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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49 views

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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Commutation relation for Dirac field

In "Quantum Field Theory" by Peskin and Schroeder, I couldn't understand the commutation relation calculation for Dirac field (pg. 53): $$ \begin{align} \psi(x) &= \int \frac{d^3p}{(2\pi)^3} ...
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In the Dirac equation, if the $\alpha$ is the mean velocity, why does it commute with $x,y,z,t$ if the velocity is related to the momentum?

In the Wikipedia talk page for the Dirac equation I found the following passage: The Dirac equation can be proved with the help of the correspondence principle. The energy and momentum of a ...
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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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What are the theoretical / mathematical problems in discarding negative solutions of Dirac equation?

I read some Q&A about it, but my question is why Dirac was so sure that he could not discard negative energy solutions. It seems so natural that energy must be positive, that I suppose that if ...
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Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: $$\...
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Relativistic corrections to first-principles Hamiltonian?

In quantum treatments of solids it is common to start off discussions by writing down the "full" first-principles Hamiltonian for a group of electrons and nuclei as $$H = \sum_i \frac{\hat{p}_i^2}{...
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Conservation of the axial current using Dirac equations of motion

Dirac equation: $$(i\gamma^\mu\partial_\mu -m)\psi=0$$ implies $$\gamma^\mu\partial_\mu \psi=-im\psi$$ Adjoint Dirac equation: $$(i\gamma^\mu\partial_\mu +m)\overline{\psi}=0$$ implies $$\...
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Lorentz covariant propagator

So the Feynman propagator for a Klein Gordon is manifestly Lorentz invariant clearly by looking at the momentum space representation written in terms of Lorentz scalars. But in the case of the Dirac/ ...
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Going from the Dirac Lagrangian to the adjoint Dirac equation

I am comfortable doing the following calculation, Derivation of the adjoint of Dirac equation, notably — going from the standard Dirac equation to the adjoint Dirac equation via using Dirac ...
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Dirac adjoint of a gamma matrix product

I was wondering if, because for generic bounded operators, anti-distributivity applies, i.e. $$(AB)^{\dagger} = B^{\dagger}A^{\dagger},$$ the same is true of gamma matrices. I was asked to prove $$\...
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How to derive the general expression for eigenvalue for the square of Pauli Lubanski operator?

After some trials, I managed to get the correct eingenvalue $(\frac{-3}{4}m^2)$ for $W^2$, where $W$ is the Pauli Lubanski pseudo vector. The expression for each $J^{\mu \nu}$ is a sum of a 4x4 ...
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Why is it hard to extend the Feynman Checkerboard to more than 1+1 dimensions?

The Feynman Checkerboard Wikipedia article states: "There has been no consensus on an optimal extension of the Chessboard model to a fully four-dimensional space-time." Why is it hard to extend ...
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Can the Dirac Hamiltonian accommodate a variable speed of light?

The Dirac Hamiltonian has the form1 $$\left[\beta m c^2+c\sum_{n=1}^3\alpha_np_n\right]$$ where $\alpha_n$ and $\beta$ are Hermitian matrices, and $c$ is the speed of light. My question: Is there a ...
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What do we mean by a Free-particle solution to Dirac equation?

I read this part in Peskin,QFT. And I'm confused about the terminology "free-particle solution", I don't know what does it mean. If by free-particle it means no interaction, then the Lagrangian from ...
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Separation of Klein-Gordon-/Dirac-equation (Bohmian-mechanics)

With the function $R{ e }^{ \frac { i }{ \hbar } S }$ one can separate the Schrödinger equation $$i \hbar \frac{\partial \psi}{\partial t}=\left(-\frac{\hbar^{2}}{2 m} \nabla^{2}+V\right) \psi$$ into ...
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Quick question on Deriving Klein–Gordon equation from Dirac equation

On page 172 of Schwatz’s QFT book, he derives the Klein–Gordon equation from Dirac equation as following: $$(i \not\partial +m) (i \not\partial -m)\psi=(-\frac{1}{2} \partial_\mu \partial_\nu {\gamma^...
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Interpretation of Dirac Spinor components in Chiral Representation?

I failed to find any book or pdf that explains clearly how we can interpret the different components of a Dirac spinor in the chiral representation and I'm starting to get somewhat desperate. This is ...
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Some questions about Dirac's sea? [closed]

Did Paul Dirac develop some way to include bosons in his formulation of the sea of particles? I have read that both electrons and anti-electrons would follow the same Dirac equation. But could there ...
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How does the negative energy solution to the Dirac equation predict the antielectron?

Please, can someone explain how the negative energy solution can be used to predict the existence of the antielectron?
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Second order relativistic corrections to Pauli equation from Dirac equation

I'm trying to derive the full and correct Hamiltonian for spin$\frac{1}{2}$ particles from Dirac equation up to second order in $v/c$. For a potential and magnetic field constant in time. In ...
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Two difficulties in Dirac's derivation of the velocity and spin of a particle

(i) In Dirac’s book, The Principles of Quantum Mechanics, there is something quite baffling about the observed velocity of a particle. On p. 262 of the fourth edition, he wrote, 'We can conclude that ...
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How do the Weyl spinors differ from dotted and undotted spinors? [duplicate]

As asked in the title, how do the Weyl spinors $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ differ from dotted and undotted spinors?
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Possible to have Dirac AND Majorana mass?

Supposing you have a lagrangian consisting of $(1/2,0)\oplus (0,1/2)$ representation. Writing in terms of Weyl fermions, the following terms are possible: $$-\frac{m_1}{2} (\psi_R^T \epsilon \psi_R -...
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Antimatter and quantum mechanics

This question could have a very simple answer but I could not find that answer anywhere. My question is since electrons, protons, etc they all have their antiparticles, why are not they mentioned in ...
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How did we know that the Dirac equation describes the electron but not the proton?

I'm suddenly getting confused on what should be a very simple point. Recall that the $g$-factor of a particle is defined as $$\mu = \frac{ge}{2m} L$$ where $L$ is the spin angular momentum. For any ...
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Fermion zero modes extra conditions?

A fermion zero mode is a zero eigenfunction, $$i\gamma^\mu(\partial_\mu-iA_\mu)\psi=0$$ The number of zero modes is apparently related to the instantons of the gauge field. But now my question is ...
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How can a Dirac field $\psi$ satisfy KG equation? What's wrong in my derivation?

The covariant form Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi(x)=0$ can be multiplied from the left with the operator $(i\gamma^\nu\partial_\nu+m)$ and 4xpanding it out, to get $$(i\gamma^\nu\...
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Confusion with the meanings of fermion fields $\hat{\Psi},\hat{\overline{\Psi}},\hat{\Psi}^C$

If $\hat{\Psi}$ is a field that annihilates an electron and creates a positron, $\hat{\overline{\Psi}}$ is a field that annihilates a positron and creates an electron. This takes all possibilities ...
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How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
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Dirac sea for Majorana equation

To resolve the problem of infinite energies, Dirac has proposed the conception of the Dirac sea. Is the same concept present for the case of Majorana spinors? How it is affected by the absence of ...
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1answer
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Is the mass an eigenvalue of Dirac equation?

Writing the Dirac equation as: $$(i \hbar\gamma^{\mu}\frac{\partial}{\partial x^{\mu}})\psi = m \psi$$ it seems that $m$ is an eigenvalue of the operator of the left side, and we need to find the ...
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Propagator for Dirac spinor field

I am currently trying to learn Quantum Field Theory through David Tong's notes which only talk about canonical quantisation for the scalar field and Dirac spinor field. In Chapter 2, the propagator ...
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How to derive the Klein-Nishina formula from the Dirac equation?

I'm looking for the simplest demonstration of the Klein-Nishina formula, from the Dirac equation without the field described as a quantum operator: https://en.wikipedia.org/wiki/Klein%E2%80%...
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Dirac equation plane with solution

Using the Dirac equation with or $p=$ zero and the $\gamma^0$ matrix defined as $$\gamma^0=\begin{pmatrix}0 & \sigma_0 \\ \sigma_0 & 0\end{pmatrix} = \begin{pmatrix}0 & \bf{I} \\ \bf{I} &...
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How to show this spinor identity

This is from page 59 of Peskin, Schroeder. Let $u^s(p)= \begin{pmatrix} \sqrt{p\cdot\sigma}\xi^s \\\sqrt{p\cdot\bar{\sigma}\xi^s} \end{pmatrix}$. On page 59 they claim that $u^s(\Lambda^{-1}p) = \...
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Dirac Lagrangian in terms of left and right handed fields

When we define the operators $P_{L,R}=\frac{1\mp\gamma_{5}}{2}$, we are able to define right- and left-handed Dirac fields: $$\psi=(P_{L}\psi+ P_{R}\psi)=:\psi_{L}+\psi_{R}$$ The corresponding Dirac-...
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Fierz identities and anticommutation relations

Let us consider the following term $$\bar\psi(p_1)M\psi(p_2) \bar\psi(p_3)N\psi(p_4)$$ According to Ortin's book (equation (D.65)), from the Fierz identity we would have something like $$\bar\psi(...
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What's the role of the Dirac vacuum sea in quantum field theory?

It's often claimed that the Dirac sea is obsolete in quantum field theory. On the other hand, for example, Roman Jackiw argues in this paper that Once again we must assign physical reality to Dirac’...
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Dirac equation in QFT vs relativistic QM

How does the Dirac equation in quantum field theory solve the existing problems in the interpretation Dirac equation (as a single-particle wave equation) in relativistic quantum mechanics? EDIT: The ...
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Dirac spinor and field quantization

Can we simplify $$ Σ_s Σ_r [b_p^s u^s(p)\mathrm e^{ipx} (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} + (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} b_p^s u^s(p)\mathrm e^{ipx}]\tag{1}$$ as $$Σ_sΣ_r[ \{b_p^s, (b_q^r)^†...
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Anti-commutator in Quantization of Dirac field

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^{...
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Can the Dirac equation for a free particle accommodate the physical scenario of one single particle in space?

The plane wave solution for the Dirac equation for a free particle is of the form $ \psi = U e^ {i (xP_x + yP_y + zP_z -Et)/\hbar}$, where $\psi$ is the Dirac spinor with four components and $U$ is a ...
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Did Dirac anticipate not just the positron but also annihilation?

Would his equation have shown that annihilation would occur or did anyone know this before the actual discovery of the positron?
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Time reversal for Dirac particles in 2+1 or 6+1 (mod 8) spacetime dimensions

It is my understanding that time reversal invariance for Dirac fermions is usually (in 3+1 dimensions at least) implemented by an antiunitary operator ${\mathfrak T}$ that acts on the Dirac field ...
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Contracting gamma matrices with explicit indices

So I was calulating the matrix element of an interaction and arrived at the following contraction $$\gamma^\mu_{ab}\gamma_{\mu\,cd}$$ With $a,b,c,d$ spinor indices that are never contracted with ...
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Symmetries and Dirac Lagrangian [closed]

For spacetime translation given by Under spacetime translations the spinor transforms as $$\delta\psi=\epsilon^\mu\partial_\mu\psi$$ The Lagrangian depends on $\partial_\mu\psi$, but not $\...

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