Some Mathematical Consequences of the Postulates

Inge Svein Helland

A recent approach towards the foundation of quantum mechanics is described in some detail. This requires some mathematics, but it is claimed that this approach is easier to understand than the usual formal quantum foundation. Several consequences of the theory are described, consequences that for example also lead to a simpler and deeper understanding of Quantum Decision Theory, which has bearings both on our daily decisions and on the decisions made by political leaders. Another consequence is in the form of arguments for the existence of God, a God that is perfectly rational and has infinite knowledge. To understand these consequences, it is not necessary to understand all the mathematics, but simply to appreciate the fact that the theory is derived from 6 postulates, postulates that can be interpreted in several different directions. Two of these postulates, when interpreted in a very general setting, form the basis for my theological arguments. Finally, it is a hope that by reading this article, and then further discussing the mentioned consequences with other people, one may contribute to a better understanding between groups of people, groups that partly may have their basis in different cultures. These cultures may be of many kinds, also based upon different scientific communities. Or upon different religious groups.

R. Shankar

Chun‐Lei Tang

Basic scientific theories usually start with a set of hypotheses or “postulates.” There is generally no logical reason, apart from internal consistency, that can be given to justify such postulates absolutely. They come from ‘revelations’ in the minds of ‘geniuses,’ most likely with hints from Nature based on extensive careful observations. Their general validity can only be established through experimental verification. If numerous rigorously derived logical consequences of a very small set of postulates all agree with experimental observations without exception, one is inclined to accept these postulates as correct descriptions of the laws of Nature and use them confidently to explain and predict other natural phenomena. Quantum mechanics is no exception. It is based on a few postulates. For the purpose of the present discussion, we begin with three basic postulates involving: the “state functions,” “operators,” and “equations of motion.” In this chapter, this set of basic postulates and some of the corollaries and related definitions of terms are introduced and discussed. We will first simply state these postulates and introduce some of the related mathematical tools and concepts that are needed to arrive at their logical consequences later. To those who have not been exposed to the subject of quantum mechanics before, each of these postulates taken by itself may appear puzzling and meaningless at first. It should be borne in mind, however, that it is the collection of these postulates as a whole that forms the foundations of quantum mechanics.

Alexander I. Lvovsky