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.