Hitoshi Kitada's Home Page

List of Works

Current investigation

Quantum mechanics is formulated on the basis of two quantities which satisfy the non-commutation relation called canonical commutation relation, and those two quantities are identified as configuration operator and momentum operator. In usual formulation of quantum mechanics, it is assumed that the third quantity called time coordinate exists in addition to these two quantities. However, a careful examination of the formulation of quantum mechanics shows that the two quantities, configuration operator and momentum operator, are sufficient in formulating the quantum mechanics, which proves that the time coordinate is a redundant quantity. The formulation of quantum mechanics based on this fact introduces the concept of time as a quantity defined in terms of configuration and momentum operators, and we see that the usual uncertainty occurs as an uncertainty of time. This formulation of quantum mechanics gives a rigorous notion of time which has been considered as an a priori given quantity in physics in somewhat ambiguous manner since the age of Isaac Newton. Consequences of these considerations are that 1) quantum mechanics must be considered as a theory describing the internal motion of a local system consisting of a finite number of particles, and 2) the concept of time is a ‘local’ notion proper to each local system having two quantities satisfying the canonical commutation relation. That every local system has its own local time means that each local system is an unceasingly changing system with the inside components always moving. This local internal motion is the origin of the local time of each local system.

The rigorous formulation of these things is given through the investigation of the asymptotic behavior of the solutions of Schrödinger equations, which region is called scattering theory. The main ingredients used in the region are pseudodifferential operators, Fourier integral operators, and Functional Analysis. The origin of the local time, namely the existence of the local motion inside each local system is explained by reducing it to the result in metamathematics of the existence of undecidable propositions.

My recent investigation is on the last part, namely on the Gödel's incompleteness theorem. This has been begun around 2003. The theorem is thought to have shown the existence of undecidable propositions. To show the Gödel's theorem, it is usual to assume the finitary standpoint on the meta level, and the object number theory is investigated by this finitary method. As an extension if we assume that the meta and object levels are symmetric or reflexive, and discuss the object set theory ZFC (set theory with axiom of choice) by assuming ZFC on the meta level, it is shown that the object theory must have an uncountably infinite number of propositions. On the other hand by the definition of formal theory, the object ZFC theory has at most countable number of propositions, and we have a contradiction. This suggests that the cause of the contradiction is that we have assumed the axiom of infinity on the both levels. If as usual we assume finitary standpoint on the meta level and discuss object ZFC theory, there would not arise contradictions. However if we take the standpoint that the meta and object levels are reflexive, we have a conclusion that usual mathematics with axiom of infinity is inconsistent. In other words, if we take the standpoint that the meta and object levels of mathematics are symmetric or reflexive, only finite mathematics is consistent. Further investigation suggests a possibility that there is a problem of identifying the meta level and object level in the proof of Gödel's theorem. The problem is seen to be the one similar to the self-referential problems which we met in Russell's paradox and others. This would suggest the possibility that we might need to restrict mathematics to finite mathematics only.