\chapter{Feynman diagrams (I)}
The platonic solids provide certain important finite subgroups of
$\SO(3)$. The vertices of a platonic solid inscribed in the unit sphere
are left invariant by subgroups of $\SO(3)$ isomorphic to the following
groups:
$$
\begin{array}{cc}
\mathrm{tetrahedron}&A_4\\
\hbox{cube/octahedron}&S_4\\
\hbox{dodecahedron/icosahedron}&A_5\\
\end{array}
$$
Up to conjugation by an element of $\SO(3)$, these are almost all the
finite subgroups of $\SO(3)$. We also have the finite cyclic groups
$\Z_n$ and the dihedral groups $D_n$, of which $\Z_n$ is a subgroup of
index $2$ and which is the group of symmetries of an
$n$-gon\footnote{Note that we are using the convention that the
subscript $n$ indicates the order of the subgroup of rotations in
$D_n$, rather than the more usual convention $D_{2n}$ where the
subscript denotes the order of the group.}. The Dihedral groups are
subgroups of $\SO(3)$ because plane reflections can be implemented by a
rotation in space.
\begin{theorem}
Every finite subgroup of $\SO(3)$ is conjugate to one of $A_4$, $S_4$,
$A_5$, $\Z_n$ or $D_n$.
\end{theorem}
\begin{theorem}
Every finite subgroup of $\SO(2)$ is $\Z_n$ for some $n$.
\end{theorem}
Note that, since $\SO(2)$ is abelian, all conjugacy classes of
subgroups consist of exactly one subgroup, so there is a notion of
``the'' $\Z_n$ in $\SO(2)$.
We could classify all subgroups of $\SO(4)$ easily. Apart from the
two-dimensional groups $D_n$ and $\Z_n$, we have the groups of
rotations of the platonic solids, and the groups obtained by adding
reflections to them (since a reflection in $3$~dimensions can be
implemented as a rotation in $4$~dimensions). Then there are the
groups of rotations of the $4$-dimensional regular polytopes.
The $4$-dimensional regular polytopes are the $4$-simplex, the
$4$-cube and the $4$-cross (analogous to the octahedron and dual to
the $4$-cube); and three ``exotic'' polytopes which can be obtained by
realizing that the group $\SU(2)$ (the double cover of $\SO(3)$) is
isometric to the $3$-sphere (on which $\SO(4)$ acts naturally) and so
the groups of rotations of the platonic solids have a good chance of
producing regular $4$-dimensional polytopes. In fact, the tetrahedron
and the dodecahedron give rise to such polytopes, and the one
associated to the dodecahedron/icosahedron has a dual. Then we have
the following
\begin{theorem}
Every finite group is a subgroup of $\SO(n)$ for sufficiently large
$n$.
\end{theorem}
To prove this, consider the real vector space of formal linear
combinations of elements of $G$, the group ring $\R[G]$. $G$ has a
natural linear action on this vector space, and the
orientation-preserving subgroup of $G$ is a subgroup of
$\SO\bigl(|G|\bigr)$. Then, reflections can be implemented in one
dimension higher, so $G$ is a subgroup of $\SO\bigl(|G|+1\bigr)$.
An interesting consequence of this is that it is hopeless to try to
obtain a general classification of the finite subgroups of $\SO(n)$ for
all $n$, since a classification of all finite subgroups is all but
impossible.
But, what does all this have to do with physics? There is in fact a
long, illustious tradition of physical theories which state that the
world is built out of the finite subgroups of $\SO(3)$.
The first such theory is due to the Pythagoreans and is known to us
from Plato's Tim{\ae}us. They thought that the platonic solids
classified the elements!
$$
\begin{tabular}{cc}
tetrahedron&fire\\
hexahedron&earth\\
octahedron&air\\
dodecahedron&quintessence\\
icosahedron&water\\
\end{tabular}
$$
The next attempt at using the platonic solids to explain the order of
the world was Kepler's theory that the known planets were carried by
concentric spheres around the Earth, nested in between platonic
solids. In this way, the five platonic solids determined six
radii. Despite the fact that there are $5!=120$ possible arrangements,
the astronomical observations at the time were accurate enough to make
it impossible to fit the model to them, and Kepler was forced to
discard it. He then moved on to the next simplest hypothesis, that the
planets moved on ellipses, and this led to Newton's law of
gravitation. At this point in history there was a shift from trying to
explain the state of the universe (as Kepler was trying to do) to
explaining its dynamical laws (as Newton did) and viewing the state as
a historical accident.
Finally, and more to our modern taste, there's quantum mechanics where
the classification of atoms involves homomorphisms not into $\SO(3)$
but out of it (or, more precisely, out of its double cover $\SU(2)$):
$$
\SU(2)\to\GL(V).
$$
That is, the quantum mechanics of atoms is all about the linear
representations of $\SU(2)$.
\section{Group representations}
\begin{definition}
A homomorphism
$$
\rho\colon G\to\GL(V)
$$
is called a representation of $G$ on $V$
\end{definition}
In Quantum Mechanics, states are described by unit vectors in a
Hilbert space and symmetries are drscribed by unitary representations,
that is, homomorphisms
$$
\rho\colon G\to \U(H),
$$
where
$$
\U(H)=\{f\colon H\to H\mid\hbox{$f$ is linear and unitary}\}.
$$
Usually, expositions of Quantum Field Theory begin with Classical
Field Theory, and then proceed to quantization prescriptions, at which
point it gets really messy. But in the end, if all works well,
thetheory turns out to be very simple and beautiful. What people never
tell you is that it is possible to get there really quickly,
circumventing all the messy parts, without having to build a
complicated scaffolding. Of course, when trying to make contact with
experiment it is useful to have the scaffolding linking the quantum
theory to a Classical Field Theory.
We are going to develop the theory of Feynman diagrams and we will see
that we can get surprisingly far just by studying group
representations.
\begin{definition}
Given two representations $(\rho,V)$ and $(\rho',V')$ of $G$, an
intertwining operator or intertwiner is a map $f\colon V\to V'$ such
that
$$
\xymatrix{V\ar[d]^{\rho(g)}\ar[r]^f&V'\ar[d]^{\rho'(g)}\\
V\ar[r]^f&V'\\}
$$
commutes for all $g\in G$.
\end{definition}
Given this definition, it is not hard to see that we have a category
where the objects are representations and the morphisms are
intertwiners.
In quantum mechanics,
\begin{itemize}
\item Hilbert spaces are used to describe states,
\item unitary representations describe how the symmetries of the
physical system affect the states, and
\item intertwiners describe processes (ways that states and systems
can change) which are covariant (compatible with the
symmetries).
\end{itemize}
We can draw intertwiners in this childishly simple way:
$$
\xy
(0,0)*++{f}*\cir{}="f";
(0,10)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"f";(0,-10)**\dir{-} ?(.75)*\dir{>}+(3,0)*{\scriptstyle V'};
\endxy
\qquad
f\colon V\to V'.
$$
Given intertwiners $f\colon(\rho,V)\to(\rho',V')$ and
$g\colon(\rho',V')\to(\rho'',V'')$, they can be composed as linear
maps and a new intertwiner $gf\colon(\rho,V)\to(\rho'',V'')$ results:
$$
\xy
(0,8)*++{f}*\cir{}="f";
(0,20)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"f";(0,-8)*++{g}*\cir{}="g";
**\dir{-} ?(.4)*\dir{<}+(3,0)*{\scriptstyle V'};
"f";"g";(0,-20)**\dir{-} ?(.75)*\dir{>}+(3,0)*{\scriptstyle V''};
\endxy
=
\xy
(0,20)*{}="v";
(0,0)*++{gf}*\cir{}="gf";
**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"gf";(0,-20)*{}="w";
**\dir{-} ?(.4)*\dir{<}+(3,0)*{\scriptstyle V''};
\endxy
\qquad
\begin{array}{rccc}
f\colon & V & \to & V'\\
g\colon & V' & \to & V''\\
gf\colon & V & \to & V''\\
\end{array}
$$
The key structure in a category, composition, is represented by the
operation of stacking the diagrams for the intertwiners on top of each
other.
We can also draw two intertwiners side by side:
$$
\xy
(0,0)*++{f}*\cir{}="f";
(0,12)**\dir{-} ?(.5)*\dir{<}+(-3,0)*{\scriptstyle V};
"f";(0,-12)**\dir{-} ?(.6)*\dir{>}+(-3,0)*{\scriptstyle V'};
\endxy
\xy
(0,0)*++{g}*\cir{}="f";
(0,12)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle W};
"f";(0,-12)**\dir{-} ?(.6)*\dir{>}+(3,0)*{\scriptstyle W'};
\endxy
=\quad
\xy
(0,0)*+{f\otimes g}*\frm<4pt>{-}="f";
(0,12)**\dir{-} ?(.5)*\dir{<}+(5,0)*{\scriptstyle V\otimes W};
"f";(0,-12)**\dir{-} ?(.6)*\dir{>}+(5,0)*{\scriptstyle V'\otimes W'};
\endxy
$$
This corresponds to the operation of tensoring. Given two
representations $(\rho,V)$ and $(\sigma,W)$, we get a new
representation $(\rho\otimes\sigma,V\otimes W)$ defined by
$$
(\rho\otimes\sigma)(g)=\rho(g)\otimes \sigma(g)
\qquad\hbox{or, in other words,}\quad
(\rho\otimes\sigma)(g)(v\otimes w)=\bigl(\rho(g)(v)\bigr)\otimes\bigl(\sigma(g)(w)\bigr).
$$
Tensor representations are used in physics to describe the states of a
composite system in terms of the states of its parts.
The motivation for the diagrammatic notation is that we can ``let the
pictures do the thinking''. Take, for example, the diagram
$$
\xy
(0,8)*++{f}*\cir{}="f";
(0,20)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"f";(0,-8)*+{f'}*\cir{}="g";
**\dir{-} ?(.4)*\dir{<}+(3,0)*{\scriptstyle V'};
"f";"g";(0,-20)**\dir{-} ?(.75)*\dir{>}+(3,0)*{\scriptstyle V''};
\endxy
\xy
(0,8)*++{g}*\cir{}="f";
(0,20)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"f";(0,-8)*++{g'}*\cir{}="g";
**\dir{-} ?(.4)*\dir{<}+(3,0)*{\scriptstyle W'};
"f";"g";(0,-20)**\dir{-} ?(.75)*\dir{>}+(3,0)*{\scriptstyle W''};
\endxy
\qquad
(f'\otimes g')(f\otimes g)=(f'f)\otimes(g'g).
$$
Not only is the diagram easier to understand and remember than the
equation, it is also rather involved to prove that the equation is
true algebraically compared to the simplicity of the corresponding
diagrammatic manipulations. Of course, a number of identities like
this have to be established before we can trust that the diagrams will
give the right answer in any given manipulation.
We can also thing about intertwiners of the form
$$
\begin{xy}
(0,0)*++{f}*\cir{}="f";
(-6,12)**\dir{-} ?(.5)*\dir{<}+(-2,-1)*{\scriptstyle V_1};
"f";(-2,12)**\dir{-} ?(.5)*\dir{<}+(2,-1);
(1,9)*{\scriptstyle\cdots};
"f";(6,12)**\dir{-} ?(.5)*\dir{<}+(2,-1)*{\scriptstyle V_m};
"f";(-6,-12)**\dir{-} ?(.67)*\dir{>}+(-2,1)*{\scriptstyle W_1};
"f";(-2,-12)**\dir{-} ?(.67)*\dir{>}+(2,1);
(1,-8)*{\scriptstyle\cdots};
"f";(6,-12)**\dir{-} ?(.67)*\dir{>}+(3,0)*{\scriptstyle W_n};
\end{xy}
\qquad
f\colon(\rho_1,V_1)\otimes\cdots\otimes(\rho_m,V_m)\to(\sigma_1,W_1)\otimes\cdots\otimes(\sigma_n,W_n),
$$
We can hook these up to get more complex intertwiners. Also, we have
the very simplest of all intertwiners:
$$
\begin{xy}
(0,6)*{};(0,-6)**\dir{-} ?(.5)*\dir{>}+(3,0)*{\scriptstyle V};
\end{xy}
\qquad
\id_V\colon V\to V.
$$
The Quantum field theory jargon is as follows:
\begin{itemize}
\item Representations
\begin{xy}
(0,3)*{};(0,-3)**\dir{-} ?(.5)*\dir{>};
\end{xy}
are called ``particles''; and
\item Intertwiners
\begin{xy}
(0,0)*{};(-1,2)**\dir{-};
(0,0)*{};(-1,-2)**\dir{-};
(0,0)*{};(3,0)**\dir{-};
\end{xy}
are called ``interactions''.
\end{itemize}
The remaining ingredients that specify a ``theory'' are a symmetry
group and the list of representations that represent physical
particles.
\chapter{Feynman diagrams (II)}
\section{The category of representations}
We have seen that for each group $G$ there is a category of linear
representations. In this category:
\begin{itemize}
\item Objects $(\rho,V)$ are representations of $G$, i.e., group
homomorphisms
$$
\rho\colon G\to\GL(V).
$$
For all $g\in G$, $\rho(g)\colon V\to V$ is a linear map, and
$\rho(gh)=\rho(g)\rho(h)$ for all $g,h\in G$.
\item Morphisms $f\colon(\rho,V)\to(\rho',V')$ are intertwiners, i.e.,
linear maps
$$
f\colon V\to V'
$$
such that
$$
\xymatrix{V\ar[d]^{\rho(g)}\ar[r]^f&V'\ar[d]^{\rho'(g)}\\
V\ar[r]^f&V'\\}
$$
commutes for all $g\in G$. That is,
$\rho'(g)\bigl(f(v)\bigr)=f\bigl(\rho(g)(v)\bigr)$ for all $v\in
V$ and $g\in G$.
\item We saw that a category has a $1$-dimensional aspect, namely
composition
$$
\xy
(0,8)*++{f}*\cir{}="f";
(0,20)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"f";(0,-8)*++{g}*\cir{}="g";
**\dir{-} ?(.4)*\dir{<}+(3,0)*{\scriptstyle V'};
"f";"g";(0,-20)**\dir{-} ?(.75)*\dir{>}+(3,0)*{\scriptstyle V''};
\endxy
=
\xy
(0,20)*{}="v";
(0,0)*++{gf}*\cir{}="gf";
**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle V};
"gf";(0,-20)*{}="w";
**\dir{-} ?(.4)*\dir{<}+(3,0)*{\scriptstyle V''};
\endxy
\qquad
\begin{array}{rccc}
f\colon & V & \to & V'\\
g\colon & V' & \to & V''\\
gf\colon & V & \to & V''\\
\end{array}
$$
The properties of composition (associativity and identity
morphisms) are automatic when composition is represented
graphically and the indentity morphism on $(\rho,V)$ is
represented by
$$
\begin{xy}
(0,6)*{};(0,-6)**\dir{-} ?(.5)*\dir{>}+(3,0)*{\scriptstyle V};
\end{xy}
\qquad
\id_V\colon V\to V.
$$
\item In addition, the category of representations has a
$2$-dimensional aspect to it, given by the tensor product of
representations. In technical terms, the category of
representations is monoidal.
$$
\xy
(0,0)*++{f}*\cir{}="f";
(0,12)**\dir{-} ?(.5)*\dir{<}+(-3,0)*{\scriptstyle V};
"f";(0,-12)**\dir{-} ?(.6)*\dir{>}+(-3,0)*{\scriptstyle V'};
\endxy
\xy
(0,0)*++{g}*\cir{}="f";
(0,12)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle W};
"f";(0,-12)**\dir{-} ?(.6)*\dir{>}+(3,0)*{\scriptstyle W'};
\endxy
=\quad
\xy
(0,0)*+{f\otimes g}*\frm<4pt>{-}="f";
(0,12)**\dir{-} ?(.5)*\dir{<}+(5,0)*{\scriptstyle V\otimes W};
"f";(0,-12)**\dir{-} ?(.6)*\dir{>}+(5,0)*{\scriptstyle V'\otimes W'};
\endxy
$$
\end{itemize}
\subsection{The braiding}
The category of representations is what is called a ``braided monoidal
category'', which gives it a three-dimensional aspect as well. Given
two representations $(\rho,V)$ and $(\rho',V')$, we get a special
intertwiner (called the ``braiding'') $B_{V,W}\colon V\otimes W\to
W\otimes V$:
$$
\begin{xy}
\xoverv~{(-4,8)}{(4,8)}{(-4,-8)}{(4,-8)}<>|>>><{V}|{W}>{V};
\end{xy}
=
\begin{xy}
(0,0)*+{B}*\cir{}="B";
(-4,8)**\dir{-} ?(.5)*\dir{<} +(-2,-1)*{\scriptstyle V};
"B";(4,8)**\dir{-} ?(.5)*\dir{<} +(2,-1)*{\scriptstyle W};
"B";(-4,-8)**\dir{-} ?(.67)*\dir{>} +(-2,1)*{\scriptstyle W};
"B";(4,-8)**\dir{-} ?(.67)*\dir{>} +(2,1)*{\scriptstyle V};
\end{xy}
\qquad
\begin{array}{rccc}
B_{V,W}\colon & V\otimes W & \to & W\otimes V\\
& v\otimes w & \to & w\otimes v\\
\end{array}
$$
The braiding satisfies laws such as
\begin{eqnarray*}
\begin{xy}
\vtwist~{(-8,18)}{(0,6)}{(-8,-18)}{(0,-6)}<>|>>><{U}|{W}>{U};
\vtwist~{(0,18)}{(8,18)}{(0,6)}{(8,-6)}<>|>>><{V}|{W}>{V};
\vtwist~{(0,-6)}{(8,-6)}{(0,-18)}{(8,-18)}<>|>>><{U}|{V}>{U};
\end{xy}
& = &
\begin{xy}
\vtwist~{(0,6)}{(8,18)}{(0,-6)}{(8,-18)}<>|>>><{U}|{W}>{U};
\vtwist~{(-8,18)}{(0,18)}{(-8,-6)}{(0,6)}<>|>>><{U}|{V}>{U};
\vtwist~{(-8,-6)}{(0,-6)}{(-8,-18)}{(0,-18)}<>|>>><{V}|{W}>{V};
\end{xy}
\\ \\
(\id_W\otimes B_{V,U})(B_{U,W}\otimes\id_V)(\id_U\otimes B_{V,W})
& = &
(B_{W,V}\otimes\id_U)(\id_V\otimes B_{U,W})(B_{U,V}\otimes\id_W)\\
\end{eqnarray*}
This equation is called the Yang-Baxter equation, and we write it only
to illustrate the power of the diagrammatic notation.
In any case, this is yet another illustration that laws such as the
Yang-Baxter equation are not only algebraic requirements, but
topological as well.
The braiding is an isomorphism, but note that our diagrammatic
notation makes it clear that the braiding is not required to be its
own inverse. We denote
$$
\begin{xy}
\xunderv~{(-4,8)}{(4,8)}{(-4,-8)}{(4,-8)}<>|>>><{V}|{W}>{V};
\end{xy}
=
\begin{xy}
(0,0)*{B^{-1}}*\cir{}="B";
(-4,8)**\dir{-} ?(.5)*\dir{<} +(-2,-1)*{\scriptstyle W};
"B";(4,8)**\dir{-} ?(.5)*\dir{<} +(2,-1)*{\scriptstyle V};
"B";(-4,-8)**\dir{-} ?(.67)*\dir{>} +(-2,1)*{\scriptstyle V};
"B";(4,-8)**\dir{-} ?(.67)*\dir{>} +(2,1)*{\scriptstyle W};
\end{xy}
$$
Indeed,
$$
\begin{xy}
\vunder~{(-4,8)}{(4,8)}{(-4,0)}{(4,0)}<>|<>><{V}|{W}>{V};
\vover~{(4,-8)}{(-4,-8)}{(4,0)}{(-4,0)}<>|<>><{V}|{W}>{V};
\end{xy}
=
\begin{xy}
\huncross~{(-4,8)}{(4,8)}{(-4,-8)}{(4,-8)}<<|<><<{W}|{V}>{W};
\end{xy}
$$
\subsubsection{Symmetry}
But the game of group representations has a $4$-dimensional aspect,
too. In $4$-dimensional space, it is possible to pass crossing strings
through each other!
$$
\begin{xy}
\xoverv~{(-4,8)}{(4,8)}{(-4,-8)}{(4,-8)}<>|>>><{V}|{W}>{V};
\end{xy}
=
\begin{xy}
\xunderv~{(-4,8)}{(4,8)}{(-4,-8)}{(4,-8)}<>|>>><{W}|{V}>{W};
\end{xy}
$$
A category in which the braiding satisfies this equation is called a
symmetric monoidal category.
\subsection{Duals and conjugates}
This is great stuff, but group representations have other features.
\subsubsection{The dual representation}
Suppose we have a group representation $\rho\colon G\to\GL(V)$. Then
we can get a representation $\rho^*$ on the dual vector space
$V^*$. Recall that
$$
V^*\colon =\{f\colon V\to\C\mid\hbox{$f$ linear}\}.
$$
We need to use the representation $\rho\colon G\to\GL(V)$ to define
the representation $\rho^*\colon G\to \GL(V^*)$. We try
$$
\rho^*(g)(f)(v)=f\bigl(\rho(g)(v)\bigr).
$$
We then have
$$
\rho^*(gh)(f)(v)=f\bigl(\rho(gh)(v)\bigr)=f\Bigl(\rho(g)\bigl(\rho(h)(v)\bigr)\Bigr)=\rho^*(g)(f)\bigl(\rho(h)(v)\bigr)=\rho^*(h)\bigl(\rho^*(g)(f)\bigr)(v),
$$
which means
$$
\rho^*(gh)\neq\rho^*(g)\rho^*(h),
$$
so this definition does not provide a representation.
To solve this problem, we need to look at the ``dual pairing'' or
``counit''
$$
\begin{array}{rccc}
\varepsilon_V\colon & V^*\otimes V & \to & \C\\
& f\otimes v & \mapsto & f(v)\\
\end{array}
$$
which must be an intertwiner from $\rho\otimes\rho^*$ into the trivial
representation. This implies that
$$
f(v)=\bigl(\rho^*(g)\otimes\rho(g)\bigr)(f\otimes
v)=\rho^*(g)(f)\bigl(\rho(g)(v)\bigr),
\qquad\hbox{or}\quad
\rho^*(g)(f)(v)=f\bigl(\rho(g^{-1})(v)\bigr).
$$
This is therefore the only sensible definition of $\rho^*$.
To draw the counit we need to realize that, since $\C\otimes\C=\C$, it
makes sense to draw $\C$ as nothing at all rather than
$\xy(0,3)*{};(0,-3)**\dir{-}?(.5)*\dir{>}+(2,0)*{\scriptstyle\C}\endxy$.
Then, we draw
$$
\xy
(-6,4)*{};(6,4)*{};
**\crv{(0,-12)} ?(.50)*\dir{>}+(0,-2)*{\scriptstyle V}
\endxy
=
\xy
(0,-4)*+{\varepsilon_V}*\cir{}="e" ;
(-6,4)**\dir{-} ?(.75)*\dir{<}+(-2,-1)*{\scriptstyle V^*};
"e";(6,4)**\dir{-} ?(.5)*\dir{<}+(2,-1)*{\scriptstyle V};
\endxy
\qquad
\begin{array}{rccc}
\varepsilon_V\colon & V^*\otimes V & \to & \C\\
& f\otimes v & \mapsto & f(v)\\
\end{array}
$$
the idea being that the ``natural'' picture on the left motivates the notation
$$
\xybox{(0,6)*{};(0,-6)**\dir{-}?(.5)*\dir{<}+(2,0)*{\scriptstyle V};}
=
\xybox{(0,6)*{};(0,-6)**\dir{-}?(.5)*\dir{>}+(3,0)*{\scriptstyle V^*};}
$$
Recall that in physics each arrow
$\xy(0,3)*{};(0,-3)**\dir{-}?(.5)*\dir{>}+(2,0)*{\scriptstyle
V}\endxy$ is called a ``particle''. Then,
$\xy(0,3)*{};(0,-3)**\dir{-}?(.5)*\dir{<}+(2,0)*{\scriptstyle
V}\endxy$ corresponds to its ``antiparticle'' and
$\xy(-3,2)*{};(3,2)*{};**\crv{(0,-6)}
?(.50)*\dir{>}+(0,-2)*{\scriptstyle V};\endxy$ represents the physical
process of ``particle-antiparticle annihilation''. We are actually
neglecting conservation of energy, which explains why a
particle-antiparticle pair can annihilate into nothing.
Come to think about it, when Feynman invented Feynman diagrams and
proposed that antiparticles are particles moving backwards in time, he
was doing nothing more than finding a graphical interpretation to the
fact that, if the space of states of a particle is $V$, then $V^*$ is
the space of states of its antiparticle.
If $V$ is finite-dimensional, we also get an intertwiner going the
opposite way, called the ``unit'':
$$
\xy
(-6,-4)*{};(6,-4)*{};
**\crv{(0,12)} ?(.50)*\dir{>}+(0,2)*{\scriptstyle V}
\endxy
=
\xy
(0,4)*+{\iota_V}*\cir{}="e" ;
(-6,-4)**\dir{-} ?(.75)*\dir{<}+(-2,1)*{\scriptstyle V^*};
"e";(6,-4)**\dir{-} ?(.5)*\dir{<}+(2,1)*{\scriptstyle V};
\endxy
\qquad
\begin{array}{rccc}
\iota_V\colon & \C & \to & V\otimes V^* \\
& 1 & \mapsto & e_i\otimes e^i
\end{array}
$$
where $\{e_i\}$ is any basis of $V$ and $\{e^i\}$ is its dual basis
(defined by $e^i(e_j)=\delta_{ij}$). It is not entirely obvious that
this definition is basis-independent, but the operator $e_i\otimes
e^i$ is simply the identity in $\End(V)$. Indeed, if $v=v^ie_i$, then
$\iota_V(v)=(e_i\otimes e^i)(v^je_j)=e_iv^j\delta^i_j=v^ie_i=v$.
If $V$ is infinite dimensional, we have that $V\otimes
V^*\subseteq\hom(V,V)$, but $\id_V\not\in V\otimes V^*$.
Now, combining the unit and counit, we can draw the following fun
diagrams:
$$
\begin{xy}
(8,9)*{};(-8,-9)*{};**\crv{(-3,8)&(0,0)&(3,-8)}?(.5)*\dir{<};
\end{xy}
=
\begin{xy}
(0,9)*{};(0,-9)**\dir{-}?(.5)*\dir{>};
\end{xy}
\qquad\hbox{and}\quad
\begin{xy}
(8,-9)*{};(-8,9)*{};**\crv{(-3,-8)&(0,0)&(3,8)}?(.5)*\dir{<};
\end{xy}
=
\begin{xy}
(0,9)*{};(0,-9)**\dir{-}?(.5)*\dir{<};
\end{xy}
$$
\subsubsection{The conjugate representation}
If we have a complex vector space $V$, there is a conjugate vector
space $\bar V$, defined as follows:
\begin{itemize}
\item As a set, $\overline V=V$, but they have different vector space
structures. To avoid confusion, we denote $\bar v\in\overline V$ for
the vector $v\in V$, regarded as an element of $\overline V$ (you
know you are in trouble when a mathematician says ``regarded
as'').
\item As far as addition is concerned, $\overline V=V$ too; that is,
$$
\bar v+\bar w=\overline{v+w}\qquad\hbox{for all}\quad v,w\in V.
$$
\item Scalar multiplication involves conjugation in $\C$:
$$
z\bar v = \overline{\bar z v}\qquad\hbox{for all}\quad v\in V,z\in\C.
$$
In particular, $i\bar v=\overline{-iv}$.
\end{itemize}
In this way $V$ sprouts a whole family of related vector spaces:
$V,V^*,\overline V,{\overline V}^*,\overline{V^*},\ldots$. In fact,
already $\overline{V^*}$ and ${\overline V}^*$ are naturally
isomorphic, so there are only four different vector spaces.