supernumberSupernumber2002-09-17 12:44:282005-10-20 15:33:00Definitionbodysoul\usepackage{amssymb}
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\renewcommand*{\bar}[1]{\overline{#1}}Supernumbers are the generalisation of complex numbers to a commutative superalgebra of commuting and anticommuting ``numbers''.
They are primarily used in the description of \PMlinkescapetext{fermionic fields} in \PMlinkescapetext{quantum field theory}.
Let $\Lambda_N$ be the Grassmann algebra generated by $\theta^i$, $i = 1 \ldots N$,
such that $\theta^i\theta^j = -\theta^j\theta^i$ and $(\theta^i)^2 = 0$.
Denote by $\Lambda_\infty$, the Grassmann algebra of an infinite number of generators $\theta^i$.
A \defn{supernumber} is an element of $\Lambda_N$ or $\Lambda_\infty$.
Any supernumber $z$ can be expressed uniquely in the form
\[
z = z_0 + z_i \theta^i + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots
+ \frac{1}{n!} z_{i_1 \ldots i_n} \theta^{i_1} \ldots \theta^{i_n} + \ldots,
\]
where the coefficients $z_{i_1 \ldots i_n} \in \Cset$ are antisymmetric in their indices.
\section{Body and soul}
The \defn{body} of a supernumber $z$ is defined as $z_\mathrm{B} = z_0$,
and its \defn{soul} is defined as $z_\mathrm{S} = z-z_\mathrm{B}$.
If $z_\mathrm{B} \neq 0$ then $z$ has an inverse given by
\[
z^{-1} = \frac{1}{z_\mathrm{B}} \sum_{k=0}^\infty \left(-\frac{z_\mathrm{S}}{z_\mathrm{B}}\right)^k.
\]
\section{Odd and even}
A supernumber can be decomposed into the even and odd parts:
\begin{eqnarray*}
z_\mathrm{even} & = & z_0 + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots
+ \frac{1}{(2n)!} z_{i_1 \ldots i_{2n}} \theta^{i_1} \ldots \theta^{i_{2n}} + \ldots, \\
z_\mathrm{odd} & = & z_i \theta^i + \frac{1}{6} z_{ijk} \theta^i\theta^j\theta^k + \ldots
+ \frac{1}{(2n+1)!} z_{i_1 \ldots i_{2n+1}} \theta^{i_1} \ldots \theta^{i_{2n+1}} + \ldots.
\end{eqnarray*}
Even supernumbers commute with each other and are called \defn{c-numbers},
while odd supernumbers anticommute with each other and are called \defn{a-numbers}.
Note, the product of two c-numbers is even,
the product of a c-number and an a-number is odd,
and the product of two a-numbers is even.
The superalgebra $\Lambda_N$ has the vector space decomposition
$\Lambda_N = \Cset_c \oplus \Cset_a$,
where $\Cset_c$ is the space of c-numbers,
and $\Cset_a$ is the space of a-numbers.
\section{Conjugation and involution}
There are two ways, one can define a complex conjugation for supernumbers.
The first is to define a linear conjugation in complete analogy with complex numbers:
\[
\bar{(z_1 z_2)} = \bar{z_1} \;\bar{z_2}.
\]
The second way is to define an anti-linear involution:
\[
(z_1 z_2)^* = z_2^* z_1^*.
\]
The \PMlinkescapetext{difference} comes down to whether the product of two real odd supernumbers is real or imaginary.