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 be the Grassmann algebra generated by , ,
such that and .
Denote by , the Grassmann algebra of an infinite number of generators .
A supernumber is an element of or .
Any supernumber can be expressed uniquely in the form
where the coefficients are antisymmetric in their indices.
1 Body and soul
The body of a supernumber is defined as ,
and its soul is defined as .
If then has an inverse given by
2 Odd and even
A supernumber can be decomposed into the even and odd parts:
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 has the vector space decomposition
,
where is the space of c-numbers,
and 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:
The second way is to define an anti-linear involution:
The comes down to whether the product of two real odd supernumbers is real or imaginary.
Title | supernumber |
---|---|
Canonical name | Supernumber |
Date of creation | 2013-03-22 13:03:27 |
Last modified on | 2013-03-22 13:03:27 |
Owner | mhale (572) |
Last modified by | mhale (572) |
Numerical id | 12 |
Author | mhale (572) |
Entry type | Definition |
Classification | msc 16W55 |
Related topic | SuperAlgebra |
Defines | body |
Defines | soul |