Single-Particle Electrodynamics - Assassination Science

distributed factor; for example,
A [α X µ Y ν Z τ
(
Bβ] + C β]
)
.
(A.36)
A.6.3
Symmetrisors
The symmetrisor of a product of n factors A, B, C, . . . , Z is defined to be
1/n! times the sum of the terms of all of the n! permutations of the factors,
and is denoted
{ABC · · · Z } .
(A.37)
For example,
{AB } ≡
2( 1 )
AB + BA ,
{ABC } ≡ 1 )
ABC + ACB + BAC + BCA + CAB + CBA .
6(
The use of the double-braces ensures complete disambiguation from the use
of braces as a binding symbol, since two identical sets of braces, without
intervening symbols, would in the latter case be completely redundant. Furthermore,
the double-braces used in (A.37) are typographically spaced closer
together than is the case when braces are used as binding symbols.
The inclusion of the factor of 1/n! in the above definition of the symmetrisor
means that it is not necessary to identify or count the number of
“truly non-commutative” factors in the product: commuting factors may be
(trivially) symmetrised over without affecting the numerical result.
Note also that the symmetrisation is a mathematical one, not a typographical
one. This is of importance for the three-vector cross-product of
Section A.9.13, namely,
A×B ≡ ε ijk A j B k ;
(A.38)
the symmetrisor of (A.38) is defined to be the symmetrisor of the right-hand
side of this definition:
{A×B } i
≡ 1 2 ε )
ijk(
Aj B k + B k A j . (A.39)
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