Single-Particle Electrodynamics - Assassination Science

where r has components (x, y, z).
The four-dimensional Dirac delta function, of four-vector argument X, is
denoted δ (4) (X), or simply δ(X) where unambiguous, and is defined as
δ (4) (X) ≡ δ(t)δ(x)δ(y)δ(z),
(A.26)
where X has components (t, x, y, z).
A.5.3
Heaviside step function
The Heaviside step function ϑ(t) is defined as the integral of the Dirac delta
function (A.23):
ϑ(t) ≡
∫ t
−∞
dt ′ δ(t ′ ).
(A.27)
The same considerations for evaluation and physicality apply to the Heaviside
step function as apply to the Dirac delta function (see Section A.5.2).
A.5.4
Alternating function
The d-dimensional alternating function of offset ω is denoted
ε (ω,d)
{i} . (A.28)
Its argument is a special collection symbol {i} of enumerations (see Section
A.4), which may be dereferenced by removing the braces and subscripting
with a unit-offset enumeration of dimension d. The collection symbol {i}
may be also be denoted either by writing its elements adjacently, in order of
the enumeration index:
{i} ≡ i 1 i 2 i 3 · · · i d−2 i d−1 i d ,
(A.29)
or by writing them in set notation:
{i} ≡ {i 1 , i 2 , i 3 , . . . , i d−2 , i d−1 , i d }.
(A.30)
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