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.

179 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
13
votes
1answer
806 views

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 ...
8
votes
0answers
411 views

Is there supersymmetry between Dirac and Klein Gordon solutions?

Usually supersymmetry is explained at the level of the action of a quantum field theory, and there are two ways to go down from QFT to relativistic quantum mechanics: either a non-covariant way where ...
6
votes
0answers
321 views

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 L=\frac{1}{2}(\partial\phi^2+m^...
5
votes
0answers
209 views

How to derive the Klein-Nishina formula from the Dirac equation?

I'm looking for the simplest demonstration of the Klein-Nishina formula, from the Dirac equation without the field described as a quantum operator: https://en.wikipedia.org/wiki/Klein%E2%80%...
5
votes
0answers
90 views

Relativistic scattering off Dirac delta potential

Consider the case of a relativistic electron on a graphene lattice described by the Hamiltonian $$ \mathcal{H} = v\begin{pmatrix} 0 & p_x+ip_y \\ p_x-ip_y & 0 \end{pmatrix}, $$ where $v$ is ...
5
votes
1answer
150 views

Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful. We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
5
votes
0answers
255 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.
4
votes
1answer
74 views

Varying the Dirac action with differential forms

The Dirac action in a curved spacetime can be written in terms of the vierbein $\{ e^a \}$ and spin connection $\{ \omega^{ab} \}$ differential forms. Let the spinor field $\psi$ be interpreted as a ...
4
votes
0answers
135 views

Should the parallel propagator appear in the point-split stress-energy tensor?

The first step in Hadamard regularization of the stress-energy tensor of a free Dirac field is to write out the point-split expression $$\langle T_{\mu \nu} \rangle \equiv \frac{1}{4} \lim_{x'\to x} \...
4
votes
2answers
415 views

Relation between spinors and anticommutation relation of fermions

I read that the state of a pair of particles is the tensor product of the single states of both, and you will get a wavefunction with the parameters of both, if you swap the parameters you will get a ...
4
votes
0answers
261 views

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 ...
4
votes
0answers
348 views

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 ...
4
votes
0answers
141 views

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, ...
4
votes
0answers
109 views

A question about the emergence of 'spin' from relativistic QM

I know that quantum-spin is not equivalent to the spinning of a classical object about an axis passing through it, although there are some similarities. I also know that spin naturally emerges out of ...
4
votes
0answers
268 views

Dirac equation in curved spacetime - found second derivatives of the metric, violation of the principle of equivalence?

I am working on the Dirac equation on curved spacetime. A Foldy-Wouthuysen transformation was applied to obtain the semiclassical limit of the equation to study the dynamics of the spin of the ...
4
votes
0answers
273 views

About Dirac equation in curved spacetime (spherical)

I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta d\...
4
votes
0answers
127 views

Path Integral on Einstein Cartan Manifold

In condensed matter, crystal with disclination and dislocation has both curvature and torsion. I am looking for a reference in which path integral quantization of Dirac equation on manifold with ...
4
votes
0answers
328 views

Dirac equation in curved space-time with Torsion

I am looking for pedagogical references in which Dirac equation in space-time with curvature and torsion were discussed.
4
votes
0answers
768 views

Relativistic genarization of Quantum Harmonic Oscillator

I am trying to find out relativistic description of a quantum harmonic oscillator. For a classical relativistic oscillator mass is a function of co-ordinates(http://arxiv.org/abs/1209.2876). $$m(x)=m-\...
4
votes
1answer
119 views

Why does the Dirac equation work for the hydrogen atom?

The Dirac equation works well for predicting the spectrum of the hydrogen atom, famously incorporating relativistic effects like fine structure. Yet, there seems to be a sense in which this is ...
4
votes
1answer
270 views

Does electron have some intrinsic ~$10^{21}$ Hz oscillations (de Broglie's clock/Zitterbewegung)?

De Broglie has postulated in 1927 that with electron's mass there comes some inner oscillation: $E=mc^2=h f=\hbar \omega$. We would get such oscillation e.g. if using $E=mc^2$ energy in stationary ...
3
votes
2answers
102 views

Why would a spinor transform under Lorentz transformations?

From my understanding of spinors, they arise as projective representations of $SO_0(1,3)$ that do not correspond to representations of $SO_0(1,3)$. But still one says here - and virtually everywhere - ...
3
votes
1answer
192 views

Second order relativistic corrections to Pauli equation from Dirac equation

I'm trying to derive the full and correct Hamiltonian for spin$\frac{1}{2}$ particles from Dirac equation up to second order in $v/c$. For a potential and magnetic field constant in time. In ...
3
votes
0answers
51 views

Conservation of $\int |\psi|^2$ for Dirac wave

When $\psi$ be Schrodinger wave $\int |\psi|^2$ is conserved even when this wave interact whit another wave say electromagnetic wave. and this is very necessary for one particle interpretation of this ...
3
votes
0answers
41 views

Is there a mathematical singularity in the relativistic contraction of valence orbitals as a function of nuclear charge?

I'm looking at Figure 1 on p. 2 of Jansen, "Effects of relativistic motion of electrons on the chemistry of gold and platinum" (2005) (should be a free download). From the bottom curve, there is such ...
3
votes
0answers
63 views

How similar are the spin states and the matter/antimatter states?

Within the Dirac formalism, we have bispinors that represent both if a particle is spin up or spin down, and if a particle is an electron or a positron. And these representations are very similar. (...
3
votes
0answers
178 views

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 $$E=\frac{v_1}{c}...
3
votes
0answers
583 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 $$ T_{\mu\nu}=...
3
votes
0answers
275 views

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 ...
3
votes
0answers
177 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 ...
3
votes
1answer
188 views

Difference between positron and electron scattering in Coulomb field

In first order of perturbation theory the S-matrix amplitude for electron scattering in the Coulomb field will be (up to normalization factors) $$ S_{fi} = \frac{iZ q^2}{\sqrt{2E_{f}2E_{i}}}\bar {u}(...
3
votes
0answers
586 views

Step in derivation of solution to Dirac equation for hydrogen

My text, when solving hydrogen in the Dirac equation, makes the claim $\varphi_{j m_j}^{(+)} = \frac{\mathbf{\sigma} \cdot \mathbf{x}}{r} \varphi_{j m_j}^{(-)}$ where $\varphi_{j m_j}^{(\pm)}$ are ...
2
votes
0answers
24 views

Lorentz invariant probability from the Dirac equation

My question is regarding a proof given in Greiner's "Relatavistic Quantum Mechanics", 3rd Edition textbook. On pg 148, he proves that the current density $j^{\nu}(x)$ is invariant to a Lorentz ...
2
votes
0answers
62 views

Re-Writing the Dirac Equation in True Covariant Form

This is a rather brief inquiry, but to get to the point it's always frustrated me that in non-relativistic and relativistic quantum mechanics spin matrices are written as a "vector of matrices" ...
2
votes
0answers
146 views

Does the Dirac sea have any mass or gravitational effect?

Dirac sea is a model for vacuum which considers the empty space as a sea full of negative-energy particles. Anti-particles are holes in this sea. Dirac sea is used to model Quantum field theory in the ...
2
votes
0answers
63 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
2
votes
0answers
77 views

Can the Dirac Hamiltonian accommodate a variable speed of light?

The Dirac Hamiltonian has the form1 $$\left[\beta m c^2+c\sum_{n=1}^3\alpha_np_n\right]$$ where $\alpha_n$ and $\beta$ are Hermitian matrices, and $c$ is the speed of light. My question: Is there a ...
2
votes
0answers
104 views

In what sense are solutions to the Dirac equation and solutions to the Laplace equation equivalent in string theory?

I have come across statements like elementary particles on a Calabi-Yau correspond to harmonic forms (or to cohomology classes, which is equivalent for a compact Kähler manifold, since every ...
2
votes
0answers
62 views

Relativistic Von Neumann equation?

In non-relativistic quantum theory, Schrödinger's equation can be re-expressed using the density matrix $\rho=|\psi\rangle \langle\psi|$ as the Von Neumann equation: $$i\partial_t \rho = \frac{1}{\...
2
votes
0answers
193 views

Calculate Chern number from band structure

When we use a tight-binding approach in order to calculate the band structure of electrons on a 2D honeycomb lattice such as Graphene we find that there are two energy bands touching in six points, ...
2
votes
0answers
109 views

Two ways of thinking about the Dirac equation

My impression is that there are two ways of thinking about the Dirac equation: Quantum Mechanically: Here we think of the spinor $\phi$ as a generalization of the Schrodinger wave function which ...
2
votes
0answers
116 views

Why are antiparticles associated with spin-flipped spinors?

In section 2.2 of Elvang and Huang's Scattering Amplitudes in Gauge Theory and Gravity (http://arXiv.org/abs/1308.1697), beneath equation (2.9), it is mentioned that $u^{\pm}=v^{\mp}$, where $u^\pm$ ...
2
votes
0answers
41 views

Why is it reasonable to recognize solutions to Dirac Equation as electron and positron?

I'm reading Griffiths' book and when it claims the first two components of plane wave solution to massive Dirac equation as electron and the “holes” of other two components as positron, it does not ...
2
votes
0answers
359 views

Spin covariant derivative of gamma matrices?

Where can I find a general expression (on curved manifolds) in local coordinates, for the following: $$\nabla^S_{\mu}\gamma^{\nu} = ?$$ $\nabla^S_{\mu} = \partial_{\mu} + \omega^S_{\mu}$ is the spin ...
2
votes
2answers
159 views

What's the path of least action for fermions off-shell?

The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following ...
2
votes
0answers
144 views

Transformation of Weyl spinors

I usually see Weyl spinor and Weyl equations as derived from Dirac equation, like in Peskin. Now, I'm following a course where the professor actually builds Weyl spinor lagrangians BEFORE talking ...
2
votes
0answers
359 views

In the Dirac equation, if the $\alpha$ is the mean velocity, why does it commute with $x,y,z,t$ if the velocity is related to the momentum?

In the Wikipedia talk page for the Dirac equation I found the following passage: The Dirac equation can be proved with the help of the correspondence principle. The energy and momentum of a ...
2
votes
0answers
149 views

Formal definition of gauge field and spinors in QFT

I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where $i$ is a color index in the fundamental ...
2
votes
0answers
94 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) ...
2
votes
0answers
131 views

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 ...