1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
|
\cdbalgorithm{join}{}
Join two fully anti-symmetrised gamma matrix
products according to the expression
\begin{equation}
\Gamma^{b_{1}\dots b_{n}}\Gamma_{a_{1}\dots a_{m}} =
\sum_{p=0}^{\text{min}(n,m)}\ \frac{n! m!}{(n-p)! (m-p)! p!}
\Gamma^{[b_{1}\ldots b_{n-p}}{}_{[a_{p+1}\ldots a_{m}}
\eta^{b_{n-p+1}\ldots b_{n}]}{}_{a_{1}\ldots a_{m-p}]} \, .
\end{equation}
This is the opposite of \subscommand{gammasplit}.
Without further arguments, the anti-symmetrisations will be left
implicit. The argument ``{\tt expand}'' instead performs the sum over
all anti-symmetrisations, which may lead to an enormous number of
terms if the number of indices on the gamma matrices is large. Compare
\begin{screen}{1,2}
\Gamma{#}::GammaMatrix(metric=g).
\Gamma_{m n} \Gamma_{p};
@join!(%);
\Gamma_{m n p} + 2 \Gamma_{m} g_{n p};
\end{screen}
with
\begin{screen}{1,2}
\Gamma{#}::GammaMatrix(metric=g).
\Gamma_{m n} \Gamma_{p};
@join!(%){expand};
\Gamma_{m n p} + \Gamma_{m} g_{n p} - \Gamma_{n} g_{m p};
\end{screen}
Note that the gamma matrices need to have a metric associated to them
in order for this algorithm to work.
In order to reduce the number somewhat, one can instruct the algorithm
to make use of generalised Kronecker delta symbols in the result;
these symbols are defined as
\begin{equation}
\delta^{r_1}{}_{s_1}{}^{r_2}{}_{s_2}\cdots{}^{r_n}{}_{s_n}
= \delta^{[r_1}{}_{s_1}\delta^{r_2}{}_{s_2}\cdots {}^{r_n]}{}_{s_n}\, .
\end{equation}
Anti-symmetrisation is implied in the set of even-numbered
indices. The use of these symbols is triggered by the ``{\tt
gendelta}'' option,
\begin{screen}{1,2}
{m,n,p,q}::Indices(position=fixed).
\Gamma{#}::GammaMatrix(metric=\delta).
\Gamma_{m n} \Gamma^{p q};
@join!(%){expand}{gendelta};
\Gamma_{m n}^{p q} + \Gamma_{m}^{q} \delta_{n}^{p}
- \Gamma_{m}^{p} \delta_{n}^{q} - \Gamma_{n}^{q} \delta_{m}^{p}
+ \Gamma_{n}^{p} \delta_{m}^{q} + 2 \delta_{n}^{p}_{m}^{q};
\end{screen}
Finally, to select only a single term (for a given $p$) in this
expansion, give the join an argument with the value of $p$.
\begin{screen}{1,2}
\Gamma{#}::GammaMatrix(metric=g).
\Gamma_{m n} \Gamma_{p};
@join!(%){expand}{3};
\Gamma_{m n p};
\end{screen}
This option can also be combined with {\tt gendelta} if required.
\cdbseeprop{GammaMatrix}
\cdbseeprop{KroneckerDelta}
\cdbseealgo{gammasplit}
|
