Contradictions in Mechanics

  1. Both the primary and secondary electromagnetic waves propagate through the interatomic void with speed of c. The resulting transmitted wave may appear to have a phase velocity less than, equal to, or even greater than c. See [Hec, p.2, r.c., l.17-l.22]. The key to this apparent contradiction: [Hec, p.94, l.9-l.13]. An explanation: [Hec, p. 93, Fig. 4.9 (a) and (b)].

  2. The creation of the d-function originates from a contradiction. With the exception of [Born, p.755, l.9-l.11], most textbooks fail to point out this important fact. In order to avoid direct confrontation with a contradiction, we circumvent the obstacle by expanding the meaning of an integral [Born, l.-15-l.-13]. The expansion is based on the concept of integration by parts.

  3. (Wave-particle duality)
        Wave-particle duality has nothing to do with the dual space of a vector space. The macroscopic phenomena such as interference and diffraction lead us to regard light as a wave. However, blackbody radiation, the photoelectric effect and the Compton effect lead us to consider electromagnetic waves as photons. In order to make these two irreconcilable concepts [Coh, Complement DI] compatible, we expand the macroscopic regime into the microscopic regime using the Planck-Einstein relation [Coh, 11, (A-1)] which allows us to treat a wave as a particle.

  4. (The inner product on R3 vs. the scalar product of 4-vectors)
        The inner product of two vectors on R3 is invariant under rotations. Similarly, the scalar product [Lan2, p.14, l.15] of 4-vectors is invariant under Lorentz transformations [Lan2, p.14, l.16-l.17]. However, the definition of the inner product and that of the scalar product are not consistent. There are two ways to eliminate this inconsistency. The first way is to define the contravariant and covariant components of a 4-vector and then define the scalar product of 4-vectors as in [Lan2, p.15, l.10]. Then the scalar product will have the same form as the inner product. The second way is to generalize the concept of an inner product in a manner as in [Spi, vol. 1, p.408, l.14]. Then both the inner product and the scalar product become special cases of the generalized inner product.

  5. Using the extra freedom of a Green function to solve the possible confliction between Dirichlet boundary condition and the Neumann boundary condition [Jack, p.39, l.5-l.12].