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RECIPROCAL SYMMETRY AND ITS APPLICATION IN DERIVING EINSTEIN'S POSTULATE, LORENTZ TRANSFORMATION AND CORRESPONDENCE BETWEEN RECIPROCITY AND PERIODICITY

Reciprocal Symmetric (RS) function is by definition a function which remains invariant when the argument is replaced by its reciprocal. We have observed that Lorentz transformation is Reciprocal symmetric since it remains invariant if velocities u/c etc. are replaced by their reciprocals. We have shown that RS requires that velocities must be bounded above. We have expressed distance covered by a moving body as a RS function, X(u/c,t), of velocity u and time t. RS requires that the time of travel must be an integral multiple of unit time. The distance covered is symmetric or ant-symmetric combination of two parts depending on whether the time of travel is odd or even multiple of unit time. We have also shown that reciprocal symmetry relates wave motion to particle motion by showing equivalence between periodicity of wave motion to reciprocity of particle motion.

Reciprocal Symmetry and Correspondence between Relativistic and Quantum Mechanical Concepts

We recall the principle of objectivity that physics should be independent of the quantities we define. We have postulated that the study of motion in terms of slowness (reciprocal of velocity) is as valid as the study in terms of velocity. We require that a kinematical theory should be reciprocal symmetric. We have given a mathematical definition of reciprocal symmetry. It follows from this definition that velocities must have an upper bound and slowness must have a lower bound. From group property requirement that the relative slowness (which is a difference between two slowness) is also a slowness, we have shown that slowness are discrete. The same motion may be represented by continuous quantities (like velocity) and by discrete quantities (like slowness). A hyperbolic transform relates a Galilean velocity to a “Lorentz algebraic velocity”, which obeys Lorentz algebra. Isomorphism between real numbers and imaginary numbers allow us to replace all real numbers by corresponding imaginary numbers. This changes all hyperbolic functions to circular functions. Circular functions are periodic and represent wave motion. Hyperbolic functions are monotonic and represent particle motion. We have shown that real-imaginary symmetry establishes the equivalence between particle representation and wave representation. We have also shown that application of reciprocal symmetry on Planck’s distribution law gives Fermi-Dirac distribution law. For this we have not invoked exclusion principle.

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