Related papers: Physical Principles Based on Geometric Properties
We describe a geometric and symmetry-based formulation of the equivalence principle in non-relativistic physics. It applies both on the classical and quantum levels and states that the Newtonian potential can be eliminated in favor of a…
To apply the abstract quantum formalism to a particular physical system, one must specify the precise form of the relevant measurement and symmetry transformation operators. These operators are determined by a set of rules, the…
We investigate the possibility that a background independent quantum theory of gravity is not a theory of quantum geometry. We provide a way for global spacetime symmetries to emerge from a background independent theory without geometry. In…
We show that the principles of a ''complete physical theory'' and the conclusions of the standard quantum mechanics do not irreconcilably contradict each other as is commonly believed. In the algebraic approach, we formulate axioms that…
The four-dimensional gauge group of general relativity corresponds to arbitrary coordinate transformations on a four-manifold. Theories of gravity with a dynamical structure remarkably like Einstein's theory can be obtained on the basis of…
The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kaehler manifold. The classical principles of Quantum Mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability…
Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of…
If one takes seriously the postulate of quantum mechanics in which physical states are rays in the standard Hilbert space of the theory, one is naturally lead to a geometric formulation of the theory. Within this formulation of quantum…
In this essay, we wish to propose a general principle: \it{the equation of motion or dynamics of a fundamental force should not be prescribed but instead be entirely driven by geometry of the appropriate spacetime manifold, and the equation…
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…
Our aim in this review article is to present the applications of Connes' noncommutative geometry to elementary particle physics. Whereas the existing literature is mostly focused on a mathematical audience, in this article we introduce the…
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. As a key feature, the laws are formulated in terms of quantum events rather than of particle states. Temporal and spatial…
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.…
Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows…
In this chapter we take up the quantum Riemannian geometry of a spatial slice of spacetime. While researchers are still facing the challenge of observing quantum gravity, there is a geometrical core to loop quantum gravity that does much to…
It is shown that the equations of relativistic Bohmian mechanics for multiple bosonic particles have a dual description in terms of a classical theory of conformally "curved" space-time. This shows that it is possible to formulate quantum…
Noncommutative or `quantum' differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and…
We study quantum statistical inference tasks of hypothesis testing and their canonical variations, in order to review relations between their corresponding figures of merit---measures of statistical distance---and demonstrate the crucial…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally,…