The details of the contents and formulations of the Born rule have changed considerably from its inception by Born in 1926 to the present day. This paper traces the early history of the Born rule 100 years ago, its generalization (essential for today's quantum optics and quantum information theory) to POVMs around 50 years ago, and a modern derivation from an intuitive definition of the notion of a quantum detector. Also discussed is the extent to which the various forms of the Born rule have, like any other statement in physics, a restricted domain of validity, which leads to problems when applied outside this domain.
The details of the contents and formulations of the Born rule have changed considerably from its inception by Born in 1926 to the present day. This paper traces the early history of the Born rule 100 years ago, its generalization (essential for today's quantum optics and quantum information theory) to POVMs around 50 years ago, and a modern derivation from an intuitive definition of the notion of a quantum detector. Also discussed is the extent to which the various forms of the Born rule have, like any other statement in physics, a restricted domain of validity, which leads to problems when applied outside this domain.
Born's rule in its conventional textbook form applies to the small class of projective measurements only. It is well-known that a generalization of Born's rule to realistic experiments must be phrased in terms of positive operator valued measures (POVMs). This generalization accounts for things like losses, imperfect measurements, limited detection accuracy, dark detector counts, and the simultaneous measurement of position and momentum. Starting from first principles, this paper gives a self-contained, deductive introduction to quantum measurement and Born's rule, in its generalized form that applies to the results of measurements described by POVMs. It is based on a suggestive definition of what constitutes a detector, assuming an intuitive informal notion of response. The formal exposition is embedded into the context of a variaety of quotes from the literature illuminating historical aspects of the subject. The material presented suggests a new approach to introductory courses on quantum mechanics.
Traditionally, some version of Born’s rule is always considered to be an indispensable part of any interpretation of quantum mechanics, either as a postulate, or as a result derived-not always on the basis of convincing reasoning-from other postulates. In this section, we have a close look at the possible forms of Born’s rule and discuss the limits of its validity. All traditional foundations of quantum mechanics heavily depend on the concept of (hypothetical, idealized) experiments-far too heavily. This is one of the reasons why these foundations are still unsettled, over 90 years after the discovery of the basic equations for modern quantum mechanics. No other theory has such controversial foundations. The main reason is that the starting point of the usual interpretations is an idealization of the measurement process that is taken too seriously, namely as the indisputable truth about everything measured. But in reality, this idealization is only a didactical trick for the newcomer to make the formal definitions of quantum mechanics a bit easier to swallow. Except in a few very simple cases, it is too far removed from experimental practice to tell much about real measurements, and hence about how quantum physics is used in real applications. In experimental physics, measurement is a very complex thing, far more complex than Born’s rule (the usual starting point) suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift, just to mention three basic examples, cannot be captured by the idealistic measurement concept used there, nor by any of the refinements of it discussed in the literature. In each of the three cases mentioned, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly, the theory must already be in place in order to do that. We do not even know what a top quark should be, whose mass we are measuring, unless we have a theory that tells us this! The two most accurately determined observables in the history of quantum physics, namely the anomalous magnetic moment of the electron and Lamb shift, are not even q-observables! To present the stage for the criticism of Born’s rule in Section 14.3, we first need to clarify the meaning of the term “Born’s rule”. To distinguish different useful meanings, we look in Sections 14.1-14.2 at the early history of Born’s rule.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
Measurement-induced interrelations between quantum mechanics and its interpretation The development of quantum mechanics The formalism of quantum mechanics was developed within the very short period of a few months in 1925 and 1926 by Heisenberg [Heis 25] and by Schrodinger [Schrö 26], respectively. Together with the contributions of Born and Jordan [BoJo 26], [BHJ 26], Dirac [Dir 26] and others, the formalism of this theory was already brought in 1926 into its final form, which is still used in present-day text books. (For all details of the historical development, we refer to the monograph by M. Jammer [Jam 74].) It is a very remarkable fact that a theory which was formulated 70 years ago has never been corrected or improved and is still considered to be valid. Numerous experiments performed during this long period to test the theory have confirmed it to a very high degree of accuracy without any exception. Hence there are good reasons to believe that quantum mechanics is universally valid and can be applied to all domains of reality, i.e., to atoms, molecules, macroscopic bodies, and to the whole universe. However, the interpretation of the new theory was at the time of its mathematical formulation still an almost open problem. Any interpretation of quantum theory should provide interrelations between the theoretical expressions of the theory and possible experimental outcomes.
Abstract A concise review of the derivation of the Born rule and Schroedinger equation from first principles is provided. The starting point is a formalization of fundamental notions of measurement and composition, leading to a general framework for physical theories known as the positive formalism. Consecutively adding notions of spacetime, locality, absolute time and causality recovers the well established convex operational framework for classical and quantum theory. Requiring the partially ordered vector space of states to be an anti-lattice one obtains quantum theory in its standard formulation. This includes Born rule and Schroedinger equation.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.