Observability and Mathematics

5 Observability and Yang–Mills equations

Classical Yang–Mills theory including Yang–Mills equations was born in the article [11].

A detailed enough description of the classical Yang–Mills theory is presented in the book [1].

Let us now consider the Mathematics with Observers point of view.

YME1

The classic definition of the Hodge star (⋆) operator, applied to $\underset{\mathit{p}}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}$, requires that

$\begin{array}{c}\mathrm{\star }:{\mathrm{\Omega }}^{\mathit{q}}\mathrm{(}\underset{\mathit{p}}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mapsto {\mathrm{\Omega }}^{\mathit{p}{-}_{\mathit{n}}\mathit{q}}\mathrm{(}\underset{\mathit{p}}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\end{array}$

is an “almost linear map” from q- to r-forms and satisfies some conditions formulated below.

Here

$\begin{array}{c}\mathit{q}=0\mathrm{,}1\mathrm{,}\dots \mathrm{,}\mathit{p};\mathit{r}=\mathit{p}{-}_{\mathit{n}}\mathit{q}\end{array}$

and “almost linear map” means as usual a linear map up to random variables.

For $\mathit{p}=3$, this means that

$\begin{array}{r}\mathrm{\star }:{\mathrm{\Omega }}^0\mathrm{(}\underset{3}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mapsto {\mathrm{\Omega }}^3\mathrm{(}\underset{3}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mathrm{,}\\ \mathrm{\star }:{\mathrm{\Omega }}^1\mathrm{(}\underset{3}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mapsto {\mathrm{\Omega }}^2\mathrm{(}\underset{3}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mathrm{,}\\ \mathrm{\star }:{\mathrm{\Omega }}^2\mathrm{(}\underset{3}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mapsto {\mathrm{\Omega }}^1\mathrm{(}\underset{3}{\underbrace{{\mathit{W}}_{\mathit{n}}\times \cdots \times {\mathit{W}}_{\mathit{n}}}}\mathrm{)}\mathrm{,}\\ \mathrm{\star }:{\mathrm{\Omega }}^3\mathrm{(}\underset{}{\underset{}{{\mathit{W}}_{\mathit{n}}\times \cdots }}\end{array}$