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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Classical Fermion and Grassmann number

In the theory of relativistic wave equations, we derive the Dirac equation and Klein-Gordon equation by using representation theory of Poincare algebra. For example, in this paper ...
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Angular momentum of the vacuum

I'm studying quantum field theory from "An introduction to Quantum field theory" by Peskin and Schroeder and from "A modern introduction to quantum field theory" by Maggiore. I've read from "An ...
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68 views

Definition of partity in quantized Dirac Theory.

I'm studying from the book "An Introduction to Quantum Field Theory" from Michael E. Peskin and Daniel V. Schroeder, and I read the following: "The operator P should reverse momentum of a particle ...
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38 views

Wave packets in Dirac equation

Gaussian wave packets remain Gaussians after evolution in case of the Schrodinger equation. It is a very useful property of these wave packets. I don't think the same is true for a Gaussian wave ...
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Dirac equation, $\alpha_i$, $\beta$ hermitian

The argument I've seen is the one given here: http://epx.phys.tohoku.ac.jp/~yhitoshi/particleweb/ptest-3.pdf under (3.10): $$H=\vec{\alpha}\cdot(-i\vec{\nabla})+\beta m$$ $H$ is hermitian, ...
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What is the effective (quantum) lagrangian of a fermion field for fixed electromagnetic field?

... or, put it another way, what are the loop corrections to the dirac equation in the presence of a fixed (external) electromagnetic field?. Background Let $\mathcal ...
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21 views

How to measure the Fermi velocity in Dirac materials?

Suppose that one has a Dirac material (e.g., graphene), i.e., a system where there exists a number $N$ of identical Dirac cones (linear dispersion) at the Fermi energy $E_F=0$. How can one measure ...
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2answers
80 views

Questions regarding the Feynman-Stueckelberg interpretaion

I am studying for an introductory particle physics exam, and I am having some problems with the Feynman-Stueckelberg interpretation of antiparticle states. Background: The course was being thaught ...
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69 views

Looking for a reference for $\gamma_a e^{a}_{\mu} D^\mu \gamma_b e^{b}_{\nu} D^\nu =D^\mu D_\mu - \tfrac{1}{4}R$

I am having trouble finding references for the following identities: Dirac Operator: $$ \gamma_a e^{a}_{\mu} D^\mu \gamma_b e^{b}_{\nu} D^\nu =D^\mu D_\mu - \tfrac{1}{4}R \tag{1} $$ QED Operator: $$ ...
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Solving Weyl Equations

In my second taking of QFT we just finished the Dirac equation. As an exercise I tried applying what I have (re-) learned to the Weyl equations. I'd like someone to check if my work is correct. For ...
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46 views

Definition of the charge conjugation operator

My question will be a bit provocative, I hope it will attract more interest (and hopefully no downvoting). I introduce the following notation: $u(p)\exp(-ipx)$ positive energy solution ...
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51 views

Hyperfine structure in hydrogen

Consider the Dirac equation for bounded electron in hydrogen atom. I am trying to get a clear physical explanation for all mathematical terms that appear in the Hamiltonian and energy spectrum. ...
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135 views

is following alternative interpretation of total energy possible? E=m'v^2 instead of E=m'c^2 [closed]

I have read the paper, http://arxiv.org/pdf/physics/0206061.pdf "Fundamental Disagreement of Wave Mechanics with Relativity", some time ago, in which the author claims that there is another way to ...
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64 views

Non-relativistic limit of the Dirac equation

How to recover non-relativistic limit of the Dirac equation $$\left( i\gamma^{\mu}\mathcal{D}_{\mu} - m \right)\Psi(x) = 0$$ where $\mathcal{D}_{\mu} = \partial_{\mu} + iqA_{\mu}$. I do not assume ...
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55 views

Dirac equation in the algebra of physical space and conservation laws

I have the following question: I was thinking, is it possible to obtain the conservation laws for the Dirac equation in the algebra of physical space? If yes, how? Can anyone show me a book for these ...
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1answer
26 views

Positive free particle Dirac equation

I've been set the task of showing that: $$ \bar{\psi^{s}}\psi^{s}=2m $$ For s=0,1. Where: $$ \psi^{0,1}=\sqrt{|E|+m}\begin{pmatrix}\chi^{0,1}\\ ...
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59 views

A question on the Dirac equation

In Quarks and Leptons by Halzen and Martin p. 105 it says: The bonus embodied in the Dirac equation is the extra twofold degeneracy. This means that there must be another observable which commutes ...
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75 views

Non-symmetry of a lagrangian

If a transformation $\Phi \rightarrow \Phi + \alpha \partial \Phi/ \partial \alpha$ is not a symmetry of the Lagrangian, then the Noether current is no longer conserved, but rather ...
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132 views

Which problem is Oppenheimer working on in this picture? [closed]

Does anybody recognize the equations on the blackboard? Above his hand, with the $\gamma_k$ term and its complex conjugate, it looks like a written out matrix representation of a Hamiltonian $U$, ...
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Transferring between field and single-particle versions of the Dirac equation

We're covering spinors in QFT class. The Lagrangian (density) $\mathcal{L} = \overline{\psi} (i \gamma^\mu \partial_\mu - m)\psi$ gives the Dirac equation, $(i \gamma^\mu \partial_\mu - m)\psi = 0$. ...
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53 views

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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53 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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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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1answer
68 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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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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36 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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74 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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75 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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53 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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188 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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36 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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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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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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53 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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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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1answer
100 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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88 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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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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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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105 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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105 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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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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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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129 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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444 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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139 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 ...