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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436 views

How did Dirac come up with the idea of using Pauli matrices?

Dirac equation in natural units is: $$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi=0$$ where $\gamma^{0}=\pmatrix{I_{2} & 0\\ 0 & -I_{2}}$ and $\gamma^{n}=\pmatrix{0 & \sigma_{n}\\ -\...
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529 views

Do $\gamma$ matrices commute with the Dirac spinor field?

I want to know if gamma matrices commute with the Dirac spinor field, i.e., are the following equalities correct? $$ \psi\gamma^{\mu}\overset{?}{=}\gamma^{\mu}\psi $$ $$ \psi^{\dagger}\gamma^{\mu}\...
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How to construct the charge conjugation matrix for any given spacetime dimension?

Generally, Gamma matrices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally ...
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Spinor Lorentz Transformation

Why should the transformation between the solutions of the Dirac equation for different inertial observers be linear?
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586 views

Algebraic solution of Dirac equation for Coulomb potential

The Runge-Lenz operator enables an algebraic solution of Coulomb potential energy levels without a solution of a differential equation. What is the analog for the solution of the Dirac equation in a ...
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227 views

Can we generate the Proca action (spin-1) from the Dirac action (spin-1/2)?

The Lagrangian for a spin-$\frac{1}{2}$ particle is the Dirac Lagrangian, while for a spin-$1$ particle is the Proca Lagrangian. But $1$ should just be $\frac{1}{2} \bigoplus \frac{1}{2}$, so is it ...
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79 views

Time evolution of a massive fermion produced in a state of definite chirality

But for massive particles like an electron, the chirality is not conserved in time i.e. if an electron is produced in the state $e_L$ at time t=0, at a later time it becomes a mixture of left-handed ...
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2answers
2k views

What happens to the Lagrangian of the Dirac theory under charge conjugation?

Consider a charge conjugation operator which acts on the Dirac field($\psi$) as $$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$ Just as we can operate the parity operator ...
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231 views

Deriving anti-commutation relation between creation/annihilation operators for Dirac fermions

Starting from Dirac fields: $$\Psi(x) = \dfrac{1}{(2\pi)^{3/2}} \int \dfrac{d^3k}{\sqrt{2\omega_k}}\sum_r\left[ c_r(k)u_r(k)e^{-ikx}+d^\dagger_r(k)v_r(k)e^{-ikx} \right]_{k_0=\omega_k}$$ $$\Psi^\...
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Why is the Lagrangian $-\partial^u\overline\Psi\partial_u\Psi-m^2\overline\Psi\Psi$ not Lorentz invariant?

Let $-\partial^u\overline\Psi\partial_u\Psi-m^2\overline\Psi\Psi$ be a Lagrangian density. Here $\Psi$ is the Dirac spinor and $\overline \Psi$ is defined to be $\Psi^\dagger \gamma^0$. It is said ...
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Wave function - Dirac Notation

Based on that notes (equation 54): https://warwick.ac.uk/fac/sci/physics/staff/academic/boyd/stuff/dirac.pdf I was reading about the wave functions and I have a question about the notation. You can ...
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1answer
110 views

Helicity in Graphene

In Graphene, there are two independent points, the Dirac points, where the conduction and the valence band touch. Let's call these points $K_+$ and $K_-$. In a low-energy description, the Hamiltonians ...
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2answers
173 views

What does it mean to act a Dirac field operator on the vacuum?

The usual interpretation from my QFT courses is that when acting the scalar field operator onto the vacuum, we create particle: $$ |x\rangle = \phi(x)|0\rangle. $$ If I have a multi-component field ...
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3answers
384 views

Mathematical proof on helicity of a massive fermion is not Lorentz invariant

What is the mathematical proof that the helicity of a massive spin-$1/2$ fermion is not Lorentz invariant? Something is Lorentz invariant (e.g., $P_\mu P^\mu$) if it commutes with all the generators ...
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47 views

Dynamics of a Dirac Wavepacket

I am interested in calculating the dynamics of a wavepacket obeying the Dirac equation, eventually with an applied Electric field, but I am stuck on the case without an applied field. I eventually ...
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1answer
158 views

Do massless spin-1/2 particles have to be Weyl spinors?

Weyl spinors are massless. Is the converse also true? Does any massless spin-1/2 fermion have to be a two-component Weyl spinor? In the Standard model, before symmetry breaking, the electron (for ...
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Calculate Chern number from band structure

When we use a tight-binding approach in order to calculate the band structure of electrons on a 2D honeycomb lattice such as Graphene we find that there are two energy bands touching in six points, ...
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384 views

What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
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Do physical results for spinors depend on the Clifford algebra representation?

As I understand it, the Dirac equation and its solutions depend on the representation of the $\gamma$ matrices one uses. So if I were to use the Dirac representation I would get different mathematical ...
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Two ways of thinking about the Dirac equation

My impression is that there are two ways of thinking about the Dirac equation: Quantum Mechanically: Here we think of the spinor $\phi$ as a generalization of the Schrodinger wave function which ...
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54 views

Understanding the substitution of a solution of Dirac equation in the Dirac equation

Dirac equation is $$(i \gamma^{\mu} \partial_{\mu}) \psi =0. $$ a solution of Dirac equation for massless fermion case is $$\psi (x) =u (\vec{p}) e^{ip^{\mu} x_{\mu}}.$$ substitution should give $$...
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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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67 views

Non-unitarity of finite dimensional Lorentz group and its implications

In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one ...
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How to obtain Dirac equation from Schrodinger equation and special relativity?

I'm reading the Wikipedia page for the Dirac equation: The Dirac equation is superficially similar to the Schrödinger equation for a free massive particle: A) $-\frac{\hbar^2}{2m}\nabla^2\...
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71 views

A question about the decoupling of Dirac equation in 1+1 dimension

It is said that in 1+1 dimension, if we take $\gamma^0=i\sigma^2$ and $\gamma^1=\sigma^1$, then the two components of dirac spinor $\psi_L$(upper component) and $\psi_R$(lower component) decouple in ...
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Deriving the massless Dirac equation in static spherical sphere spacetime?

I am a physics postgraduate and was curious if what I've done below was the correct equation for neutrinos in curved space time? Deriving the massless dirac equation using differential geometry ...
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560 views

Fine structure Hamiltonian from Dirac equation

The Hamiltonian for fine structure (the atom with $\text{Z}$ protons and with electron interaction terms included) is $$H=\frac{\text{Z}^2}{ r}+\underbrace{\frac{p^2}{m}+\frac{p^4}{m^3}}_{\text{...
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Is proton a Dirac fermion? If yes, does it also have a Lande-g factor $g=2$?

Proton is a spin-1/2 particle but composite i.e., it's a bound state of three quarks. Protons have partner called anti-proton which is also composite. Is it not a Dirac fermion? If not, why? In other ...
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582 views

The dimension of the energy-momentum tensor and the Einstein-Hilbert action

I have been thinking recently what will happen if one uses the energy momentum tensor of the Dirac field as a source in the Einstein Field equations. It is well known that in this case $$ T_{\mu\nu}=...
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On the Pauli-Lubansky vector and spin

Lahiri's A First Book on Quantum Field Theory states on problem 4.24 that from the Pauli-Lubansky vector $$W_\mu=-\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}P^\nu J^{\lambda\rho}$$ one can prove that for ...
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Why are antiparticles associated with spin-flipped spinors?

In section 2.2 of Elvang and Huang's Scattering Amplitudes in Gauge Theory and Gravity (http://arXiv.org/abs/1308.1697), beneath equation (2.9), it is mentioned that $u^{\pm}=v^{\mp}$, where $u^\pm$ ...
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Why is it reasonable to recognize solutions to Dirac Equation as electron and positron?

I'm reading Griffiths' book and when it claims the first two components of plane wave solution to massive Dirac equation as electron and the “holes” of other two components as positron, it does not ...
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1answer
75 views

Moving on from Dirac's equation as an undergrad

In my QM class, we're covering the basics of attempting to reconcile QM with special relativity. From what I understand, Dirac took the definition $E=\sqrt{p^2c^2+m^2c^4}$ and used it for the ...
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A few doubts with showing Lorentz invariance of Dirac equation and probability current

Trying to understand some about Lorentz invariance and representation theory, I thought that the best way is with an example of application: Show the Lorentz invariance of the Dirac Equation $$(i \...
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1answer
275 views

Using ladder operators to solve for Landau levels of graphene

I had recently been studying the Dirac equation, and as an example of how the equation is used, I was given a problem about the Landau levels of graphene (but I personally have no knowledge about ...
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Homework and exercises - Relating fermion chain and the Dirac Lagrangian in 1+1d

The question is as follow: Consider a chain of free fermions with $H=\sum_{n}c_n^{\dagger}c_{n+1}+h.c.$. Show that the low-energy excitations at a generic value of the filling are described by the ...
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Dirac action and conventions

I have a (possibly) fundamental question, which is driving me crazy. Notation When considering the Dirac action (say reading Peskin's book), one have $\int dV\;\bar{\psi}\left(\imath\not\partial-m\...
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Relation of Berry phase and winding number

I am reading the following article dealing with the properties of Dirac fermion in condensed matter physics : https://arxiv.org/abs/1410.4098 In the page 5 of this article, the formula for the ...
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Dirac spinor parity

I'm not sure I understand the effect of a parity transform on a Dirac spinor $\left( \begin{array}{c} \psi_R\\ \psi_L\\ \end{array} \right)$. I've been given the definitions $P\psi=\gamma_0\psi$, ...
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Quantisation of the Dirac field in curved space-time

I would like to know how to quantize the Dirac field in curved space-time. The Dirac equation in curved space-time has the following form: $i \gamma^a e_a^{\mu} D_\mu \Psi(x) -m \Psi(x) =0$ ...
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Cross product of the quantum mechanical operators $\textbf{p}$ and $\textbf{A}$

While reading Advanced Quantum Mechanics by J.J. Sakurai, chapter: Relativistic Quantum Mechanics of Spin-1/2 Particles, section 3.2 the Dirac Equation, the author states the following identity: $$\...
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1answer
240 views

How does the Dirac equation show the existence of anti-particles?

I want to understand the following "The Dirac equation for a charged massive fermion predicts, correctly, the existence of an antiparticle of the same mass and spin, but opposite charge, and ...
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Antimatter: Dirac's Sea Interpretation or Feynman's Backward in Time Interpretation

I know that these are the 2 interpretations about antiparticles, hypothesized to handle negative solutions to energy multiplied with time. What are the superiorities and drawbacks of each, compared ...
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35 views

Dirac Equation Dimensionality

In Griffith's Introduction to Elementary Particles, the Dirac equation is given during its derivation as (Equation 7.19): $$ \gamma ^ \mu p_\mu - mc = 0 $$ However, the dimensions don't seem to make ...
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440 views

Casimir's Trick in QED

In Griffith's Introduction into Particle Physics (p. 251, eq. 7.125) we derive Casimir's trick $$ \sum_{s_1,s_2}[\bar{v}(s_1,p_1)\Gamma_1 v(s_2,p_2)][\bar{v}(s_a,p_a)\Gamma_2v(s_b,p_b)]^* = \text{Tr}[...
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Why would Klein-Gordon describe spin-0 scalar field while Dirac describe spin-1/2?

The derivation of both Klein-Gordon equation and Dirac equation is due the need of quantum mechanics (or to say more correctly, quantum field theory) to adhere to special relativity. However, excpet ...
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651 views

Why can't the Klein-Gordon equation explain the hydrogen atom but the Dirac equation does?

Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom? Why can the first order Dirac equation explain it? What are the failures?
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Why aren't purely Dirac neutrinos ruled out?

It is common knowledge that in neutrinos can be Dirac particles without any Majorana masses as given a mass matrix, \begin{equation} \left( \begin{array}{cc}\nu _L & \nu _R \end{array} \right) \...
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Is a Dirac Engine in the realms of the conceivable?

I'm quite happy to delete this question if it is felt to be rubbish or if someone can say "nothing original". I have an idea about how humans might explore the universe and obtain ridiculously huge ...
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Does electron have some intrinsic ~$10^{21}$ Hz oscillations (de Broglie's clock/Zitterbewegung)?

De Broglie has postulated in 1927 that with electron's mass there comes some inner oscillation: $E=mc^2=h f=\hbar \omega$. We would get such oscillation e.g. if using $E=mc^2$ energy in stationary ...