Problem in Understanding notation of distributional section

In summary, Urs Schreiber discusses the space of distributional sections and defines it as ##\Gamma_{\Sigma}^{\prime}\left(E^*\right)##. He also presents proposition 7.10 which states that a compactly supported smooth section of the dual vector bundle can be regarded as a functional on sections with a specific support condition. This operation is a dense subspace inclusion into the topological vector space of distributional sections. The map ##u_{()}## is a map from the space of sections of the dual bundle to the space of distributional sections, ##\Gamma_{\Sigma, s}^{\prime}(E)##. This raises the question of whether ##u_{()}## should be
  • #1
amilton
5
0
In this post [Observables][1] By Urs Schreiber he denotes the space of distributional sections in defenition 7.9 by ##\Gamma_{\Sigma}^{\prime}\left(E^*\right) ##

That is if ##u \in \Gamma_{\Sigma}^{\prime}\left(E^*\right)## than ##u## is a linear functional that takes as argument sections of a vector bundle ##E##

In the same post he has proposition 7.10

> Let ##E \stackrel{f b}{\rightarrow} \Sigma## be a smooth vector bundle over Minkowski spacetime and let ##s \in\{c p, \pm c p, s c p, t c p\}## be any of the support conditions from def. 2.36.
Then the operation of regarding a compactly supported smooth section of the dual vector bundle as a functional on sections with this support property is a dense subspace inclusion into the topological vector space of distributional sections from def. 7.9:
$$
\begin{array}{ccc}
\Gamma_{\Sigma, \mathrm{cp}}\left(E^*\right) & \stackrel{u_{(-)}}{\longrightarrow} & \Gamma_{\Sigma, S}^{\prime}(E) \\
b & \mapsto & \left(\Phi \mapsto \int_{\Sigma} b(x) \cdot \Phi(x) \operatorname{dvol}_{\Sigma}(x)\right)
\end{array}
$$

In my understanding ##u_{()}## is a map from the space of sections of the dual bundle to the space of the distributional section .Why ##u_{()} \in \Gamma_{\Sigma, s}^{\prime}(E)## ? Shouldn't we have ## u_{()} \in \Gamma_{\Sigma, s}^{\prime}(E^*)## [1]: https://www.physicsforums.com/insights/newideaofquantumfieldtheory-observables/
 
Last edited by a moderator:
Physics news on Phys.org
  • #2


It is correct that in proposition 7.10, ##u_{()}## is a map from the space of sections of the dual bundle ##E^*## to the space of distributional sections ##\Gamma_{\Sigma, s}^{\prime}(E)##. This can be seen from the notation ##\Gamma_{\Sigma, s}^{\prime}(E)## which indicates that we are considering distributional sections of ##E##, not ##E^*##.

The notation may be confusing because in definition 7.9, ##\Gamma_{\Sigma}^{\prime}\left(E^*\right)## is used to denote the space of distributional sections of ##E^*##. However, in this case, the notation is being used to emphasize that the distributional sections are taken with respect to the dual bundle.

Therefore, in proposition 7.10, we are considering distributional sections of the original bundle ##E##, not its dual ##E^*##. This is why ##u_{()}## is an element of ##\Gamma_{\Sigma, s}^{\prime}(E)##, not ##\Gamma_{\Sigma, s}^{\prime}(E^*)##.
 

1. What is the distributional section in scientific notation?

The distributional section in scientific notation refers to the part of a scientific paper or report where the data is presented and analyzed. It includes tables, graphs, and other visual representations of the data.

2. Why is understanding notation in the distributional section important?

Understanding notation in the distributional section is important because it allows for accurate interpretation and analysis of the data. Notation can convey important information about the distribution, variability, and relationships within the data.

3. What are some common notations used in the distributional section?

Common notations used in the distributional section include mean (μ), standard deviation (σ), median (M), and interquartile range (IQR). Other notations may be specific to the field or type of data being analyzed.

4. How can I improve my understanding of notation in the distributional section?

One way to improve understanding of notation in the distributional section is to familiarize yourself with common notations used in your field of study. Additionally, practice interpreting and analyzing data using different notations and seek guidance from experts or resources if needed.

5. Can notation in the distributional section vary between different studies or reports?

Yes, notation in the distributional section can vary between different studies or reports, especially if the data being analyzed is from different fields or disciplines. It is important to carefully read and understand the notation used in each study or report to accurately interpret the data.

Similar threads

  • Differential Geometry
Replies
14
Views
3K
Replies
11
Views
1K
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Replies
3
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
Replies
1
Views
696
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
Back
Top