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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Invertibility of Dirac matrices

Suppose we have Dirac equation in the following form : $i\partial_t\psi = (-i\vec{\alpha}\cdot \nabla +m\beta)\psi$ and assume that the Klein-Gordon equation is satisfied, i.e. $\partial_t^2 \psi ...
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The true dimension of Dirac field

In natural units with $\hbar=1$ and $c=1$, as we know, the energy dimension of the Dirac field $\psi(x)$ in QED is $\frac{3}{2}$. But in cgs units, what is the true dimension of the Dirac field ...
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Why is the unitary matrix relating the gamma matrices and their complex conjugates antisymmetical?

In Messiah's Quantum Mechanics Vol. II, properties of the Dirac matrices are derived. There is so-called fundamental theorem, which states that, Let $\gamma^\mu$ and $\gamma^{'\mu}$ be two systems of ...
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Question about the Dirac equation

Energy and momentum of a particle can be expressed by equation $$E^2=p_1^2c^2+p_2^2c^2+p_3^2c^2+m^2c^4\hspace{40pt}(1)$$ Equation (1) can be divided into $E$ on both sides. We obtain ...
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Adjoint Dirac equation (in momentum form) from Dirac eq in momentum form method

I just wanted to check the method I have formulated for the derivation for the adjoint Dirac equation using Gamma matrice notation. This is a problem from the very excellent "Modern Particle Physics" ...
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Energy in free Dirac equation [duplicate]

In one text after general solution of free Dirac equation, I read: for consistency in contribution to the energy both from particles and antiparticles we need anti commutator, and particle and ...
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202 views

Are there negative energy states in QED?

I was reading Weinberg I, when I came upon the following statement$^1$ (slightly edited by me): \begin{align} (\not p+m)u=ie\not A\\ (\not p-m)v=ie\not A \tag{1} \end{align} The minus sign on ...
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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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67 views

Plane Wave Solutions of the Dirac Equation

I'm trying to understand the plane wave solutions of the Dirac Equation. But I'm still a newbie on indices notation and contravariant and covariant objects. What I don't understand is how to get: ...
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Plane Wave Solutions to the Majorana Equation with Zero Momentum

My question concerns the plane wave solutions to the Majorana equation. First, recall the Dirac equation: $$(i\gamma^\mu \partial_\mu-m)\psi=0$$ I suggest a solution in the form of a plane wave with ...
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48 views

Preference of Chirality

I was interested to see that , $$ \gamma^5 \psi = \psi_R - \psi_L $$ By the definition of chirality projection operator and that $\psi = \psi_R + \psi_L$. since $\gamma^5 \psi$ pops up a lot in ...
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Behaviour of Dirac Bilinears

Dirac bilinears transform in the Lorentz indices as, $\bar{\psi}\psi$ scalar $\bar{\psi}\gamma^\mu\psi$ vector $\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor ...
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Adjoint momentum Dirac equation

So we have the commonly quoted momentum space version of the Dirac equation and the adjoint Dirac equation: $$ (\gamma^{\mu}p_{\mu}-m)u=0 $$ Often, we are asked to show that the adjoint momentum ...
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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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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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Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
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Confusion over trying to understand spinor components

I've been reading about the quantisation of the Dirac field $\psi(x)$ and it is stated that the general solution to the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi(x)=0$ is given by the ...
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86 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 $$ ...
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Confusion with chirality eigenstates

In the Weyl/chiral basis, the four components of the Dirac spinor represent left-chirality spin up, left-chirality spin down, right-chirality spin up, and right-chirality spin down, respectively. When ...
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Effective theory of topological insulator in coulomb impurity

I am trying to solve for the Haldane model with a coulomb impurity at one site in the effective theory approach and look for some topology in the solutions of the wave functions. The Hamiltonian near ...
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Intervalley scattering in graphene in presence of impurities

A long range impurity like coulomb impurity does not induce an inter valley scattering between the two Dirac points. Is there any mathematical explanation for the same although this is explained by ...
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88 views

Hermitian properties of Dirac operator

I am trying to understand the Hermiticity of the (massless) Dirac operator in both (flat) Minkowski space and Euclidean space. Let us define the Dirac operator as $D\!\!\!/=\gamma^\mu D_\mu$, where ...
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How do you derive the Dirac Lagrangian density for spinor fields?

I know the how the Dirac Lagrangian is written but I don't understand how to derive it from the general definition $L=T-V$. So I guess I would also like to know what the Kinetic and Potential energies ...
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Exact energies of spherical harmonic oscillator in Dirac equation

The potential is given by: $$ V(r) = {1\over 2} \omega^2 r^2 $$ and we are solving the radial Dirac equation (in atomic units): $$ c{d P(r)\over d r} + c {\kappa\over r} P(r) + Q(r) (V(r)-2mc^2) = E ...
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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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Understanding Dirac equation notation

I'm trying to recover the Einstein energy-momentum relation from the Dirac equation. I'm given a solution wavefunction, $$\psi = u(E,\vec p) e^{i(\vec p\cdot\vec x - Et)}$$ with $$\vec u = ...
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Justifying commutation and anticommutation relations in lattice QCD

The article "Construction of a selfadjoint, strictly positive transfer matrix for euclidean lattice gauge theories" (Lüscher 1977), about lattice QCD, says the following: The fermion Hilbert space ...
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QFT: prove Dirac lagrangian is invariant under C, P, T separately

As it is stated in Peskin, $\mathcal{L}=\bar\Psi(i\gamma_{\mu}\partial^{\mu}-m)\Psi$ is invariant under C,P and T transformation separately. I have some problems to see how the partial derivative is ...
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Why are spin-1/2 particles the simplest particles? [closed]

Paul Dirac, in his interview with Friedrich Hund, mentioned that it was to his surprise that his equation automatically incorporated spin. He said that he thought the simplest theory, for which he was ...
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Dirac equation from a vierbein operator?

Klein-Gordon equation can be derived straightforwardly by getting the mass-energy relation from special relativity in tensorial form, $$\eta^{\mu\nu}p_{\mu}p_{\nu} = m^2c^2$$ and promoting the ...
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Charge conjugation and the conserved charge for the Dirac field

So, while reading Peskin & Schroeder's chapter on the Dirac field, they claim that the charge conjugation operator has the following properties: $$ \mathcal{C}\psi(x) \mathcal{C} = -i \gamma^2 ...
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Angular Momentum of the Dirac field

I'm going through the Peskin & Shroeder's discussion on the Dirac field, and I am struggling with a couple of claims they make about angular momentum. First of all, the angular momentum operator ...
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Dirac or Schrödinger equation for higher spin?

Given a fermion or boson with an arbitrary integer or half integer spin, then what would be its Dirac or Klein-Gordon equation? Dirac equation for an equation with arbitrary spin 0, 1/2 , 1 , 2 , 3/2 ...
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Is it possible to decompose into eigenstates of Dirac Hamiltonian?

If we have the Hilbert space $\mathcal H = L^2(\mathbb R^3, \mathbb C^4)$ and a Hamiltonian: $$H=\gamma^i p_i + m \gamma^0$$ where $\gamma^i$ are matrices and $\{\gamma^i,\gamma^j\}=\delta^{ij}$. A ...
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Derivation of Gordon identity from Srednicki [closed]

On srednicki page 240 (print) there is a derivation of the Gordon identity, and it starts with stating that $$ \require{cancel} \gamma^{\mu}\cancel{p} = \frac{1}{2} \big\{\gamma^{\mu},\cancel{p} ...
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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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A question about the Dirac mass and Majorana mass

I am sorry if my question seems to be naive. For the free Dirac field, the Lagrangian is $$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m_D)\psi$$ or expressed in the Weyl spinor, the mass term ...
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Chiral tunneling in Weyl Equation

I am trying to understand perfect tunneling of particles obeying Weyl equation through a potential barrier at normal incidence. I know that this has something to do with chirality, but I am not ...
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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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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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Parity violating Dirac particle

We normally write down the Dirac Lagrangian as \begin{equation} {\cal L} _D = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi \end{equation} but are the Lagrangian's, \begin{equation} ...
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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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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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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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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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Unitary Lorentz transformation on quantized Dirac spinor

I am stuck again on page 59 of Peskin and Schroeder. In particular, I do not know how they get equation (3.110). Let me first give some background in the way that I understand it (but I might be ...
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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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Can we treat $\psi^{c}$ as a field independent from $\psi$?

When we derive the Dirac equation from the Lagrangian, $$ \mathcal{L}=\overline{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\overline{\psi}\psi, $$ we assume $\psi$ and ...
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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[ ...