The Structure of the Real Line in Quantum Mechanics and Cosmology

Bielas, Krzysztof; Klimasara, Paweł; Król, Jerzy

doi:10.5506/aphyspolb.46.2375

Cited by 7 publications

(7 citation statements)

References 8 publications

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“…Without any cutoff imposed, this integral is UV-divergent. However, in our model, it holds that [8,9] Theorem 2. Zero-energy modes of quantum fields have vanishing contributions to the gravitational vacuum energy.…”

Section: The Physics Of Forcingmentioning

confidence: 84%

“…Moreover, forcing can be regarded as a technique for exploring the structure of the real line rather than proving independence results in formal set theory. Set-theoretic forcing as a tool in QM has already been identified and used by several authors, such as P. Benioff [4], G. Takeuti [5], M. Ozawa [6,7], W. Boos [2], R. Van Wesep [1], and the present authors [8][9][10][11][12].…”

Section: Classical Large-scale Coordinates From Quantum Regimementioning

confidence: 99%

“…The case of the energy operator discussed above shows that some forms of energy-namely those whose self-adjoint operators give rise to the non-atomless Boolean algebras as in Proposition 2-propagate in spacetime parametrized by the reals not extended by forcing. This cosmological model was indeed recently studied [8][9][10], where it was proposed that zero-energy modes of energy of quantum fields propagate in spacetime which is parametrized in a ZFC model not extended by forcing. Here, however, B Q , B P lead to the nontrivial extension of the model and the real line.…”

Section: The Physics Of Forcingmentioning

confidence: 99%

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From Quantum to Cosmological Regime. The Role of Forcing and Exotic 4-Smoothness

et al. 2017

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Abstract:Recently, a cosmological model based on smooth open 4-manifolds admitting non-standard smoothness structures was proposed. The manifolds are exotic versions of R 4 and S 3 × R. The model has been developed further and proven to be capable of obtaining some realistic cosmological parameters from these exotic smoothings. The important problem of the quantum origins of the exotic smoothness of space-time is addressed here. It is shown that the algebraic structure of the quantum-mechanical lattice of projections enforces exotic smoothness on R n . Since the only possibility for such a structure is exotic R 4 , it is found to be a reasonable explanation of the large-scale four-dimensionality of space-time. This is based on our recent research indicating the role of set-theoretic forcing in quantum mechanics. In particular, it is shown that a distributive lattice of projections implies the standard smooth structure on R 4 . Two examples of models valid for cosmology are discussed. The important result that the cosmological constant can be identified with the constant curvature of the embedding (exotic R 4 ) → R 4 is referred. . The calculations are in good agreement with the observed small value of the dark energy density.

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Section: The Physics Of Forcingmentioning

confidence: 84%

Section: Classical Large-scale Coordinates From Quantum Regimementioning

confidence: 99%

Section: The Physics Of Forcingmentioning

confidence: 99%

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From Quantum to Cosmological Regime. The Role of Forcing and Exotic 4-Smoothness

et al. 2017

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“…refs. 10,[74][75][76] ). We expect to develop systematically this approach, which again should have an impact (through forcing) on QM and gravity as phenomena reconciled in physical spacetime.…”

Section: Future Perspectives and Researchmentioning

confidence: 99%

Random World and Quantum Mechanics

Król¹,

Bielas²,

Asselmeyer-Maluga³

2021

Preprint

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Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

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“…Since it have been shown an unexpected connection of some set theoretical concepts with the quantum mechanics as well as in cosmology [33,34], the treatment in this scenario via this ideas is the following steps to explore, taking the formal models in ZFC (Zermelo-Fraenkel-Axiom of Choice) and hence the forcing as that special tool to make the shift from the micro to macro scale. Finally, others scenarios in quantum cosmology can be analyzed in order to explore the feasibility of the mathematical methods presented in this paper, however this is work that will be done elsewhere.…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

Asymptotically equivalent functions and ultrafilters applied to noncommutative quantum cosmology

2019

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Using the semi-classical approximation to the Wheeler-DeWitt equation obtained via Arnowitt-Deser-Misner (ADM) formalism in the Friedmann-Lemaître-Robertson-Walker (FLRW) model coupled to a scalar field and positive cosmological constant, and in the Kantowski-Sachs (KS) Universe, we introduced a deformation on the commutation relation for the minisuperspace variables and find an explicit semiclassical expression equivalent, in an adequate limit, to the solution with the aid of asymptotically equal functions and the theory of Ultrafilters, offering a suggestive alternative to sketch the behavior of the dynamical system involved without the need to solve it numerically.

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The Structure of the Real Line in Quantum Mechanics and Cosmology

Cited by 7 publications

References 8 publications

From Quantum to Cosmological Regime. The Role of Forcing and Exotic 4-Smoothness

From Quantum to Cosmological Regime. The Role of Forcing and Exotic 4-Smoothness

Random World and Quantum Mechanics

Asymptotically equivalent functions and ultrafilters applied to noncommutative quantum cosmology

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