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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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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388 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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68 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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32 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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1answer
102 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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343 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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38 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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1answer
146 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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248 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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37 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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2answers
113 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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28 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
89 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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1answer
78 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
83 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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54 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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72 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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933 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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103 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
57 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
106 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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68 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
79 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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1answer
100 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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4answers
743 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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1answer
222 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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68 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
158 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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161 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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81 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
684 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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1answer
119 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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118 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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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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1answer
138 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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2answers
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
67 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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67 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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120 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 ...
2
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1answer
254 views

Lorentz transformations and gamma matrices

I am reading Zee's QFT in a nutshell, 2nd ed. On pg. 97 below eq. 14 he writes: $$ S \gamma^{\lambda } S^{-1} = \omega_{\,\, \mu }^{\lambda } \gamma ^{\mu }+\gamma ^{\lambda }. $$ Building ...
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3answers
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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1answer
93 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
60 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}, ...
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1answer
95 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
72 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
214 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 ...
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2answers
282 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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2answers
121 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" ...