Ultraviolet dimensional reduction of spacetime with zero-point length

Abstract

Among the many disparate approaches towards quantum gravity, the reduction of spacetime dimension in the ultraviolet regime is expected to be a common thread. The spectral dimension of spacetime is defined in the context of diffusion processes on a manifold. We show that a spacetime with zero-point length has spectral dimension 3.5 when the characteristic diffusion time has the size of the zero-point length. The spectral dimension (negatively) diverges for an even shorter diffusion time, thus preventing reliable physical interpretation in the deep ultraviolet regime. The thermodynamic dimension is defined by realizing that the free energy $$F(\beta )$$ F ( β ) of a free field or ideal gas at finite temperature ($$\beta ^{-1}$$ β - 1 ) in D dimensions scales as $$F\sim \beta ^{-D}$$ F ∼ β - D . Using Schwinger’s proper time formalism, we show that for spacetime incorporating a zero-point length, the thermodynamic dimension reduces to 1.5 near the Planck scale and then to 1 in the deep ultraviolet regime. This signifies a “phase-transition” in which a (massless) bosonic ideal gas in four dimensions essentially behaves like radiation ($$w=1/3$$ w = 1 / 3 ) at low energies, whereas near the Planck scale, it behaves equivalently to having an equation of state parameter $$w=2$$ w = 2 . Furthermore, dimension can be deduced from the potential $$V_D(r)$$ V D ( r ) of interaction between two point-like sources separated at a distance r as its scaling depends on D. Comparing with the scaling behavior of conventional Yukawa-like potentials at short distances, we show that the ultraviolet dimension appears to be either 2 or 3 depending on the use of massive or massless force carriers as probes.

Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A ∗

“…However, T-duality is not an exclusive property of string theory, and has been observed to emerge quite generally in a number of other systems [1530], wherein the path integral is invariant under exchange of the infinitesimal path length Dx → 1/Dx. Recently, an important application of T-duality consisted in the formulation of a finite electrodynamics theory driven by a T-dual photon propagator [1531,1532], which naturally incorporates a minimum length ℓ 0 . String-inspired T-duality corrections to BH space-times have been studied in [1533], and subsequently in [1534] by one of us.…”

Discrete time heat kernel and UV modified propagators with dimensional deconstruction

“…In recent years, various ideas have been discussed to consider [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] how to take changes in the Schwinger's proper-time parameter of the heat kernel [21,22] for a certain Laplace operator as a way to improve ultraviolet (UV) behavior in quantum field theory. In our previous paper [19], we proposed a method to derive propagators (Green's functions) with moderate behavior at an infinitesimally short distance by discretizing the proper-time parameter and adjusting the range of summation.…”

Charged black holes from T-duality

“…The second application, has been the formulation of a finite electrodynamics governed by a T-dual photon field propagator [34,35]. The corresponding electrostatic potential has been found to be V ∼ 1/r D−3 .…”