Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMissing the point in noncommutative geometry
63
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale - and ultimately the concept of a point - makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal-Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.
rate research
Read More
We study some consequences of noncommutativity to homogeneous cosmologies by introducing a deformation of the commutation relation between the minisuperspace variables. The investigation is carried out for the Kantowski-Sachs model by means of a comparative study of the universe evolution in four different scenarios: the classical commutative, classical noncommutative, quantum commutative, and quantum noncommutative. The comparison is rendered transparent by the use of the Bohmian formalism of quantum trajectories. As a result of our analysis, we found that noncommutativity can modify significantly the universe evolution, but cannot alter its singular behavior in the classical context. Quantum effects, on the other hand, can originate non-singular periodic universes in both commutative and noncommutative cases. The quantum noncommutative model is shown to present interesting properties, as the capability to give rise to non-trivial dynamics in situations where its commutative counterpart is necessarily static.
A classic problem in general relativity, long studied by both physicists and philosophers of physics, concerns whether the geodesic principle may be derived from other principles of the theory, or must be posited independently. In a recent paper [Geroch & Weatherall, The Motion of Small Bodies in Space-Time, Comm. Math. Phys. (forthcoming)], Bob Geroch and I have introduced a new approach to this problem, based on a notion we call tracking. In the present paper, I situate the main results of that paper with respect to two other, related approaches, and then make some preliminary remarks on the interpretational significance of the new approach. My main suggestion is that tracking provides the resources for eliminating point particles---a problematic notion in general relativity---from the geodesic principle altogether.
We discuss the notion about physical quantities as having values represented by real numbers, and its limiting to describe nature to be understood in relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values on a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutating algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex number, the idea makes reasonable sense theoretically as well as practically.
We review a gravitational model based on noncommutative geometry and the spectral action principle. The space-time geometry is described by the tensor product of a four-dimensional Riemanian manifold by a discrete noncommutative space consisting of only two points. With a specific choice of the finite dimensional involutive algebra, the noncommutative spectral action leads to the standard model of electroweak and strong interactions minimally coupled to Einstein and Weyl gravity. We present the main mathematical ingredients of this model and discuss their physical implications. We argue that the doubling of the algebra is intimately related to dissipation and the gauge field structure. We then show how this noncommutative spectral geometry model, a purely classical construction, carries implicit in the doubling of the algebra the seeds of quantization. After a short review on the phenomenological consequences of this geometric model as an approach to unification, we discuss some of its cosmological consequences. In particular, we study deviations of the Friedmann equation, propagation of gravitational waves, and investigate whether any of the scalar fields in this model could play the role of the inflaton.
We illustrate an isomorphic description of the observable algebra for quantum mechanics in terms of functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product with explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base to essentially translate the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry, hence obtaining the latter from the physical theory itself. We have essentially an extended formalism of the Schrodinger versus Heisenberg picture which we try to describe mathematically as a coordinate map from the phase space, which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry as coordinated by the six position and momentum operators. The observable algebra is taken as an algebra of functions on the latter operators. We advocate the idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of quantum (phase) space. Issues about the kind of noncommutative geometry obtained are also explored.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
