Related papers: Some problems in mathematics and mathematical phys…
A discussion of different criteria of consistency of quantum field theory from the point of view of physics and mathematics.
These notes offer a basic introduction to the primary mathematical concepts of quantum physics, and their physical significance, from the operator and Hilbert space point of view, highlighting more what are essentially the abstract…
An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds.…
Theory of Riemann Extensions of the spaces with constant affine connection for the studying of the properties of nonlinear the first order systems of differential equations is proposed. Quadratic planar system of equations and the Lorenz…
We attempt to contribute some novel points of view to the "foundations of quantum mechanics", using mathematical tools from "quantum probability theory" (such as the theory of operator algebras). We first introduce an abstract algebraic…
The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures --such as triad and area operators-- exhibit a non-commutativity. At first…
This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
The development of Noncommutative Geometry is creating a reworking and new possibilities in physics. This paper identifies some of the commutation and derivation structures that arise in particle and field interactions and fundamental…
We develop a mathematical theory of quantization of multidimensional variational principles, and compare it with traditional constructions of quantum field theory. We conjecture that mathematical realization of quantum field theory axioms,…
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate…
What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and…
The Weltanschauung emerging from quantum theory clashes profoundly with our classical concepts. Quantum characteristics like superposition, entanglement, wave-particle duality, nonlocality, contextuality are difficult to reconcile with our…
A mathematically rigorous Hamiltonian formulation for classical and quantum field theories is given. New results include clarifications of the structure of linear fields, and a plausible formulation for nonlinear fields. Many mathematical…
Geometrical aspects of quantum computing are reviewed elementarily for non-experts and/or graduate students who are interested in both Geometry and Quantum Computation. In the first half we show how to treat Grassmann manifolds which are…
This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and…
We provide a rather extended introduction to the group field theory approach to quantum gravity, and the main ideas behind it. We present in some detail the GFT quantization of 3d Riemannian gravity, and discuss briefly the current status…
This talk is an introduction to ideas of non-commutative geometry and star products. We will discuss consequences for physics in two different settings: quantum field theories and astrophysics. In case of quantum field theory, we will…
We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector…
The central theme of this thesis is to study some aspects of noncommutative quantum mechanics and noncommutative quantum field theory. We explore how noncommutative structures can emerge and study the consequences of such structures in…