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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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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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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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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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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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}...
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Can a wavefunction be written as an active transformation of the metric tensor?

Let's examine the Dirac Lagrangian of quantum mechanics: $$\bar{\psi}\left(i\gamma^{\mu}\overrightarrow{\partial}_{\mu}-mc^{2}\right)\psi=0$$ Where the arrow implies direction of action for the ...
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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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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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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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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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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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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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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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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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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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A more general completeness relation for Dirac spinors

Assume that we have two 1/2-spin particles with four-momenta $p$ and $p'$. Particle Dirac spinors satisfy the completeness relation $$ \sum_{s=1}^2u_s(p)\overline{u}_s(p)=\not p+m $$ My goal now is to ...
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What was missing in Dirac's argument to come up with the modern interpretation of the positron?

When Dirac found his equation for the electron $(-i\gamma^\mu\partial_\mu+m)\psi=0$ he famously discovered that it had negative energy solutions. In order to solve the problem of the stability of the ...
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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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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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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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Do the equations of motion simply tell us which degrees of freedom are superfluous?

A massless spin $1$ particle in 4D has 2 degrees of freedom. However, we usually describe it using four-vectors, which have four components. Hence, somehow we must get rid of the superfluous degrees ...
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153 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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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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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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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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258 views

Dirac Lagrangian after decomposing the Dirac spinor into Weyl spinors

Consider the Dirac Lagrangian, $$L=\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi$$ and take the Dirac spinor chiral decomposition with $\psi_{L}=\frac{1}{2}\left(1-\gamma^{5}\right)\...
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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 Zitterbewegung an artefact of single-particle theory?

I have seen a number of articles on Zitterbewegung claiming searches for it such as this one: http://arxiv.org/abs/0810.2186. Others such as the so-called ZBW interpretation by Hestenes seemingly ...
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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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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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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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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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$4\times4$ 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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Why does transformation matrix commute with $\gamma$ matrices?

In Paul Langacker's The Standard Model and Beyond, equation 3.80 says the following $$ \mathcal{L}' = \overline{\psi} \mathrm{i} \partial ^{\mu} \gamma _{\mu} e^{- \mathrm{i} \beta ^i L^i} e^{\mathrm{...
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How to understand the Dirac Lagrangian?

I am having some basic questions about how to interpret Lagrangians, lets start with Dirac: $L = \bar{\Psi} (i \gamma^{\mu} \partial_{\mu} -m) \Psi$, where $\Psi$ is a Dirac-Spinor, $m$ is the mass,...
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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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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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649 views

Weyl transformation of Dirac equation

The Dirac Equation is given by $$\left(i\gamma^\mu\partial_\mu- \frac{mc}{\hbar}\right)\Psi_D = 0,$$ where $\gamma^\mu$ are the Dirac $\gamma$-matrices and $\Psi_D$ is a Dirac spinor. I would like to ...
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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 = ...
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Derivation of Casher-Banks relation

Consider two-point function $\langle \bar{\psi}\psi\rangle$ in a model with massive fermions $\psi$ and gauge field: $$ \langle \bar{\psi}\psi\rangle =\frac{1}{V}\sum_{n} \frac{1}{\lambda_{n} +im}, $$...
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Relation between Dirac spinors, quaternions, and bicomplex numbers

Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ...
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Mathematical proof for antimatter?

I've heard (BBC Documentary Atom-The Illusion of Reality) that the Dirac Equation implies the existence of antimatter. Can someone tell me how that mathematical proof is done. Just for knowledge. ...
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Why is a fermion field complex?

The Lagrangian of a fermion field is \begin{equation} \mathcal{L} = \overline{\psi} (i\gamma_{\mu} \partial^{\mu} - m)\psi \end{equation} It is said that the fermion field $\psi$ is necessarily ...
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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 ...
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Breaking of a commutator involving Dirac spinors and gamma matrices

I'm trying to understand a particular step in the solution to problem 27 in THIS solution sheet. By the middle of the page, they start with the simplification of this expression $$\left[s^{\mu}\left(...
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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%...
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Is it possible to circumvent matrices in Dirac's coup by looking at alternative factorizations in the momentum domain?

Question: I was looking at Dirac's factorization of the Klein Gordon Equation and became inspired to see if there was an alternative way to yield some of its results without resorting to 4x4 matrices....
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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 \hbar ...