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 ...

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280 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 ...
4
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347 views

Interpretation of Dirac Spinor components in Chiral Representation?

I failed to find any book or pdf that explains clearly how we can interpret the different components of a Dirac spinor in the chiral representation and I'm starting to get somewhat desperate. This is ...
4
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179 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
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0answers
102 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
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541 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
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442 views

Dirac action and conventions

I have a (possibly) fundamental question, which is driving me crazy. Notation When considering the Dirac action (say reading Peskin's book), one have $\int dV\;\bar{\psi}\left(\imath\not\partial-m\...
3
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102 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
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89 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
187 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^...
3
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80 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.
3
votes
0answers
100 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
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0answers
88 views

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 ...
3
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0answers
240 views

Proof of equivalence of different representations of the $\gamma$-matrices in the Dirac equation

This question concerns the Dirac equation and the $4\times4$ $\gamma$-matrices. The task is to prove that a similarity transformation of the standard $\gamma$-matrix conserves the commutation relation ...
3
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175 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\...
3
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0answers
236 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.
2
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64 views

Intervalley scattering in graphene in presence of impurities

A long range impurity like coulomb impurity does not induce an inter valley scattering between the two Dirac points. Is there any mathematical explanation for the same although this is explained by ...
2
votes
0answers
61 views

Hyperfine structure in hydrogen

Consider the Dirac equation for bounded electron in hydrogen atom. I am trying to get a clear physical explanation for all mathematical terms that appear in the Hamiltonian and energy spectrum. ...
2
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0answers
84 views

Non-relativistic limit of the Dirac equation

How to recover non-relativistic limit of the Dirac equation $$\left( i\gamma^{\mu}\mathcal{D}_{\mu} - m \right)\Psi(x) = 0$$ where $\mathcal{D}_{\mu} = \partial_{\mu} + iqA_{\mu}$. I do not assume ...
2
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93 views

How one can write $\bar{\psi}$ in odd dimension?

I know that the Dirac equation in general dimensions has a form of $$ (i\gamma_{\mu} \nabla_\mu - m ) \psi =0 $$ and the action for that is written as $$ S = \int d^d x \bar{\psi} (i\gamma_{\mu} \...
2
votes
0answers
140 views

Spinor Commutator in Peskin and Schroeder

In (3.87, page. 53) Peskin and Schroeder write $$\psi(\vec{x}) = \int\frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{\vec{p}}}} e^{i\vec{p} \cdot \vec{x}} \sum_{s=1,2} (a_{\vec{p}}^{s}u^{s}(\vec{p}) + b_{...
2
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0answers
282 views

The anapole moment, derivation from Dirac current density

Basically I am looking for a way to expand the electromagnetic interaction energy $W = A_{\mu}j^{\mu}$ (both $A$ and $j$ obtained from the Dirac equation) similar to the classical expansion in ...
2
votes
0answers
78 views

Zitterbewegung for massless particle

Is it possible for a massless particle to undergo zitterbewegung? In massive Dirac theory the Zitterbewegung frequency comes out to be $2mc^2/\hbar$. It looks like the effect will vanish for a ...
2
votes
0answers
48 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}(...
2
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0answers
158 views

The proof that Dirac's hamiltonian commutes with inversion operator

I tried to check the statement that Dirac free Hamiltonian commutes with inversion operator. For $$ \hat {P}\Psi(\mathbf r , t) = i\hat {\gamma}_{0}\Psi (-\mathbf r , t), \quad \hat {H} = (\hat {\...
2
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155 views

Calculating the dispersion relation of dirac lagrangian in curved spacetime

I am trying to calculate the dispersion relation for a fermion in a gravitational field. So far, I have computed the equation of motion, but I am stuck trying to figure out a determinant I just can't ...
2
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137 views

Majorana equation and non-invariance of spinor representation under discrete Lorentz transformations

Here I asked about getting an equation for two-component spinor as the alternative for Dirac equation. It was found that it is called Majorana equation. It may be easily derived by using historical ...
2
votes
0answers
132 views

WKB expression for Dirac equation?

given the one dimensional Schroedinger equation $$ - \frac{\hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}\Psi(x)+ V(x) \Psi(x) =E_{n}\Psi (x) $$ the WKB method for the energies is $$ (n+1)2\pi \hbar =\int_{a}...
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78 views

How to arrive at the Dirac Equation from Poincare Algebra?

For the case of Galilean group, the time translation is given by the generator $H$. Hence, $$\mid\psi(t)\rangle\to \mid\psi(t+s)\rangle =e^{-iHs}\mid\psi(t)\rangle$$ Which immediately is the ...
1
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0answers
41 views

How do you derive the Dirac Lagrangian density for spinor fields?

I know the how the Dirac Lagrangian is written but I don't understand how to derive it from the general definition $L=T-V$. So I guess I would also like to know what the Kinetic and Potential energies ...
1
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0answers
18 views

Chiral tunneling in Weyl Equation

I am trying to understand perfect tunneling of particles obeying Weyl equation through a potential barrier at normal incidence. I know that this has something to do with chirality, but I am not ...
1
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0answers
114 views

Classical Fermion and Grassmann number

In the theory of relativistic wave equations, we derive the Dirac equation and Klein-Gordon equation by using representation theory of Poincare algebra. For example, in this paper http://arxiv.org/...
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74 views

Definition of partity in quantized Dirac Theory.

I'm studying from the book "An Introduction to Quantum Field Theory" from Michael E. Peskin and Daniel V. Schroeder, and I read the following: "The operator P should reverse momentum of a particle ...
1
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0answers
46 views

Wave packets in Dirac equation

Gaussian wave packets remain Gaussians after evolution in case of the Schrodinger equation. It is a very useful property of these wave packets. I don't think the same is true for a Gaussian wave ...
1
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0answers
61 views

How to measure the Fermi velocity in Dirac materials?

Suppose that one has a Dirac material (e.g., graphene), i.e., a system where there exists a number $N$ of identical Dirac cones (linear dispersion) at the Fermi energy $E_F=0$. How can one measure ...
1
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0answers
77 views

Dirac Spinors as Eigenvalues of Helicity Matrix

I am trying (unsuccessfully) to verify this relation regarding the helicity of Dirac spinors: $$ { \sigma }_{ \vec { p } }u_{ r }\left( \vec { p } \right) =\frac { \vec { \Sigma } \cdot \vec { p } ...
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0answers
118 views

Physical significance of Dirac equation in (2+1)-D

What's is the physical significance of the two inequivalently irreducible-represented Dirac equations in (2+1)-D? As it is known, all the $4\times 4$ matrix representations of the Dirac algebra $\...
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0answers
73 views

What is really interacting in weak interactions?

Only particles with chirality $-1$ do interact weakly. The corresponding eigenstate in the Dirac basis is $ \Psi_L = \begin{pmatrix}f \\ -f \end{pmatrix} = \begin{pmatrix}u_r {\mathrm{e}}^{-imt} \\ ...
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167 views

Dirac fermion in curved space

What is the connection between Dirac equation in curved space-time and effective Hamiltonian for Dirac fermion in curved space (topological insulators)? I am trying to find this connection but I am ...
1
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0answers
25 views

How to find full energy of field of an arbitrary half-integer spin?

Let's have arbitrary half-integer spin $n + \frac{1}{2}$ representation: $$ \Psi_{\mu_{1}...\mu_{n}} = \begin{pmatrix} \psi_{a, \mu_{1}...\mu_{n}} \\ \kappa^{\dot {a}}_{\quad \mu_{1}...\mu_{n}}\end{...
1
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0answers
151 views

How do I prove the equivalence of chirality and helicity operators acting on a massless Dirac spinor?

I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf p)}{|\...
1
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0answers
179 views

From Dirac to Klein-Gordon in curved spacetime

Is there an easy/elegant way of showing that "squaring" the Dirac equation in curved spacetime yields the Klein-Gordon equation, just like it happens in Minkowski space? A brute force approach would ...
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0answers
149 views

Two pairs of projection operators of the Dirac equation

The Dirac equation may be interpreted as the action of projection operator $\frac{1 - \Delta}{2}\Psi = 0$, where $$ \Delta = \begin{pmatrix} 0 & \Delta_{b \dot {a}} \\ \Delta^{\dot {b}a} & 0 ...
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0answers
368 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 ...
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0answers
129 views

Lagrangians for non-local equations of motion

Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion, $(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p d^3q}{...
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0answers
235 views

Matrix manipulation for Dirac matrices

From the Dirac equation in gamma matrices, we know that $$\gamma^i=\begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}$$ and $$\gamma^0=\begin{pmatrix} I & 0 \\ 0 & -I ...
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99 views

Translate a two dimensional classical Dirac theory to a (1+1)-dim quantum theory

Suppose I have a two dimensional classical Dirac Hamiltonian with $\Psi=(\psi_1,\psi_2)^T$: $$ H=\int \mathrm{d}x \mathrm{d}y \Psi^\dagger(\sigma^x i\partial_x+\sigma^y i\partial_y+m\sigma^z)\Psi. $$ ...
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0answers
46 views

Local Phase Transformation of the Dirac equation

The Dirac Equation ("free Dirac") is a relativistic Equation of Motion (EoM) for a free ($V=0$) Spin $1/2$ particle (like an electron). The free Dirac equation is invariant under global phase ...
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55 views

Plane Wave Solutions to the Majorana Equation with Zero Momentum

My question concerns the plane wave solutions to the Majorana equation. First, recall the Dirac equation: $$(i\gamma^\mu \partial_\mu-m)\psi=0$$ I suggest a solution in the form of a plane wave with $...
0
votes
0answers
39 views

Confusion over trying to understand spinor components

I've been reading about the quantisation of the Dirac field $\psi(x)$ and it is stated that the general solution to the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi(x)=0$ is given by the ...
0
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0answers
51 views

Understanding Dirac equation notation

I'm trying to recover the Einstein energy-momentum relation from the Dirac equation. I'm given a solution wavefunction, $$\psi = u(E,\vec p) e^{i(\vec p\cdot\vec x - Et)}$$ with $$\vec u = N\begin{...