Remarks on Mathematical Foundations of Quantum Mechanics

Abstract

We show that the set-theoretic forcing is the essential part of the continuous measurement of a suitably rich Boolean algebra of quantum observables. The Boolean algebra structure of quantum observables enables us to give a classical and geometric meaning to the results of measurements of the observables. The measurement takes place in the semiclassical state of the system which is the generic filter added by a forcing to the ZFC model based on the Borel measure algebra. The analogue of the semiclassical state (the pseudoclassical state) was described by Wesep in 2006 in his studies of the local hidden variables program in quantum mechanics.

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

“…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.…”

“…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].…”

“…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.…”

The Structure of the Real Line in Quantum Mechanics and Cosmology

“…This section is the summary of our previous article regarding the forcing emerging from QM. An interested reader is referred to [6] for a more detailed treatment. A forcing can be seen as the deriving property of some Boolean algebra 1 .…”

“…A forcing can be seen as the deriving property of some Boolean algebra 1 . By B, we denote some Boolean algebra of projections chosen from the lattice of projections L(H) on a Hilbert space H. The spectral theorem, in general, gives the correspondence between the algebra of self-adjoint (s-a) operators which are in the Boolean algebra B [6,7] and the measure algebra defined on the Borel subsets of X = R 3 . Actually, there exists the isomorphism between the algebra B generating the family of s-a operators {A α } and the measure algebra of the space 2 (X, µ).…”