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

Abstract

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.

How to obtain a cosmological constant from small exotic R4

“…In addition exotic R 4 's can be described by operator algebras naturally appearing in QM [37]. Hence, if one agreed that the evolution of the universe began with a quantum regime where the standard QM description by a Hilbert space of states applies, the large scale differentiable structure should be an exotic R 4 rather than the standard 4-space as shown in [46]. Keeping this line of argumentation in mind, we conjecture that this exotic R 4 is central for the understanding of the small as well the large scale structure of the universe.…”

“…was described as the boundary of the compact subset K whereas Y n is given by 0−framed surgeries along nth untwisted Whitehead double of the pretzel knot 9 46 . The entire embedding E : R 4 → R 4 and the sequence {Y n } were determined from the failure of the h-cobordism and the explicit example of the non-smoothcobordant pair of 4-manifolds.…”

“…Using the result [20], every Whitehead double is also a hyperbolic link because of the splitting of the knot/link complements. The complement of a Whitehead double of 9 46 is the sum of C(K 9 46 ) and C(L W h ) both admitting a hyperbolic structure. Finally, the iteration of this case leads to the general case that C(K + ) admits also hyperbolic structure, i.e.…”

“…pretzel knot (−3, 3, −3) or the knot 946 in Rolfson notation producing the 3-manifold Y 1 by 0−framed Dehn surgery stages Casson handle, will result in the nth untwisted Whitehead double of the pretzel knot (−3, 3, −3) as in…”

Dimension 4: Quantum Origins of Spacetime Smoothness

“…[7,8]). Impressively, one can derive the value of cosmological constant (CC) which reasonably well matches the Planck experimental data [7][8][9][10]. Consequently, we are interested whether it is possible to formally justify the choice of exotic smoothness structure of spacetime.…”