Questions tagged [dirac-equation]

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 spectrum.

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2answers
262 views

Is there a bi-4-vector representation of the Dirac gamma matrices and the spinor?

I learned recently that if you have the Dirac spinor represented in the Weyl (chiral) basis $\Psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}$, then given a Lorentz Transformation $\Lambda = exp[\...
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The strange behave of wave function of the hexagonal lattice. Not single-valued, strange at K and K' Points

I want to know a wave function should be periodic or not in the Brillouin Zone. I calculated the eigenvalues and wave function of the hexagonal lattice by the tight-binding approach. The wave ...
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93 views

Dirac operator with torsion

On a spin manifold $M$, (in local coordinates) the Dirac operator $D_M$ is of the form $$-i\gamma^{\mu}(\partial_{\mu}-\frac{1}{4}\tilde{\Gamma}^b_{\mu a}\gamma^a\gamma_b),$$ where the (torsion-free) ...
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How this spinor identity is shown?

In one QFT problem it is asked to prove the following identity: $$\overline{u}_\sigma(p)\gamma^\mu u_{\sigma'}(p)=2\delta_{\sigma\sigma'}p^\mu.$$ Considering $u_\sigma$ the basis solutions to the ...
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Has the relative sign in the Dirac equation any meaning?

I know there are many questions about this topic and also various answers, but it's never stated explicitly, why there is a certain sign before the mass term in the Dirac Lagrangian. I'm also confused,...
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What does “Relativistic” mean in Quantum Mechanical Terms?

I was reading recently how the compatibility of quantum mechanics with special relativity was initially a problem for physicists and then Dirac succeeded in formulating a relativistic, quantum-...
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1answer
198 views

Charge Conjugation of massive Dirac spinor in 3 dimensions with Euclidean signature

In 2+1 dimensional massive Dirac equation (Minkowski signature), we can define the charge conjugation operator so that the equation can be symmetric under it. However, the charge conjugation does not ...
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1answer
156 views

Green's function for Dirac operator on $S^4$

Let $S^4$ be a round sphere of radius 1 (with the standard Riemannian round metric), and let $D_\text{F} \equiv \gamma^\mu \nabla_\mu$ be the Dirac operator on $S^4$, acting on the usual spinors for ...
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109 views

Question about the Dirac Field: determining the scalars inside the integral on the spinors

You can read only this first paragraph to understand my problem. In the second one I explain how I tried to understand but it is not successful. In my QFT course we wrote : $$ \psi(x)= 2m \int d\...
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2answers
103 views

Can the necessity of using anti-commutators for Dirac fields and commutator for Klein-Gorden be deduced from the field equations?

We all learned to use the commutator for quantizing the KG field and the anti-commutator for the Dirac field. We are told (which is correct) so that KG-excitations are bosons and Dirac-excitations ...
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1answer
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Charge conjugation operator and gamma matrices

The gamma matrices are defined by their anticommutation relations, which are symmetrical in permutations of $\gamma_1, \gamma_2, \gamma_3$. Given this symmetry, why is the change conjugation operator $...
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312 views

Plane wave solutions to massless Dirac equation

I'm revising for my QFT exam and have encountered some issues with something I haven't seen before: finding the plane wave solutions to a massless Dirac equation: $$i\gamma^\mu\partial_{\mu} \psi = 0$...
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How to find Bilinears of a theory?

I'm trying to understand how one finds the bilinears of a given theory. In most litterature the bilinears are not really derived but rather taken as fundamental. The dirac bilinears are of course: $$\...
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Interpretation of diferent chirality components as diferent particles

In many questions like What is chirality?, Confusion with chirality eigenstates and What is the relation between the Higgs field and chirality?; and also here http://www.quantumdiaries.org/2011/06/19/...
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1answer
142 views

Confusing signs in solutions to Dirac Equation

In the Dirac-Pauli Representation $\gamma^{0}= \left( \begin{smallmatrix} I&\ \ 0\\ 0&-I \end{smallmatrix} \right)$ , $\gamma^{i}= \left( \begin{smallmatrix} 0&\sigma_i\\ -\sigma_i&0 \...
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What is negative energy and how does it predict antimatter?

According to multiple online sources, antimatter was discovered through the Dirac equation because there were multiple solutions; a positive energy solution, to be expected and a negative energy ...
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189 views

Interpretation of the conserved current in classic Klein-Gordon and Dirac equations

The conserved current in KG is $$j^{\mu}=i(\phi^*\partial^{\mu}\phi-\phi\partial^{\mu}\phi^*) =2p^{\mu}|N|^2$$ where N is a normalization factor. This current can't be understood as a probability ...
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263 views

Dirac equation for $\bar{\psi}$ [closed]

I know that the Dirac equation is $$i\gamma^{\mu}\partial_{\mu}\psi=m\psi$$. How do I use this to show that $$(\partial_{\mu}\bar{\psi})\gamma^{\mu}=im\bar{\psi}?$$
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1answer
262 views

Why do we say that in non-relativistic limit we need only two component spinor?

Why do we say that in non-relativistic limit we need only two component spinor? (As in Schrödinger equation, we do not even talk of spinors,... they are one component object) I have read this ...
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247 views

Obtaining curved space Dirac equation from action (tetrad formalism)

I'm reading the book Covariant Loop quantum gravity by C.Rovelli where in 3.2 the action of a dirac fermion is presented in the tetrad formalism: $$S= \int \bar{\psi} \gamma^{I} D\psi \wedge e^J \...
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Nature of Chirality: Additive or multiplicative?

What kind of quantum number is Chirality? Helicity, being the projection of spin in the direction of the momentum, is like a component of spin, and therefore, additive in nature. For a process, $A\to ...
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Spin Connection derivation in dirac equation

I'm learning about the spin connection, more specifically how it is derived in the dirac equation through this notes: http://web.phys.ntnu.no/~mika/CPP/ch15.pdf In the third page of the document it ...
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843 views

Helicity of Antiparticles

I'm really confused by the helicity and handeness of antiparticles. Consider the particle case, the plane wave solution is $\psi(x) = u(p)e^{-ip\cdot x}$, where $u^s(p) = \begin{pmatrix} \sqrt{p\...
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Spontaneous Creation of Electron and Positron Pair

In second quantization, we prove the existence of positrons due to the necessity of negative frequency. Later in the book it talks about motion in a centrally symmetrical field and says that Dirac's ...
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1answer
277 views

Meaning of terms in Dirac Lagrangian

If I have a look at the most generalized version of the Dirac Lagrangian I can not identify all terms with there contrubution in the Feynman diagrams. $$\mathcal{L} = \sum_a (\overline{\psi}_a {\not}...
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Is the conserved current probability or charge in Klein-Gordon and Dirac equations?

The conserved currents in KG and Dirac are: K-G : $j^{\mu}=i(\phi^*\partial^{\mu}\phi-\phi\partial^{\mu}\phi^*)$ Dirac: $j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$ $j^0$ is positive definite in Dirac's ...
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Misconceptions about the Klein-Gordon and Dirac equations and their problems [duplicate]

In my course on particle physics the KG and Dirac equations were explained in an historical way. However, it was more confusing than clarifying. The problems with the KG equation were The probability ...
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How is a particular form of the Dirac Spinor derived by a boost from the system of rest?

In Peskin-Schroeder Chapter 3.3: Free Particle Solutions of the Dirac Equation the form of the general Dirac Spinor $u(p)$ along 3-direction is derived from $u(p_0)$ in the system of rest by applying ...
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77 views

Spin states in a finite potential well

i have a question concerning an electron in an attractive potential well. Let's suppose the potential function is defined as $$V = \left\{ \begin{array}{cl}0, & \mbox{for } z < 0\\ ...
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1answer
104 views

Is the Cosmological constant part of or predicted by or supported by the Dirac Equation?

By Cosmological constant here I mean the constant as would be predicted by the vacuum. The one and the same that if compared to the actual expansion of the universe it is off by a factor of $10^{120}$....
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Dirac equation and particle probability density

I do know that nowadays dirac equation is interpreted as field-theoretic equation, not as particle equation. But is there any sense that dirac equation can still be said to have wavefunction that ...
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384 views

Lagrangian density for Majorana particles

The Majorana fermion satisfies the equation $$(i\tilde\gamma^\mu\partial_\mu -m)\psi=0,$$ where the matrices $\tilde\gamma^\mu$ satisfy the standard Dirac relations $$\{\tilde\gamma^\mu,\tilde\gamma^\...
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135 views

Dirac equation, why not unitary, why not single-particle formalism?

I am reading the first chapter of Akhiezer, Berestetskii QED (1981). They state that Dirac was wrong to assume that the evolution of the wave function is described by $\psi(t) = e^{-iHt} \psi(t_0)$ ...
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The determinant of the Dirac operator in Euclidean signature

Suppose the Dirac operator determinant in Euclidean space-time with manifold $\mathbb R^{4}$: $$ d = \text{det}(iD), \quad iD = i\gamma^\mu (\partial_\mu +A_{\mu}) $$ The Dirac operator is elliptical, ...
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1answer
573 views

Pauli matrices and Lorentz transformations

Consider the Weyl equations: \begin{align} i\sigma^{\mu} \partial_{\mu} \psi_{L} & = 0 \\ i\overline{\sigma}^{\mu} \partial_{\mu} \psi_{R} & = 0, \end{align} where $\sigma^{\mu} = \left ( \...
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1answer
243 views

Problem in deriving Schrodinger and Pauli equation from Dirac's

Working out the non relativistic limit of the Dirac equation, we encounter this quantity: $(\vec{\sigma} \cdot \vec{p})$ and in my notes it says that $$ (\vec{\sigma} \cdot \vec{p})^2 = p^i p^j\sigma^...
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Why is the trivial analogous expression for Feynman's checkerboard approach to Dirac's equation in 3+1 dimensions (as described below) not correct?

Feynman's checkerboard approach to Dirac's equation in 1+1 space says that a half spin particle can be assumed to be traveling at speed of light and switching directions only after discrete intervals ...
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595 views

Lagrangian for a free Dirac field equal zero?

The Lagrangian (density) for a free Dirac field is given as $${\mathcal L}_\mathrm{Dirac} = \bar{\psi} \left( i \gamma^{\mu} \partial_{\mu} - m \right) \psi,$$ but given that $\psi$ obeys the Dirac ...
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Derivative with respect to a spinor of the free Dirac lagrangian

When we derived Dirac Equation starting form the lagrangian, our QFT professor said: "let's take the free lagrangian $$\mathscr L = i\bar\Psi\gamma^\mu\partial_\mu\Psi - m\bar\Psi\Psi$$ and perform $...
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1answer
495 views

Gauging the chiral symmetry: Is there a vector field that couples to chiral current?

I'm trying to understand the consequences of massless Dirac field $$\mathcal{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi\tag{1}$$ when the chiral symmetry is made local i.e., $$\psi\rightarrow\psi^\prime=...
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646 views

Why can't the Klein-Gordon equation explain the hydrogen atom but the Dirac equation does?

Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom? Why can the first order Dirac equation explain it? What are the failures?
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Solution of Dirac equation-Positive and Negative energy

For particles defined with positive energy, we use $$\phi= \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ or $$ \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix} $$...
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319 views

Need help with solution of the Dirac equation

$$\left(\vec\sigma \cdot \vec{p} \right)^2=\left(\vec\sigma \cdot \vec{p}\right) \left(\vec\sigma \cdot \vec{p} \right)=\vec{p} \cdot \vec{p}+\mathrm{i}\left(\vec\sigma \cdot \left[ \vec{p} \times \...
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1answer
751 views

How to prove that Weyl spinors equations are Lorentz invariant? [duplicate]

The Dirac equation is given by: $[iγ^μ ∂_μ − m] ψ(x) = 0$ . We can prove that it's Lorentz invariant when: $ψ(x) \to S^{-1} \psi'(x')$ and $\partial_\mu \to \Lambda^\nu_\mu \partial'_\nu$, where ...
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1answer
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A question about covariant derivative on spinor field (Vierbein formalism)

Let's use Latin letters $a,b,c,\cdots$ for local Minkowski frame indices, and Greek letters $\mu,\nu\,\lambda,\cdots$ for coordinate indices. On one hand, we all know that the covariant derivatives ...
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1answer
91 views

Solution of Dirac equation: arbitrary polarizations

In my lecture notes (signature $-+++$) we find the free Dirac equation solutions. We proceed in this way: Dirac equation: $$ (i\,\displaystyle{\not} p +m)\psi(x) = 0 $$ We make the following ansatz:...
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Factorization of the energy/mass relation (as used to obtain the Dirac equation) applied to the invariant interval

In considering the Klein-Gordon equation Dirac was examining the energy equation: $$m^{2}c^{2}=g_{\mu\nu}P^{\mu}P^{\nu}$$ (though he only was considering the flat space case $g_{\mu\nu}=\eta_{\mu\nu}...
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1answer
404 views

Covariance of the Dirac equation

In the Ashok Das book "Lectures on quantum field theory" , it's written that in page 76 : therefore, the matrix $ S~ \gamma^0~ S^\dagger ~ \gamma^0$ must be proportional to the identity matrix (this ...
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How does chirality of a massive fermion change with time?

For a massive fermions, like the electron, chirality is not conserved in time. It is not a good quantum number although it is Lorentz invariant. The Dirac Hamiltonian (in particular, the mass term in ...
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Cross product of the quantum mechanical operators $\textbf{p}$ and $\textbf{A}$

While reading Advanced Quantum Mechanics by J.J. Sakurai, chapter: Relativistic Quantum Mechanics of Spin-1/2 Particles, section 3.2 the Dirac Equation, the author states the following identity: $$\...