Theory for Quantum Spacetime

TL;DR

The paper presents a bold unification of matter and spacetime within a 'variant' framework, where 1st-order variability of a higher-dimensional Clifford-structure space generates curved geometry and quantum dynamics. It develops two complementary representations—geometry/topological lattices and matrix/gauge networks—to describe quantum curved spacetime and derives a BF-style reformulation of the Einstein-Hilbert action, with the Planck and lattice scales linked via l0 = 2 l_p. By introducing non-unitary variants and AdS/CFT-like dualities, it extends the framework to AdS/NGT equivalence and a non-Hermitian holographic correspondence, offering a route to integrate gravity with quantum mechanics through a generalized symmetry and spacetime-charge language. The work also outlines a detailed scaffold for gravitational waves, causal structure, and holographic entanglement within this unified, variant-based picture, aiming to address long-standing puzzles in quantum gravity and AdS/CFT phenomenology.

Abstract

Quantum gravity (or quantum spacetime) is to unify general relativity and quantum mechanics into a single theoretical framework and presented as the most important open puzzle in fundamental physics. The development of a microscopic theory of quantum spacetime becomes the key problem about quantum gravity. This paper is the solution to this problem. The starting point of this paper is very simple -- physical variant with higher-order variability (see the below discussion). Based on this simple starting point, a microscopic theory for quantum spacetime is developed, including its matrix representation for quantum states, its time evolution, its geometry quantization, its generalized symmetry, its canonical quantization, and the uncertainty principle, black hole, AdS/CFT correspondence, scattering amplitudes of gravitons... The result leads to a great unification of matter and spacetime -- the particles constitute the basic blocks of spacetime and spacetime is really a multi-particle system that is made of matter. As a result, this work would help researchers to understand the mysteries in quantum gravity.

Figures (27)

• Figure 1: Invariance/symmetry can be regarded as shadow of variability
• Figure 2: An illustration for 1+1D flat quantum spacetime: (a) is geometry representation with 2D uniform topological lattice that is denoted by solid red spots. The lattice distance along spatial/tempo direction is Planck length/time ($l_{0}$/$t_{0}$). During an spatial/tempo shifting Planck length $l_{p}=l_{0}/2$ (or $t_{0}/2$), the phase change of the vacuum is $\pi$; (b) is the matrix representation with 2D uniform matrix network. The matrix network is described by $\Gamma_{\mathrm{flat}}^{\{N^{\mu},M^{\mu}\}}$ (or $\Gamma_{x}$ and $\Gamma_{t}$) on all links between two nearest-neighbor lattice sites (solid blue arrows).
• Figure 3: An illustration for 1+1D curved spacetime: (a) is the geometry representation with 2D deformed topological lattice that is denoted by solid red spots; (b) is the matrix representation with 2D deformed matrix network that are described by $\Gamma_{\mathrm{curved}}^{\{N^{\mu},M^{\mu}\}}$ (or $\Gamma_{x}^{\prime }(x,t)$ and $\Gamma_{t}^{\prime}(x,t)$) on all links between two nearest-neighbor lattice sites (solid blue arrows).
• Figure 4: An illustration of the triangular equivalence principle in quantum spacetime. This is an intrinsic relationship between “ Dirac (elementary particle) particle” (or the matter), “ changing of 3-volume” (or the quantum spacetime itself) and “ magnetic monopole” (or the topological defect of quantum spacetime). Here, $N_{F}$ denotes the number of particles, $q_{m}$ denotes the “ magnetic charge” in gauge representation of quantum spacetime, $\Delta V$ denotes the changing of 3-volume in 3D space of a quantum spacetime. $l_{0}$ is the lattice constant of the topological lattice with $l_{0}=2l_{p}$ where $l_{p}$ is Planck length.
• Figure 5: Classification of changes of a quantum spacetime -- shape changes (or the processes of curving spacetime) and contraction/expansion changes (or the processes of particle annihilation/generation).