Geometry and the Quantum: Basics

Abstract

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.

33 pages, 2 figures

References in corpus (6)

• A Short Survey of Noncommutative Geometry
• Why the Standard Model
• Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification
• The Uncanny Precision of the Spectral Action
• 4-manifolds as covers of the 4-sphere branched over non-singular surfaces
• Universal measurability and the Hochschild class of the Chern character

Cited by in corpus (26)

• Quanta of Geometry: Noncommutative Aspects
• Spectral truncations in noncommutative geometry and operator systems
• Grand Unification in the Spectral Pati-Salam Model
• Asymptotically Free Mimetic Gravity
• A survey of spectral models of gravity coupled to matter
• Ghost Free Mimetic Massive Gravity
• The Unified Standard Model
• Understanding truncated non-commutative geometries through computer simulations
• Discrete Gravity
• Noncommutativity and Physics: A non-technical review
• Noncommutative geometry and stochastic processes
• A new algebraic structure in the standard model of particle physics
• Unification of gravity and quantum field theory from extended noncommutative geometry
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Quanta of Space-Time and Axiomatization of Physics
• Quanta of Geometry and Unification
• The spectral torsion for the Connes type operator
• Quantized Noncommutative Riemann Manifolds and Stochastic Processes: The theoretical foundations of the square root of Brownian motion
• Implications of the complex Singlet field for Noncommutative Geometry model
• The Möbius Symmetry of Quantum Mechanics
• Computational explorations of a deformed fuzzy sphere
• A Novel Approach To Particle Representations
• Noncommutative Geometry and Structure of Space-Time
• Parcels of Universe or why Schrödinger and Fourier are so relatives?
• Strict Almost Non-Abelian Deformations
• Origin of fermion generations from extended noncommutative geometry