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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16 views

What would be the effect of a complex vector field in the dirac equation?

Say your dirac equation had the normal vector boson intereaction term: $i\overline{\psi}\gamma^\mu A_\mu\psi + i\psi \gamma^\mu A_\mu \overline{\psi}$ What would the effect be of another field B ...
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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} \...
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93 views

How does the negative energy solution to the Dirac equation predict the antielectron?

Please, can someone explain how the negative energy solution can be used to predict the existence of the antielectron?
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33 views

Perturbation of Diracs equation (first order)

I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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81 views

What do the matrices $\alpha_k$ represent in the Dirac equation?

I have been scouring the internet for an answer. All I have managed to find are the matrices for $k=1,2,3,4,5$. However, I still have no idea they represent, within the equation. Am I correct in ...
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60 views

Does the electro-dynamical lagrangian contain a (Dirac) wave-function?

Consider a lagrangian for quantum electro-dynamics. It contains the two fields: the vector $A$-potential inside $F_{\mu\nu}$ and the matter field $\psi$ (Dirac's spinor). A series of questions arise ...
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21 views

Question about Spinors and Probability Densities

So I was toying around attempting to simulate some relativistic wave equations for a recreational project. Now I have never studied spinors in dept and the knowledge I have is from reading online (...
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Interpretation of tilted energy dispersion cones in a Dirac Semimetal

The energy dispersion of a Dirac semimetal with an effective Dirac Hamiltonian of the form $$H=v_x \sigma_xk_x+v_y\sigma_yk_y+v_t\sigma_0k_y$$ is tilted in the y direction and the tilting increases ...
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284 views

How does canonical quantization work with Grassmann variables?

Every quantum field theory textbook I've encountered seems to have the same logical oversight, because of the particular order they cover topics. First, the books introduce the Dirac Lagrangian, $$\...
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166 views

Why is the Dirac Sea concept taught in physics courses without explaining that is fundamentally flawed? [closed]

Many physics text books reference to the concept of the Dirac sea as explanation of negative frequency solutions of the Dirac equation. It is supposed to be a bottomless "sea" of filled electron ...
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Showing hermiticity properties of Dirac matrices using hamiltonians

I want to show $${\gamma^0}^\dagger=\gamma^0\\ {\gamma^i}^\dagger=-\gamma^i.$$ To do this I consider the Dirac equation $$ (i\gamma^\mu\partial_\mu-m)\psi=0$$ and I write it as $$ i\partial_t \psi=(...
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57 views

Peskin and Schroeder: derivation of Dirac fields commutator

I'm perplexed by the following non numbered equation at page 54 of Peskin & Schroeder, right between $(3.92)$ and $(3.93)$ $$ [\psi_a(x),\overline{\psi}_b(x)]=\int\frac{d^3p}{(2\pi)^3}\frac{1}{...
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98 views

Is the Dirac Lagrangian locally gauge invariant without gauge field $A$?

When it comes to the check of the invariance of the Lagrangian of the Dirac equation under local $U(1)$-transformations I have made the following observation: $$L = \bar{\psi} (i\gamma^{\mu}\...
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142 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 ...
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83 views

How does the Dirac operator change under adjoint transformation?

I'm trying to demonstrate an identity $$\int \overline{\psi}D\phi = \int \overline{D\psi}\phi$$ by substituting in the dirac operator as $D = i\gamma^{a}\partial_{a}$ and $\overline{\psi} = \psi^{\...
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110 views

C and T Symmetry of Free Dirac Lagrangian

I want to show the $C$ and $T$ symmetry of the free Dirac Lagrangian $$\mathcal{L}=\overline{\psi}\left(i\gamma^\mu\partial_\mu-m\right)\psi.$$ Following the notation of Peskin, Schroeder, we have ...
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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, ...
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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 ...
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Zero momentum in Non relativistic Quantum Mechanics and about Dirac matrices

In relativistic quantum mechanics, we can solve the Dirac's equation with an added condition that the momentum of the particle is $0$. However, such independence isn't provided by the Schrodinger's ...
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73 views

Dirac equation and Hamiltonian for a collection of magnetic monopoles

I am trying to understand a mathematical comment by Eugene Wigner in some old lecture$,^{[1]}$ "The Hamiltonian of the Dirac equation for two oppositely charged monopoles is not self-adjoint." What ...
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58 views

Equivalence between Dirac and Majorana action in CFT

In Mussardo's Statistical field theory Chapter 12, section 12.3 about the conformal field theory of a free fermion field he talks about the complex fermion field (Dirac field) $$ \Psi(z,\bar{z}) = \...
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Massless Dirac fermions vs helical Dirac fermions

Some papers when, dealing with graphene, write about charge carriers called helical Dirac fermions that have a conical energy–dispersion relation and a conserved quantity $\sigma\cdot k$ (pseudospin–...
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80 views

Majorana fermions

If you write the Majorana spinors as $$\chi = \begin{pmatrix}\psi_L\\ i\sigma_2\psi_L^* \end{pmatrix} \tag1$$ It satisfies the Dirac equation that leads you to the Majorana equation $$i\bar{\sigma}^\...
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48 views

Weyl basis gamma matrix identity

In finding the scattering amplitude matrix $|\mathcal{M}|^2$, I see the solutions get a way nicer calculation by using that (using Peskin & Schroeder notation): $$(\bar v \gamma^\mu u)^*= \bar u\...
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183 views

Derivation of the adjoint of Dirac equation

My goal is to deduce the adjoint of Dirac equation: $$ \overline \psi (i\gamma^\mu \partial_\mu+m)=0 \tag{1} $$ My process: I started with Dirac equation $(i\gamma^\mu \partial_\mu-m)\psi=0$. ...
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Quantum wave function evolution and time dilation [closed]

We know that spin state evolves with time...but in non relativistic QM time dilation is not accounted ...so in Dirac equation does evolution of spin state with time depend on speed...i.e does time ...
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104 views

Could there be a pseudovector kinetic term for fermions?

Could there be a kinetic term of the form $\bar{\Psi} \gamma_5 \gamma^\mu \partial_\mu \Psi $ in addition to the usual one? Or is this forbidden by a symmetry?
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109 views

Meaning of the subscripts $L,R$ for the two component Weyl spinors $\phi_{L,R}$

For a Dirac spinor $\psi$, its chiral projections are $\psi_{L,R}$ are defined as $$\psi_{R,L}=\frac{1}{2}(1\mp\gamma^5)\psi.\tag{1}$$ Acting with the chirality operator $\gamma^5$, we find $$\gamma^5\...
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Does the positron prove that Dirac's electron sea must exist? [duplicate]

Does the fact that the positron exists in cosmic rays prove the existence of Dirac' s 'sea' of spin-paired electrons in space.
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180 views

What is the relationship between the Lorentz group and the $CL(1,3)$ algebra?

In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ...
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83 views

Dirac propagator causality

I was studying the Dirac propagator and came across an excelent article which includes all the derivation, and interestingly we can conclude that the anticommutator is zero for space-like intervals. ...
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125 views

Why don't we “see” the classical Dirac field?

The electromagnetic field describes photons. If there are many photons then things become classical and we can use classical electromagnetism to describe the EM field. We can also measure the EM field ...
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151 views

Can the mass term be responsible for creation and destruction of particles?

In an interacting quantum field theory, for example, QED, the Dirac mass $m\bar{\psi}\psi$ is a piece of the free Dirac Lagrangian. On the other hand, the interaction term $j^\mu A_\mu=e\bar{\psi}\...
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175 views

What is the exact relation between anti-matter and relativity?

I have seen in relativistic QM that, when trying to create the Dirac Equation, it only make sense to be acting on -- at minimum -- a 4-component vector (actually a bi-spinor). I guess this is because ...
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Confusion about signs in the Dirac equation for an external electromagnetic field

I'm working through Maggiore's A Modern Introduction to Quantum Field Theory, and I'm studying the Dirac equation in an external electromagnetic field given by: $$ \left[\gamma^{\mu} \left(i\partial_{...
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142 views

Interpretation of the adjoint Dirac equation

Adjoint spinors $\bar{\psi}$ satisfy the adjoint Dirac equation $$ \bar{\psi} \left( \gamma^{\mu} \, p_{\mu} - mc \right) = 0 \; $$ (I'm using the terminology and units of Griffiths' "Intro. to ...
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167 views

Plugging Majorana Spinor into Dirac Lagrangian does not give Majorana Lagrangian?

This seems like it should be simple but somehow I do not see how. The Majorana Lagrangian can be written in terms of a left handed Weyl spinor $\psi_L$ as $$ \mathcal{L}_M= i \psi_L^\dagger \bar{\...
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168 views

What is the physical interpretation of chirality / chiral anomaly?

I'm dealing with this paper from C. Bär and A. Strohmeier about a rigorous derivation of the chiral anomaly. I'm not quite familiar with the physical context of chirality and its anomaly. What ...
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151 views

Lorentz covariant propagator

So the Feynman propagator for a Klein Gordon is manifestly Lorentz invariant clearly by looking at the momentum space representation written in terms of Lorentz scalars. But in the case of the Dirac/ ...
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Can the existence of antimatter be inferred from Matrix Mechanics?

It is well known that Antimatter was first predicted by interpreting the matrices that show up in the Dirac Equation as indicating its existence. Dirac factorizes $E^2=p^2+m^2$ ($c=1,\hbar=1$) into $...
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41 views

Replacing a squared potential by a position-dependent mass

I'm studying the solutions of the Klein-Gordon and Dirac equations for a relativistic particle in a potential of the form $$V(x)=\left\lbrace\begin{array}{ll} 0, & x\in[0,L]\\V_0, & x\not\in[0,...
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103 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 ...
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Metrics and Spinors

this might be better posed in mathematics but I'll ask here anyway. So the Lagrangian for the spinor field can be viewed as follows. Let $(M,g,\nabla)$ denote a locally Minkowskian spacetime, Where $\...
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81 views

$4\times4$ Dirac Hamiltonian in Graphene

When linearizing the Hamiltonian of Graphene in reciprocal space around $\vec{q} = \vec{k}-\vec{K}_\pm = \vec{0}$, where $\vec{K}_\pm$ are two independent Dirac points, one can get two Hamiltonians, ...
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171 views

Shouldn't the infinite sea of electrons contribute to gravity?

According to my understanding of the dirac equation, there's an infinite sea of electrons occupying all negative energy states which prevents an electron from dropping to lower and lower energy states ...
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59 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}{\...
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111 views

Missing identity element in the Clifford relation

While studying the Dirac equation, $$\left(i\gamma^{\mu} \partial_{\mu} - m\right)\psi = 0.$$ I have been finding difficulty understanding the following summarisation of the algebra that the $\gamma$-...
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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 ...
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46 views

Density of states after scattering

I need some help with a probably simple question because I'm not sure whether my approach is correct. Let the free Green's function of a system on a discrete lattice described by a massless Dirac ...
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84 views

Dyson Equation and scattering amplitude

For scattering theory in quantum mechanics, one can use the Dyson Equation which states that the Green's function which is a solution to the equation $$ (E - H_0 - V)G = 1$$ is given by $$ G = ...