Full-text resources of CEJSH and other databases are now available in the new Library of Science.
Visit https://bibliotekanauki.pl

Results found: 5

first rewind previous Page / 1 next fast forward last

Search results

Search:
in the keywords:  noncommutative geometry
help Sort By:

help Limit search:
first rewind previous Page / 1 next fast forward last
1
Publication available in full text mode
Content available

Matematyka i kosmologia

100%
Heller M.
Zagadnienia Filozoficzne w Nauce
|
2012
|
issue 50
63-74
PL
The mathematical and cosmological works of a group associated with the Copernicus Center for Interdisciplinary Studies in Cracow are summarized. The group consists mainly of M. Heller, L. Pysiak, W. Sasin, Z. Odrzygóźdź and J. Gruszczak. The first paper by members of the group was published in 1988, and research has been continued to the present day. The main mathematical tool used in the first part of the group’s activity was the theory of differential spaces and, in the second, methods of noncommutative geometry. Among the main topics investigated have been classical singularities in relativistic cosmology and the unification of general relativity with quantum mechanics.
2
Publication available in full text mode
Content available

The existence of singularities and the origin of space-time

100%
Heller M.
Zagadnienia Filozoficzne w Nauce
|
2008
|
issue 43
35-43
EN
Methods of noncommutative geometry are applied to deal with singular space-times in general relativity. Such space-times are modeled by noncommutative von Neumann algebras of random operators. Even the strongest singularities turn out to be probabilistically irrelevant. Only when one goes to the usual (commutative) regime, via a suitable transition process, space-time emerges and singularities become significant.
3
Publication available in full text mode
Content available

Geneza prawdopodobieństwa

75%
Heller M.
Zagadnienia Filozoficzne w Nauce
|
2006
|
issue 38
61-75
PL
After briefly reviewing classical and quantum aspects of probability, basic concepts of the noncommutative calculus of probability (called also free calculus of probability) and its possible application to model the fundamental level of physics are presented. It is shown that the pair (M, *), where M is a (noncommutative) von Neumann algebra, and a state on it, is both a dynamical object and a probabilistic object. In this way, dynamics and probability can be unified in noncommutative geometry. Some philosophical consequences of such an approach are indicated.
4
Publication available in full text mode
Content available

Nieprzemienne rachunki prawdopodobieństwa

75%
Heller M.
Zagadnienia Filozoficzne w Nauce
|
2010
|
issue 47
38-53
PL
The paper can be regarded as a short and informal introduction to noncommutative calculi of probability. The standard theory of probability is reformulated in the algebraic language. In this form it is readily generalized to that its version which is virtually present in quantum mechanics, and then generalized to the so-called free theory of probability. Noncommutative theory of probability is a pair (M, φ) where M is a von Neumann algebra, and φ a normal state on M which plays the role of a noncommutative probability measure. In the standard (commutative) theory of probability, there is, in principle, one mathematically interesting probability measure, namely the Lebesgue measure, whereas in the noncommutative theories there are many nonequivalent probability measures. Philosophical implications of this fact are briefly discussed.
5
Publication available in full text mode
Content available

Quantum geometry, logic and probability

75%
Majid S.
Zagadnienia Filozoficzne w Nauce
|
2020
|
issue 69
191-236
EN
Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = (−Δθ + q − p)f for the graph Laplacian Δθ, potential functions q, p built from the probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ı(−Δ + V )ψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations.
first rewind previous Page / 1 next fast forward last
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.