\chapter{Quantum Gravity (I)}

Last time we did a lot of work to show how to calculate a number of
linear operators based on spin networks. In particular, a closed spin
network is a linear operator from $\C$ to $\C$ (i.e. a number). We
concentrated on
$$
\xy
 (0,0)*\dir{*}="Y";
 (-7,4)*\dir{*}="a";
 **\dir{-} ?(.5)+(-1,-1)*{\scriptstyle\mathbf{i}};
 (0,-9)*\dir{*}="b";
 "Y"**\dir{-} ?(.5)+(2,0)*{\scriptstyle\mathbf{q}};
 (7,4)*\dir{*}="c";
 "Y"**\dir{-} ?(.5)+(-1,2)*{\scriptstyle\mathbf{m}};
 "a";"b"**\crv{(-7,-4)} ?(.5)+(-2,-1)*{\scriptstyle\mathbf{p}};
 "c";"b"**\crv{(7,-4)} ?(.5)+(2,-1)*{\scriptstyle\mathbf{k}};
 "c";"a"**\crv{(0,9)} ?(.5)+(0,2)*{\scriptstyle\mathbf{j}};
\endxy
$$ 
which associates a complex number to each tetrahedron. It turns out
that (up to fudge factors),
$$
\xy
 (0,9)*\dir{*}="b";(7,4)*\dir{*}="c"**\dir{-};
 (7,-4)*\dir{*}="d"**\dir2{-};
 (0,-9)*\dir{*}="e"**\dir{-};
 (-7,0)*\dir{*}="a"**\dir{-};
 "b"**\dir{-};
 "d"**\dir{-};
 "a"**\dir2{-};
 "c"**\dir2{.};
 "e"**\dir{.};
\endxy
\quad=\sum_k\quad
\xy
 (0,9)*\dir{*}="b";(7,4)*\dir{*}="c"**\dir2{-};
 (7,-4)*\dir{*}="d"**\dir{-};
 (0,-9)*\dir{*}="e"**\dir2{-};
 (-7,0)*\dir{*}="a"**\dir2{-};
 "b"**\dir2{-};
 "d"**\dir2{-};
 "a"**\dir{-};
 "c"**\dir{.};
 "e"**\dir2{.};
 "b"**\dir3{.} ?(.5)+(2,0)*{\scriptstyle\mathbf{k}};
\endxy
$$
Now the following theorem suggests an intimate relationship between
topology and spin networks:
\begin{theorem}[Pachner]
Any two triangulations of a compact $3$-manifold can be obtained from
one another by a sequence of ``Pachner moves'':
$$
\displaylines{
\xy
 (5,1)*\dir{*}="1";
 (1,5)*\dir{*}="2"**\dir{-};
 (-5,1)*\dir{*}="3"**\dir{-};
 (1,-5)*\dir{*}="4"**\dir{-};
 "1"**\dir{-};
 "3"**\dir{.};
 "2";"4"**\dir{-};
\endxy
\quad\Leftrightarrow\quad
\xy
 (0,0)*\dir{o}="0";
 (5,1)*\dir{*}="1"**\dir{.};
 (1,5)*\dir{*}="2"**\dir{-};
 (-5,1)*\dir{*}="3"**\dir{-};
 (1,-5)*\dir{*}="4"**\dir{-};
 "0"**\dir{.};
 "2"**\dir{.};
 "4"**\dir{-};
 "1"**\dir{-};
 "3"**\dir{.};
 "0"**\dir{.};
\endxy
\qquad\hbox{($1-4$ move)}\cr
\xy
 (0,9)*\dir{*}="b";(7,4)*\dir{*}="c"**\dir{-};
 (7,-4)*\dir{*}="d"**\dir2{-};
 (0,-9)*\dir{*}="e"**\dir{-};
 (-7,0)*\dir{*}="a"**\dir{-};
 "b"**\dir{-};
 "d"**\dir{-};
 "a"**\dir2{-};
 "c"**\dir2{.};
 "e"**\dir{.};
\endxy
\quad\Leftrightarrow\quad
\xy
 (0,9)*\dir{*}="b";(7,4)*\dir{*}="c"**\dir2{-};
 (7,-4)*\dir{*}="d"**\dir{-};
 (0,-9)*\dir{*}="e"**\dir2{-};
 (-7,0)*\dir{*}="a"**\dir2{-};
 "b"**\dir2{-};
 "d"**\dir2{-};
 "a"**\dir{-};
 "c"**\dir{.};
 "e"**\dir2{.};
 "b"**\dir3{.};
\endxy
\qquad\hbox{($2-3$ move)}\cr}
$$
\end{theorem}

To the wise, this suggests that we should use spin networks to
generate {\it three-dimensional topological quantum field
theories\/}. So, our objective for the next series of lectures is:
let's get wise!  

\section{Topological Quantum Field Theories (I)}

In the struggle to reconcile quantum mechanics and general relativity,
we are helped by an analogy between two otherwise very different
subjects.

General relativity is about space and space-time, and space is very
flexible object. Space will be any $(n-1)$-dimensional manifold, and
space-time will be an $n$-dimensional manifold limited by two
``choices of space''. So spacetime is a manifold with boundary,
technically a cobordism between two disconnected parts of its
boundary, which are labelled ``input'' and ``output''.
$$
\xymatrix{
S\ar[d]^M\\
S'\\}
\qquad
\xy
 (-7,7)*\xycircle(3,2){};
 (7,7)*\xycircle(3,2){};
 (0,-7)*\xycircle(3,2){};
 (-10,7)*{};(-3,-7)**\crv{(-11,3)&(-2,-3)};
 (10,7)*{};(3,-7)**\crv{(11,3)&(2,-3)};
 (-4,7)*{};(4,7)**\crv{(0,-1)};
\endxy
$$

In Quantum mechanics, on the other hand, we describe the possible
states of a system using a vector space (Hilbert space) and we
describe the passage of time by linear operators. In other words, the
space of states is a Hilbert space, and processes are described by
linear operators.
$$
\xymatrix{
\psi\rlap{$\in{\cal H}$}\ar[d]^T\\
T(\psi)\rlap{$\in {\cal H}'$}\\}
$$ 

In sum, in general relativity and quantum mechanics we have notions of
what things can be, and how things can change to become other
things. A topological quantum field theory will be a way to go from
general relativity to quantum mechanics, i.e. given a manifold called
``space'', it will spit out a Hilbert space, and given a spacetime it
will spit out a linear operator. Therefore, we are looking for some
kind of map between the world of manifolds and cobordisms and the
world of Hilbert spaces and linear operators. This was the approach
taken by Atiyah in his axiomatisation of topological quantum field
theories.

\subsection{Category Theory (I)}
Those in the know will have realised that, in the above exposition, by
``world'' we mean ``category'', which we now define.

\begin{definition}
A {\bf Category\/} $\cal C$ consists of
\begin{itemize}
\item a collection\footnote{This collections is not in general a set, but a
      proper class. Consider the category $\Set$, in which the
      collection of all objects cannot be a set because of the famous
      Russell paradox.} of \textbf{objects};
\item given two objects $x,y$, a set $\Hom(y,x)$ of
      \textbf{morphisms}. Generalizing from the categories where
      $\Hom(y,x)$ is a set of functions, we denote $f\in\Hom(y,x)$ by
      $f\colon x\rightarrow y$. Morphisms satisfy the following
      properties:
      \begin{itemize}
      \item given morphisms $f\colon x\rightarrow y$ and $g\colon
            y\rightarrow z$, we can compose them and
            obtain\footnote{At this point, category theorists split
            into warring factions, depending on the order in which
            they write the composition of morphisms.} $g\circ f\colon
            x\rightarrow z$. When there is no possibility of confusion
            $g\circ f$ is abbreviated $gf$.
            $$
            \xymatrix{&y\ar[dl]_g\\
                      z&&x\ar[ul]_f\ar[ll]^{gf}\\}
            $$ 
      \item for any $x$, there is an \textbf{identity} morphism
            $1_x\colon x\rightarrow x$ such that, for any $f\colon
            x\rightarrow y$, we have $f1_x=f=1_yf$. For example,
            $$
            \xymatrix{S\ar[d]^{[0,1]\times S}\\
                      S\\} 
            \qquad
            \xy
             (0,7)*\xycircle(3,2){};
             (0,-7)*\xycircle(3,2){};
             (-3,7)*{};(-3,-7)**\dir{-};
             (3,7)*{};(3,-7)**\dir{-};
            \endxy
            $$
      \end{itemize}
\end{itemize}
\end{definition}

This definition captures the most primitive notions of ``things'' and
the ``processes'' that things can undergo, in other words, the ways
that things can ``be'' and the ways that things can ``happen''. 

Examples of categories are:
\begin{itemize}
\item $\Set$, where objects are sets and morphisms are functions.
\item $n\Cob$,where objects are $(n-1)$-dimensional compact manifolds, and
    morphisms are $n$-dimensional cobordisms.
        $$
        \xymatrix{S\amalg S\ar[d]^{M^*\amalg 1_S}\\
                   S\amalg S\amalg S\ar[d]^{1_S\amalg M}\\
                   S\amalg S\\}
        \qquad
        \xy
         (-7,0)*\xycircle(3,2){};
         (7,0)*\xycircle(3,2){};
         (-14,-14)*\xycircle(3,2){};
         (0,-14)*\xycircle(3,2){};
         (14,-14)*\xycircle(3,2){};
         (-7,-28)*\xycircle(3,2){};
         (7,-28)*\xycircle(3,2){};
         (-4,0)*{};(3,-14)**\crv{(-5,-4)&(4,-10)};
         (4,0)*{};(11,-14)**\crv{(3,-4)&(12,-10)};
         (10,0)*{};(17,-14)**\crv{(10,-4)&(19,-10)};
         (4,-28)*{};(-3,-14)**\crv{(5,-24)&(-4,-18)};
         (-4,-28)*{};(-11,-14)**\crv{(-3,-24)&(-12,-18)};
         (-10,-28)*{};(-17,-14)**\crv{(-10,-24)&(-19,-18)};
         (-10,0)*{};(-17,-14)**\crv{(-9,-4)&(-18,-10)};
         (10,-28)*{};(17,-14)**\crv{(9,-24)&(18,-18)};
         (-11,-14)*{};(-3,-14)**\crv{(-7,-6)};
         (11,-14)*{};(3,-14)**\crv{(7,-22)};
        \endxy
        $$
\item $\Vect$, where objects are (finite-dimensional, complex) vector spaces,
    and morphisms are linear operators.
\item $\Hilb$, where objects are (finite-dimensional, complex) Hilbert spaces,
    and morphisms are linear operators.
\end{itemize}

Quantum mechanics uses $\Hilb$ rather than $\Vect$ because (among
other things)
\begin{itemize}
\item given state vectors (i.e. unit vectors) in a Hilbert space, say
    $\phi$ and $\psi$, then $\bracket\phi\psi$ is the \textbf{amplitude} and $\left|\bracket\phi\psi\right|^2$ is the \textbf{probability} that a system prepared in state $\psi$ will be
    found in state $\phi$. There is no such structure in $\Vect$.
\item given an operator $T\colon {\cal H}\rightarrow {\cal H}'$, the
    condition $\bracket{T^*\phi}\psi=\bracket\phi{T\psi}$ defines an
    \textbf{adjoint} operator $T^*\colon {\cal H}'\rightarrow {\cal
    H}$. In $\Vect$, the best we can get is the dual $T^*\colon{\cal
    H'}^*\rightarrow{\cal H}^*$.
\item \textbf{observables} in quantum mechanics are represented by
    self-adjoint operators $A\colon {\cal H}\rightarrow {\cal H}$,
    where ${\cal H}$ is the space of states of the system and
    $A=A^*$. Such an operator\footnote{More generally, any \textbf{normal}
    operator, i.e. any operator such that $NN^*=N^*N$, has an
    orthonormal basis of eigenvectors with complex eigenvalues.} has
    associated an orthonormal basis $\{\psi_i\}$ of ${\cal H}$ such
    that $A\psi_i=a_i\psi_i$ with $a_i\in\R$. The interpretation is
    that $\psi_i$ is a state in which $A$ will always be measured to
    be $a_i$.
\end{itemize}
The fact that in $\Hilb$ we have a canonical antiisomorphism ${\cal
H}\rightarrow {\cal H}^*$ induced by $\bracket\cdot\cdot$ is very
different from $\Vect$ or $\Set$, but a lot like $n\Cob$, where the
``dual'' of a space is the same space with the opposite orientation,
and the ``adjoint'' of an $n$-cobordism is its time-reversal. Time
reversal is of utmost importance in physics.

\chapter{Quantum Mechanics from a Category-Theoretic Viewpoint (I)}
We have come short of defining topological quantum field theories
because we still haven't explained just what is a map between
categories. A topological quantum field theory is, among other things,
a {\it functor\/} $Z\colon n\Cob\leftarrow \Hilb$. What this means is:
\begin{definition}
If $\cal C$ and $\cal D$ are categories, a {\bf functor\/} $F\colon
{\cal C}\rightarrow {\cal D}$ consists of:
\begin{itemize}
\item an object $F(x)\in{\cal D}$ for each $x\in{\cal C}$;
\item a morphism $F(f)\colon F(x)\rightarrow F(y)$ for each $f\colon
    x\rightarrow y$ such that 
    \begin{itemize}
    \item $F(1_x)=1_{F(x)}$ for all $x\in{\cal C}$ 
    \item $F(gf)=F(g)F(f)$ for all $f\colon x\rightarrow y$ and $g\colon
        y\rightarrow z$.  
    \end{itemize}
\end{itemize}
\end{definition}

This looks a lot like a group homomorphism, and that should be no
surprise because a group is a special kind of category. In fact, for
any object $x$ in a category $\cal C$, $\Hom(x,x)$ is a monoid and
$F\colon\Hom(x,x)\leftarrow\Hom\bigl(F(x),F(x)\bigr)$ is a monoid
homomorphism.

\section{Schr\"odinger's Equation}
In ordinary quantum mechanics we don't talk about how the topology of
space changes, and also time is a parameter (there is some kind of
fixed clock which ticks to ``universal time''). So we assume that
there is a single Hilbert space, not a whole collection of them. Also,
for each time $t\in\R$ we have an operator $U(t)\colon {\cal
H}\rightarrow {\cal H}$ which describes time evolution in such a way
that, if $\psi$ is the state of the system at time $t=0$,
$\psi(t)=U(t)\psi$ is the state of the system at time
$t$. Time-translation symmetry is expressed by
$U(t)U(s)=U(s+t)=U(s)U(t)$, and if $U(t)$ is defined for $t<0$ we have
a group rather than a semigroup homomorphism. Moreover, we require
that\footnote{By the polarisation identities, knowledge of
$\bracket\phi\phi$ for all $\phi$ determines $\bracket\phi\psi$ for
all $\phi,\psi$} $\bracket{\phi(t)}{\phi(t)}=\bracket\phi\phi=1$. From
this we can see that $U(t)$ must be unitary as follows.

First, observe that $U(0)^2=U(0+0)=U(0)$ implies that $U(0)$ is a
projector. If ${\cal H}'$ is the subspace onto which $U(0)$ projects, the
image of $U(t)=U(0)U(t)U(0)$ is in ${\cal H}'$ for all $t$, so we can assume
without loss of generality that ${\cal H}={\cal H}'$ and $U(0)=1_{\cal H}$.

Then,
$\bracket\psi\psi=\bracket{U(t)\psi}{U(t)\psi}=\bracket{U^*(t)U(t)\psi}\psi$
implies $U^*(t)U(t)=1_{\cal H}$ for all $t$. Since
$U(t)U(-t)=U(0)=1_{\cal H}$, we conclude that
$U^*(t)=U^*(t)U(t)U(-t)=U(-t)$.

If we add the continuity assumption that $\lim_{t\rightarrow
s}\left\|U(s)\psi-U(t)\psi\right\|=0$ for all $\psi$, we have that
$U(t)$ is a {\bf strongly-continuous, one-parameter unitary
group\/}. Quite a mouthful. We then have
\begin{theorem}[Stone]
If ${\cal H}$ is a Hilbert space and $U(t)$ is a strongly-continuous,
one-parameter unitary group, then $U(t)=\exp\bigl(-itH\bigr)$, where
$H$ is self-adjoint.
\end{theorem}

The operator $H$ in Stone's theorem is called the Hamiltonian operator
and it corresponds to the energy observable. Stone's theorem and
$\psi(t)=U(t)\psi$ imply the abstract Schr\"odinger equation,
$i{\partial\over\partial_t}\psi(t)=H\psi(t)$. 

Normally, the way physicists approach a quantum-mechanical problem is,
given the Hamiltonian, solve for the evolution of the system. In
contrast, in quantum field theory and quantum gravity the hard part is
to figure out the Hilbert space and Hamiltonian of the theory. 




