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Typesetting rules in physics (was: Re: Binary Relations, draft 1)

Hans Aberg:

> The first rule about typesetting rules of pure math I think is that
> there is none such rule! :-)

The second rule about typesetting rules of physics is that even when
there are clearly-stated rules, they are never strictly observed. ;-)

Anyway, here's a discussion paper I began writing a long time ago,
but somehow never quite finished it.

Cheers, Ulrik.
(still extremely busy finishing my thesis for submission)

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\documentclass[preprint]{ltugboat}
\usepackage{enumerate}
\usepackage{url}
\usepackage{cite}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage{amsmath,amssymb}

\overfullrule0pt

%% from revtex.sty

\makeatletter
\def\tensor#1{\protect\@ontopof{#1}{\leftrightarrow}{1.15}\mathord{\box2}}
\def\overstar#1{\protect\@ontopof{#1}{\ast}{1.15}\mathord{\box2}}
\def\overdots#1{\protect\@ontopof{#1}{\cdots}{1.0}\mathord{\box2}}
\def\overcirc#1{\protect\@ontopof{#1}{\circ}{1.2}\mathord{\box2}}
\def\loarrow#1{\protect\@ontopof{#1}{\leftarrow}{1.15}\mathord{\box2}}
\def\roarrow#1{\protect\@ontopof{#1}{\rightarrow}{1.15}\mathord{\box2}}

\def\@ontopof#1#2#3{%
{\mathchoice
{\@@ontopof{#1}{#2}{#3}\displaystyle\scriptstyle}%
{\@@ontopof{#1}{#2}{#3}\textstyle\scriptstyle}%
{\@@ontopof{#1}{#2}{#3}\scriptstyle\scriptscriptstyle}%
{\@@ontopof{#1}{#2}{#3}\scriptscriptstyle\scriptscriptstyle}%
}}
\def\@@ontopof#1#2#3#4#5{%
\setbox0=\hbox{$#4#1$}%
\setbox1=\hbox{$#5#2$}%
\setbox2=\hbox{}\ht2=\ht0 \dp2=\dp0 %
\ifdim\wd0>\wd1 %
\setbox1=\hbox to\wd0{\hss\box1\hss}%
\mathord{\rlap{\raise#3\ht0\box1}\box0}%
\else   %
\setbox1=\hbox to.9\wd1{\hss\box1\hss}%
\setbox0=\hbox to\wd1{\hss$#4\relax#1$\hss}%
\mathord{\rlap{\copy0}\raise#3\ht0\box1}%
\fi
}

\def\lambdabar{\protect\@lambdabar}
\def\@lambdabar{%
\relax
\bgroup
\def\@tempa{\hbox{\raise.73\ht0
\hbox to0pt{\kern.25\wd0\vrule width.5\wd0
height.1pt depth.1pt\hss}\box0}}%
\mathchoice{\setbox0\hbox{$\displaystyle\lambda$}\@tempa}%
{\setbox0\hbox{$\textstyle\lambda$}\@tempa}%
{\setbox0\hbox{$\scriptstyle\lambda$}\@tempa}%
{\setbox0\hbox{$\scriptscriptstyle\lambda$}\@tempa}%
\egroup
}
\makeatother

%% from nucetc.sty

\def\iotabar%
{\raisebox{-1pt}{$\mathchar'40$}\mkern-5.43mu\iota}

%% from ioplppt.sty

\newcommand{\mat}[1]{\underline{\underline{\bf #1}}}

% \newcommand{\e}{{\mathrm e}}
% \let\ii=\i
% \renewcommand{\i}{{\mathrm i}}
% \let\du=\d
% \renewcommand{\d}{{\mathrm d}}

%% from elsart.cls

% \def\d{\,\mathrm{d}}
% \def\e{\mathop{\mathrm{e}}\nolimits}

%% private hacks

\DeclareSymbolFont{vectors}  {OML}{cmm}{b}{it}
\DeclareSymbolFont{uptensors}{OT1}{cmss}{bx}{n}
\DeclareSymbolFont{tensors}  {T1}{cmss}{bx}{it}
\DeclareSymbolFontAlphabet{\mathvec}   {vectors}
\DeclareSymbolFontAlphabet{\mathtensup}{uptensors}
\DeclareSymbolFontAlphabet{\mathtens}  {tensors}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Requirements for typesetting physics}
\author{Ulrik Vieth}
\address{%
  Heinrich-Heine-Universit{\"a}t\\
  Institut f{\"u}r Theoretische Physik~II\\
  Universit{\"a}tsstra\ss{}e 1\\
  D-40225 D{\"u}sseldorf\\
  Germany}
\netaddress{vieth@thphy.uni-duesseldorf.de}
\personalURL{http://www.thphy.uni-duesseldorf.de/~vieth/}

\def\rtitlex{MFG discussion document}
\def\midrtitle{{\sl Task 2: Requirements analysis\/}}
\setcounter{page}{1}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\maketitle

\begin{abstract}
  This paper attempts to summarize the requirements for
  typesetting math-mode material in physics and related sciences
  in accordance with the established typesetting rules applicable
  in those fields.  Particular emphasis is placed on pointing out
  the subtle differences from \AllTeX{}'s default settings for
  math-mode, which are primarily geared towards the requirements
  for mathematics and computer science.
  
  This is an unfinished draft of a discussion paper which was
  begun back in May~1997.  Unfortunately, as of \today, a number
  of sections still remain very sketchy and incomplete and need 
  to be fleshed out eventually.
\end{abstract}


\section{Introduction}

Besides mathematics and computer science, physics is probably one
of the most important areas in which \AllTeX{} is heavily used for
typesetting math-mode material (as opposed to applications in the
humanities for typesetting body text).

While the default math-mode typesetting rules implemented in
\AllTeX{} are primarily geared towards the conventions applicable
in North American math typesetting, as specified in authorative
handbooks such as \emph{The Chicago Manual of Style}
\cite{Chicago:1982} or the style guides of well respected math
publishers \cite{AMS:1979}, some slightly (or not so slightly)
different conventions are applicable in physics and related
disciplines, such as physical chemistry or engineering sciences.
  
This discussion paper attempts to summarize the requirements for
typesetting physics, following the recommendations of \acro{IUPAP},
the International Union for Pure and Applied Physics
\cite{IUPAP:1978,IUPAP:1987,ISO:ISO31,NIST:SP330,NIST:SP811}, and it
also discusses various alternative conventions commonly found in
physics journals or textbooks produced by publishers
\cite{APS:REVTeX:1996,APS:CompuscriptPRL:1996,APS:StyleGuide:1993,%
  IOP:ioplau:1994,IOP:iopfts:1994,Elsevier:instraut:1995,%
  IAEA:nucfus:1996} 
who adhere to these guidelines to a greater or lesser degree.

An earlier review of the requirements for typesetting physics was
presented by J{\"o}rg Knappen at the Euro\TeX{}~'92 conference
\cite{Knappen:EuroTeX:1992}.  The specific requirements in
high-energy physics with particular emphasis on elementary
particle notation were summarized in a \TUB{} article by Michel
Goossens \cite{Goossens:TB13-2-201} and also in Chapter~6.1 of
\emph{The \LaTeX{} Graphics Companion} \cite{GMR:TLGC:1997}.
Another very extensive review of the requirements for typesetting
mathematics in science and technology including references to the
relevant \acro{ISO} standards appeared in a recent \TUB{} article
by Claudio Beccari \cite{Beccari:TB18-1-39}.
 

\section{Requirements for math alphabets}

\subsection{Symbols for physical quantities}

To quote directly from the \acro{IUPAP} recommendations
\cite[sec.~1.2.1--2]{IUPAP:1978}, the general rules say:
\begin{quotation}
  1. Symbols for physical quantities should be single letters of
  the Latin or Greek alphabets with or without modifying signs.
\end{quotation}
and 
\begin{quotation}
  2. Symbols for physical quantities should be printed in italic 
  (or sloping) type.
\end{quotation}
Since this conforms exactly to the default font shape in \TeX{}'s
math mode (accessible by \cs{mathnormal}), no special markup is
needed, and there is little to worry about in this case, except
for the fact that there is some disagreement concerning the
preferred shape of uppercase Greek letters.

While the examples in the \acro{IUPAP} document seem to indicate
that uppercase Greek should be set in math italics type as well,
just like anything else, this convention isn't always observed in
publications, and one frequently finds the use of upright (roman)
type for uppercase Greek letters instead, whether it be for
laziness, lack of suitable fonts, or a policy of following a
traditional house style.
  
To achieve the greatest possible flexibility, both conventions
need to be supported at the font encoding level, and a switching
mechanism at the macro level should be provided, if possible.


\subsection{Subscripts to physical quantities}

As a guiding principle for the printing of indices, it is recommended
in \cite[sec.~1.2.2]{IUPAP:1978} that:
\begin{quotation}
  Only indices which are symbols for physical quantities should
  be printed in italic (sloping) type.
\end{quotation}
This implies that normal (upright) type should be used otherwise,
especially for textual indices.

Thus, for example, in the energy confinement time~$\tau_{E}$ the
subscript~`$E$' should be printed in math italics type because it
refers to the physical quantity ``energy'', whereas in the particle
confinement time~$\tau_{\mathrm{p}}$ the subscript~`$\mathrm{p}$'
should be printed in roman (upright) type because it stands as an
abbreviation for a textual expression ``particle''.

Some publishers' style guides
\cite{APS:REVTeX:1996,APS:CompuscriptPRL:1996,APS:StyleGuide:1993}
try to simplify these guidelines by suggesting to use upright type
for multi-letter subscripts (which almost always represent textual
material) while using math italics for single-letter subscripts
(which may or may not represent a physical quantity).  It is
probably not a good idea to follow such a simplification since it
may be misleading at times.

Concerning the font used to denote textual subscripts, care should
be taken to ensure that this should always be upright (roman) type
regardless of the surrounding text font, since the accidental use
of text italics or slanted type (e.g.\ within a theorem
environment) might only cause confusion.  Thus, a \cs{text} macro
like in \AmSLaTeX{} which has the effect of selecting the
surrounding text font rather than the default roman font may only
be suitable for the mark-up of intervening words between formulas,
but not for textual subscripts within formulas.

Using \cs{mathrm} for the latter purpose might be a better
alternative, provided that the roman letters are taken from a font
with normal text spacing.  What would happen if math roman letters
were taken from a hypothetical \texttt{MC}-encoded roman font with
math spacing hasn't been studied so far, but it is likely to
assume that some incompatibilities might arise.


\subsection{Symbols for vectors and tensors}

Concerning the representation of vectors and tensors the the
\acro{IUPAP} recommendations \cite[sec.~1.2.3]{IUPAP:1978} suggest:
\begin{quotation}
  To avoid the excessive usage of subscripts, it is often
  convenient to indicate vectors and tensors of the second rank by
  letters of a special type.
\end{quotation} 
which in \LaTeX{} lingo means using special math alphabets loaded
into extra math groups.  In particular, it is recommended that
\begin{enumerate}[(a)]
\item 
  vectors should be printed in bold italic (sloping) type,
  e.g.\ $\mathvec{A}$,~$\mathvec{a}$,
\item 
  tensors of the second rank should be printed in bold sans serif
  italic (sloping) type, e.g.\ $\mathtens{S}$,~$\mathtens{T}$.
\end{enumerate}
Although the recommended choice of fonts is fairly specific, the
conventions used in practice often vary, and one frequently finds
the use of bold upright type (\cs{mathbf}) for vectors,
e.g.\ $\mathbf{A}$,~$\mathbf{a}$, and bold sans serif upright type
for tensors, e.g.\ $\mathtensup{S}$,~$\mathtensup{T}$.

Still other publishers \cite{IOP:ioplau:1994,IOP:iopfts:1994}
resort to replacement notations in the preprint styles, such as
bold upright type combined with double underlining for tensors,
e.g.~$\mat{A}$, while using the proper typefaces in the final
publications.  

To some extent, all these deviations from the standard may, in the
past, have been dictated by the lack of suitable fonts or the
difficulties of accessing such unusual font shapes.  However, if
such technical constraints no longer persist, it is probably best
to try to avoid any such deviations in the future and use the
proper notations wherever possible.

Since, in general, a full set of Latin and Greek letters may be
needed to denote vectors and tensors, a number of additional
\texttt{MC}-encoded fonts are needed to provide suitable math
alphabets, whereas using \texttt{T1}-encoded fonts will be
insufficient for this purpose.  To allow catering for all commonly
used conventions, at least four additional \texttt{MC}-encoded
fonts in bold upright, bold math italics, bold sans serif upright,
and bold sans serif oblique would be needed.  However, it may be
sufficient to provide only a subset of the \texttt{MC}-encoding
restricted to Greek and Latin letters or letter-like symbols,
while omitting non-changeable symbols that are included in the
basic \texttt{MC}-encoded font only for technical reasons.

Finally, if no suitable math alphabets are available, or if their
use is not desirable for whatever reason, it is suggested in
\cite[sec.~1.2.3]{IUPAP:1978} that:
\begin{quotation}
  A vector may be indicated by an arrow and a tensor by a double
  arrow above the symbol, e.g. $\vec{A}$,~$\Vec{\Vec{S}}$.
\end{quotation}
Whereas the use of a single arrow for vectors is well-established
and corresponds to the default \cs{vec} macro in \TeX{}'s math
mode, there is a somewhat wider variety of conventions used for
tensors.  While the \acro{IUPAP} document suggests a double arrow
consisting of two arrows stacked on top of each other, the
\acro{REV}\TeX{} package \cite{APS:REVTeX:1996} provides a
\cs{tensor} macro that uses a left-right over-arrow like this:
$\tensor{S}$.

In any case, establishing a consistent mark-up syntax and
providing a flexible switching mechanism at the macro level to
allow selecting the preferred representation of vectors and
tensors would be very desirable.  In particular, this would be 
a significant advantage with regard to document portability and
reusability despite the absence of previously established standard
macros to access extra math alphabets.  If so, adopting already
existing input notations, such as \cs{vec} for vectors and
\cs{tens} for tensors, to be used regardless of their eventual
representation might be the most preferable choice.


\subsection{Symbols for physical units}

Acoording to the \acro{IUPAP} document \cite[sec.~2.1]{IUPAP:1978}:
\begin{quotation}
  Symbols for units of physical quantities should be printed in
  roman (upright) type.
\end{quotation}
In general, physical units are represented by single or multiple
Latin letters, usually in lowercase only except for units derived
from proper names which are in uppercase or mixed-case.

There are only a very small number of exceptions from this rule
that require the use of Greek letters, specifically the multiplier
micro (\textmu), the unit Ohm~(\textohm) and the inverted
Ohm~(\textmho), all of which are available from a
\texttt{TS1}-encoded text companion font.  Finally, there is the
non-SI unit \AA{}ngstr{\o}m (\AA{}) consisting of an accented
letter which could always be accessed by resorting to \cs{textrm}
within math mode if nothing else is provided.

We can conclude that, in general, a \texttt{T1}-encoded text font
accessible with \cs{mathrm} or a synonym such as \cs{unit} would
be sufficient to typeset physical units if provisions are taken to
handle the exceptional cases by special macros.

Knappen \cite{Knappen:EuroTeX:1992} suggested a \cs{Ohm} macro for
\textohm{} and similarly one might introduce a \cs{micro} macro
for~\textmu{}.  Although, some similar macros are already provided
in the \textsf{textcomp} package as \cs{textohm} and \cs{textmu},
such macros should also be included in a physics-specific
math-mode package to make it unnecessary to rely on other \LaTeX{}
packages.

As for the typographical treatment of physical units, it should be
noted that it is frequently necessary to insert a little space
between numbers and units or in between physical units where
multiplication is implied.  While \emph{The \TeX{}book} suggests
using a thinspace, some other sources even suggest a thickspace or
an unbreakable word space.  Whether such spacing should be
inserted automatically by the macros used to mark up physical
units seems highly questionable, as it is often highly dependent
on the context whether or not extra spacing is needed.

As for the formatting of numbers, nearly all publishers' style
guides remind the author to avoid dangling decimal points, i.e.\ 
to write out $1.0$ or $0.1$ instead of $1.$ or $.1$ to avoid
confusion.


\subsection{Symbols for chemical elements, nuclides, and particles}

\begin{itemize}
\item to be filled in, for details see \cite[sec.~4]{IUPAP:1978}
\item symbols for chemical elements and elementary particles 
  should be set in upright roman (\cs{mathrm})
\item requires access to a full set of upper- and lowercase 
  Latin and Greek, i.e.\ a \texttt{MC}-encoded font
\item suggested mark-up (\cs{chem}, \cs{particle})
\end{itemize}


\subsection{Quantum states and spectroscopic notations}

\begin{itemize}
\item to be filled in, for details see \cite[sec.~5]{IUPAP:1978}
\item symbols for quantum states should be set in upright roman 
  (\cs{mathrm})
\item requires access to a full set of upper- and lowercase 
  Latin and Greek, i.e.\ a \texttt{MC}-encoded font
\end{itemize}


\subsection{Dimensionless numerical parameters}

\begin{itemize}
\item to be filled in, for details see \cite[sec.~7.14]{IUPAP:1978}
\item two-character mixed-cased expressions like `$\mathit{Re}$' 
  for the Reynolds number, `$\mathit{Ma}$' for the Mach number, etc.
\item to be set in italic (sloping) type, preferably in text italics
  (\cs{mathit}) for better kerning in two-character expressions
\end{itemize}


\subsection{Other math alphabets}

\paragraph{Script/Calligraphic letters.}  
These are used only in a few specific contexts, typically in cases
where two different physical quantities would normally be
represented by the same letter.  Examples include:
\begin{itemize}
\item 
  `$\mathcal{E}$' when used for energy in contexts where `$E$'
  is already used to represent the electric field.
\item 
  `$\mathcal{L}$' and `$\mathcal{H}$' when used to denote the
  Lagrangian or Hamiltonian density in field theory, where 
  `$L$'~and~`$H$' represent the Lagrangian or Hamiltonian function.
\item 
  `$\mathcal{O}$' when used to denote the order of neglected terms
  in series expansions.  Some authors (including DEK himself)
  prefer to use a regular math italics~`$O$' for this purpose, so
  the use of a caligraphic~`$\mathcal{O}$' might be regarded as
  controversial.
\end{itemize}
There may be some disagreement among true believers as to whether
\TeX{}'s default caligraphics letters are sufficiently scripty for
this purpose or whether a different script font, such as
\texttt{rsfs}, should instead be substituted for it.  

There is probably no need to introduce any special mark-up tags
for these relatively few cases of script or calicgraphic letters
used in physics.  However, it may, of course, be very convenient
to define private shorthands, such as `\cs{L}'~or~'\cs{H}', when
dealing with material where these symbols are frequently used.



\paragraph{Blackboard Bold letters.}
These are sometimes used to denote number sets, such as
`$\mathbb{N}$', `$\mathbb{R}$' or `$\mathbb{C}$' exactly as
they're normally used in mathematics.  Again, there may be some
disagreement among true believers as too their prefered shape.
Also, it may again seem convenient to define private shorhands,
such as `\cs{N}' or `\cs{R}', if these symbols are used rather
frequently.
 

\paragraph{Fraktur letters.}
These are no longer used in modern physics textbooks, but they are
sometimes found in reprints of historic texts (e.g.\ Einstein's
lectures of 1922) where Fraktur was used to denote vectors,
 


\section{Requirements for mathematical symbols}

\subsection{Mathematical constants}

Certain mathematical constants should always be printed in upright
(roman) type, especially when there is a danger that they might be
confused with a similar symbol denoting a physical quantity.

In particular, this convention applies to the following mathematical
constants:
\begin{itemize}
\item `$\pi$' in its usual meaning ($\pi = 3.14159\dots$).  This
  convention is rarely observed at present, presumably for lack of
  suitable fonts.
  
\item `$\mathrm{e}$' when used as the base of the exponential function
  or natural logarithm ($\mathrm{e} = 2.718\dots$).  In this context,
  `$\mathrm{e}$' is almost invariably followed by a possibly
  non-trivial exponent, but never by a subscript.  It is sometimes set
  off from the preceeding material by a little space at the author's
  or publisher's discretion.

  Several publisher's macro packages define a control sequence
  \cs{e} for this purpose, usually defined simply as a synonym 
  for \verb|\mathrm{e}|.  Elsevier's macros, however, define
  \begin{displaymath}
    \verb|\def\e{\mathop{\mathrm{e}}\nolimits}|
  \end{displaymath}
  which also affects the spacing.
  
\item `$\mathrm{i}$' when used as the imaginary unit ($\mathrm{i} =
  \sqrt{-1}$).  Some sub-fields in physics such as electrical
  engineering may use `$\mathrm{j}$' instead to avoid confusion with
  the AC-current~$i(t)$.
  
  Some publisher's macro packages define a control sequence \cs{i} as
  a synonym for \verb|\mathrm{i}|, thereby replacing or overwriting
  the usual meaning of text mode dotless-i, which is not applcable in
  math mode anyway.  It may be possible to retain both meanings by
  employing a suitable switching macro depending on the mode.
\end{itemize}


\subsection{Derivatives, increments, variations, etc.}

In addition to mathematical constants, several Latin letters should be
printed in upright (roman) type, when used in a well-defined context
as mathematical operators.  In particular, this applies to:
\begin{itemize}
\item `$\mathrm{d}$' when used to denote the integration variable in
  an integral, a total derivative, or as a total (exact) differential.
  
  Some publisher's macro packages define a control sequence \cs{d} as
  a synonym for \verb|\mathrm{d}|, thereby replacing or overwriting
  the usual command for the dot-under accent in text mode. It may be
  possible to retain both meanings by employing a suitable switching
  macro.

\item `$\mathrm{D}$' when used a differential along geodesics
  in general relativity (relatively rare).
  
\item `$\mathrm{\partial}$' when used as a partial derivative.  This
  convention is rarely observed so far, presumably for lack of
  suitable fonts.  If an upright \cs{partial} becomes available in a
  new math font encoding, it may be reasonable to make this the
  default choice when in a physics context.
  
\item `$\mathrm{\delta}$' when used as a variational derivative, or
  as an inexact differential, e.g. $\mathrm{d}U = \delta Q + \delta W$.

\item `$\mathrm{\Delta}$' when used to denote a finite increment.

\item `$\eth$' and `\th' when used as a weighted derivative
  in quantum field theory (rare).
\end{itemize}
Some slashed or barred variants of `$\mathrm{d}$' or
`$\mathrm{\delta}$' are ocassionally used in thermodynamics to
distinguish between (exact or inexact) differental quantities in
reversible or irreversible processes. However, such a notation
presumably might be an author's invention and thus might be 
considered non-standard.


\subsection{Operators in vector caculus}

\begin{itemize}
\item Nabla (gradient, divergence, rotation/curl)
\item Laplace operator (Delta)
\item d'Alembert operator (``Quabla'')
\end{itemize}


\section{Requirements for special notations}

\subsection{Special brackets, braces, etc.}

\begin{itemize}
\item (anti-)commutator in quantum theory: 
  $[p,q]_{\pm}$
\item Poisson- or Lie-brackets in theoretical mechanics:
  $[p,q]$ or $\{p,q\}$
\item bra-ket notation in quantum theory,
  quantum states, averages, projection operator:
  $\langle x|H|x \rangle$
\item ensemble averages, time averages:
  $\langle x \rangle$, $\bar{x}$
\item directions, planes, coordinates in crystals
\end{itemize}


\subsection{Barred letters}

\begin{itemize}
\item $\hbar$ (\cs{hbar}) and/or $\hslash$ (\cs{hslash}), both
  representing a short-hand notation for $h/2\pi$ where $h$ is
  Planck's constant.  This notation is very well-established 
  and widely used in quantum theory.  
  
\item $\lambdabar = lambda/(2\pi)$, representing a wavelength
  over $2\pi$.  A suitable macro is provided as \cs{lambdabar}
   in the \acro{Rev\TeX} macro package.
  
\item $\iotabar$, used in a sub-field of fusion plasma physics to
  denote $\iota/2\pi$ where $\iota$ is the rotational transform of
  magnetic field lines.  Another sub-field of plasma physics prefers
  to use a ``safety factor'' defined as $q = 2\pi/\iota = 1/\iotabar$
  instead, thus making it unnecesary to use unusual notation.

\end{itemize}


\subsection{Slashed letters in QED}

\begin{itemize}
\item Feynman slash notation
\end{itemize}


\subsection{Operators in quantum mechanics}

\begin{itemize}
\item Operators vs. classical quantities
  
\item Complex conjugate states, denoted by `$a^{*}$' in quantum
  mechanics where `$\bar{a}$' would normally be used for the 
  complex conjugate in a math context.

\item (Self-)adjoint operators, denoted by `$a^{+}$' or `$a^{\dag}$'.
  This example indicates the use of the dagger as a math symbol,
  not just as a footnote symbol.
\end{itemize}


\section{Summary}

The following math alphabets are needed in the \texttt{MC}-encoding
providing access to both Latin and Greek letters in a given shape:
\begin{itemize}
\item italic (sloping) type (\cs{mathnormal}),\\
  for symbols denoting physical quantities
\item upright (roman) type (\cs{mathrm}),\\
  for particle symbols or chemical elements
\item bold italic (sloping) type,\\
  for vectors (recommended notation)
\item bold sans serif italic (sloping) type,\\
  for tensors of the second rank
\item bold upright type (\cs{mathbf}),\\
  for vectors (widely-used alternative notation)
\item bold sans serif upright type,\\
  for tensors (widely-used alternative notation)
\end{itemize}
The following math alphabets are needed only for Latin letters 
and may thus be implemented either using a \texttt{T1}-encoded 
or \texttt{MC}-encoded font:
\begin{itemize}
\item upright (roman) type (\cs{mathrm}),\\
  for textual subscripts

\item upright (roman) type (\cs{mathrm}),\\
  for physical units (with few exceptions)

\item text italic type (\cs{mathit}),\\
  for dimensionless parameters
\end{itemize}

\begin{thebibliography}{10}

\bibitem{Chicago:1982}
{University of Chicago Press}.
\newblock {\em The Chicago Manual of Style}.
\newblock University of Chicago Press, Chicago and London, 13th revised
  edition, 1982.

\bibitem{AMS:1979}
Ellen Swanson.
\newblock {\em Mathematics into Type}.
\newblock American Mathematical Society, revised edition, 1979.

\bibitem{IUPAP:1978}
{International Union of Pure and Applied Physics, S.U.N. Commission}.
\newblock Symbols, units and nomenclature in physics.
\newblock {\em Physica~A}, 93:1--60, 1978.
\newblock Document U.I.P. 20 (1978).

\bibitem{IUPAP:1987}
{International Union of Pure and Applied Physics, S.U.N. Commission}.
\newblock Symbols, units and nomenclature in physics.
\newblock {\em Physica~A}, 146:1--67, 1987.

\bibitem{ISO:ISO31}
ISO.
\newblock Quantities and units.
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\end{thebibliography}

\makesignature
\end{document}