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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.