The spinor space [complex vector] is closer to physical reality [Go2,
p.158, l.2] because [Go2, p.150, l.!1] determines
the position of a particle only, while [Go2, p.151, l.2] allows room for wave
functions and probability to describe its wave picture.
Physical motivation for the Fourier transform [Mer2, p.31, l.-3-p.32, l.2].
For a plane wave in
position space with definite momentum p,
y(r,0)=exp(ipCr/S).
Via the similarity between òd(p_{0})d^{3}p=1
and òf^{*}(p,t)f(p,t)d^{3}p=1,
we generalize the concept of a plane wave to that of a wave packet. That is,
y(r,t)
is designed to distribute some proportion
of probability amplitude f(p,t)
in momentum space for each plane wave with
definite momentum p [Mer2, p.30, (3.18)].
Then
y(r,t)
is the Fourier
transform of
f(p,t).
Similarly, for a plane wave in momentum space with definite position r,
f(p,0)=exp(-ipCr/S).
In the general case, f(p,t)
is designed to distribute some proportion of
probability amplitude
y(r,t)
in position space for each plane wave in momentum space with definite position
r [Mer2, p.30, (3.19)].
(Symmetry [Mer2, p.197, l.20]) The dual of a dual is the original.
vector space [Halm,
§15, §16
& §67].
Lattice [Ashc, p.87, l.-19-l.-15].
The Fourier coefficients of a periodic function can be expressed in terms of the reciprocal lattice [Ashc, p.764, (D.11)].
Treating waves as particles: [Wu, p.41, (1-14)].
Treating particles as waves: [Wu, p.43, (1-15)].
[Lan2, p.17, l.31 & l.32; p.18, l.14 & l.16] fail to use the concept of
inner product and determinants to define the dual of tensors, so it is difficult to
recognize the dual relationships. In contrast, [Spi1, p.84, l.2] defines the
cross product using the concepts of inner product and determinants.
Consequently, the dual relationship in Spivak's version of the dual of tensors is
explicit and obvious.
field transformations from electric dipole radiation to magnetic dipole radiation: [Wangs, p.483, (28-77)].