Scalar quantum field theory on fractals Original Research Article

We investigate scale invariant measures over multiple variables for scalar field theories by imitating Wiener’s construction of the measure on the space of functions of one variable. We assign random fields values on the vertices of simple geometric shapes (triangles, squares, tetrahedra) which are subdivided into a finite number of similar shapes. We find several Gaussian measures with anomalous scaling associated with these field variables. A non-Gaussian fixed point arises from the Ising model on a fractal. In the continuum limit, we construct correlation functions that vary as a power of the distance. It is either a positive power (analogous to the Wiener process) or a negative power depending on the subdivision scheme used; however it is an irrational number for all the examples. This suggests that in the continuum limits it corresponds to quantum field theories (random fields) on spaces of fractional dimension.