Single-Particle Electrodynamics - Assassination Science

and G are four-tensors, then
ε(A, B, F ) ≡ ε αβµν A α B β F µν ,
ε(A, F, B) ≡ ε αβµν A α F βµ B ν ,
ε(F, A, B) ≡ ε αβµν F αβ A µ B ν ,
ε(F, G) ≡ ε αβµν F αβ G µν .
If one of the entries in the ε( ) symbol is vacant, then the overall symbol represents
a four-vector with the vacant entry as the vector index; for example,
ε(, B, C, D) has components
( ) α
ε(, B, C, D) ≡ ε
α
βµν B β C µ D ν .
If two entries in the ε( ) symbol are vacant, then the overall symbol represents
a four-tensor, where the first vacant entry represents the first index, and the
second vacant entry the second index; for example, ε(, B, , D) has components
( ) αµ
ε(, B, , D) ≡ ε
α µ
β ν B β D ν .
A.8.10
Cross-products
An alternative notation for the epsilon product is that of the four-crossproduct.
It is defined, for four-vectors B, C and D, as
B×C ×D ≡ ε(, B, C, D).
(A.53)
(This is analogous to the three-vector case; see Section A.9.13.)
Either pair of adjacent four-vectors may be replaced by a four-tensor;
e.g., if F is a four-tensor, then
F ×D ≡ ε(, F, D),
B×F ≡ ε(, B, F ).
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