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supernumber

(Definition)

Supernumbers are the generalisation of complex numbers to a commutative superalgebra of commuting and anticommuting “numbers”. They are primarily used in the description of fermionic fields in 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 supernumber is an element of $\Lambda_N$ or $\Lambda_\infty$ .

Any supernumber can be expressed uniquely in the form

$\displaystyle z = z_0 + z_i \theta^i + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots + \frac{1}{n!} z_{i_1 \ldots i_n} \theta^{i_1} \ldots \theta^{i_n} + \ldots,$

where the coefficients $z_{i_1 \ldots i_n} \in \mathbb{C}$ are antisymmetric in their indices.

Body and soul

The body of a supernumber 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 has an inverse given by

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

Odd and even

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

$\displaystyle z_\mathrm{even}$	$\displaystyle =$	$\displaystyle z_0 + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots + \frac{1}{(2n)!} z_{i_1 \ldots i_{2n}} \theta^{i_1} \ldots \theta^{i_{2n}} + \ldots,$
$\displaystyle z_\mathrm{odd}$	$\displaystyle =$	$\displaystyle z_i \theta^i + \frac{1}{6} z_{ijk} \theta^i\theta^j\theta^k + \ld... ...2n+1)!} z_{i_1 \ldots i_{2n+1}} \theta^{i_1} \ldots \theta^{i_{2n+1}} + \ldots.$

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.

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:

$\displaystyle \bar{(z_1 z_2)} = \bar{z_1} \;\bar{z_2}.$