\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{physical unit}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{UnitsOfLengthInLagrangianFieldTheory}{Units of length in Lagrangian field theory}\dotfill \pageref*{UnitsOfLengthInLagrangianFieldTheory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

When formulating a [[theory of physics]] in terms of [[mathematics]] one typically models the range of certain physical quantities by [[torsors]] over some [[group]] of transformations.

For instance a [[wavelength]] would be identified as element in the [[real line]] minus its origin, $\mathbb{R}-\{0\}$ , being a torsor over the [[multiplicative group]] $\mathbb{R}^\times$ of [[real numbers]].

In order for the ``[[coordination]]'' of the mathematical theory with physical [[experiment]] to take place, one needs to choose an identification of this abstract torsor with the (idealized) one that it is supposed to model in nature. Such a choice is equivalent to a choice of [[unit]] (in the mathematical sense), hence a choice of element of the torsor. In this context this is then a \emph{physical unit}.

For instance picking an element in $\mathbb{R}-\{0\}$ and declaring this to be length of the path travelled by light in a vacuum in 1/299 792 458 second means defining a \emph{physical unit of length} (in this example: of the [[meter]]).

Physical units are often called \emph{physical constants}. But by definition physical units are arbitrary choices made in the desciption of a physical system. Of course once made, one wants to keep these choices constant, such as to be useful.

The actual \emph{constants of nature} are instead quotients of physical units. For instance the [[fine structure constant]] is the [[quotient]]

\begin{displaymath}
\alpha \coloneqq \frac{e^2}{ (4 \pi \epsilon_0) \hbar c} \in \mathbb{R}
\end{displaymath}
where $e$ is the [[electric charge]] of the [[electron]] expressed in physical units of charge (such as [[coulomb]]s), $\hbar$ is [[Planck's constant]] etc. The resulting quotient is then independent of any choices and is hence a [[real number]] characterizing nature independently of any conventions about how to parameterize it.

Notice that \emph{choice of unit} is also called \emph{choice of gauge}. This is indeed the same ``[[gauge]]'' as in ``[[gauge theory]]'', as it is how (\href{gauge#Weyl23}{Weyl 23}) introduced the concept of gauge theory: as a theory in which the choice of unit of length may change along paths in space.

\hypertarget{UnitsOfLengthInLagrangianFieldTheory}{}\subsection*{{Units of length in Lagrangian field theory}}\label{UnitsOfLengthInLagrangianFieldTheory}

\begin{quote}%
under construction


\end{quote}
Let $\Sigma \simeq \mathbb{R}^{p,1}$ be [[Minkowski spacetime]] and let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]] thought of as a [[field]] bundle. Write $\{\phi^a\}$ for local [[coordinates]] on the typical fiber of this bundle.

The total space of the corresponding [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb^\infty}{\to} \Sigma$ carries an [[action]]

\begin{displaymath}
sc \;\colon\; \mathbb{R}^\times \times J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)
\end{displaymath}
of the multiplicative [[group of units]] $\mathbb{R}^\times$ of the [[real numbers]], given on the induced jet coordinates by

\begin{displaymath}
\begin{aligned}
    x^\mu & \mapsto r x^\mu
    \\
    \phi^a & \mapsto \phi^a
    \\
    \phi^a_{,\mu_1, \cdots \mu_k} & \mapsto r^{-k} \phi^a_{,\mu_1 \cdots \mu_k}
  \end{aligned}
  \,.
\end{displaymath}
Let then

\begin{displaymath}
\mathbf{L}_{(-)}
   \;\colon\;
  \mathbb{R}^n \longrightarrow \mathbf{\Omega}^{p+1}(J^\infty_\Sigma(E))
\end{displaymath}
be a smoothly $n$-parameterized collection of [[Lagrangian densities]], equipped with an $R^\times$-action

\begin{displaymath}
scp \;\colon\; \mathbb{R}^\times \times \mathbb{R}^n \longrightarrow \mathbb{R}^n
\end{displaymath}
on $\mathbb{R}^n$.

Observe that the [[Euler-Lagrange equations]] induced by a Lagrangian density $\mathbf{L}$ equal those induced by the rescaled Lagrangian $r \mathbf{L}$, and that the [[presymplectic current]] $\Omega_{BFV}$ induced by $\mathbf{L}$ scales linearly with $r$ itself. Upon [[quantization]], this rescaling of $\Omega_{BFV}$ may be absorbed in [[Planck's constant]]. In conclusion, as long as Lagrangian densities scale \emph{homogeneously} the rescaled Lagrangian induces the same physics.

Hence we require that the combined scaling action of $\mathbb{R}^\times$ on $J^\infty_\Sigma(E)$ via $sc$ and on the parameters in $\mathbb{R}^n$ via $scp$ is homogeneous on $\mathbf{L}$ in that there exists $dim \in \mathbb{Z}$ such that for every $r \in \mathbb{R}^\times$ we have

\begin{displaymath}
sc_r^\ast \mathbf{L}_{( scp(-))} = r^{dim} \mathbf{L}_{(-)}
  \,.
\end{displaymath}
Then a parameter $a \colon \mathbb{R}^n \to \mathbb{R}$ such that there exists $w \in \mathbb{Z}$ with

\begin{displaymath}
scp_r^\ast a = r^{w} a
\end{displaymath}
is said to have \emph{dimension} $[length]^{w}$.

For example the Lagrangian density for the [[free field theory|free]] [[scalar field]]

\begin{displaymath}
\mathbf{L}_{(-)} 
     \;\colon\; 
   \mathbb{R}^1 
     \longrightarrow 
   \mathbf{\Omega}^{p+1}(J^\infty_\Sigma(\Sigma \times \mathbb{R})
\end{displaymath}
given by

\begin{displaymath}
\mathbf{L}_{m}
  \;\coloneqq\;
  \tfrac{1}{2}
  \left(
    \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu} 
    -
    m^2 \phi^2
  \right)
  dvol
\end{displaymath}
is parameterized by the [[mass]] $m$. For the Lagrangian to scale homogenously with $r^{p-1}$ the mass parameter has to have dimension $[length]^{-1}$. To indicate this action one writes the mass in the combination $m c / \hbar$, called the inverse \emph{[[Compton wavelength]]}, so that the homogenously scaling collection of Lagrangians is

\begin{displaymath}
\mathbf{L}_{m c / \hbar}
  \;\coloneqq\;
  \tfrac{1}{2}
  \left(
    \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu} 
    -
    \left(
      \tfrac{m c}{\hbar}
    \right)
   \phi^2
  \right)
  dvol
\end{displaymath}
(\ldots{})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[meter]], [[femtometer]]


\item [[electronvolt]], [[MeV]], [[GeV]], [[TeV]]


\item [[speed of light]]


\item [[Planck's constant]]


\item [[gravitational constant]]


\item [[Compton wavelength]]


\item [[Schwarzschild radius]]


\item [[Planck mass]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion in the context of [[philosophy of science]] includes

\begin{itemize}%
\item [[Georg Hegel]], \href{Science+of+Logic#TheMeasureFirstChapter}{Book I, third section, first chapter} of \emph{[[Science of Logic]]}

\end{itemize}
[[!redirects physical unit]] [[!redirects physical units]]



\end{document}