Single-Particle Electrodynamics - Assassination Science

AB denotes the four-tensor that is the outer-product of the four-vectors A
and B:
(AB) αβ ≡ A α B β .
(A.42)
A.8.7
Inner-products, dot-products
A nondereferenced four-vector with a central dot either preceding or following
it denotes the operation variously known as the contraction, inner-product
or dot-product of that four-vector with the quantity lying to the other side
of the central dot; for example, given two four-vectors A and B, then
A·B ≡ A α B α .
(A.43)
A nondereferenced four-tensor with a central dot preceding (following) it
denotes the dot-product of that four-tensor with the quantity lying to the
left (right) of the central dot, over the first (second) index of the four-tensor.
For example, if A and B are four-vectors and F a four-tensor, then A·F has
components
(A·F ) α ≡ A β F α β ,
(A.44)
and F ·B has components
(F ·B) α ≡ F α βB β . (A.45)
Four-tensors may be dot-producted on both sides at once; for example,
A·F ·B ≡ A α F α β B β ,
(A.46)
and may be “chained” together into longer dot-products:
A·F ·F ·B ≡ A α F α β F β µ B µ .
(A.47)
In any of the above dot-products using the central dot, in which all indices
are fully contracted (i.e., the result is a Lorentz scalar), parentheses are often
used to visually delineate the product; e.g.,
D(A·F ·B) ≡ DA·F ·B,
(A.48)
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