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Duality in Mechanics

  1. 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.

  2. 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 d(p0)d3p=1 and f*(p,t)f(p,t)d3p=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)].

  3.  
    1. (Symmetry [Mer2, p.197, l.20])
          The dual of a dual is the original.
      1. vector space [Halm, 15, 16 & 67].
      2. Lattice [Ashc, p.87, l.-19-l.-15].
    2. The Fourier coefficients of a periodic function can be expressed in terms of the reciprocal lattice [Ashc, p.764, (D.11)].

  4.  

    1. Treating waves as particles: [Wu, p.41, (1-14)].
    2. Treating particles as waves: [Wu, p.43, (1-15)].

  5. [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.

  6. field transformations from electric dipole radiation to magnetic dipole radiation: [Wangs, p.483, (28-77)].