Quantum Mechanics for Observer Systems

Hilary Putnam

Shabnam Siddiqui

This chapter discusses the mathematical formalism of the theory of quantum mechanics and present the differences between the theory of classical mechanics and quantum mechanics. The mathematics that is used to account for the indeterminacy of quantum mechanics renders the mathematical approach of quantum mechanics distinct from that of classical mechanics. In quantum mechanics, any dynamical variable that is an observable that can be measured experimentally is described as a linear Hermitian operator. In quantum mechanics, it is not only necessary to define an observable, but it is also necessary to define the states associated with it. When a system is in a linear superposition of states of an observable, the number of different values of an observable observed in an experiment is equal to the number of states that a quantum mechanical system can jump into upon measurement of an observable.

S. M. Blinder

Dirk Aerts

We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.

George Jaroszkiewicz

18.1 Introduction The Feynman path-integral (PI) approach to quantum mechanics (QM) has proven to be an invaluable technical tool with applicability far beyond the non-relativistic QM from which it sprang (Feynman and Hibbs, 1965). The PI approach is based on a fundamental quantum principle, which is that nothing can or should be said with certainty about a system under observation (SUO) when it is not actually being observed. This is in complete contrast to classical mechanics (CM), which asserts that an SUO exists in a well-defined state at all times, regardless of whether it is being observed or not. It is a remarkable fact that what appears to be a metaphysical point actually underpins the principles of quantum physics. The quantum principle that we cannot say precisely in classical terms what an SUO is doing when we are not observing it has a powerful corollary: if we cannot say what the SUO is doing when we are not observing it, then we have to allow for the possibility that it could be doing anything, or at least anything that would be consistent with required conservation laws, behind our backs. Following a lead from Dirac (1933), Feynman developed and popularized a mathematical way of expressing this idea: the path integral. We shall discuss his approach in the case of an SUO consisting of a Newtonian point particle moving in one spatial dimension. Discrete time (DT) comes into the picture right at the start of the formulation.

John P. Lowe