We study transition probabilities of generalized functions as introduced by Colombeau and Gsponer. We formally introduz the study of H. Bohr almost periodic functions in the generalized context and use them to give exact values of transition probabilities in dimension 2. We also prove the existence of transition probabilities, in the generalized sense, for any moderate net $T = (T_{\varepsilon})$ with $T_{\varepsilon} : \mathbb H \rightarrow \mathbb H$ , $\mathbb H$ a Hilbert space. In particular, we consider the case of selfadjoint Hilbert-Schmidt operators. This alludes to the possibility of existence of transition probabilities in the Fock space, in the sense introduced by Colombeau and Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. This was already indicated in two papers coauthored by J. Aragona and J.F. Colombeau et. al.We study transition probabilities of generalized functions as introduced by Colombeau and Gsponer. We formally introduz the study of H. Bohr almost periodic functions in the generalized context and use them to give exact values of transition probabilities in dimension 2. We also prove the existence of transition probabilities, in the generalized sense, for any moderate net $T = (T_{\varepsilon})$ with $T_{\varepsilon} : \mathbb H \rightarrow \mathbb H$, $\mathbb H$ a Hilbert space. In particular, we consider the case of selfadjoint Hilbert-Schmidt operators. This alludes to the possibility of existence of transition probabilities in the Fock space, in the sense introduced by Colombeau and Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. This was already indicated in two papers coauthored by J. Aragona and J.F. Colombeau et. al.

arXiv:2305.08862Jan 1, 20230 citations

First-order quantum perturbation theory and Colombeau generalized functions

André Gsponer

The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a `Dirac delta-function,' an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to `Fermi's golden rule.' In this paper these conventional calculations are carefully reviewed, and their crucial parts reformulated in a Colombeau algebra -- in which the product of distributions is mathematically well defined. The conclusions are: (1) The Dirac delta-function insuring energy conservation in first order perturbation theory belongs to a particular subset of representatives of the Schwartz distribution defined by the Dirac measure. These particular representatives have a well-defined square, and lead to a physically meaningful result in agreement with the data. (2) A truly consistent mathematical interpretation of these representatives is provided by their redefinition as Colombeau generalized functions. This implies that their square, and therefore the quantum mechanical rule leading from amplitudes to probabilities, is rigorously defined.The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a `Dirac delta-function,' an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to `Fermi's golden rule.' In this paper these conventional calculations are carefully reviewed, and their crucial parts reformulated in a Colombeau algebra -- in which the product of distributions is mathematically well defined. The conclusions are: (1) The Dirac delta-function insuring energy conservation in first order perturbation theory belongs to a particular subset of representatives of the Schwartz distribution defined by the Dirac measure. These particular representatives have a well-defined square, and lead to a physically meaningful result in agreement with the data. (2) A truly consistent mathematical interpretation of these representatives is provided by their redefinition as Colombeau generalized functions. This implies that their square, and therefore the quantum mechanical rule leading from amplitudes to probabilities, is rigorously defined.

arXiv:0809.2576Sep 15, 20085 citations

MARKOV STATES ON QUASI–LOCAL ALGEBRAS

Francesco Fidaleo

Quantum Probability and Infinite Dimensional AnalysisJul 1, 20070 citations

Quantum information V : proceedings of the fifth conference, Meijo University, Japan, 17-19 December 2001

武幸飛田, Kimiaki Saitô

On Gaussian and Poisson White Noises (N Asai) Renormalization, Orthogonalization, and Generating Functions (I Kubo et al.) Insider Trading in Continuous Time (E Barucci et al.) Existence, Uniqueness, Consistency and Dependency on Diffusion Coefficients of Generalized Solutions of Nonlinear Diffusion Equations in Colombeau's Algebra (H Deguchi) On Mathematical Treatment of Quantum Communication Gate on Fock Space (W Freudenberg et al.) A Frontier of White Noise Analysis (T Hida) An Interacting Fock Space with Periodic Jacobi Parameter Obtained from Regular Graphs in Large Scale Limit (A Hora & N Obata) Error Exponents of Codings for Stationary Gaussian Channels (S Ihara) White Noise Analysis on Classical Wiener Space Revisited (Y-J Lee & H-H Shih) Fractional Brownian Motions and the L vy Laplacian (K Nishi et al.) Recognition and Teleportation (M Ohya) Jump Finding of a Stable Process (Si Si et al.) On Entropy Production of a One-Dimensional Lattice Conductor (S Tasaki) Quantum Information and Spacetime Structure (I V Volovich).On Gaussian and Poisson White Noises (N Asai) Renormalization, Orthogonalization, and Generating Functions (I Kubo et al.) Insider Trading in Continuous Time (E Barucci et al.) Existence, Uniqueness, Consistency and Dependency on Diffusion Coefficients of Generalized Solutions of Nonlinear Diffusion Equations in Colombeau's Algebra (H Deguchi) On Mathematical Treatment of Quantum Communication Gate on Fock Space (W Freudenberg et al.) A Frontier of White Noise Analysis (T Hida) An Interacting Fock Space with Periodic Jacobi Parameter Obtained from Regular Graphs in Large Scale Limit (A Hora & N Obata) Error Exponents of Codings for Stationary Gaussian Channels (S Ihara) White Noise Analysis on Classical Wiener Space Revisited (Y-J Lee & H-H Shih) Fractional Brownian Motions and the L vy Laplacian (K Nishi et al.) Recognition and Teleportation (M Ohya) Jump Finding of a Stable Process (Si Si et al.) On Entropy Production of a One-Dimensional Lattice Conductor (S Tasaki) Quantum Information and Spacetime Structure (I V Volovich).

WORLD SCIENTIFIC eBooksJan 1, 20060 citations

T = (T_{\varepsilon})

with

T_{\varepsilon} : \mathbb H \rightarrow \mathbb H

\mathbb H

a Hilbert space. In particular, we consider the case of selfadjoint Hilbert-Schmidt operators. This alludes to the possibility of existence of transition probabilities in the Fock space, in the sense introduced by Colombeau and Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. This was already indicated in two papers coauthored by J. Aragona and J.F. Colombeau et. al.