Wiener Process of Fractals and Path-Integrals I: Emergent Lorentz Symmetry in Stochastic Process of Quantum Fields
This is the first paper of a series of researches (that is followed by [42, 43]) that aims to interpret the gravitational effects of nature within some consistent stochastic fractal-based (intrinsically conformal) path-integral formulation. In this paper, we initially study the asymptotic behaviors of fractal structure of Weierstrass-like functions by means of Hardy's criteria for nowhere differentiability. It is proved that the asymptotic behavior of Fourier-Laplace coefficients of such functions leads to a non-linear differential equation which in its turn gives rise to an exponentially increasing norm, the so-called fractal norm, on the phase space. Then, using the fractal norm the Wiener Brownian process is accomplished for fractal functions on a flat space. By substituting non-local terms with approximated local ones within the derived formula the dAlembertian operator emerges automatically in the Gaussian terms of the Wiener measure. Hence, it is established that the Lorentz symmetry would be regarded as the first-order approximate symmetry of nature on a flat space-time manifold based on the stochastic essence of Brownian motion of the background fractal geometry. ...