All Questions
15
questions
1
vote
1
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101
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Question on the spinor Indices, in non-relativistic quantum mechanics
I've caught by a loop of:
Standard texts of Non-Relativistic Quantum Mechanics $\to$ Representation theory of Lie groups and Lie algebras of $SO(3)$ and $SU(2)$ $\to$ Discussions of infinitesimal ...
- 315
0
votes
0
answers
561
views
Bilinear covariants of Dirac field
In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
- 59
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
1
vote
1
answer
219
views
How to contract spinor indices?
In normal vector representation, vectors can be contracted as follows:
$$v^\mu v_\mu$$
with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
- 1,324
5
votes
2
answers
451
views
What is the idea behind 2-spinor calculus?
In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
- 7,980
0
votes
2
answers
371
views
Riemannian and Weyl tensors as spinor representation
There is the way of converting vector indices to spinor indices, for example, Maxwell stress tensor $F_{[\mu\nu]}$ can be decomposed to $(1,0) \oplus (0,1)$ irreducible representations of $\mathfrak{...
- 5,707
6
votes
1
answer
674
views
The spinor metric, basic spinor calculations and spinor indices
I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24.
Background
Scharf introduces the index-raising symbol (spinor metric)...
- 144
1
vote
0
answers
702
views
Relation between Levi-Civita tensor and the trace of Lorentz transformations
here is this tricky identity to prove in an appendix of W.B. Supersymmetry and Supergravity that's driving me crazy. Some premises first:
This book use the van der Waerden's convention for spinor ...
1
vote
0
answers
84
views
Converting field equation from position to momentum space
Question
I would like to convert the following equation on position space to an algebraic equation on momentum space. $$\partial^{\alpha\dot{\alpha}}\Phi_{\alpha\alpha_1\cdots\alpha_{A-1}\dot{\alpha}...
- 780
3
votes
1
answer
347
views
General definition of an $n$-rank spinor
I have been looking around for a formal (and easy to comprehend) definition of a general $n$-rank spinor. I have had no luck trying to find such a definition, or any definition for that matter.
So ...
9
votes
2
answers
478
views
If $v_{a \dot{b}}$ transforms like a four-vector, what does $v_{a}^{\dot{b}}$ describe?
The $( \frac{1}{2}, 0)$ representation of the Lorentz group acts on left-chiral spinors $\chi_a$, the $( 0,\frac{1}{2} )$ representation on right-chiral spinors $\chi^{\dot a}$.
The $( \frac{1}{2}, \...
- 1,872
2
votes
4
answers
1k
views
Nature of Fields in QFT
I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere!
There are specific types of fields used in physics: scalar fields (i.e. as ...
- 10.8k
4
votes
1
answer
1k
views
Interpretation of rank 2 spinors
While inspecting the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group and defining a right-handed spinor with upper dotted index and a left-handed spinor with lower undotted index and ...
- 10.1k
2
votes
2
answers
627
views
Recovering 4-vector Lorentz transformation from spinor formalism
I'm trying to recover the 4-vector transformation laws using spinors. I have defined
$$v^{\dot{a}b} = v^{\nu} \sigma_{\nu}^{\dot{a}b}$$
as usual, with $\sigma_0=1$.
Now with the rules for dotted ...
- 10.1k
1
vote
0
answers
220
views
Direct sum of the spinors and EM field tensor
EM field tensor refer to the direct sum of $(1, 0), (0, 1)$ spinor representation of the Lorentz group. How to show it?
Each of these spinor representations corresponds to the symmetrical spinor ...
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