The origin of quantum behaviour (or equivalently, wave-particle duality) is an important problem for physics; moreover, Euclidean geometry and Riemannian geometry may be invalid if the small scales of real universe exhibit fractal structure. With this purpose, we attempt to develop a mathematical framework -call it the "non-local geometry"- and meanwhile propose a set of non-local calculus theory for analytically describing fractal (Euclidean geometry and Riemannian geometry are two special cases of fractal whenever the dimension equals an integer). Our study shows that the "Heisenberg Uncertainly Principle" and "non-local entanglement" would naturally emerge in the theoretical framework of non-local geometry. More interestingly, using the non-local geometry we show that if the dimension of time axis is slightly less than 1, then we can directly derive Planck's formula of energy quantum. This means that non-zero Planck's constant itself requires that the dimension of space-time is slightly less than 4; thus, our theory presents a natural explanation for the dimensional regularization of quantum field theory. Our further study shows that all computing results obtained by quantum field theory can be reproduced in the theoretical framework of non-local geometry. To discriminate our theory from current quantum field theory, we suggest a method of measuring the dimension of time axis.