Skip to main content

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.

Filter by
Sorted by
Tagged with
8 votes
3 answers
604 views

How does special relativity lead to anti-particles?

Anti-particles and spinors pop out of the Dirac equation very naturally, yet the Dirac equation is only a modified version of the Schrödinger equation which includes the relativistic energy-momentum ...
eli morhayim's user avatar
0 votes
0 answers
31 views

Relation between four-components of the Dirac wavefunction and the four solutions of the Dirac equation

Recently I've been studying the Dirac equation. I've studied that the Dirac wavefunction must have four components ($4\times 1$ column matrix) since the Dirac Hamiltonian must be a $4\times4$ matrix ...
A.M.M Elsayed 马克's user avatar
0 votes
0 answers
33 views

Verifying transformation law for Dirac field

I am trying to verify the Lorentz covariance of the Dirac field \begin{equation}\label{transformation}U(A)\psi(x)U(A^{-1})=D(A^{-1})\psi(\Lambda(A)x)\end{equation} where $A\in SL(2,\mathbb{C})$, $\...
user609020's user avatar
2 votes
1 answer
105 views

Derivation of Dirac Hamiltonian

In Minkowski spacetime with signature $(-,\;+,\;+,\;...,\;+)$ the Dirac Lagrangian reads $$ L=\int d^dx\;\mathcal{L}=\int d^dx\;\psi^\dagger\left(i\gamma^0\gamma^\mu\partial_\mu-im\gamma^0\right)\psi. ...
TopoLynch's user avatar
  • 495
1 vote
1 answer
59 views

Dirac field coupling to gauge fields

I've seen in couple sources that the gauge invariant Lagrangian for the Dirac field being written as follows: $$\mathcal{L} = \frac{i}{2}[\bar{\psi}\gamma^{\mu}D_{\mu}\psi-(\bar{D}_{\mu}\bar{\psi})\...
physics_2015's user avatar
1 vote
1 answer
52 views

Why do the edge states of a topological insulator always have zero energy?

Recently, I am learning the topological insulator, I learned about SSH model, and found out that the edge states of topological insulator always have zero energy, but on the other side,we define edge ...
Susstring W's user avatar
0 votes
0 answers
29 views

Can a particle have properties of both Dirac and Majorana particles?

Dirac fermions have antiparticles of opposite properties, and only a particle and an antiparticle can annihilate. Majorana fermions have no antiparticles because they can annihilate with themselves. ...
哲煜黄's user avatar
  • 1,435
0 votes
1 answer
82 views

Do gamma matrices commute with 4-vectors?

One of my exercises was to prove the identity $$\gamma^\mu\displaystyle{\not}a\gamma_\mu=-2\displaystyle{\not}a.$$ Which is trivial if $\gamma^\mu a_\nu=a_\nu \gamma^\mu$, as follows $$\gamma^\mu\...
agaminon's user avatar
  • 1,607
4 votes
1 answer
276 views

Why Dirac's spin-$\frac12$ theory proves wrong Kaluza-Klein theory?

I recently saw Sabine's short video that mentions that Dirac's spin-$\frac12$ theory proves wrong Kaluza-Klein theory unless supersymmetry is amended to the Kaluza-Klein. Is there a more detailed ...
Markoul11's user avatar
  • 4,155
0 votes
0 answers
50 views

Trying to solve the energy levels of a spin 1/2 particle in a one-dimensional box using Dirac Equation

I was studying the problem I asked above in the title and found the article P Alberto et al 1996 Eur. J. Phys. 17 19. The wave function inside the walls is: $$ \psi(z)=B\ exp(ikz) \left[\begin{array}{...
Joao Pedro Medeiros's user avatar
1 vote
1 answer
84 views

Non-Hermiticity of the Dirac Hamiltonian in curved spacetime

In flat spacetime, Dirac fermions are classically described by the action $$ S=\int d^Dx\;\bar\psi(x)\left(i\gamma^a\partial_a-m\right)\psi(x). $$ One can generalize this to a general curved spacetime ...
TopoLynch's user avatar
  • 495
2 votes
0 answers
96 views

How to motivate spinors from the Dirac equation? [closed]

I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator). Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
Integral fan's user avatar
0 votes
0 answers
38 views

Proof that commuting Dirac fields violate causality

What is the proof that commuting Dirac fields violate causality? All sources I could find just quote this result, but I couldn't find an explicit derivation anywhere. In particular, the case I am ...
pll04's user avatar
  • 337
1 vote
1 answer
35 views

Does the anticommutator of two spinors affect the transpose of their product?

My lecture notes claim that for an anticommutation relation $$[ \psi_{\mu}(\bf{x},t),{\psi_{{\nu}}^{*}}(\bf{y},t)] = \delta_{\mu \nu} \delta^3(\bf{x}-\bf{y})$$ between two spinors, the transpose of ...
pll04's user avatar
  • 337
0 votes
1 answer
45 views

Fermi tetrad field: Fermi-Walker tetrad formalism?

These days I'm reading Dirac Eq in GR, and I'm confused about "Fermi tetrad field". Is it Fermi-Walker tetrad formalism?
Lou TY's user avatar
  • 1
1 vote
0 answers
64 views

History/motivation: road from wavefunction QM to QFT

I am trying to give concise motivation (to math students) for why we have QFT as the fundamental theory of matter and forces, I may start like this: Schrodinger's equation (in the sense of ...
Integral fan's user avatar
0 votes
0 answers
65 views

How does the simple description of spin relate to the Dirac equation?

Coming from a chemistry background, spin (the spin 1/2 case at least) is usually introduced in QM by saying there are abstract $\alpha$ and $\beta$ states together with rules for how operators act on ...
FusRoDah's user avatar
  • 679
0 votes
0 answers
60 views

Going to momentum space from Hamiltonian equation 17.4 chapter 17 in Schwartz

I'm reading the chapter 17 on the anomalous magnetic moment in QFT and the SM (Schwartz). In the section "17.1 Extracting the moment" he says "Going to momentum space,the Dirac equation ...
Andrea's user avatar
  • 11
0 votes
0 answers
26 views

Evaluating expressions involving Dirac spinors

Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi=0$. This equation describes the free Dirac field. Consider the plane wave solutions $\psi(x)=u(\vec{p})e^{-ip\cdot x}$ and $\psi(x)=v(\...
Anant Badal's user avatar
0 votes
2 answers
66 views

Transition from positive energies to negative energies in Dirac equation

In textbooks, the criticism of the Dirac equation is that it may allow transitions from the positive energy state $m c^2$ to its corresponding negative energy state by releasing one or more photons. ...
Francesco's user avatar
0 votes
1 answer
96 views

How to derive this form of helicity spinor (massless/high energy limit)

Srednicki 50.7 says that in the massless limit, we can express $$u_-(\textbf{p})\bar{u}_-(\textbf{p}) = \begin{pmatrix} 0&-p_{a\dot{a}}\\0&0\end{pmatrix}$$ This comes from a previously ...
JohnA.'s user avatar
  • 1,713
2 votes
2 answers
309 views

The dimension of the Clifford algebra for the Dirac equation

The Dirac algebra contains sixteen linearly independent elements. In general, a Clifford algebra $\mathcal{C}\!\ell(V,Q)$ generated from a vector space $V$ equipped with a quadratic form $Q$ has ...
Nada Band's user avatar
1 vote
1 answer
51 views

Obtaining the 16 elements of the Clifford algebra from the $\gamma^\mu$ generators

In my study of the Dirac equation, I have fully understood the "linearization" of the relativistic energy to obtain a matrix-valued equation that reduces to the Klein-Gordon equation if the ...
Nada Band's user avatar
1 vote
1 answer
115 views

Charge conjugation is a symmetry for the quantized free Dirac action?

I am self-studying QFT on "A modern introduction to quantum filed theory" by Maggiore, and on page 95 he states: "For the free Dirac action, one immediately sees that C,P and T are ...
Andrea's user avatar
  • 509
2 votes
0 answers
70 views

Interpretation of "spin-1/2" in classical Dirac field

I emphasize that the proceeding is purely classical physics. Consider the Grassmann-valued field (where $\mathcal{N}$ is a Grassmann number), which is a solution to the Dirac equation, given by $$\psi(...
Silly Goose's user avatar
  • 2,548
2 votes
1 answer
71 views

Product of spinors in Dirac field anticommutators

I am reading a "A modern introduction to quantum field theory" by Maggiore and on page 88 it shows the anticommutators of the Dirac field: $$ \{\psi_a(\vec{x},t),\psi_{b}^{\dagger}(\vec{y},t)...
Andrea's user avatar
  • 509
0 votes
0 answers
24 views

Electrostatic potential created by a relativistic bounded electron (i.e a 4-spinor)

Let's take the Gordon solution of the central field Dirac equation for the Hydrogen atom and look at the wave functions. There is bounded functions inside the spinor, which represents here the full ...
fefetltl's user avatar
0 votes
2 answers
67 views

Understanding derivation of Klein-Gordon equation from Dirac equation

Above is Tong's notes which shows how the Klein-Gordon equation is derived from Dirac equation. But I don't get why: $\gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} = \frac{1}{2} \{\gamma^{\mu},\...
Stallmp's user avatar
  • 665
-2 votes
1 answer
142 views

Why do the Schrödinger and Dirac equations contain the mass?

I know the Schrödinger equation is bascially the "quantized" Hamiltonian formalism from classical mechanics, and the Dirac equation is the special-relativistic version. But these equations ...
ldfjglfkgj's user avatar
-2 votes
1 answer
65 views

Non-relativistic limit of time-dependent Dirac equation

Can someone point me to the derivation of the non-relativistic limit of the time-dependent Dirac equation? I'm presuming that the limit is nothing but the time-dependent Schrodinger equation. I ...
John Doe's user avatar
0 votes
0 answers
33 views

Left-handed fermion oscillating into right-handed fermion

Given a Dirac fermion $\psi$ $$\mathcal{L} = \bar{\psi} \gamma^\mu \partial_\mu \psi - m \bar{\psi}\psi \ ,$$ which can be written in terms of chiral left and right handed fields as $$\mathcal{L} = \...
Rudyard's user avatar
  • 770
0 votes
0 answers
50 views

Dirac equation: Green's function specified for only one dimension

Normally, the Dirac equation for the Green's function reads: $$(i\gamma^\mu\partial_\mu - m)S_F(x,y) = \delta^{(4)}(x-y)$$ Is it possible to define a Green's function describing the propagation ...
Lê Dũng's user avatar
  • 938
1 vote
1 answer
69 views

Dirac Lagrangian under charge conjugation

I am trying to understand why the Dirac Lagrangian is invariant under charge conjugation. The Dirac Lagrangian is: $$\mathcal{L} = i\bar{\psi}\gamma^\mu \partial_\mu\psi - m \bar{\psi}\psi $$ I know ...
Joe's user avatar
  • 413
1 vote
0 answers
34 views

Taking the non-relativistic limit of the Dirac Lagrangian

I know the usual derivation (as well as the Foldy–Wouthuysen derivation) to obtain Schrödinger equation from the Dirac equation. See for example U Alberta Phys 512. But it it possible to go from $$i\...
Mauricio's user avatar
  • 5,488
1 vote
1 answer
144 views

How particles interact with the electromagnetic potential $A^\mu$?

It is well known that one reason quantum mechanics started to being developed, was because scientist wanted a model to explain electron orbits in atoms. Borh interpreted that the for orbits to exist ...
Álvaro Rodrigo's user avatar
1 vote
1 answer
62 views

Parity operator action on quantized Dirac field

I am stuck on equation 3.124 on p.65 in Peskin and Schroeder quantum field theory book. There they are claiming that: $$P\psi(x)P=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf p}}}\...
Joe's user avatar
  • 413
0 votes
0 answers
109 views

Are eigenvalues of slashed covariant derivative real?

I am trying to demonstrate that the slashed covariant derivative $$ \gamma^\mu D_\mu = \gamma^\mu(\partial_\mu -iA_\mu) $$ has real eigenvalues: $$ \gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x)...
Gorga's user avatar
  • 161
0 votes
1 answer
39 views

Covariant derivative property

I am trying to demonstrate this propertie $$ \not{D}^2= \mathcal{D}^\mu \mathcal{D}_\mu-\frac{i}{4}\left[\gamma^\mu, \gamma^\nu\right] F_{\mu \nu} $$ where $\not{}~$ is the Feynmann slash, and $D_\mu ...
Gorga's user avatar
  • 161
0 votes
1 answer
92 views

Why is the derivative necessary to connect left and right-hand spinors?

I am studying Weyl and Dirac spinors. Suppose we have two Weyl fermions $\eta, \chi$ transforming under $(1/2,0)$ representation of the Lorentz group. I learned that to construct Lorentz invariant ...
IGY's user avatar
  • 1,783
2 votes
0 answers
60 views

What is the difference between a twistor and bispinor?

Reading the book on General Relativity written by R.M. Wald I (tags according to Wald's book) encountered the concept of a twistor $$ Z = (\omega^A, \pi_{A'}) \tag{14.1.9} $$ which looks very much as ...
Frederic Thomas's user avatar
-1 votes
1 answer
43 views

Calculating time dilation from a solution to the Dirac equation

Suppose I have a Muon in a potential well. Its wavefunction is a solution of the Dirac equation, a relativistic version of the Schrodinger equation for spin 1/2 particles. Because the particle is &...
Kevin Kostlan's user avatar
0 votes
1 answer
107 views

Derivation of Dirac equation in curved spacetime by varying the action

I want to derive the Dirac massless equation in curved spacetime from the action. I have the symmetric form of the Dirac action: $$S = \frac{1}{2} \int \bigg[i\bar{\psi} \gamma^\mu D_\mu \psi - i D_\...
Yaezir's user avatar
  • 23
2 votes
1 answer
63 views

Spinor Components, Helicity, and Chirality in Dirac Theory

In the Dirac the spinor components are defined by fermion/antifermion (here labeled as $+,−$) and spin component $S_z$ ($↑,↓$): \begin{pmatrix} \psi_-^\uparrow \\ \psi_-^\downarrow \\ \psi_+^\uparrow \...
Julián Oviedo's user avatar
1 vote
0 answers
75 views

What does a quantized field in QFT do? [duplicate]

I'm studying for an exam called Introduction to QFT. One of the main topics in this class is the quantized free fields. I can now find the fields that solve the Klein-Gordon equation and the Dirac ...
BBBZZZ's user avatar
  • 19
0 votes
1 answer
81 views

Is there an equivalent to the Klein-Gordon and Dirac Equations for Vector and other fields?

The Klein-Gordon equation describes a scalar field, and the Dirac Equation describes a spinor field. Is there an equivalent equation for a vector field? As well as spin 3/2 and spin 2 tensor fields? ...
zion does math weird's user avatar
0 votes
0 answers
43 views

Finding the bound state energies in the MIT bag model numerically for a general bag shape

The MIT Bag Model is a simple model used to describe the properties of bound quarks in Hadrons, without considering the strong interaction between the quarks with the following boundary condition $$(1+...
Amirhossein Rezaei's user avatar
6 votes
1 answer
141 views

Obscure Calculations in Foldy-Wouthuysen Transformation (electron in EM field)

I'm studying the Foldy-Wouthuysen Transformation on Bjorken-Drell's book and I got stuck strying to replicate some calculations. First of all, introducing the transformation $\psi'=e^{iS}\psi$ we get ...
WindUpBird's user avatar
0 votes
1 answer
74 views

The Dirac-Hestenes equation as an eigenvalue equation: Interpretation of the $m \psi \gamma_0$ term and the wavevector

This is somewhat of a follow-up question to my previous question on the Dirac-Hestenes equation. In that question, I asked whether the equation could be written in a form that omits the dangling ...
brainandforce's user avatar
1 vote
0 answers
80 views

Product of Dirac $\gamma^0$ and $\gamma^\mu$ generate a representation of some algebra?

I need your help with an issue about Dirac gamma matrices. Precisely, I need to know if $\gamma^0\gamma^\mu$ generates an irreducible representation of some algebra. This problem has come out in the ...
dallla's user avatar
  • 59
1 vote
2 answers
138 views

Charge conjugated Dirac equation

I would very much like to understand the motivation behind the correlation between: $(i\partial\!\!/-eA\!\!/-m)\psi=0$ and $(i\partial\!\!/+eA\!\!/-m)\psi_c=0$ when dealing with the derivation of the ...
Xhorxho's user avatar
  • 151

1
2 3 4 5
24