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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Demonstration of identities appearing in Dirac spinors in the chiral representation

Using the chiral representation of the gamma matrices, Peskin and Schroeder arrive in some expressions for the 4-component spinors $u(p)$ and $v(p)$ in terms of a square root of the Pauli matrices ...
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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}(...
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Dirac spinor for arbitrary momentum

In many treatments of the Dirac equation (e.g. Peskin and Schroder, pages 45-46) after subbing in $\psi(x) = e^{-ix_\mu p^\mu}u(\vec p)$, with $u$ a constant spinor, into the Dirac equation, we ...
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Hamilton's equations for Dirac Hamiltonian [duplicate]

The Dirac Lagrangian $$\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_\mu \psi - m \bar{\psi}\psi$$ gives a Hamiltonian $$\mathcal{H}(\Pi,\bar{\Pi},\psi,\bar{\psi})=\Pi \dot{\psi}-\mathcal{L}=-\bar{\...
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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 ...
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Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?

For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be Grassmann variable but can also be complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion) We can ...
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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 ...
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Derivation of Klein Gordon equation from Dirac equation; what does it mean?

In Dirac field (Peskin and Schroeder), there is one equation in which it multiples the Dirac operator $$(-i\gamma^{\mu}\partial_{\mu}-m )$$ by $$(i\gamma^{\nu}\partial_{\nu}-m ),$$ obtaining $\...
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single chirality electron and photon interaction

I asked a similar question about QED Lagrangian but I guess the question wasn't clear enough since I didn't get any correct answers. So, I'll try to ask the question in a different way: how does one ...
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QED Lagrangian in terms of Weyl spinors

Let's say the electron field can be written in term of its left and right handed Weyl spinors: \begin{equation} \psi_{e} = \begin{pmatrix} \chi \\ \eta^{\dagger} \end{pmatrix} \end{equation} In ...
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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 ...
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Negative probability distribution function for Dirac equation

People say that the probability density function of the continuity equation for the Dirac equation is definite positive. I wanted to see it myself and followed the same path as Bjorken & Drell's ...
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Peskin and Schroeder spinor high-energy limit (5.26 and A.20)

P&S say the high-energy limit of spinor $u^s (p)$ is $ \sqrt{2E} {1 \over 2} (1-\widehat{p} . {\sigma}) \xi^s $ and similar for the right-handed spinor (formulae 5.26 and A.20). I can't ...
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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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Time evolution using the Dirac equation

In non-relativistic qantum mechanics, the energy eigenstates (i.e.e eigensattes of the hamiltonian) evolve in phase according to their eigenenergies $\phi_(t) = e^{-iE_nt}\phi_n(0)$ using natural ...
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Why is the Dirac mass term Hermitian if Grassmann-valued fields anticommute?

Let's have Dirac mass term in lagrangian: $$ L_{M} = \bar{\Psi}\Psi $$ Lagrangian must be real-valued, i.e., its Hermitian conjugation doesn't change it. But due to Grassmann nature of spinor fields, $...
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Microcausality for Dirac's current

I`m supposed to show as an exercises that for the Dirac field's associated current: $$j^\mu=\bar{\Psi}\gamma^\mu\Psi$$ The microcausality relation holds: $$ [j^\mu(x),j^\nu(y)]=0 \text{ for } (x-y)^...
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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 ...
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Gordon decomposition in Cheng and Li p. 422

In the $\mu \rightarrow e+\gamma$ calculation in Cheng and Li "Gauge theory of elementary particle physics" p.422 they have $$ T=A\bar{u}_e(p-q)(1+\gamma_5)i\sigma_{\lambda\nu}q^\nu\epsilon^\lambda ...
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How to find propagator for domain wall fermions

I am working on domain wall fermions right now and I am trying to understand how Luescher finds the propagator for the domain wall fermions in this review https://arxiv.org/abs/hep-th/0102028 on pages ...
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Generally Covariant Dirac equation: The spin connection

Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
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Quantum field theory, Dirac field interaction Yukawa theory

From this Phys.SE question: Please can someone answer me to get the scattering amplitude and the cross section
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Maximal anticommuting sets of Dirac matrices

At the end of this webpage, it is said that there exist 6 maximal anticommuting sets each consisting of 5 Dirac $\Gamma$-matrices. I couldn't find anything more in the book cited there, either. I ...
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Is the partial derivative in the Dirac equation in curved space contracted with a tetrad?

The Dirac Equation in Curved spacetime makes a difference between Lorentzian indicies and Covariant indicies. In the equation we find a $\partial_\mu$. Is this actually $e^a_\mu\partial_a$ where $e$ ...
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Understanding solutions of the Dirac equation

In one of the lectures that I'm currently taking we encountered the Dirac equation. The general solution was given as $$\psi ( x ) = \sum _ { s } \int \frac { d ^ { 3 } \bf { p } } { ( 2 \pi ) ^ { 2 }...
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Deriving the Pauli-Schrödinger equation from the Dirac equation

Since the Schrödinger Pauli equation describes a non-relativistic spin ½ particle. This equation must be an approximation of the Dirac equation in an electromagnetic field. I was trying to derive this ...
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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" ...
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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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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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Vanishing of a solution of Dirac equation

Let $\psi(x,t)$ be a solution of the free Dirac equation. Assume that $$\psi(\vec x,0)=\delta^{(3)}(\vec x) u,$$ where u is a fixed spinor. (In other words $\psi(\vec x,0)$ is assumed to be supported ...
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Unphysical degrees of freedom for the Weyl spinor?

I am attempting to solve the Weyl equation: $$\bar\sigma^{\mu}\partial_{\mu}\phi=0$$ Where $\bar\sigma^{\mu}=(-1,\vec{\sigma})$ in my convention, and $\phi$ is a two component Weyl spinor. I consider ...
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What is the physical meaning of a pseudo-vector potential in a Dirac equation?

Consider a Dirac action with a pseudo-vector potential: $$S = \overline{\psi}(\gamma^\mu(\partial_\mu + i\gamma_5 A_\mu) + m_e)\psi$$ i.e. exactly like a Dirac equation with an electromegnetic ...
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Dirac solution with coulomb-field (perturbation theory)

The dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$ The solution up to first order is $...
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Aharonov-Casher effect vs Spin-Orbit coupling

The Aharonov-Casher phase is the electromagnetic dual of the Aharonov-Bohm phase. It arises when a neutral particle with a magnetic moment encircles, for example, a line charge, or moves on a closed ...
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Orthogonality relations for spinors of plane wave solution

I was looking for an explanation that doesn't depend on the representation of the gamma matrices that shows that the orthogonality relations are fulfilled. Let me situate my problem: So the Dirac ...
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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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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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Energy-momentum tensor of the Dirac field

I'm trying to compute the energy momentum tensor for the dirac field $$\mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi $$$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\partial^...
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Time reversal for fermionic fields

I have some doubts about the way we apply time reversal to Dirac's Lagrangian in QFT. Looking for the transformed field, $\psi^t(x)$, I've found sources (see below) that claims: $$\psi^t(x) = \gamma^...
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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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expanding a Dirac spinor in Weyl basis

For a massless electron Dirac spinor in Weyl basis (where $\chi$ is the left-handed spinor and $\eta$ is the right-handed spinor): \begin{equation} \begin{pmatrix} \chi \\ \eta \end{pmatrix} \end{...
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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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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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How did Paul Dirac predict the existence of antiproton?

The existence of the antiproton with -1 electric charge, opposite to the +1 electric charge of the proton, was predicted by Paul Dirac in his 1933 Nobel Prize lecture. Quotation by Wikipedia. ...
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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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Sign in front of QFT kinetic terms

I'd like to know if the sign in front of a kinetic term in QFT important. For the scalar field we conventionally write (in the $ + --- $ metric), \begin{equation} {\cal L} _{ kin} = \frac{1}{2} \...
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Diffeomorphisms and the Dirac action

I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach. I heard that there is a special way of ...
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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 ...