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 ...

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Anti-commutation relation for the dirac field [closed]

The anti-commutation relation for the dirac field is: $$ \{\Psi_a(t,\vec x),\Psi_b^{\dagger}(t,\vec y)\}=\delta (x-y) \delta_{ab} $$ Where: $$ \Psi (x)=\int \frac{dp^3}{(2\pi)^3} ...
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59 views

Quadratic terms in QED lagrangian density

I recently learned that when we speak about a "free lagrangian", this actually means that the lagrangian is quadratic in the fields. When considering the Lagrangian density describing the coupling to ...
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271 views

Is there supersymmetry between Dirac and Klein Gordon solutions?

Usually supersymmetry is explained at the level of the action of a quantum field theory, and there are two ways to go down from QFT to relativistic quantum mechanics: either a non-covariant way where ...
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84 views

What do the density function and the current vector in the continuity eq. form of the Klein-Gordin eq. mean?

In explaining Dirac's attempt to arrive at a relativistic Schrodinger equation, it is claimed that Dirac has started from $$\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right)\phi = ...
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38 views

Variable $r$ in Dirac equation

Solutions of Dirac equation for an electron in orbit of an atom are usually expressed in terms of spherical coordinates $r,\theta$ and $\phi$. For a point ($r,\theta,\phi$) the variable $r$ represents ...
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Spinor field normalisation from poles in the propagator

In the theory of free scalar bosons (KG field) it is a basic result that the propagator $\Delta(p)$ has poles at $p^2=m^2$, with residue $1$ (or any other constant, depending on conventions). Thinking ...
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40 views

Simple question about gamma five and re-writing the Dirac lagrangian

I'm working a problem (Zee, p. 100) asking me to rewrite the Dirac lagrangian in terms of the left and right projections, and along the way I run into: $$\overline{\psi} i \gamma_µ \partial^µ \psi - ...
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88 views

Derivation of Dirac equation in curved spacetime

In all the Literature I have read, the covariant Dirac equation in curved spacetime is given as \begin{equation} ...
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77 views

Four-momentum and Dirac equation in curved spacetime

Norm of four momentum in Minkowski spacetime is proportional to the square of rest mass as \begin{equation}|P|^2= P^\alpha \eta_{\alpha\beta}P^\beta= (E/c)^2 - p^2 = (mc)^2 \end{equation} While in ...
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56 views

What is the relation between energy levels of hydrogen atom in Bohr's solution to that of Dirac solution

In Dirac solution for hydrogen atom, the energy levels are calculated as positive \begin{equation} ...
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281 views

Klein paradox for bosons and fermions

I am reading this paper about the Klein paradox, i.e. transmission of relativistic particles incident on a potential step of height $V_0 > E + mc^2 > 2mc^2$ with $E$ the energy of the incident ...
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37 views

Regarding properties of matrices involved in Dirac equation

In this document, after equation 62 on page 9, the author says that we can rewrite $\alpha^i \alpha^j \partial_i \partial_j$ as $\frac{1}{2} (\alpha^i \alpha^j + \alpha^j ...
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25 views

Dirac equation in the presence of a defect

The 1D Dirac equation in the presence of a defect is described by a position dependent mass term known as a "kink" or "soliton". It is sign changing and tends to a constant at positive and negative ...
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52 views

Hamiltonian describing a 3D Weyl point: seperability

A 3D analogue of a Dirac point is a Weyl point, with first quantized Hamiltonian $H = \sigma_x p_x + \sigma_y p_y + \sigma_z p_z $ where $\sigma_i$ are Pauli's matrices and $p_i$ are momentum ...
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57 views

One stupid question about Dirac mass term and Grassmann numbers

Let's have Dirac mass term in lagrangian: $$ L_{M} = \bar{\Psi}\Psi $$ Lagrangian must be real-valued, i.e., its Hermitian conjugation doesn't change it. But due to Grassmann nature of spinor fields, ...
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231 views

Is the Dirac equation equivalent to the Klein-Gordon equation for its left handed component?

The Dirac equation $$(i\gamma^a\partial_a - m)\psi=0\tag{0}$$ is given by a first order operator acting on a Dirac spinor, which is the direct sum of a left handed spinor and a right handed spinor. ...
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113 views

Interpretation of negative mass in condensed matter physics

I am reading the book "Topological insulator: Dirac equation in condensed matters" by Shun-Qing Sheng. I do not know much about this topic and this is the first time I am confronted with it, so this ...
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147 views

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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Any relationship between semi-conductor holes and the Dirac Hole Theory

Isn't the idea of a hole in semi-conductors synonymous to that of anti-matter? I.e. the Dirac Hole Theory and Dirac Sea? I'm aware the Dirac Sea is a theoretical model. But the concepts are almost ...
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116 views

Why does the Dirac equation require matrices to be rotationally invariant?

Why does the Dirac equation derivation require matrices? Starting from $$i\hbar \frac{\partial \psi}{\partial t} = \left(\frac{\hbar c}{i}\alpha^k\partial _k + \beta m_0 c^2 \right) \psi =H \psi.$$ ...
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317 views

Is there a reason why a relativistic quantum theory of a single fermion exists, but of a single scalar not?

When we try to construct the relativistic generalization of non-relativistic time dependent Schroedinger equation, there are at least two possible completions - Klein-Gordon equation and Dirac ...
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109 views

Definition of probablity current in dirac space not including spatial dimension?

I'm currently reviewing (basic) relativistic quantum mechanics and stumbled upon the probability current in "dirac space", defined as $j^μ = (j^0,\vec j)^\mathrm T$ with $j^0 = c\,ρ = c\,ψ^+ψ$ and ...
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117 views

Hermiticity of Dirac operator in curved spacetime

The Dirac Lagrangian in curved spacetime is usually given by \begin{equation} \mathcal{L} = i\bar{\Psi}\gamma^a e^{\mu}_a(\partial_\mu + \frac{1}{4}\omega_{\mu bc}\gamma^b\gamma^c)\Psi \end{equation} ...
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62 views

The chirality of (2+1)D Dirac equation

Are there any definitions about the chirality of (2+1)D Dirac equation? For the (3+1)D Dirac equation, the Dirac field can be written as the sum of left- and right-hand Weyl field. Can this be reduced ...
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108 views

Non-hermiticity of Dirac Lagrangian: null momentum?

The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are $$ \pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\ \bar \pi=\frac{\partial ...
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Future causality of Dirac charge current spinor

I am trying to solve following problem: The Dirac equation reads \begin{equation} \nabla^{AA'} \psi_{A} = \mu \chi^{A'}, \quad \nabla_{AA'} \chi^{A'} = -\mu \psi_{A} \end{equation} where $ \mu ...
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171 views

Plane wave solutions of Dirac equation

I'm reading chapter 3 in Peskin on the Dirac equation. First of all, they say since Dirac satisfies Klein Gordon it can be written as a linear combination of plane waves. This is fine. So a general ...
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489 views

How can it be derived that particles described by the Dirac equation must have spin 1/2?

I am reading some lecture notes that unfortunately don't seem to be available online, but that are quite close in spirit in their treatment of the Dirac equation to Sakurai's "Advanced Quantum ...
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69 views

Dirac Eqn: why separate operators

At some point Dirac writes: (OpA)(OpB)Y = 0 where OpA and OpB are those two brackets that differ only in the sign of m, then he deduces: (OpA)Y = 0 OR (OpB)Y = 0 (or is that AND). I don't get ...
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149 views

Does charge conjugation affect parity?

Notice that these transformations do not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino, which does not interact in ...
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Dirac Spinors as Eigenvalues of Helicity Matrix

I am trying (unsuccessfully) to verify this relation regarding the helicity of Dirac spinors: $$ { \sigma }_{ \vec { p } }u_{ r }\left( \vec { p } \right) =\frac { \vec { \Sigma } \cdot \vec { p } ...
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177 views

Wrong sign anticommutation relation for the Dirac field?

Consider the Dirac Lagrangian $$\mathcal{L}=\psi ^{\dagger }\gamma ^{0}\left( \mathrm{i}\gamma ^{\rho }\partial _{\rho }-m\right) \psi .$$ The conjugate momenta to $\psi ^{a}$ are defined, as usual, ...
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218 views

Where does the Lorentz boost for a Dirac spinor come from?

I have read, that if you have a Dirac spinor \begin{equation} \psi = \begin{pmatrix} \phi_R\\ \phi_L \end{pmatrix} \end{equation} that you can apply a Lorentz boost along the $z$-direction with ...
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75 views

What is the physical meaning of $\overline{\Psi} \Psi$ in the Dirac current's “Gordon Decomposition”?

When writing the Dirac (charge) current out in a way that resembles the (charge) current in the Pauli/Schrödinger theories, one obtains the following: $ j^\mu = -\frac{\mathrm{i} e\hbar}{2m} \left[ ...
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1answer
208 views

Why does the Dirac equation reduce the fermionic degree of freedom by half

We know that in 4D a Dirac spinor has 4 complex components or 8 real components meaning 8 real off shell degrees of freedom (please correct me if I say something wrong here). When we go on-shell i.e ...
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1answer
239 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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3answers
121 views

Spin in Schrodingers Equations

With the usage of Dirac notation we've gotten around a large amount of of inconvenience that would be dealing with spin. But, I was wondering, do we merely do this because it is inconvenient to try ...
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65 views

How do you derive the Dirac equation for momentum space?

$\require{cancel}$ \begin{align} 0 &= i \gamma^\mu \partial_\mu \psi(x) - m \psi(x) \\ &= \int \frac{d^4 k}{(2\pi)^4}e^{-i k x}\left( \gamma^\mu k_m \tilde{\psi}(k) - m \tilde{\psi}(k) ...
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147 views

How to solve the Dirac equation numerically?

The effective Hamiltonian for my system is: $$ H=\nu_{F} \sigma\cdot\left(q-By\hat x\right) $$ where $\sigma$ and $q$ are the Pauli matrices and the momentum operator respectively and $\nu_{F}$ and ...
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1answer
71 views

Two-component formalism and four-component formalism [closed]

When deriving the Dirac equation for spin-1/2 particles, we realize that the wave function must be four-component. In some works, people use two-component wave function for calculation. So, my ...
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1answer
143 views

Covariant formulation of physical equations?

Is it possible to rewrite equations like the Klein-Gordon, the Dirac or the Proca equation in a generally covariant way? And if yes, how and how can the general covariance be shown? (I searched ...
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197 views

Is there a 2D manifold on which the Dirac equation has a zero mode?

The two-dimensional (2D) Dirac equation $(\sigma_1iD_1+\sigma_2 iD_2)\psi=E\psi$ admits zero mode ($E=0$) solutions on a non-trivial gauge background, such as the zero mode at the core of a U(1) gauge ...
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89 views

How one can write $\bar{\psi}$ in odd dimension?

I know that the Dirac equation in general dimensions has a form of $$ (i\gamma_{\mu} \nabla_\mu - m ) \psi =0 $$ and the action for that is written as $$ S = \int d^d x \bar{\psi} (i\gamma_{\mu} ...
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1answer
140 views

Most general separable solution of free Dirac equation

In relativistic quantum mechanics, the solution of the free Dirac equation is assumed to be $$\Psi(\textbf{r},t)=u(\textbf{p})e^{i(\textbf{p}\cdot \textbf{r}-Et)}$$ How do I know that this is the most ...
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1answer
329 views

Dirac operator in curved spacetime in 2 dimensions – hermitian?

I'm currently trying to learn about the Dirac equation in curved spacetime and have come across an odd remark in Nakahara's well-known textbook "Geometry, Topology and Physics" that I would like to ...
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76 views

The Dirac equation for helium?

How to write down the Dirac equation for the two electrons in the helium atom? The problem is the interaction term, as $1/|r_1 - r_2|$ is apparently not Lorent-covariant.
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1answer
120 views

what does Peskin's square root of a matric mean?

Peskin (Intro to QFT) is using the next symbols when discussing dirac fields - $\sqrt{p\sigma}$ with $\sigma = (1,\sigma^1,\sigma^2,\sigma^3)$ (unit & Pauli). For example he represents the dirac ...
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236 views

Why do we need matrices in the Dirac equation?

Consider the following equation: \begin{equation} \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A ...
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1answer
296 views

Describe proton and electron by one wavefunction

when I was new into quantum mechanics, I thought we can describe helium atom by two wavefunctions - one for every electron. After some time I discovered how wrong I was - first, because electrons are ...
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1answer
113 views

Charge operator for Dirac spinor

In QED, the gauge transformation which acts upon a fermionic field $\psi$ is $$\psi'(x)= e^{i \alpha(x) Q}\psi(x)$$ where $Q$ is the charge operator. Most of the time it's just written as $$\psi'(x)= ...