Based on the space spherical symmetry of 3-dimensional and the translational symmetry of time and the uncertainty principle, a 4-dimensional space-time cylinder model of quarks and leptons is established. With this model, equations of the special relativity can be extended more perfectly, thereby achieving a unity of the special relativity and quantum mechanics in deeper level. New equations can not only interpret issues explained by old equations but also solve several important pending problems. For example, a formula to strictly calculate the coefficient ξ of Lorentz invariance violation (LIV) is derived, to above 4 × 10¹⁹ eV UHECR protons the calculated |ξ| < 4.5 × 10^-30, although there is the LIV effect it is too weak to change the GZK cutoff, which is consistent with observations of HiRes and Auger; Also, a relation formula between the Hubble constant and several basic constants is derived, thus theoretically calculated H₀ = 70.937 km·s^-1·Mpc^-1, which is well consistent with the final observation result of HST Key Project. In addition, an unusual effect predicted by new equations can be experimentally tested in the electron storage ring; a preliminary experiment result has hinted its signs of existence.