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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 $ z$ 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 $ 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

$\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}. $
The second way is to define an anti-linear involution:
$\displaystyle (z_1 z_2)^* = z_2^* z_1^*. $
The difference comes down to whether the product of two real odd supernumbers is real or imaginary.



"supernumber" is owned by mhale.
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See Also: superalgebra

Also defines:  body, soul
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Cross-references: imaginary, real, involution, analogy, complete, conjugation, complex conjugation, decomposition, vector space, product, odd, even, inverse, indices, antisymmetric, coefficients, generators, number, infinite, generated by, Grassmann algebra, superalgebra, commutative, complex numbers
There are 9 references to this entry.

This is version 9 of supernumber, born on 2002-09-17, modified 2005-10-20.
Object id is 3463, canonical name is Supernumber.
Accessed 8457 times total.

Classification:
AMS MSC16W55 (Associative rings and algebras :: Rings and algebras with additional structure :: ``Super'' structure)

Pending Errata and Addenda
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