
\section{The Heisenberg Algebra, \gdo, and the Alexander Polynomial}

\draftcut \subsection{Gaussian Differential Operators} \label{ssec:GDO}

In the examples we care about (see Motivation~\ref{mot:PBW} and
Section~\ref{ssec:realistic}) the generating functions turn out
to be perturbed Gaussians, whose perturbations are in some sense
``docile''\footnotemark. Hence we seek to define a category \dpg\ of
docile perturbed Gaussian generating functions, with ``differential
operator'' compositions as in Proposition~\ref{prop"LMcomposition}. We
start with the unperturbed version, \gdo:

\footnotetext{Or perhaps, we care about those examples precisely because their generating functions are
docile perturbed Gaussians.}

\begin{definition} \label{def:GDO}
\gdo\ is the category with
objects finite sets and, if $A$ and $B$ are finite, with $\mor(A\to B)$
the set of ``Gaussians in $\zeta_A\cup z_B$'':
\[ \mor(A\to B) = \left\{\omega\bbe^Q\right\}, \]
where $\omega\in\Qh$ is a scalar and where $Q$ is a
``small'' quadratic expression in $\zeta_A\cup z_B$ with
coefficients in $\Qh$. To define ``small'' and the composition law, we
decompose quadratics in $\zeta_A\cup z_B$ into a Greek-Latin part $E$,
and Greek-Greek part $F$, and a Latin-Latin part $G$:
\[ Q = \sum_{i\in A,j\in B}E_{ij}\zeta_iz_j
  + \frac12\sum_{i,j\in A}F_{ij}\zeta_i\zeta_j
  + \frac12\sum_{i,j\in B}G_{ij}z_iz_j.
\]
With this, ``small'' means that $G$ must be a multiple of
$\hbar$.\footnote{The formulas below make sense either if the $G$
terms are always small or if the $F$ terms are always small. Mostly,
in the applications $G$ will be small and so we made the condition ``$G$
is small'' be a part of the definition of \gdo. Rarely we will encounter
cases where $F$ is small but $G$ isn't. See Discussion~\ref{}.} Also,
we define the composition of $\omega_1\bbe^{Q_1}\in\mor(A\to B)$
and $\omega_2\bbe^{Q_2}$ to be $\omega\bbe^Q$, with
\begin{equation} \label{eq:gdocompositions} \begin{aligned}
  E &= E_1(I-F_2G_1)^{-1}E_2, &
  F &= F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T, \\
  G &= G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2, &
  \omega &= \omega_1\omega_2\det(I-F_2G_1)^{-1/2},
\end{aligned} \end{equation}
where $(E,F,G)$ and $(E_i,F_i,G_i)$ are the Greco-Roman decompositions of
$Q$ and of $Q_i$ as above.  Finally, the identity morphism in $\mor(A\to
A)$ is declared to be $\bbe^{\zeta_A\cdot z_A}$.\endpar{\ref{def:GDO}}
\end{definition}

\begin{comment} \label{com:QGood}
The formulas in Definition~\ref{def:GDO} may appear
unfriendly. But appearances are deceiving. Note that the rank of the
space of quadratics in a certain set of variables is in itself quadratic
in the number of variables, and quadratics grow very slowly relative to
exponentials. Hence the storage and time requirements to store and compute with elements of \gdo\ are much milder
than those for many other computations in quantum algebra, which tend to be exponential.
\end{comment}

\begin{theorem} \label{thm:GDO}
(i) \gdo\ is indeed a category (the composition law is associative, the identity morphisms are identity morphisms).
\newline (ii) The explicit composition law of~\eqref{eq:gdocompositions} agrees with the ``differential
operator'' one of~\eqref{eq:LMcomposition}.
\end{theorem}

\begin{proof} Part (i) can be verified by explicit matrix computations. It
can also be implemented and tested, and seeing that we are committed
to computability, we do that in Appendix~\ref{app:GDOCompositions}. Finally,
part (i) follows from part (ii) and the fact that the composition law
of~\eqref{eq:LMcomposition} is obviously associative. Hence we concentrate
on proving (ii). We do it in two ways: pictorial, right below, for those who
are familiar with diagrammatic algebra, and pure algebraic, on page~\pageref{pf:GDO:algebraic}.
\end{proof}

\parpic[r]{\input{figs/GDOMorphism.pdf_t}}
\noindent{\it Pictorial proof of Theorem~\ref{thm:GDO}, (ii).} This proof assumes familiarity with the kind of
diagrammatics that occurs with Feynman diagrams in quantum field theory and/or with exponentials of connected
diagrams as they occur in, say,~\cite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}. Pictorially, we view
morphisms in $\mor_\gdo(A\to B)$ as in the picture on the right: we put the Greek input variables corresponding
to $A$ on the left, the Latin output variable corresponding to $B$ on the right, we indicate the scalar
coefficient $\omega$ at the top, and we use the bulk of the picture to indicate $Q$ and its Greco-Roman
decomposition, with an obvious ``Greek facing'' placement of $F$, ``Latin facing'' placement of $G$, and
``across the divide'' placement of $E$. Note that $Q$ is exponentiated and that exponentials are ``reservoirs
of multiple copies'' $\bbe^x=1+x+xx/2+xxx/6+\ldots$. We emphasize this by drawing $E$, $F$, and $G$ as
having multiple shadows.

\Needspace{23mm} % 22mm is not enough.
\parpic[r]{\input{figs/GDOComposition.pdf_t}}
With this language, a composition as in~\eqref{eq:LMcomposition} of
a pair of morphisms as on the right is interpreted as ``sum over all
possible contractions of Latin-side ends in $\bbe^{Q_1}$ with Greek-side
ends in $\bbe^{Q_2}$ (provided their labels, which are elements of $B$,
agree)''. Thus to figure out, say, the $E$ part of the output, we need to
figure out all the ways to travel from $A$ to $C$ across the composition
of $\bbe^{Q_1}$ and $\bbe^{Q_2}$ by carrying out such contractions.

\Needspace{31mm} % 31mm is not enough.
\parpic[r]{\input{figs/E.pdf_t}}
The most obvious way to travel across is the direct route: contract $E_1$
with $E_2$. This contributes a term proportional to $E_1E_2$ to the
output $E$. Another possibility is to travel along $E_1$, then $F_2$, then $G_1$, then $E_2$, producing a term
proportional to $E_1F_2G_1E_2$. Another possibility is to take the $F_2G_1$ detour twice, producing a term 
proportional to $E_1(F_2G_1)^2E_2$. In general, and with proper accounting of the combinatorial factors (it
turns out that all proportionality factors are $1$), we get 
\[ E = \sum_{r=0}^\infty E_1(F_2G_1)^rE_2 = E_1(I-F_2G_1)^{-1}E_2, \]
where the last equality was obtained by summing a geometric series, and where convergence is assured by the
``smallness'' condition on $G$ in Definition~\ref{def:GDO}.

Similar reasonings justify the formulas for $F$ and for $G$.

\Needspace{25mm}
\parpic[r]{\input{figs/FGCycles.pdf_t}}
Yet there is one further contribution to $\bbe^{Q_1}\act\bbe^{Q_2}$,
coming from closed $F_2G_1$ cycles as on the right (but of an arbitrary
length $r$). This contribution is a scalar that modifies $\omega_1\omega_2$,
and it is
$ \exp\left(\sum_{r=1}^\infty\frac{1}{2r}\tr(F_2G_1)^r\right)
  = \exp(-\frac12\tr\log(1-F_2G_1))
  = \det(1-F_2G_1)^{-1/2},
$
justifying the last part of Equation~\eqref{eq:gdocompositions}. Note that
in the last formula we used the familiar quantum field theory dictum to
``divide each diagram by the order of its symmetry group'' to get the
$1/2r$ factor, and that throughout the proof we regarded only connected
diagrams and exponentiated the result, as per the dictum ``the logarithm
of the partition function is generated by connected diagrams''.
\endpar{pictorial}

{\red MORE. Write in the Alexander polynomial.}
