Support of the free measure for quantum field on fractal space-time

In constructive quantum theory, the free field is constructed based on a Gaussian measure on the space of tempered distributions. We generalize the classic results about support property of the Gaussian measure from Euclidean space-time to fractal space-time \(\mathbb {R}\times F\). More precisely, we show that the set \((I-\Delta _F)^{(d_s-1)/4+\alpha }(1+| x| ^2)^{(d_H+1)/4+\beta }L^2(\mathbb {R}\times F)\) is of the Gaussian measure one if \(\alpha >0\) and \(\beta >0\), while the set is of the Gaussian measure zero if \(\alpha >0\) and \(\beta <0\). Here, \(\Delta _F\) is the Laplacian on the underlying fractal space F, \(d_s\) is the spectral dimension of \(\Delta _F\), and \(d_H\) is the Hausdorff dimension of F.