# supernumber

Supernumbers are the generalisation of complex numbers to a commutative superalgebra of commuting and anticommuting “numbers”. They are primarily used in the description of in .

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 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 $z_{i_{1}\ldots i_{n}}\in\mathbb{C}$ are antisymmetric in their indices.

## 1 Body and soul

The body of a supernumber $z$ is defined as $z_{\mathrm{B}}=z_{0}$, and its soul is defined as $z_{\mathrm{S}}=z-z_{\mathrm{B}}$. If $z_{\mathrm{B}}\neq 0$ then $z$ has an inverse given by

 $z^{-1}=\frac{1}{z_{\mathrm{B}}}\sum_{k=0}^{\infty}\left(-\frac{z_{\mathrm{S}}}% {z_{\mathrm{B}}}\right)^{k}.$

## 2 Odd and even

A supernumber can be decomposed into the even and odd parts:

 $\displaystyle z_{\mathrm{even}}$ $\displaystyle=$ $\displaystyle z_{0}+\frac{1}{2}z_{ij}\theta^{i}\theta^{j}+\ldots+\frac{1}{(2n)% !}z_{i_{1}\ldots i_{2n}}\theta^{i_{1}}\ldots\theta^{i_{2n}}+\ldots,$ $\displaystyle z_{\mathrm{odd}}$ $\displaystyle=$ $\displaystyle z_{i}\theta^{i}+\frac{1}{6}z_{ijk}\theta^{i}\theta^{j}\theta^{k}% +\ldots+\frac{1}{(2n+1)!}z_{i_{1}\ldots i_{2n+1}}\theta^{i_{1}}\ldots\theta^{i% _{2n+1}}+\ldots.$

Even supernumbers commute with each other and are called c-numbers, while odd supernumbers anticommute with each other and are called 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 $\Lambda_{N}=\mathbb{C}_{c}\oplus\mathbb{C}_{a}$, where $\mathbb{C}_{c}$ is the space of c-numbers, and $\mathbb{C}_{a}$ is the space of a-numbers.

## 3 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:

 $\overline{(z_{1}z_{2})}=\overline{z_{1}}\;\overline{z_{2}}.$

The second way is to define an anti-linear involution:

 $(z_{1}z_{2})^{*}=z_{2}^{*}z_{1}^{*}.$

The comes down to whether the product of two real odd supernumbers is real or imaginary.

Title supernumber Supernumber 2013-03-22 13:03:27 2013-03-22 13:03:27 mhale (572) mhale (572) 12 mhale (572) Definition msc 16W55 SuperAlgebra body soul