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Among the many disparate approaches towards quantum gravity, the reduction ofspacetime dimension in the ultraviolet regime is expected to be a commonthread. The dimensionality of spacetime can be defined in various contexts. Thespectral dimension of spacetime is defined in the context of diffusionprocesses on a manifold. We show that a spacetime with zero-point length hasspectral dimension $3.5$ when the characteristic diffusion time has the size ofthe zero-point length. The spectral dimension (negatively) diverges for an evenshorter diffusion time, thus preventing any reliable physical interpretation inthe deep ultraviolet regime. The thermodynamic dimension is defined byrealizing that the free energy $F(\beta)$ of a free field or ideal gas atfinite temperature ($\beta^{-1}$) in $D$ dimensions scales as $F\sim\beta^{-D}$. Using Schwinger's proper time formalism at finite temperature, weshow that for spacetime incorporating a zero-point length, the thermodynamicdimension reduces to $1.5$ near the Planck scale and then to $1$ in the deepultraviolet regime. This signifies a "phase-transition" in which a (massless)bosonic ideal gas in four dimensions essentially behaves like radiation($w=1/3$) at low energies, whereas near the Planck scale, it behavesequivalently to having an equation of state parameter $w=2$. Furthermore,dimension can be deduced from the potential $V_D(r)$ of interaction between twopoint-like sources separated at a distance $r$ as its scaling depends on $D$.Comparing with the scaling behavior of conventional Yukawa-like potentials atshort distances, we show that the zero-point length spacetime's ultravioletdimension appears to be either $2$ or $3$ depending on the use of massive ormassless force carriers as probes.

Ultraviolet dimensional reduction of spacetime with zero-point length