or the classical version. ...
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A common standard model Lagrangian mistake?

A common standard model lagrangian is written in a cup like this. It appears in many places also on a T shirt. But isnt that there is an obvious mistake? That the Dirac lagrangian is already itself ...
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Why didn't the Klein-Gordon equation suggest antimatter like the Dirac equation did?

I have heard the story that the Dirac equation suggested the existence of antimatter due to the existence of negative energy solutions. The Klein-Gordon equation also has negative energy solutions. ...
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About Einstein's sum rule and Dirac equation

I am studying the Dirac equation and I'm having some trouble about something that I think should be trivial. I'm working in a (1+1)-dimensional Minkowski spacetime with signature $(+, -)$, i.e., $ds^2=...
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Charge Conjugation of Dirac equation

In contituation of this question In answers of this question people mentioned charged conjugation and formula below $\bar{\psi}\gamma^\mu\psi=u^2-v^2$ With $u$ for particles and $v$ for antiparticles ...
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Initial value formulation of Dirac equation for spin $>1$ - modern resources?

At the end of Wald Section 13.2, he mentions that the natural generalization of Dirac equation to curved spacetime does not have a well-posed initial value formulation for $s>1$ , and refers to ...
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How many field components are there in vector-spinor field?

I am trying to find out the degrees of freedom of the vector-spinor field ($s=3/2$). The degrees of freedom are given by $N=\frac{1}{2}\left(N_{F}-N_{C}\right)$ for this spin where $N_F$ is the number ...
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What is the meaning of a propagator of a Dirac field and how to get a probability of a process from it?

Let me first present what is my understanding of a propagator. What we measure in the experiment is a probability of scattering. We try to construct a theory predicting these measurements. What we are ...
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A problem with QED

I have a small problem with the understanding of QED. The equations of motion in QED are $\square A^\mu=e\bar{\psi}\gamma^\mu\psi$ $\left(i\gamma^\mu\partial_\mu-m\right)\psi=e\gamma^\mu A_\mu\psi$ If ...
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Symmetry arguments and derivation for product of gamma matrices and derivatives

I am trying to work with the Dirac equation and the solution for the Klein-Gordon equation for some derivation and I stomped on the following problem in my derivation. $\gamma^{\mu} \gamma^{\nu} \...
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Derivation of the Binding Energy of Atoms without Kinetic Considerations and Relativistic Corrections

The question Relativistic corrections to the binding energy of atoms is answered with a brief chronology of ideas about the binding energy of the electron to the atomic nucleus. (The emphasis in the ...
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What is the deep connection between the two ways to get Pauli matrices?

The first way is the one in which we start with the commutator relations $[J_x, J_y]=ihJ_z$, etc. We consider the simultanelus eigenbasis of $J^2$ and $J_z$ : $|j,m\rangle$. We then obtain their ...
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Simple time evolution for quantum relativistic particle

I am considering the time evolution of a relativistic particle in 1D, with the time-evolution governed by the equation: \begin{eqnarray} \left(\begin{matrix} \mathrm{i}\partial_{x} && m_{0}\...
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Does the Dirac equation put charge and spin on the same footing?

I read somewhere that, when represented as 4 complex numbers, the wavefunction in the Dirac equation can be thought of as the respective probabilities of (1) spin up electron, (2) spin down electron, (...
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Parity of object made up of Dirac spinors and gamma matrices

I am reading Introduction to Elementary Particles by Griffiths, specifically the chapter on Dirac equation. Griffiths states without proof, that the expression $\bar{\psi}\gamma^\mu\gamma^5\psi$ is a ...
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Why is helicity in this context well-defined?

I am reading these notes on helicity. One of the definitions (see page 5) that is used is the following (here $J^{a}=\bar{\psi} \gamma^{a} \psi$ is the probability current and $K_{a}=\bar{\psi} \...
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Dirac spinor definition

is it right to say that the Dirac spinor is a mathematical representation of a wave-function that satisfy the Dirac equation? or are there more requirements to it?
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Issue in deriving non-relativistic Dirac Equation

In natural units, the Dirac Equation is $$i \frac{\text{d}}{\text{d}t} \psi = \left[\vec \alpha \cdot \vec P +\beta m + e \Phi\right]\psi.$$ I use Pauli-Dirac basis for matrices, \begin{align*} \...
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(L&L vol. 4, sec. 39) Relativistic Correction to the Wavefunction of an Electron in a Coulomb Field

At the end of section 39 of Quantum Electrodynamics by Landau & Lifshitz there is a problem which obtains the relativistic correction to the wavefunction of an electron in a Coulomb field (in the ...
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Current in the Dirac equation

In the Dirac Hamiltonian the current that couples with the vector potential is: \begin{equation} j^{\mu} = \bar{\psi}(x)\gamma^{\mu}\psi(x) \end{equation} However, in the non relativistic context, the ...
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Quantum Field Theory Unitary Transformations

I am currently reading through Itzyskon and Zuber for my quantum field theory class, and I came across this regarding the unitary transformations of the Dirac bispinors in chapter 2. They show that ...
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Obtaining Dirac spectrum on unorientable manifold ($RP^n$) from orientable manifold

The Dirac spectrum for $S^n$ is well known along with its multiplicities. In Appendix D of https://arxiv.org/abs/1510.05663 author computes dirac spectrum of $RP^4$ from that of $S^4$. The argument ...
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Relation between rank-2 antisymmetric tensor and other bilinear covariants

Given a spinor $\psi$, if one defines the bilinear covariants $J=\bar{\psi} \psi$, $J_{5}=i \bar{\psi} \gamma_{5} \psi$, the current $J_{\mu}=i \bar{\psi} \gamma_{\mu} \psi$, the axial current $J_{5 \...
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Error showing the invariance of Dirac equation

Error showing the invariance of Dirac equation Starting from the following equation: $$a’\Psi’(x’)-\partial' \Psi'(x')=0 \tag{1}$$ Where, $a’=a$ is a constant, $\Psi’=A_{4 \times 4}\Psi$, is a spinor ...
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What is the relation between the wave function in Born's rule and the wave function in Dirac's equation? [duplicate]

The wave function for spins without position can be seen as a complex wave vector $\psi=(\psi_1,\psi_2,\ldots)$ and the probability to measure a state $\psi^{(A)}$ in another state $\psi^{(B)}$ is $$ ...
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Lowering the spacetime index of a Dirac matrix

$\gamma_\mu\partial^\mu$=$\gamma^\nu\partial_\nu$ Does the above equation hold for Gamma matrices? If so, why?
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Is Dirac equation valid only for spin-$\frac{1}{2}$ particles?

It is usually said that Dirac got his equation by looking for the square root of the 4-momentum norm (see Dirac’s coop here). The relativistic 4-momentum norm is $$(E)^2-(\mathbf{p}c)^2=(mc^2)^2 \tag{...
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