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\begin{document}
\begin{center}
{\large Classical Mechanics, Lecture 19 \\}
{\small March 13, 2008 \\}
{\small lecture by John Baez \\}
{\small notes by Alex Hoffnung}
\end{center}
\section{The Non-Cartesianness of Classical Mechanics}
Last time we introduced a category $\Poiss$, with
\begin{itemize}
\item Poisson manifolds $X$ as objects
\item Poisson maps $\phi\maps X\to Y$ as morphisms:
smooth maps such that $\forall f,g\in C^\infty(Y)$
\[\phi^*\{f,g\} = \{\phi^*f,\phi^*g\}\]
where $\phi^*f = f\phi$.
\end{itemize}
There can be various ways to ``glom together" objects in a category -
disjoint union, tensor products, Cartesian products, etc.$\ldots$
For example: $\Set$ is the category with:
\begin{itemize}
\item sets $X$ as objects
\item functions $\phi\maps X\to Y$ as morphisms.
\end{itemize}
This has `Cartesian product' $X\times Y$ as a way of glomming together
sets. Here are the key properties of the Cartesian product, written
so as to make sense in any category: we say the {\bf product} $X\times
Y$ is an object with morphisms
\[p_1\maps X\times Y\to X\]
\[p_2\maps X\times Y\to Y\]
such that: given any morphisms
\[f\maps Z\to X \textrm{ and } g\maps Z\to Y,\]
there exists a unique morphism $\langle f,g\rangle\maps Z\to X\times Y$ such that
\[
\xymatrix{
&Z\ar[dl]_{f}\ar[dr]^{g}\ar@{-->}[d]^{\langle f,g\rangle}&\\
X & X\times Y\ar[l]^{p_1}\ar[r]_{p_2}& Y
}
\]
commutes:
\[f = p_1\langle f,g\rangle\]
\[g = p_2\langle f,g\rangle.\]
In a `cartesian' category, every pair of objects has a product.
Quantum theory uses not $\Poiss$ but the a category $\Hilb$ where:
\begin{itemize}
\item objects are Hilbert spaces
\item morphisms are bounded linear operators,
\end{itemize}
Again, we use objects in this category
to describe physical systems and morphisms to describe physical
processes.
One reason quantum theory seems `weird' to some people is that in this
theory, we 'glom together' two physical systems using the tensor
product of Hilbert spaces, which is {\it not} the `product' in
the sense just described!
I.e., given Hilbert spaces $X$ and $Y$, we have this new Hilbert space $X\otimes Y$, but there are generally {\it not} any interesting morphisms
\[p_1\maps X\otimes Y\to X\]
\[p_2\maps X\otimes Y\to Y\]
For example, we use the vector $\psi \otimes \phi \in X \otimes Y$ to
describe to describe a state of the system $X \otimes Y$ where the first
subsystem is in the state $\psi$ and the second subsystem is in the
state $\phi$. But, there are no linear operators as above that pick out
these states:
\[p_1(\psi\otimes\phi) = \psi\]
\[p_2(\psi\otimes\phi) = \phi\]
for all $\psi\in X$, $\phi\in X$. Even more importantly, we can't
find $p_1, p_2$ making $X \otimes Y$ into the product of $X$ and $Y$:
that is, operators such that for all $f\maps Z\to X$, $g\maps Z\to Y$,
$\exists\langle f,g\rangle\maps Z\to X\otimes Y$ such that
\[
\xymatrix{
&Z\ar[dl]_{f}\ar[dr]^{g}\ar@{-->}[d]^{\langle f,g\rangle}&\\
X & X\otimes Y\ar[l]^{p_1}\ar[r]_{p_2}& Y
}
\]
commutes.
This has important consequences. For example,
in a category with products, we can always ``duplicate" a system:
i.e. we have a morphism
\[\Delta_X\maps X\to X\times X.\]
We get this as follows:
\[
\xymatrix{
&X\ar[dl]_{1_x}\ar[dr]^{1_x}\ar@{-->}[d]^{\Delta_x}&\\
X & X\times X\ar[l]^{p_1}\ar[r]_{p_2}& X
}
\]
In the case of $\Set$, we have
\[\Delta_X\maps X\to X\times X\]
\[x\mapsto (x,x).\]
But in $\Hilb$ we do not have any interesting linear operators
\[\Delta_X\maps X\to X\otimes X.\]
For example,
\[\psi\mapsto\psi\otimes\psi\]
is not linear. Wooters and Zurek proved a theorem making this issue
precise: ``you can not clone a quantum".
In fact, the right way of glomming together \underline{classical}
systems is also not the Cartesian product, but some kind of `tensor
product' of Poisson manifolds!
For example, if $X = T^*\R^n$ and $Y = T^*\R^m$ then
\[X\otimes Y \cong T^*\R^{n+m}\]
where all three have their usual Poisson brackets. As \underline{manifolds}
\[T^*\R^{n+m}\cong T^*\T^n\times T^*\R^m\]
\[(\q,\q^\prime,\p,\p^\prime)\mapsto ((\q,\p),(\q^\prime,\p^\prime))\]
and this is a product in the category of manifolds and smooth maps.
{\it But, it is not a product in the category of Poisson manifolds!}
I believe the non-Cartesian nature of this product means there's no
classical machine that can `duplicate' states of a classical system:
\[\textrm{picture of classical machine where you feed a system into \\
the hamper and two identical copies come out the bottom}\]
But, strangely, this issue has been studied less than in the quantum
case!
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