- Tytuł:
- Quantum geometry, logic and probability
- Autorzy:
- Majid, Shahn
- Powiązania:
- https://bibliotekanauki.pl/articles/1047619.pdf
- Data publikacji:
- 2020-12-29
- Wydawca:
- Copernicus Center Press
- Tematy:
-
logic
noncommutative geometry
digital geometry
quantum gravity
duality
power set
Heyting algebra - Opis:
- 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.
- Źródło:
-
Zagadnienia Filozoficzne w Nauce; 2020, 69; 191-236
0867-8286
2451-0602 - Pojawia się w:
- Zagadnienia Filozoficzne w Nauce
- Dostawca treści:
- Biblioteka Nauki
Artykuł