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.
1,112
questions
0
votes
0
answers
19
views
Explain this step (related to gamma matrices and parity operator)
I am having hard time reproducing a step from the textbook "Lecture Notes on Quantum Field Theory", by Ashok Das. On page 429 ( above equation 11.72), the author is talking about the parity ...
- 291
0
votes
0
answers
36
views
Wave function fermions and boson, Peskin & Schroeder [closed]
We have, from peskin (the results are right, just citing) the wave function of fermions and bosons, but they have a different sign:
Fermions, page 45:
Bosons, page 20:
The issue here, is that the ...
- 31
0
votes
0
answers
38
views
Why Schrodinger equation for graphene electrons is a Dirac equation for massless particles when apparently C atoms & electron cloud aren't massless?
Recently, I was attending in a colloquium and the speaker, quite reputable btw, mentioned shortly that electrons in graphene are governed by Dirac equation instead of Schrodinger equation. However, ...
0
votes
0
answers
91
views
Why do we need the Dirac's hole picture?
When I have to quantize a Dirac field I have to start by the usual classical Lagrangian and find the associated Lagrange equations, then quantize the solutions promoting them to quantum operators. In ...
- 319
2
votes
1
answer
81
views
Imaginary part of the Dirac Lagrangian density as a total derivative
Show that the imaginary part of the Dirac Lagrangian [density] is a total derivative.
My attempt:
The Dirac Lagrangian density is given by,
$$ \mathcal{L} \ = \ -\bar{\psi}(\gamma^{\mu}\partial_{\mu} ...
- 130
0
votes
1
answer
65
views
Basis vectors for quantum electrodynamics
The following unnormalized vectors are solutions to the Dirac equation.
\begin{align*}
u_1&=\begin{pmatrix}E+m\\0\\p_z\\p_x+ip_y\end{pmatrix}
\exp\left(\frac{i\phi}{\hbar}\right)
%
& v_1&=\...
- 11
1
vote
1
answer
54
views
Confusion about left/right-handed spinor notations
Peskin & Schroeder eq (3.78) states that
$$(\bar{u}_{1R}\sigma^\mu u_{2R})(\bar{u}_{3R}\sigma_\mu u_{4R})=\cdots$$
But I don't understand what the $u_{1R}$ means. Since 4-component Dirac spinor ...
- 23
1
vote
3
answers
139
views
Difference between classical fields and wavefunctions? Relation between QM and Classical field theory
We all know that Dirac equation comes up naturally by trying to build a relativistic generalization of the Schrödinger equation
$$(i\gamma^{\nu}∂_{\nu}-m)\psi=0$$
and that the associated solutions are ...
- 319
0
votes
1
answer
87
views
How to solve the Dirac Equation and find its eigenfunctions?
Imagine you want to solve the Dirac equation:
$$
(i\gamma^\mu \partial_\mu - m )\psi=0 \\
$$
Where $\gamma^\mu$ are the $4 \times 4$ gamma matrices, and $\psi$ is a 4 component spinor.
$$
\psi = \...
- 1,203
0
votes
1
answer
40
views
Do $p_\mu$ and $\gamma^\mu$ commute?
So I am trying to derive the relation $\bar{u}_{(s)} (\displaystyle{\not}{p} -m)=0$ from the conjugate dirac equation $(i \partial_\mu\bar{\psi}\gamma^\mu+m\bar\psi) = 0$ but I am running into issue. ...
- 410
0
votes
3
answers
98
views
Uniqueness of Dirac matrices
I am trying to understand the motivation behind the Dirac equation for a free particle
$$
i\gamma^\mu\partial_\mu \psi-m\psi=0 \tag{1}
$$
I am wondering how to get the concrete form of the matrices $\...
- 2,002
1
vote
1
answer
46
views
Does electric potential give electron mass as same as higgs field?
Inside negatively charged ball there is constant electric potential. If electron will be placed inside ball with constants potential will it give mass from electric potential as same as higgs field ...
- 51
-1
votes
2
answers
88
views
Square root of the wave operator
How are the Dirac matrices the square root of the wave operator? I keep seeing it mentioned as such but never explained.
0
votes
0
answers
42
views
Continuous wave function and non-continuous probability density. Is this possible?
I'm currently dealing with the 2D Dirac equation with an inhomogeneous mass term ($-M$ for $x<0$ and $+M$ for $x>0$) with an external magnetic field applied.
If I impose continuity at $x=0$ for ...
0
votes
0
answers
24
views
Understanding the Dirac Spinor under Lorentz Transformation
Recently I have been studying SR and QFT (in the beginnings). Initially it was a bit hard for me to understand, conceptually the Lorentz transformations. I will give a brief description of my current ...
- 1,284
1
vote
0
answers
21
views
Foldy-Wouthuysen (FW) Transformation alternative derivation
In my lecture, when considering the FW representation of the Dirac Equation, for the unitary matrix the following form was given:
$U=U^\dagger=(c\beta +\frac{\lambda}{m}\vec\alpha \vec p)$ with $c=\...
- 1,284
-1
votes
0
answers
30
views
Letting Dirac wave function evolve from an initial wave function
How does one go about calculating the general wave function (which solves the Dirac equation) spanning the entire Minkowski spacetime when only an initial wave function, confined to a constant-time ...
- 129
0
votes
0
answers
23
views
Understanding Wave Function Evolution in the Dirac Equation for Different Geometries
I've come to understand that for Schrödinger's equation, you can select an initial wave function within a constant-time space. By allowing it to evolve according to the Schrödinger equation, you can ...
- 129
0
votes
1
answer
35
views
Gordon identity in QED [closed]
To derive Gordon identity
$$ \bar u_2 (q^\mu_1 + q^\mu_2) u_1 = 2m \bar u_{2} \gamma^{\mu} u_{1} + i (q_{1,\nu} - q_{2,\nu} ) \bar u_2 \sigma^{\mu\nu} u_1 $$
we use the relation
$$\gamma^\mu q_\mu u(q)...
- 3
0
votes
0
answers
34
views
Adjoint of the Dirac equation, and hermiticity of the momentum operator
I'm trying to derive the adjoint of the Dirac equation in standard relativistic quantum mechanics. We have the Dirac equation as follows :
$$(i\gamma^{\mu}\partial_\mu -m)\psi=0$$
To find it's adjoint,...
- 1,640
0
votes
1
answer
55
views
How to compute the amplitude of a Feynman diagram with a loop containing a fermion and a scalar?
I know that when we have a Feynman diagram with a fermion loop, we must take the trace and, by doing so, we get rid of the $\gamma$ matrices.
What if we have a diagram like the one in the picture ...
- 193
0
votes
1
answer
43
views
How to do the non-relativistic limit of an Energy expression?
I solved for the bound state energies of a system using the Dirac equation, I was told that to compare my result to the energies obtained with the Schrodinger equation I need to do the non-...
- 45
0
votes
0
answers
26
views
Allowed energies for relativistic and non-relativistic bound states in quantum mechanics
I'm studying the Dirac equation in one spatial dimension, more specifically bound states. My question is: given a quantum system (a potential well for example), whose bound states have the allowed ...
- 45
1
vote
1
answer
37
views
Proving multiplication with Dirac adjoint spinor is Lorentz scalar
I'm studying QFT with the book "Quantum Field Theory" by Dr. David Tong, and I have difficulties with understanding some used transformations.
On the page 88 the author calculates Hermitian ...
0
votes
2
answers
82
views
How did the number of unknowns change in the Dirac equation?
So I haven't seen this argument addressed in any textbook which makes me doubt it's legitimacy. Here goes:
Since Newton's $F=ma$ is essentially a second-order differential equation. Any equation of ...
- 1,221
2
votes
0
answers
57
views
Spinor Lorentz generators in curved spacetime
The Dirac matrices in curved spacetime are written as $\gamma_{\mu}=e^{a}_{\mu}\gamma_{a}$ where $e^{a}_{\mu}$ are the vielbein fields and $\gamma_{a}$ are the constant Dirac matrices. Given that the ...
- 163
0
votes
1
answer
191
views
What is the proper description of an electron?
Is it described by a trajectory with a spin $x^\mu(\tau)$, $S^\mu(\tau)$ OR a Grassmann-valued spinor field (i.e. the Dirac field a la path integral QFT) $\psi_a(x^\mu)$? If so, how are these two ...
0
votes
0
answers
73
views
Unitarity of time-evolution operator of Dirac's equation
Let $\Psi(t)$ be state of Dirac's electron in context of Dirac's equation and consider time-evolution operator
$$\Psi(t) = U(t)\Psi(0)$$
is or is not $U$ an unitary (preserving length) operator?
(note ...
- 1,325
1
vote
2
answers
58
views
Integration from Schrodinger and Dirac solution in momentum space
I asked myself this questions to understand wave equations but couldn't solve them:
(I hope answer of first question be yes, and the second one no, as a result of relativistic effect of probability of ...
- 1,325
0
votes
1
answer
45
views
Free and non-free Dirac's particles
In trying to understand evolution of Dirac particle in relativistic QM (not QFT) I have following guess that I don't know I'm correct or wrong? edited after
solution of akhmeteli:
It is well known ...
- 1,325
1
vote
1
answer
36
views
Why does Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ is frequently written with a factor of $i$?
The tensor Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ has the matrix tensor $\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^\mu,\gamma^\nu\right]$.
I can understand that the factor of $\frac{1}{2}$ is a ...
- 907
1
vote
0
answers
91
views
How we know that Dirac equation in curved spacetime is the correct one?
Dirac equation in curved spacetime is given by
$$\left(i \gamma^\mu D_\mu-m\right) \Psi=0$$
How does we know that this is the correct equations to describe spinors in curved space time?
Is there any ...
- 439
3
votes
0
answers
45
views
Negative momentum arguments in Feynman amplitude
I encountered the below Feynman amplitude with Dirac spinors and photon vectors that have negative momentum arguments:
$$M=\epsilon^\alpha_r(-k)\bar\nu_{s'}(-\vec p')( ie\gamma_\beta) iS_F(p-k)ie\...
- 4,095
0
votes
1
answer
57
views
Is Dirac neutrino ruled out by current experimental observation?
I have read the neutrino mass problem. The unnatural smallness of neutrino mass implies the existence of new physics so the seesaw mechanism is introduced to solve this theoretical problem.
I ...
- 31
1
vote
0
answers
32
views
Question about a construction of a 1+1 free Dirac field
I have a question about the example Ron Maimon gave here of a (1+1) dimensional free Dirac field. In the original wording:
In two dimensions (one space one time), there is a nice dimensionally ...
- 1,253
1
vote
0
answers
48
views
Calculation of the source term for the Einstein-Dirac equation in the weak field limit
I have seen the same being done for Einstein- Klein Gordon equations quite successfully. However, I'm struggling with it in the case of the E-D equations. I know that the einstein equations in the ...
- 23
0
votes
0
answers
39
views
How the Dirac spinor change under the Lorentz transformation?
My question is, what form do Dirac spinors take under Lorentz transformations?
I would be grateful if you could send me the answer this year
2
votes
1
answer
121
views
If a flagpole visualizes a Pauli spinor, what visualizes a Dirac spinor?
@AndrewSteane writes in his textbook that a Pauli spinor is a flagpole - with length, direction, flag orientation - and a sign.
A Dirac spinor is a more complicated object. What is the most intuitive ...
- 617
0
votes
1
answer
64
views
Dimensional analysis in QFT
I have the Lagrangian density:
$$\mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - M)\psi$$
How do I know what dimensions $\bar{\psi}$ and $\psi$ have?
- 17
1
vote
2
answers
108
views
Negative energy solutions not a problem for Klein-Gordon equation?
I already posed this question Negative energy solutions in Klein-Gordon and Dirac equations but I am not satisfied with the answers.
Trying to be very sharp:
does Klein-Gordon equation have negative ...
- 1,062
0
votes
0
answers
28
views
Dirac 4-current for orbital transition
The conserved 4-current is defined as
$j^\mu=\bar{\Psi} \gamma^\mu \Psi$
where $\Psi$ is the 4-component wave function. To get the wavefunctions, if we look at the Dirac orbital spinor solution for ...
1
vote
0
answers
58
views
Dirac/Weyl/Majorana Basis and their Importance
I have just begun studying Dirac equations and was confused by the physical significance of Dirac Basis. In principle, we can have as many representations of Clifford Algebra as we wish by conducting ...
0
votes
1
answer
32
views
Two normalization constants for Dirac plain wave function
I stumbled across two different expressions for a Dirac plane wave function, namely
$$\psi=\sqrt{\frac{m}{EV}}ue^{-ip\cdot x}$$
and
$$\psi=\frac{1}{\sqrt{2EV}}ue^{-ip\cdot x}$$
where $u$ is the Dirac ...
- 141
1
vote
0
answers
42
views
Are the massive spinors (left and right Weyl spinors) helicity eigenstates?
For massless spinor, with the help of Dirac equation, I can show that both left-handed and right-handed Weyl spinors are helicity eigenstates with eigenvalues +1 and -1 (respectively). I am unsure ...
- 67
2
votes
1
answer
107
views
Representing the Dirac equation in spacetime algebra without leftover indices
The Dirac equation as derived by Hestenes is
$$
\hbar \nabla \psi I \sigma_3 = mc \psi \gamma_0
$$
where $I \sigma_3 = \gamma_2 \gamma_1$. The equation is claimed to be Lorentz invariant, because the ...
3
votes
0
answers
154
views
Dropping spinor into a black hole? [closed]
Are there any purely formal, algebraic calculations that show what happens when one drops a spinor into a black hole? No hand-waving allowed!
To make the problem precise, several definitions are ...
- 233
3
votes
1
answer
85
views
How does the $\not{\partial}$ work in the Dirac Lagrangian?
The Dirac Lagrangian (Density) is defined in the text "Quantum Field Theory, An Integrated Approach" by Fradkin as:
$$\mathcal{L}=\bar{\Psi}\left(i\not{\partial}-m\right)\Psi\equiv \frac{1}{...
- 89
0
votes
0
answers
76
views
Vague argument on alpha, beta matrices in Bjorken Drell sec 1.3 on Dirac Equation
My question concerns a very vague argument given in section 1.3 (specifically, page 8, right before equation 1.17) in the Bjorken and Drell "Relativistic Quantum Mechanics" book. The ...
- 139
1
vote
1
answer
45
views
Conserved Noether current [closed]
I want to prove that in the massless limit $m=0$, Noether's current is conserved:
$$ \partial_{\mu} J^{\mu} = \partial_{\mu} (\psi^{\dagger} \gamma_0 \gamma^{\mu} \gamma_5 \psi ) =0$$
I don't really ...
- 1,094
0
votes
0
answers
40
views
What’s the Fermi energy of a Dirac fermion?
Knowing that the energy of a Dirac fermion is $$
\varepsilon(\vec{k})= \pm \sqrt{\left(m c^2\right)^2+(\hbar c k)^2},
$$
Why is the chemical potential $0$ at $T=0$? Since the density of state is $0$ ...
- 119