All Questions
14
questions
2
votes
0
answers
75
views
How do Maxwell's equations follow from the action of Lorentz generators on field strength?
Following Warren Siegel's book on Field theory (pg. 223), one might derive the action of Lorentz generators $S_{ab}$ on an antisymmetric 2-tensor field strength $F_{cd}$ which arises for example in ...
- 785
0
votes
0
answers
29
views
Unclear passage in Lorentz generators derivation
It's not clear to me a passage, in the extraction of the generators of Lorentz's group acting on the Minkowksi's space points: we have
\begin{equation*}
\begin{split}
x^{' \alpha} & = \Lambda^{\...
- 361
0
votes
1
answer
83
views
How can I calculate the square of Pauli-Lubanski vector in a rest frame?
recently I've been trying to demonstrate that, $$\textbf{W}^2 = -m^2\textbf{S}^2$$ in a rest frame, with $W_{\mu}$ defined as $$W_{\mu} = \dfrac{1}{2}\varepsilon_{\mu\alpha\beta\gamma}M^{\alpha\beta}p^...
- 1
0
votes
1
answer
209
views
Formulation of the Bianchi identity in EM
I'm trying to understand, as a self learner, the covariant formulation of Electromagnetism. In particular I've been stuck for a while on the Bianchi identity. As I've come to understand, when we ...
- 540
1
vote
0
answers
66
views
Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory
1) Introduction
From standard tensor calculus, here restricted to Minkowski spacetime, we learned that:
A scalar field is a object that transforms as:
$$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$
A ...
- 3,085
0
votes
1
answer
158
views
Einstein summation convention in deriving Coulomb's law
Schwartz's QFT equation (3.43) reads
$$ \mathcal{L} = - \frac{1}{4} (\partial_\mu A_\nu - \partial_\nu A_\mu)^2 - A_\mu J_\mu. \tag{3.43}$$
Does the contraction of $\mu$ on the last term carry over to ...
0
votes
3
answers
115
views
Why projection operator is not equal to zero, as we can write 1st term as 2nd term or vice versa via raising or lowering index with metric?
$$k^2g^{\mu\nu}-k^\mu k^\nu=k^2P^{\mu\nu}(k)$$
Here 1st term can be written as 2nd term via breaking square term and then raising index.
- 107
2
votes
1
answer
57
views
From "Matrix" form to "Component" (tensor) form
Given
$\omega=-\eta\omega^T\eta^{-1}=-\eta\omega^T\eta$,
where $\eta$ is the usual Minkowski metric.
Is the following logic correct?:
$$
{\omega^{~\mu}}_{\nu}= -{\eta_{\varepsilon\nu}}{\left(...
- 43
3
votes
1
answer
211
views
Is $\displaystyle{\not} p$ a Lorentz Scalar?
I am a bit confused about something. $\gamma^\mu$ is a (Lorentz) vector (c.f. Pesking & Schroeder chapter 3), and so is $p^\mu$, therefore I’d expect their product $\displaystyle{\not}p \triangleq ...
- 700
0
votes
0
answers
144
views
How to find Bilinears of a theory?
I'm trying to understand how one finds the bilinears of a given theory. In most litterature the bilinears are not really derived but rather taken as fundamental.
The dirac bilinears are of course:
$$\...
- 849
1
vote
2
answers
162
views
Confusion about the mathematical nature of Elecromagnetic tensor end the E, B fields
I have quite a lot of confusion so the question may result not totally clear cause of that. I'll take any advice to improve it and I'll try to be as clear as possible. Everything from now on is what I ...
- 2,347
2
votes
1
answer
241
views
Klein-Gordon inner product: how to make it real
While building its way up to the construction of an inner product, one stumbles upon the following equation:
\begin{equation} \partial_i(\varphi_2^*(x)\overleftrightarrow{\partial^i}\varphi_1(x))=\...
- 245
1
vote
1
answer
431
views
Covariant and contravariant derivatives in Klein-Gordon equation
Whilst exposing how a scalar product for the solutions of the Klein-Gordon equation (written as $(\Box + m^2)\varphi(x)=0$) can be derived, my textbook starts from the following system
\begin{cases} \...
- 245
3
votes
0
answers
923
views
The connection between classical and quantum spins
I have two questions, which are connected with each other.
The first question.
In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of ...
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