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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135 views

Wave function - Dirac Notation

Based on that notes (equation 54): https://warwick.ac.uk/fac/sci/physics/staff/academic/boyd/stuff/dirac.pdf I was reading about the wave functions and I have a question about the notation. You can ...
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376 views

What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
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52 views

Sign of energy and frequency in a propagation amplitude

The following paragraph is from pag. 55 of Peskin and Schroeder's An Introduction to Quantum Field Theory: First consider the propagation amplitude $\langle 0|\psi(x) \bar{\psi}(y)|0 \rangle$, ...
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106 views

What is the meaning of $\not{p}$ in physics?

I am reading Srednicki's QFT book in physics. On page 286, the formula $(45.16)$ has a notation $\not{p}$. What is the meaning of $\not{p}$ in physics? Thank you very much.
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203 views

Weyl and Majorana-Weyl spinors why need commutation?

Let $\psi$ denote a Dirac spinor then Weyl spinors are defined by: $$\psi_{L,R}=\frac{1}{2} (I\pm \gamma)\psi$$ on even dimensions $\gamma$ commutes with $\sigma_{\mu \nu}$ (generators used to define ...
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139 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 ...
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78 views

Is a (Dirac) Particle Where $\vec{p} = (p^1,0,0)$ in an Eigenstate of Helicity? [closed]

Is a particle where $\vec{p} = (p^1,0,0)$ an eigenstate of the helicity operator? First, can I determine this without doing the math? Second, I also wanna prove it mathematically but doing the math ...
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1answer
145 views

Hermiticity of Dirac Operator $\gamma^{\mu}D_{\mu}$ and Expansion in eigenmodes

I'm interested to know under what conditions $\gamma^{\mu}D_{\mu}$ is a hermitian operator. I am studying the Fujikawa method of anomalies and I see that many sources have different answers for this. ...
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216 views

Antiparticle solution of the Dirac Equation

I'm really confused by the antiparticle solution of the Dirac equation. I follow Chapter 11 of Schwartz's book "Quantum Field Theory and the Standard Model" and find a couple of problems. In ...
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636 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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87 views

Dirac operator identity separating out gamma matrices

I am trying to show that $$(i\gamma^\mu\partial_\mu -e\gamma^\mu A_\mu)^2 = (i\partial_\mu -eA_\mu)^2 -\frac{e}{2} \sigma^{\mu\nu}F_{\mu\nu},$$ where $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\...
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Spin Up with Indefinite Helicity

Imagine we are studying the spin quantization along the same axis as the momentum. What if I have a Dirac spinor with a spin up but no definite helicity ($\psi_L,\psi_R\neq0$): $$ u(p)= \left(\begin{...
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393 views

How do you derive the spin orbit coupling classically?

In our lecture on atomic and molecular physics we are currently dealing with spin orbit coupling. Our prof showed us a derivation where he approximated the dirac equation to the non-relativistic case ...
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142 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 ...
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793 views

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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1answer
93 views

What is the definition of $\overleftrightarrow{\partial}$ in Dirac Lagrangian?

In my course, the teacher wrote the Dirac Lagrangian as : $$ \mathcal{L}=\frac{i}{2} \bar{\psi}\gamma^{\mu}\overleftrightarrow{\partial_\mu} \psi -m \bar{\psi} \psi $$ I just would like to ...
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1answer
501 views

Dirac particle in potential

I want to consider the Dirac equation in one spatial dimension $$ H_D = c\alpha_ip^i +\beta mc^2+\mathbb I_4 V(x),$$ where the potential is given by $$ V(x) = \left\{ \begin{array}{ll}0 & |x|>...
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1answer
275 views

How to commutate angular momentum and spin in the Dirac equation?

In my problem I'm looking at a spin-$\frac{1}{2}$ particle witch charge q, which is represented by some Dirac spinor $\Psi$ solving the Dirac equation $$i\partial_t\Psi = \hat{H}\Psi$$ with given ...
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279 views

Spin expectation values in Dirac theory

The wave function of a free Dirac particle moving with momentum $\vec{p}=(p,0,0)^T$ is given in the rest frame and the laboratory frame as $$\Psi_r=N_r\left(\begin{array}{c}1\\0\\0\\0\end{array}\...
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252 views

Conserved current in Dirac Equation coupled with electromagnetic field

I want to find the conserved current $j^\mu$ associated with Dirac equation coupled with a external electromagnetic field, i.e, $$\left(i\hbar \gamma^\mu \partial_\mu - \frac{e}{c} \gamma^\mu A_\mu -...
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198 views

Quantum Field Theory (Weinberg): lifting the Weyl equation for a massless particle

I am trying to construct the massless quantum field (Weinberg style) for the $(0,\frac{1}{2})$ representation. So I want to right moving Weyl spinor. My massless quantum field in the general case is ...
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269 views

Using ladder operators to solve for Landau levels of graphene

I had recently been studying the Dirac equation, and as an example of how the equation is used, I was given a problem about the Landau levels of graphene (but I personally have no knowledge about ...
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423 views

How did Dirac come up with the idea of using Pauli matrices?

Dirac equation in natural units is: $$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi=0$$ where $\gamma^{0}=\pmatrix{I_{2} & 0\\ 0 & -I_{2}}$ and $\gamma^{n}=\pmatrix{0 & \sigma_{n}\\ -\...
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Dirac Fields and Derivatives (Am I gaining extra minus signs?)

I've given myself a severe headache jumping between East/West Coast sign conventions; I have picked up an extra minus sign and could do with a hand. I am currently using $\eta=\textrm{Diag}[-,+,+,+]$ ...
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407 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 ...
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413 views

Which one is more fundamental in nature: matter or radiation?

I am following a geometric perspective on abelian gauge theory as done in the lecture notes by Timo Weigand, chapter 6, pp 165-167, here: http://www.thphys.uni-heidelberg.de/~weigand/QFT1-13-14/...
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150 views

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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How to expand the Dirac equation?

I've been reading a little bit of the Dirac Equation and I'm a little confussed about how it shall be expaned. The dirac equation has the form of $$i\hbar \gamma^{\mu} \partial_{\mu} \Psi -mc \Psi = ...
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438 views

Is there a “square root” version of the Einstein field equation?

It is well known that the Klein-Gordon equation have a kind of "square root" version : the Dirac equation. The Maxwell equations can also be formulated in a Dirac way. It is also well known that ...
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Interpretation of Dirac equation states for moving electron

I try to understand a physical interpretation of the four components of the Dirac 4-spinor for a moving electron (in the simplest case, a plane wave). There is a very good question and answer about ...
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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 ...
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51 views

A question in Dirac article about Dirac equation about a sentence

Why it is said $W$ should be linear partial time derivative so that wave function could be determined by initial wave function?
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189 views

Formal motivation of using the Dirac equation

As is well-known, in 1928 Dirac has derived the same name equation by using the requirement of constructing a relativistic covariant equation describing the function $\psi$ with corresponding positive ...
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337 views

Relativistic uncertainty principle derivation

The uncertainty principle is a fundamental law of nature, arising from the mathematics of quantum theory. In it's most general form, it reads: $$\sigma_A^2 \sigma_B^2 ≥ (\frac{1}{2i} \langle [\hat A,\...
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Electromagnetic Dirac equation in CGS units

The Hamiltonian of the Dirac equation in with an electromagnetic field in SI units is $$H=\gamma^0\left[ mc^2+c\gamma^k\left( p_k-\frac{q}{c}A_k \right) \right]+qA^0$$ (from https://en.wikipedia.org/...
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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 ...
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205 views

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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What do you lose if you rewrite the Dirac equation in terms of $\mid\Psi\mid^{2}=\Phi$?

Taking a look at the Dirac equation (taking $\hbar$ to be unity): $$\bar{\Psi}(i\gamma^{a}e_{a}^{\mu}\partial_{\mu}-m)\Psi=0$$ The operator is Hermitian and and hence we may rewrite it as: $$\Psi(i\...
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Current density indentity form Dirac equation [closed]

How to prove the following identity $\bar{u}\gamma ^{\mu}u=\frac{p^\mu}{m}$ where u(p), a unit spinor, satisfies the free Dirac equation $(\gamma\cdot p-m)u(p)=0$ Edit: This how far I have ...
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388 views

Energy-momentum tensor of a fermion

I'm wondering how does one write the energy momentum tensor of a fermion (like an electron)? I've seen the formula for deriving it through the action. What I'm specifically asking is how would each of ...
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202 views

Spin part of the Dirac spinors

I am reading Peskin's book on QFT and he defines the spin component associated with the particle initially by $\xi^1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\xi^2=\begin{...
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784 views

Fermion propagator as derivative of scalar propagator

I've seen this expression in two spacetime dimensions, $$ \langle \bar{\psi}(x) \psi(0) \rangle = \gamma^\mu{\partial_\mu} \langle \phi(x) \phi(0) \rangle $$ The LHS is the fermion propagator, and ...
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Can I replace the derivative operator in the Dirac equation with momentum?

Can I replace the derivative operator in the Dirac equation with momentum? $$ (i\gamma^\mu\partial_\mu - m)\psi(x) = 0 $$ $$ (\gamma^\mu p_\mu - m) \tilde{\psi}(p) = 0$$
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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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Dirac vs KG propagation amplitude

Can someone explain to me the physical meaning of $\bar{\psi}=\psi^\dagger\gamma^0$ in the Dirac equation? I understand it is obtained as one of the solutions of Dirac equation and it is used to build ...
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77 views

Why $g$ is given by this coefficient of a term in the matrix element?

In the non-relativistic limit of the Dirac equation, we are led to the hamiltonian $$H = \dfrac{P^2}{2m}+V(R)+\dfrac{e}{2m}\mathbf{B}\cdot \mathbf{L}+\dfrac{ge}{2m}\mathbf{B}\cdot \mathbf{S}$$ and ...
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839 views

Does the creation/annihilation operator commute with the spinors?

I am self-studying QFT and I came to the point of quantizing the Dirac field. The Dirac field expanded in terms of creation/annihilation operators is: $$\psi(\vec{x})=\sum_{s=1}^{2}\int{\frac{d^3p}{(...
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2answers
261 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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225 views

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) ...