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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36 views

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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34 views

Why only the spin $j=\frac{1}{2}$ has relativistic wave function equation that conserves positive probability?

The Dirac wave function can be thought as a relativistic wave equation, where the solution has a positive definite norm. I know that this same equation can't be thought so seriously as a wave equation ...
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1answer
140 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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2answers
111 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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24 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
67 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
74 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
81 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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53 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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71 views

Indices of a Pauli matrix transformed in the Lorentz representation

When Peskin and Schroeder want to prove a Fierz identity on page 51, they make use of the identity $$(\sigma^{\mu})_{\alpha \beta} (\sigma_{\mu})_{\gamma\delta} = 2 \epsilon_{\alpha \gamma} ...
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97 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
53 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
100 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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1answer
76 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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65 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
97 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
211 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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66 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
76 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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3answers
156 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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31 views

Energy levels for nonlinear Dirac equation

Let's have equation $$ \left( i\gamma^{\mu}\partial_{\mu} - m + a\bar{\psi}\gamma_{5}\gamma_{\mu}\psi \gamma_{5}\gamma^{\mu}\right)\psi = 0. $$ Here $a$ is constant. I need to evaluate energy levels ...
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1answer
114 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
84 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)= ...
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133 views

What specifically is incorrect about the Dirac Sea interpretation?

So taking the square root of $E^2 = (m_oc^2)^2 + p^2c^2$ yields two solutions. The Dirac Sea treats the negative solution as an infinite space of electrons with negative energy. All the observable ...
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66 views

Transformation of spinors due to Lorentz group

Assume we have a Dirac spinor $\psi(x)$ which satisfies the Dirac equation: $$(i\gamma^{\mu}\partial_{\mu} - m)\psi(x) = 0.$$ If we boost our spacetime coordinates to a new system with a Lorentz ...
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2answers
66 views

Showing that a bilinear variation is Lorentz invariant

Let $\psi, \chi$ be a spinor (say Dirac). Then the infinitesimal Lorentz variation is given by $$\delta \psi = -\frac{1}{4}\lambda^{\mu \nu} \gamma_{\mu \nu}\psi$$ then I think that the conjugate is ...
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64 views

Physical significance of Dirac equation in (2+1)-D

What's is the physical significance of the two inequivalently irreducible-represented Dirac equations in (2+1)-D? As it is known, all the $4\times 4$ matrix representations of the Dirac algebra ...
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1answer
149 views

Solutions to Dirac Equation in Weyl Representation

Reading a into QFT I recently came across basically this (Kaku p.94): If $\Psi (x)$ is a solution to the massless Dirac equation in Weyl representation, also $\Phi (x) = \exp(i \Lambda \gamma^5) ...
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1answer
40 views

Help with a vector-spinor equation

How can I show that the equation $$\gamma^{abc}\partial_{b}\psi_c=0$$ leads to $$\partial_{b}\psi_{c}-\partial_{c}\psi_{b}=0?$$ I know that $$\gamma^{abc}= \frac{1}{2}\{ \gamma^{a}, \gamma^{bc} \}$$ ...
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117 views

How to prove explicitly that by including Dirac fermions into the Einstein-Hilbert action we make torsion to be non-zero?

Recently I've heard the statement that by including Dirac fermions into the Einstein-Hilbert action we make torsion be non-zero, so that is one of problem of quantum gravity. How to prove that ...
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1answer
87 views

Where do the quantum fields encode the spin information?

I know basically the difference between Klein-Gordon and Dirac field is spin. But I am not sure where we need to implement this info. The solutions of both equations are the wave packets which ...
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1answer
59 views

Trying to understand the symmetries of higher dimensional $\gamma$-matrices

I am reading that there exists a unitary matrix $C$ (the charge conjugation) matrix such that each matrix $C\Gamma^{A}$ is either symmetric or anti-symmetric. Now, $\Gamma^{A} = \{ {\bf 1}, ...
6
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1answer
85 views

Substitution $\partial_\mu \to D_\mu \equiv \partial_\mu + ieA_\mu$ allows the introduction of electromagnetic interactions [duplicate]

I want to show that the substitution $\partial_u \to D_\mu \equiv \partial_\mu + ieA_\mu$, or equivalently $p_\mu \to p_\mu - eA_\mu$ allows the introduction of electromagnetic interactions. Here $e$ ...
3
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0answers
69 views

Substitution $\partial_\mu \to D_\mu \equiv \partial_\mu + ieA_\mu$ allows the introduction of electromagnetic interactions [closed]

I want to show that the substitution $\partial_u \to D_\mu \equiv \partial_\mu + ieA_\mu$, or equivalently $p_\mu \to p_\mu - eA_\mu$ allows the introduction of electromagnetic interactions. Here $e$ ...
2
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1answer
203 views

What are the relative limitations of the Schrödinger, Pauli, and Dirac Equations?

I know there are significant differences in the nature of the Schrödinger, Pauli, and Dirac equations. Although I know a bit about how each works, I don't understand the relative limitations of each ...
2
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2answers
114 views

Is light emitted in zitterbewegung? [duplicate]

Recently I heard of Zitterbewegung, a trembling motion of the electrons in atoms that arises from Dirac's equation. I know that, according to Bohr's model, light is emitted when the electron "jumps" ...
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2answers
246 views

Feynman-Stueckelberg interpretation

My question is related to the interpretation of antiparticles. According to the so called Feynman-Stueckelberg interpretation a negative energy solution of the Dirac equation corresponds to a positron ...
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1answer
50 views

How to get second order equation for spinor (derivation from Dirac equation)?

Dirac equation with an Abelian symmetry can be written as $$(\gamma^{\mu}D_{\mu} - m)\psi = 0$$ where $$D_{\mu}\psi = (\partial_{\mu} - iqA_{\mu})\psi$$ Then how do we get this second order equation ...
2
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1answer
97 views

Do Dirac field states belong to a Hilbert space with spinor coefficients?

The quantized Dirac field at a certain space-time point can be written (roughly) as a linear combination of creation operators acting on the Hilbert space of physical states, with coefficient that are ...
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1answer
313 views

Making sense of the canonical anti-commutation relations for Dirac spinors

When doing scalar QFT one typically imposes the famous 'canonical commutation relations' on the field and canonical momentum: $$[\phi(\vec x),\pi(\vec y)]=i\delta^3 (\vec x-\vec y)$$ at equal times ...
2
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1answer
135 views

Calculating probability of finding the particle using Dirac notation

An electron can be in one of two potential wells that are so close that it can ‘tunnel’ from one to the other. Its state vector can be written $|ψ\rangle = a|A\rangle + b|B\rangle$, where ...
6
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3answers
237 views

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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181 views

Are there eight or four independt solutions of the Dirac equation?

I edited the question as a result of the discussion in the comments. Originally my quesiton was how to interpret the four discarded solutions. Now I'm making a step back and hope that someone can ...
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65 views

What is really interacting in weak interactions?

Only particles with chirality $-1$ do interact weakly. The corresponding eigenstate in the Dirac basis is $ \Psi_L = \begin{pmatrix}f \\ -f \end{pmatrix} = \begin{pmatrix}u_r {\mathrm{e}}^{-imt} \\ ...
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29 views

Everything moves at the speed of light? [duplicate]

Whatever happened to that idea? Presumably it came from a concept known as Zitterbewegung. As wiki says, a theoretical rapid motion of elementary particles, in particular electrons, that obey the ...
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152 views

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 ...
3
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1answer
64 views

Dirac operator partial integration

When you have an action with bosonic $X$ and fermionic $\psi$ (Majorana) fields and perform a SUSY transformation $\epsilon$ (the constant, infinitesimal parameter of transformation, a real, ...
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1answer
58 views

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 ...
3
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68 views

Tunneling from Dirac material into Schrodinger material?

When a Dirac material, like graphene or TI, has a connection with a normal metal which Schrodinger equation govern on their carriers, how could we manipulate the tunneling of electron from Dirac side ...
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1answer
450 views

How can pseudospin be a vector? (Graphene)

In graphene science, I don't understand how one interprets pseudospin as a vector. I thought 'pseudospin' was the vector of Pauli matrices. So how can it be a vector that one can plot for example in ...