标题： 时间与空间的维度是整数还是分数？ 显示英文标题

标题： What Dimensions Do the Time and Space Have: Integer or Fractional?

摘要： 一种具有时间与空间分数维（FD）的理论（定义在多重分形集上的$d_{\alpha}, \alpha=t,{\bf r})$被提出。FD 通过已知物理场的拉格朗日量的能量密度（使用 FD 泛函最小原理）确定。为了描述定义在多重分形集上的函数的行为，引入了分数阶 Riemann-Liouville 导数$D_{t}^{d(t)}$的推广，其微分阶数（取决于时间和坐标）等于分数维的值。对于$d_{t}=const$，广义分数阶导数（GFD）退化为普通的 Riemann-Liouville 积分泛函，当$d_{t}$接近整数时，GFD 可以用整数阶导数表示。对于具有分数维的时间和空间，提出了一种利用 GFD 研究理论物理广义方程的方法。使用 FD 泛函最小原理得到了定义在时间和空间多重分形集上的欧拉方程。作为例子，考虑了一个广义牛顿方程，并表明该方程与广义相对论的经典极限方程对于$d_{t} \to 1$相一致。讨论了关于排斥引力存在的几个备注。展示了在分形时间与空间理论框架下几何化所有已知物理场和力的可能性。 显示英文摘要

摘要： A theory of time and space with fractional dimensions (FD) of time and space ($d_{\alpha}, \alpha=t,{\bf r})$ defined on multifractal sets is proposed. The FD is determined (using principle of minimum the functionals of FD) by the energy densities of Lagrangians of known physical fields. To describe behaviour of functions defined on multifractal sets the generalizations of the fractional Riemann-Liouville derivatives $D_{t}^{d(t)}$ are introduced with the order of differentiation (depending on time and coordinate) being equal the value of fractional dimension. For $d_{t}=const$ the generalized fractional derivatives (GFD) reduce to ordinary Riemann-Liouville integral functionals, and when $d_{t}$ is close to integer, GFD can be represented by means of derivatives of integer order. For time and space with fractional dimensions a method to investigate the generalized equations of theoretical physics by means of GFD is proposed. The Euler equations defined on multifractal sets of time and space are obtained using the principle of the minimum of FD functionals. As an example, a generalized Newton equation is considered and it is shown that this equation coincide with the equation of classical limit of general theory of relativity for $d_{t} \to 1$. Several remarks concerning existence of repulsive gravitation are discussed. The possibility of geometrization all the known physical fields and forces in the frames of the fractal theory of time and space is demonstrated.