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A geometric approach to some quantum statistical systems (including the harmonic oscillator) is presented. We regard the (N+1)-dimensional Euclidean {\it coordinate} system (X$^i$,$\tau$) as the quantum statistical system of N quantum…
In a wide range of quantum gravity theories, quasiclassical geometries, which are solutions to the Einstein field equations approximately, are described by "coherent states." Here we propose a Hamiltonian formalism for gravitational…
The Hamiltoinian analysis of the vector-tensor theory of gravity is performed. The resulting geometrical dynamics is reformulated into the connection dynamics, with the real SU(2)-connection serving as one of the configuration variables.…
We start with the Hamiltonian formulation of the first order action of pure gravity with a full $\mathfrak{sl}(2,\mathbb C)$ internal gauge symmetry. We make a partial gauge-fixing which reduces $\mathfrak{sl}(2,\mathbb C)$ to its…
The symplectic quantization scheme proposed for matter scalar fields in the companion paper "Symplectic quantization I" is generalized here to the case of space-time quantum fluctuations. Symplectic quantization considers an explicit…
Mechanical lattices support topological wave phenomena governed by geometric phases. We develop a compact Hilbert space description for one-dimensional elastic chains, expressing intra-cell motion as a normalized superposition of orthogonal…
Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the…
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a…
An algebraic framework was introduced in our previous works to address the covariance issue in spherically symmetric effective quantum gravity. This paper extends the framework to the electrovacuum case with a cosmological constant. After…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
In a novel application of the tools of topological data analysis (TDA) to nonperturbative quantum gravity, we introduce a new class of observables that allows us to assess whether quantum spacetime really resembles a ``quantum foam" near…
Treating the gravitational force on the same footing as the electroweak and strong forces, we present a quantum field theory of gravity based on spin and scaling gauge symmetries. A biframe spacetime is initiated to describe such a quantum…
Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature, but with global degrees of freedom related to the topology of the universe. We present an exact Hamiltonian lattice theory for topological…
This paper presents the extension from flat spacetime into curved spacetime of the area of theoretical investigation that has been known as topological gauge field theory. The extension here presented is based upon a new derivation of the…
We present a complete quantization of an approximately homogeneous and isotropic universe with small scalar perturbations. We consider the case in which the matter content is a minimally coupled scalar field and the spatial sections are…
These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by…
So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the…
We develop a systematic classical framework to accommodate canonical quantization of geometric and matter perturbations on a quantum homogeneous isotropic flat spacetime. The existing approach of standard cosmological perturbations is…
We consider quantum phase transitions with global symmetry breakings that result in the formation of topological defects. We evaluate the number densities of kinks, vortices, and monopoles that are produced in $d=1,2,3$ spatial dimensions…
In this paper, we address a foundational challenge in quantum field theory on curved spacetime by developing a consistent framework within loop quantum gravity. We introduce a methodology for defining meaningful superpositions of quantum…