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 ${\mathrm{\Lambda }}_N$ be the Grassmann algebra generated by ${\theta }^i$, $i=1\mathrm{\dots }N$,such that ${\theta }^i{\theta }^j=-{\theta }^j{\theta }^i$ and ${({\theta }^i)}^2=0$.Denote by ${\mathrm{\Lambda }}_{\mathrm{\infty }}$, the Grassmann algebra of an infinite number of generators ${\theta }^i$.A supernumber is an element of ${\mathrm{\Lambda }}_N$ or ${\mathrm{\Lambda }}_{\mathrm{\infty }}$.

Any supernumber $z$ can be expressed uniquely in the form

$$z=z_0+z_i{\theta }^i+\frac{1}{2}{z}_{ij}{\theta }^i{\theta }^j+\mathrm{\dots }+\frac{1}{n!}{z}_{i_1\mathrm{\dots }{i}_n}{\theta }^{i_1}\mathrm{\dots }{\theta }^{i_n}+\mathrm{\dots },$$

where the coefficients $z_{i_1\mathrm{\dots }{i}_n}\in ℂ$ 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 $z_{\mathrm{B}}\ne 0$ then $z$ has an inverse given by

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

2 Odd and even

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

$z_{\mathrm{even}}$$=$$z_0+{\displaystyle \frac{1}{2}}{z}_{ij}{\theta }^i{\theta }^j+\mathrm{\dots }+{\displaystyle \frac{1}{(2n)!}}{z}_{i_1\mathrm{\dots }{i}_{2n}}{\theta }^{i_1}\mathrm{\dots }{\theta }^{i_{2n}}+\mathrm{\dots },$

$z_{\mathrm{odd}}$$=$$z_i{\theta }^i+{\displaystyle \frac{1}{6}}{z}_{ijk}{\theta }^i{\theta }^j{\theta }^k+\mathrm{\dots }+{\displaystyle \frac{1}{(2n+1)!}}{z}_{i_1\mathrm{\dots }{i}_{2n+1}}{\theta }^{i_1}\mathrm{\dots }{\theta }^{i_{2n+1}}+\mathrm{\dots }.$

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 ${\mathrm{\Lambda }}_N$ has the vector space decomposition${\mathrm{\Lambda }}_N={ℂ}_c\oplus {ℂ}_a$,where ${ℂ}_c$ is the space of c-numbers,and ${ℂ}_a$ 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.