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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42 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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41 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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65 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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96 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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26 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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97 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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12 views

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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88 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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99 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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89 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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46 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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111 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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55 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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780 views

Dirac equation as Hamiltonian system

Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
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926 views

How did one get the defining equation of probability current and conservation of probability current and density?

I'm reading the Wikipedia page for the Dirac equation: $$\rho=\phi^*\phi$$ and this density is convected according to the probability current vector $$J = ...
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29 views

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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47 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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400 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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359 views

Why do we need $2^\text{nd}$ quantization of the Dirac equation

As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion, about it. So let me start with its definition: The Dirac equation, is given by $ i ...
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40 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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153 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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257 views

How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum?

How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum? I used the Gordon Decomposition to split the Probability Current of the Dirac Field into its orbital current and ...
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38 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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119 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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32 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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93 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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93 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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2answers
235 views

Finding wave-fuctions of a Dirac particle for given 4-momentum and spin 4-vector

I've been reading through various materials on relativistic quantum mechanics, but I find the lack of simple examples disturbing. I'm acquainted with the general form the solutions to the Dirac ...
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3answers
94 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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59 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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80 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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970 views

How to construct the charge conjugation matrix for any given dimension?

Generally, Gamma matrices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally ...
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110 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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60 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
114 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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70 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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81 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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106 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
238 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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69 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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164 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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185 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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89 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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1answer
725 views

Dirac field and stress-energy tensor density

I read somewhere that stress-energy tensor density is a symmetric tensor. But if I take the Dirac Field tensor: $$T^{\mu \nu}=i \psi^\dagger \gamma^0 \gamma^\mu \partial^\nu \psi $$ How could I ...
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135 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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119 views

How do I prove the equivalence of chirality and helicity operators acting on a massless Dirac spinor?

I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf ...
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41 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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148 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 ...