A physical theory aims to give a description of the properties and behaviour of a physical system, both qualitatively and quantitatively. Qualitatively a theory should explain the behaviour of a given physical system in terms of some fundamental properties of the system. The study of the mathematical aspects of theoretical physics is often referred to as mathematical physics. Apart from the requirement of mathematical consistency, a theory is judged by how well its predictions agree with empirical observations, especially by its ability to make new and experimentally verifiable predictions. A physical system possesses many physical quantities which reflect the properties of the system. A physical quantity is a property of the system which can be physically observed and measured to yield numerical values. To emphasise their measurability, these quantities are called observables. The relationship between states and observables can be complicated, depending on whether we are dealing with classical systems or quantum systems.
A physical theory aims to give a description of the properties and behaviour of a physical system, both qualitatively and quantitatively. Qualitatively a theory should explain the behaviour of a given physical system in terms of some fundamental properties of the system. The study of the mathematical aspects of theoretical physics is often referred to as mathematical physics. Apart from the requirement of mathematical consistency, a theory is judged by how well its predictions agree with empirical observations, especially by its ability to make new and experimentally verifiable predictions. A physical system possesses many physical quantities which reflect the properties of the system. A physical quantity is a property of the system which can be physically observed and measured to yield numerical values. To emphasise their measurability, these quantities are called observables. The relationship between states and observables can be complicated, depending on whether we are dealing with classical systems or quantum systems.
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
The term measurement theory refers to that part of a physical theory in which the empirical and operational content of the concepts of the theory is determined. Measurements are analyzed both as operational procedures defining the → observables of the theory and as physical processes which are themselves subject to the laws of physics. In classical physics, measurements are performed in order to deter-mine the values of one or several observables of the physical system un-der consideration. Classical physics allowed the idealized notion that every physical quantity has a definite value at any time, and that this value can be determined with certainty by measurement without influ-encing the object system in a significant way. By contrast, in quantum mechanics both features fail to hold without strong qualifications. Ac-cordingly, in their seminal paper of 1935 [1], Einstein, Podolsky and Rosen used elements of this description as a sufficient criterion of phys-ical reality, applicable both in classical and quantum mechanics:
Mature theories of physical science associate states with systems, for instance with particles or waves in classical physics. The quantities whose values comprise states in classical mechanics, namely position and momentum, are chosen, first of all, for predictive purposes. Everything that one would want to say about classical systems can, it seems, be said in terms of properties: specification of the properties of a system gives a complete specification of its state, and conversely. Quantum mechanics associates states with physical systems; whatever else these might be, systems are the particulars which bear physical properties. The former is non-reduced state, the quantum state proper, while the latter is the reduced state, the class of quantitative properties possessed by the system. The state therefore determines which quantities and which properties are instantiated. The non-reductive account of states was initially appealing because it seemed to promise some additional explanatory scope.
A physical theory, in the narrow sense of the word, is a logical structure based on assumptions and definitions which permits one to predict the outcome of a maximum number of different experiments on the basis of a minimum number of postulates. One usually desires the postulates to be as self-evident as possible; one demands simplicity, beauty, and even elegance. A physical theory seems to have only a finite lifetime. Newtonian mechanics was replaced by relativistic mechanics, thermodynamics by statistical mechanics, classical by quantum mechanics. There is a logical order in theoretical physics. Indeed, such an order is necessary if the endeavor of theoretical physics is to be meaningful and is to succeed in the direction of more and more general axiomatic deductive systems. The relationship between classical and quantum mechanics is especially interesting. The chapter also presents an overview of the key concepts discussed in this book.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
A simplified scheme is used to discuss the idea of the mathematical formalism of physical theory. The physical theory is associated with a certain mathematical structure which makes it possible to define objects, i.e., the subject of investigation, and some of their characteristics. The task of the theory is to find the characteristics of an object from its input data that implicitly define it, on the basis of the laws from the nature of the basic structure. It stands to reason that the task includes the finding of the indicated laws dictated by the structure, the elucidation of the general properties of the objects, the establishment of the relationship between the characteristics, etc. The finiteness of the models used to illustrate the given scheme is brought about by the replacement of the space or the space-time continuum of "real" physics by a finite set.