Mathematics and Interpretation

Euich Miztani

Chong Wang

The ensemble interpretation attributes the wave appearances of particles to their statistical characteristics. This has increasingly interested scientists. However, the ensemble interpretation is still not a scientific theory based on mathematics. Here, based on the double-slit experiment, a mathematical framework for the ensemble interpretation is constructed. The Schrodinger equation and the de-Broglie equation are also deduced. Analysis shows that the wave appearance of particles is caused by the statistical properties of these particles; the nature of the wave function is the average least action for the particles in a position.

Roland L. Omnes

Selçuk Ş. Bayın

Bihui Li

A standard approach towards interpreting physical theories proceeds by first identifying the theory with a set of mathematical objects, where such objects are defined according to mathematicians’ standards of rigor. In making this identification, philosophers rule out the relevance of many inferential methods that physicists use, as these often do not meet mathematicians’ standards of rigor. Philosophers thus sanitize physical theories of all math- ematically messy or ambiguous parts before interpreting them. My dissertation argues against this sanitized approach towards interpreting theories using the example of quantum field theory (QFT). When we look at the details of QFT, we find that the mathematical objects it requires differ according to the specific systems the theory is being applied to in ways that advocates of the sanitized approach do not anticipate. Furthermore, the mathematical objects required for successful application are still being developed in some applicational contexts, so it would be unwise to determine in advance which objects constitute the theory. During this ongoing developmental process, physicists interpret the mathematics using strategies that violate the standards of pure mathematics. In contrast to the sanitized approach, these strategies are more sensitive to the ways in which the mathematics required for the relevant contexts is still under development. I argue that these strategies are not merely instrumental. They suggest alternative approaches to interpretation that philosophers should take into account.

Casey Blood

The linear mathematics of quantum mechanics gives many versions of reality instead of the single version we perceive, with the perceived version chosen at random according to a probability law. Because of these peculiarities, the theory requires an interpretation to be fully understood. Over 50 years ago, Everett proposed in his many-worlds interpretation that these characteristics could be accounted for if the mathematics itself, with no collapse or hidden variables, was carefully analyzed. We show this is incorrect; the linear mathematics cannot account for the probability law. Thus the many-worlds interpretation is not viable. Some mechanism, such as collapse or hidden variables, must be added to obtain a satisfactory understanding of the physical universe.