Is mathematics consistent?

A question is proposed whether or not set theory is consistent.Comment: 8 pages, LaTe

Fundamental solution global in time for a class of Schr\"odinger equations with time-dependent potentials

Fundamental solution for a Schr\"odinger equation with a time-dependent potential of long-range type is constructed. The solution is given as a Fourier integral operator with a symbol uniformly bounded global in time, when measured in natural semi-norms of a symbol class.Comment: To appear in Communications in Mathematical Analysis, Volume 1 No. 2 (2006

An implication of G\"odel's incompleteness theorem

A proof of G\"odel's incompleteness theorem is given. With this new proof a transfinite extension of G\"odel's theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the G\"odel numbering. An implication of these considerations is stated.Comment: LaTeX, 50 page

Does Church-Kleene ordinal ω1CK\omega_1^{CK} exist?

A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal $\omega_1^{CK}$ really exists.Comment: 4 pages, LaTe

Quantum mechanical time contradicts the uncertainty principle

The a priori time in conventional quantum mechanics is shown to contradict the uncertainty principle. A possible solution is given.Comment: 3 page

Quantum Mechanical Clock and Classical Relativistic Clock

A cyclic nature of quantum mechanical clock is discussed as ``quantization of time." Quantum mechanical clock is seen to be equivalent to the relativistic classical clock.Comment: LaTeX, 8 page

Scattering Spaces and a Decomposition of Continuous Spectral Subspace

We introduce the notion of scattering space $S_b^r$ for $N$-body quantum mechanical systems, where $b$ is a cluster decomposition with $2\le |b|\le N$ and $r$ is a real number $0\le r\le 1$. Utilizing these spaces, we give a decomposition of continuous spectral subspace by $S_b^1$ for $N$-body quantum systems with long-range pair potentials V_\alpha^L(x_\alpha)=O(|x_\al|^{-\ep}). This is extended to a decomposition by $S_b^r$ with $0\le r\le 1$ for some long-range case. We also prove a characterization of ranges of wave operators by $S_b^0$.Comment: 36 pages, AmSTe

Quantum Mechanics and Relativity --- Their Unification by Local Time ---

A quantum-mechanical Hamiltonian with a gravitational potential is derived in the framework of local times. This Hamiltonian is the one used by E. H. Lieb (Bull. Amer. Math. Soc. 22(1990), 1-49) in his explanation of stability and instability of cold stars. Our procedure deriving the Hamiltonian is based on an analysis of the process of the observation with the background of the notion of quantum-mechanical local times compatible with general theory of relativity.Comment: 31 pages, AmSTe

A possible solution for the non-existence of time

A possible solution for the problem of non-existence of universal time is given by utilizing Goedel's incompleteness theorem.Comment: LaTeX, 7 pages, English corrected and a miscellaneous change in section

Theory of Local Times II. Another formulation and examples

The model of a stationary universe and the notion of local times presented in [10] are reviewed with some alternative formulation of the consistent unification of the Riemannian and Euclidean geometries of general relativity and quantum mechanics. The method of unification adopted in the present paper is by constructing a vector bundle $X\times R^6$ or $X\times R^4$ with $X$ being the observer's reference frame and $R^6$ or $R^4$ being the unobservable inner space(-time) within each observer's local system. Some applications of our theory to two concrete examples of human size and of cosmological size are discussed, as well as the uncertainty of time in our context is calculated.Comment: 22 pages, LaTeX, revised in minor description