Expressive macro for tensors; raised and lowered indices

Question

Does anyone have a satisfying solution to the problem of typesetting tensors with raised/lowered indices? For example, I can write the following equation: \ddot x^\mu = \Gamma^{\mu}{}_{\alpha}{}_{\beta} \dot x^\alpha \dot x^\beta

When writing a lot of tensors, this is cumbersome.

What I'm looking for is a way to construct a command which can produce tensors like \Gamma above with more expressive syntax. For example, a command \tens that operates like this would be ideal:

\ddot \tens{x}{\mu} = \tens{Gamma}{\mu}[\alpha][\beta] \dot \tens{x}{\alpha} \tens{x}{\beta}

The key quality of my desired syntax is that there is an arbitrary number of arguments of two distinct types. Argument encased in { } are raised indices, while arguments encased in [ ] are lowered indices. I don't need a command that looks exactly like this; I am looking for something that is similarly expressive.

Does anyone have a solution to this problem? An implementation of a \tens command that works as above? I don't know how to do this.

Answer 1

In my opinion, the subscripts and superscripts are a single argument.

You can use the tensor package, without reinventing the wheel: it has a very handy syntax.

I also provide a \tens command according to your preferences.

\documentclass{article}
\usepackage{tensor}
%\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\tens}{mo}
 {
  #1
  \IfNoValueTF { #2 } 
   {
    __myridium_tens_up_lookup:
   }
   {
    __myridium_tens_down_lookup: [ #2 ]
   }
 }
\cs_new_protected:Nn __myridium_tens_down_lookup:
 {
  \peek_charcode_ignore_spaces:NTF [
   {
    __myridium_tens_down:w
   }
   { \kern2\scriptspace }
 }
\cs_new_protected:Npn __myridium_tens_down:w [ #1 ]
 {
  {\mathstrut}
  \sb{#1}
  \kern-\scriptspace
  __myridium_tens_up_lookup:
 }
\cs_new_protected:Nn __myridium_tens_up_lookup:
 {
  \peek_catcode_ignore_spaces:NTF \c_group_begin_token
   {
    __myridium_tens_up:n
   }
   { \kern2\scriptspace }
 }
\cs_new_protected:Nn __myridium_tens_up:n
 {
  {\mathstrut}
  \sp{#1}
  \kern-\scriptspace
  __myridium_tens_down_lookup:
 }
\ExplSyntaxOff
\begin{document}
\subsection*{With \texttt{tensor}}
[
\tensor{\ddot{x}}{^\mu}=
\tensor{\Gamma}{^\mu_\alpha_\beta}
\tensor{\dot{x}}{^\alpha} \tensor{\dot{x}}{^\beta}
]
[
\tensor{\Gamma}{\mu^\nu^\rho\alpha^\nu^\rho}
\tensor{\dot{\Gamma}}{\mu^\nu^\rho\alpha^\nu^\rho}
]
\subsection*{With the hand-made macro}
[
\tens{\ddot{x}}{\mu}=
\tens{\Gamma}{\mu}[\alpha\beta]
\tens{\dot{x}}{\alpha} \tens{\dot{x}}{\beta}
]
[
\tens{\Gamma}[\mu]{\nu\rho}[\alpha]{\nu\rho}
\tens{\dot{\Gamma}}[\mu]{\nu\rho}[\alpha]{\nu\rho}
]
\end{document}

Answer 2

I wouldn’t use such a syntax, but SemanTeX can be set up to accomplish something resembling this (disclaimer: I am the author). Note that you will need a recent update of SemanTeX (October or later, I think) for this example to work. Note that I also prefer defining keys dot and ddot instead of directly using the commands \dot and \ddot.

\documentclass{article}
\usepackage{semantex}
\NewVariableClass\tens[
    output=\tens,
    define keys={
        {dot}{ command=\dot },
        {ddot}{ command=\ddot },
        {pre index}{ right return, symbol put right={{}} },
        {post index}{ right return, symbol put right=\kern-\scriptspace },
    },
    define keys[1]={
        {default}{ pre index, lower={#1}, post index },
        {arg}{ pre index, upper={#1}, post index },
    },
]
\begin{document}
$ \tens{\dot x}{\mu} = \tens{\dot{\Gamma}}{\mu}[\alpha][\beta] 
    \tens{\dot{x}}{\alpha} \tens{\dot{x}}{\beta} $
$ \tens{\ddot x}{\mu} = \tens{\dot{\Gamma}}{\mu}[\alpha][\beta]
    \tens{\dot{x}}{\alpha} \tens{\dot{x}}{\beta} $
$ \tens{x}[ddot]{\mu} = \tens{\Gamma}[dot]{\mu}[\alpha][\beta]
    \tens{x}[dot]{\alpha} \tens{x}[dot]{\beta} $
\end{document}

---

Personally, I would prefer to use a more keyval-based syntax, as below:

\documentclass{article}
\usepackage{semantex}
\NewVariableClass\Tensor[
    output=\Tensor,
    define keys={
        {dot}{ command=\dot },
        {ddot}{ command=\ddot },
        {pre index}{ right return, symbol put right={{}} },
        {post index}{ right return, symbol put right=\kern-\scriptspace },
    },
    define keys[1]={
        {up}{ pre index, upper={#1}, post index },
        {low}{ pre index, lower={#1}, post index },
    },
]
\begin{document}
$ \Tensor{x}[dot,up=\mu] = \Tensor{\Gamma}[dot,up=\mu,low=\alpha,low=\beta]
    \Tensor{x}[dot,up=\alpha] \Tensor{x}[dot,up=\beta] $
$ \Tensor{x}[dot,up=\mu] = \Tensor{\Gamma}[dot,up=\mu,low=\alpha,low=\beta]
    \Tensor{x}[dot,up=\alpha] \Tensor{x}[dot,up=\beta] $
\NewObject\Tensor\tGamma{\Gamma}
\NewObject\Tensor\tx{x}
$ \tx[dot,up=\mu] = \tGamma[dot,up=\mu,low=\alpha,low=\beta]
    \tx[dot,up=\alpha] \tx[dot,up=\beta] $
$ \tx[dot,up=\mu] = \tGamma[dot,up=\mu,low=\alpha,low=\beta]
    \tx[dot,up=\alpha] \tx[dot,up=\beta] $
\end{document}

Answer 3

Here is something that works like you describe but with round brackets instead of curly ones. As usual such things can be a bit fragile, so occasionally you need to \relax a bit to mark it fully work, as can be seen in the second example.

\documentclass{article}
\makeatletter
\edef\tens@u{(}
\edef\tens@l{[}
\def\tens@U#1)#2{{}^{#1}\expandafter\tens@i#2\relax}
\def\tens@L#1]#2{{}_{#1}\expandafter\tens@i#2\relax}
\def\tens@i#1#2{\edef\tens@t{#1}%
\ifx\tens@t\tens@u
\expandafter\tens@U#2
\else
\ifx\tens@t\tens@l
\expandafter\tens@L#2
\else
#1#2
\fi
\fi}
\def\tens#1#2{#1\expandafter\tens@i#2}
\makeatother
\begin{document}
\begin{tabular}{rl}
works: &
$\tens{\Gamma}[\mu](\nu\rho)[\alpha](\nu\rho) \dot\tens{x}(\alpha)
\dot\tens{x}(\beta)$ \\[2em]
does not work: & 
$\tens{\Gamma}\mu\alpha \dot\tens{x}(\alpha)
\tens{x}(\beta)$ \[2em]
relax and it works again: &
$\tens{\Gamma}\mu\alpha \dot\tens{x}(\alpha)\relax
\tens{x}(\beta)$ \
\end{tabular}
\end{document}

To be clear: such macros are mainly for recreation purposes and not for the real world. These days the LaTeX world has enough other problems...

Answer 4

The definition of your desired macro \tens using TeX primitives follows:

\def\tens#1{#1\futurelet\next\tensA}
\def\tensA{\def\tensX{}%
   \ifx\next[\def\tensX[##1]{{}_{##1}\futurelet\next\tensA}\fi 
   \ifx\next\bgroup \def\tensX##1{{}^{##1}\futurelet\next\tensA}\fi
   \tensX}
%% test:
$\tens\Gamma [\mu]{\nu\rho}[\alpha]{\nu\rho}$
$\ddot\tens{x}{\mu} = \tens{\Gamma}{\mu}[\alpha][\beta] 
                      \dot\tens{x}{\alpha} \dot\tens{x}{\beta}$