Scaling-Based Quantization of Spacetime Microstructure

TL;DR

The paper proposes a scale-based quantization of spacetime, promoting direction-dependent scale factors $a_eta$ (and second-order $b_eta$) as the fundamental quantum degrees of freedom on a two-tier scale geometry: a first-order scale manifold $( ext{M}^{(a)},\nhat{g})$ with coordinates $X^eta$ and a second-order amplitude manifold $( ext{M}^{(b)}, ilde{g})$ with coordinates $a_eta$. It develops a covariant differential-geometry framework on this structure, derives deformed commutators and locally scaled field equations, and carries out a canonical quantization of the first-order fluctuations, resulting in discrete harmonic-oscillator modes with a scale-dependent zero-point energy. A micro-area operator constructed from scale fluctuations yields a state-counting that reproduces the leading Bekenstein-Hawking entropy and provides a microscopic mechanism linking spacetime microstructure to black-hole thermodynamics. The framework also introduces a geometric renormalization-group flow for scale variables, suggesting a dynamical suppression of UV contributions and potential insights into the cosmological constant problem. Overall, the work offers a self-consistent, covariant route to encode Planck-scale spacetime fluctuations and to connect microscopic geometry with phenomenology, while outlining substantial avenues for further mathematical and experimental exploration.

Abstract

Planck-scale physics challenges the classical smooth-spacetime picture by introducing quantum fluctuations that imply a nontrivial spacetime microstructure. We present a framework that encodes these fluctuations by promoting local scale factors, rather than the metric tensor, to fundamental dynamical variables while preserving general covariance. The construction employs a two-tiered hierarchy of scale manifolds, comprising a first-order manifold of scale coordinates and a second-order manifold of fluctuation amplitude coordinates. On the first-order manifold, we formulate differential geometry, field equations, and a canonical quantization procedure. The theory yields a geometric renormalization-group flow for scale variables and implies spacetime discreteness at the microscopic level. By constructing a quadratic action and performing spectral decomposition with a stabilizing potential, we obtain discrete modal degrees of freedom quantized as harmonic oscillators. The framework proposes a microscopic description for zero-point energy of spacetime and explores implications for vacuum energy and ultraviolet regularization, suggesting a potential dynamical mechanism that could ameliorate the cosmological constant problem. Main results include a generalized uncertainty relation with scale-dependent coefficients, locally scaled Klein-Gordon and Dirac equations, geodesic equations for scale spacetime, and a microscopic area operator whose state counting is consistent with the Bekenstein-Hawking entropy. This work develops a scale-based quantization procedure, providing a foundation for further mathematical analysis and phenomenological tests of spacetime quantization.

Figures (5)

• Figure 1: (Color online) Curves of (a) $\zeta^\alpha(X^\alpha)\sim X^\alpha$, (b) $L^\alpha(X^\alpha)\sim X^\alpha$, and (c) $\bar{X}^\alpha(X^\alpha)\sim X^\alpha$ for $\bar{X}^\alpha_0=0$ and an arbitrary choice of $|\bar{X}^\alpha_0|=1.6$.
• Figure 2: (Color online) The schematic diagram of microscopic spacetime fluctuation measurements when $\bar{X}^\alpha_0<0$ and $X^\alpha>0$. Where, $\bar{X}^\alpha_0/2<L_{-}\leq\bar{X}^\alpha_0$ and $L_{+}\geq\bar{X}^\alpha_0$. For $\bar{X}^\alpha_0>0$, the situation is the same but opposite direction.
• Figure 3: (Color online) Curves of (a) $\beta(L^\alpha(X^\alpha))\sim L^\alpha(X^\alpha)$, (b) $\beta(L^\alpha(X^\alpha))\sim X^\alpha$ and (c) $\beta(L^\alpha(X^\alpha))\sim X^\alpha$ of zoomed scale for $\bar{X}^\alpha_0=0$ and an arbitrary choice of $|\bar{X}^\alpha_0|=1.6$.
• Figure 4: (Color online) Curves of (a) $d\beta(L^\alpha)/dL^\alpha\sim X^\alpha$, (b) $d\beta(L^\alpha)/dL^\alpha\sim L^\alpha$ for $\bar{X}^\alpha_0=0$ and an arbitrary choice of $|\bar{X}^\alpha_0|=1.6$.
• Figure 5: (Color online) Curves of (a) $a_\alpha\sim X^\alpha$, (b) $b_\alpha\sim X^\alpha$, (c) $a_\alpha b_\alpha\sim X^\alpha$, (d) $dX^\alpha\sim X^\alpha$, and (e) $-d a_\alpha\sim X^\alpha$, for an arbitrary choice of $|\bar{X}^\alpha_0|=1.6$.