| Version current |
Version 8 |
| Supernumbers are the generalisation of complex numbers to a commutative superalgebra of commuting and anticommuting ``numbers''. |
Supernumbers are the generalisation of complex numbers to a commutative superalgebra of commuting and anticommuting ``numbers''. |
| They are primarily used in the description of \PMlinkescapetext{fermionic fields} in \PMlinkescapetext{quantum field theory}. |
They are primarily used in the description of \PMlinkescapetext{fermionic fields} in \PMlinkescapetext{quantum field theory}. |
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| Let $\Lambda_N$ be the Grassmann algebra generated by $\theta^i$, $i = 1 \ldots N$, |
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 $(\theta^i)^2 = 0$. |
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 $\theta^i$. |
Denote by $\Lambda_\infty$, the Grassmann algebra of an infinite number of generators $\theta^i$. |
| A \defn{supernumber} is an element of $\Lambda_N$ or $\Lambda_\infty$. |
A \defn{supernumber} is an element of $\Lambda_N$ or $\Lambda_\infty$. |
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| Any supernumber $z$ can be expressed uniquely in the form |
Any supernumber $z$ can be expressed uniquely in the form |
| \[ |
\[ |
| z = z_0 + z_i \theta^i + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots |
z = z_0 + z_i \theta^i + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots |
| + \frac{1}{n!} z_{i_1 \ldots i_n} \theta^{i_1} \ldots \theta^{i_n} + \ldots, |
+ \frac{1}{n!} z_{i_1 \ldots i_n} \theta^{i_1} \ldots \theta^{i_n} + \ldots, |
| \] |
\] |
| where the coefficients $z_{i_1 \ldots i_n} \in \Cset$ are antisymmetric in their indices. |
where the coefficients $z_{i_1 \ldots i_n} \in \Cset$ are antisymmetric in their indices. |
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| \section{Body and soul} |
\section{Body and soul} |
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| The \defn{body} of a supernumber $z$ is defined as $z_\mathrm{B} = z_0$, |
The \defn{body} of a supernumber $z$ is defined as $z_\mathrm{B} = z_0$, |
| and its \defn{soul} is defined as $z_\mathrm{S} = z-z_\mathrm{B}$. |
and its \defn{soul} is defined as $z_\mathrm{S} = z-z_\mathrm{B}$. |
| If $z_\mathrm{B} \neq 0$ then $z$ has an inverse given by |
If $z_\mathrm{B} \neq 0$ then $z$ has an inverse given by |
| \[ |
\[ |
|
z^{-1} = \frac{1}{z_\mathrm{B}} \sum_{k=0}^\infty \left(-\frac{z_\mathrm{S}}{z_\mathrm{B}}\right)^k.
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z^{-1} = \frac{1}{z_\mathrm{B}} \sum_{k=0} \left(-\frac{z_\mathrm{S}}{z_\mathrm{B}}\right)^k.
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| \] |
\] |
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| \section{Odd and even} |
\section{Odd and even} |
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| A supernumber can be decomposed into the even and odd parts: |
A supernumber can be decomposed into the even and odd parts: |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| z_\mathrm{even} & = & z_0 + \frac{1}{2} z_{ij} \theta^i\theta^j + \ldots |
z_\mathrm{even} & = & 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, \\ |
+ \frac{1}{(2n)!} z_{i_1 \ldots i_{2n}} \theta^{i_1} \ldots \theta^{i_{2n}} + \ldots, \\ |
| z_\mathrm{odd} & = & z_i \theta^i + \frac{1}{6} z_{ijk} \theta^i\theta^j\theta^k + \ldots |
z_\mathrm{odd} & = & 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. |
+ \frac{1}{(2n+1)!} z_{i_1 \ldots i_{2n+1}} \theta^{i_1} \ldots \theta^{i_{2n+1}} + \ldots. |
| \end{eqnarray*} |
\end{eqnarray*} |
| Even supernumbers commute with each other and are called \defn{c-numbers}, |
Even supernumbers commute with each other and are called \defn{c-numbers}, |
| while odd supernumbers anticommute with each other and are called \defn{a-numbers}. |
while odd supernumbers anticommute with each other and are called \defn{a-numbers}. |
| Note, the product of two c-numbers is even, |
Note, the product of two c-numbers is even, |
| the product of a c-number and an a-number is odd, |
the product of a c-number and an a-number is odd, |
| and the product of two a-numbers is even. |
and the product of two a-numbers is even. |
| The superalgebra $\Lambda_N$ has the vector space decomposition |
The superalgebra $\Lambda_N$ has the vector space decomposition |
| $\Lambda_N = \Cset_c \oplus \Cset_a$, |
$\Lambda_N = \Cset_c \oplus \Cset_a$, |
| where $\Cset_c$ is the space of c-numbers, |
where $\Cset_c$ is the space of c-numbers, |
| and $\Cset_a$ is the space of a-numbers. |
and $\Cset_a$ is the space of a-numbers. |
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| \section{Conjugation and involution} |
\section{Conjugation and involution} |
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| There are two ways, one can define a complex conjugation for supernumbers. |
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 first is to define a linear conjugation in complete analogy with complex numbers: |
| \[ |
\[ |
| \bar{(z_1 z_2)} = \bar{z_1} \;\bar{z_2}. |
\bar{(z_1 z_2)} = \bar{z_1} \;\bar{z_2}. |
| \] |
\] |
| The second way is to define an anti-linear involution: |
The second way is to define an anti-linear involution: |
| \[ |
\[ |
| (z_1 z_2)^* = z_2^* z_1^*. |
(z_1 z_2)^* = z_2^* z_1^*. |
| \] |
\] |
| The \PMlinkescapetext{difference} comes down to whether the product of two real odd supernumbers is real or imaginary. |
The \PMlinkescapetext{difference} comes down to whether the product of two real odd supernumbers is real or imaginary. |
