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 \(\mathbb {R}\times F\) $\mathbb{R}\times F$ . More precisely, we show that the set \((I-\Delta _F)^{(d_s-1)/4+\alpha }(1+| x| ^2)^{(d_H+1)/4+\beta }L^2(\mathbb {R}\times F)\) ${(I-{\mathrm{\Delta }}_F)}^{(d_s-1)/4+\alpha }{(1+|x|}^2{)}^{(d_H+1)/4+\beta }{L}^2(\mathbb{R}\times F)$ is of the Gaussian measure one if \(\alpha >0\) $\alpha >0$ and \(\beta >0\) $\beta >0$ , while the set is of the Gaussian measure zero if \(\alpha >0\) $\alpha >0$ and \(\beta <0\) $\beta <0$ . Here, \(\Delta _F\) ${\mathrm{\Delta }}_F$ is the Laplacian on the underlying fractal space F, \(d_s\) $d_s$ is the spectral dimension of \(\Delta _F\) ${\mathrm{\Delta }}_F$ , and \(d_H\) $d_H$ is the Hausdorff dimension of F.