The abstract theory of probability and its interpretation are briefly reviewed, and it is explicitly demonstrated that the formalism of quantum mechanics satisfies the axioms of probability theory. This refutes the suggestions which have occasionally been made that ‘‘classical’’ probability theory does not apply to quantum mechanics. Several erroneous applications of probability theory to quantum mechanics are examined, and the nature of the errors are exposed. It is urged that more attention be given to probability theory in the physics curriculum.
The abstract theory of probability and its interpretation are briefly reviewed, and it is explicitly demonstrated that the formalism of quantum mechanics satisfies the axioms of probability theory. This refutes the suggestions which have occasionally been made that ‘‘classical’’ probability theory does not apply to quantum mechanics. Several erroneous applications of probability theory to quantum mechanics are examined, and the nature of the errors are exposed. It is urged that more attention be given to probability theory in the physics curriculum.
Abstract This study calls attention to the current state of the probability ( P ) domain which presents weak points at the mathematical level and more significant flaws at the application level. Popper noticed how fundamental issues raised in quantum mechanics (QM) directly derive from unresolved probabilistic questions. Endless philosophical debates create more problems than solutions, so the author of this research suggests going directly to the root of the issues and searching for the probability theory which formalizes the multifold nature of P . This study offers a brief overview of the structural theory of probability , recently published in a book, and applies it to QM in order to show its completeness. The whole probability-based interpretation of QM goes beyond the limits of a paper and these pages condense a few aspects of this theoretical scheme. The double slit experiment is used to corroborate the theorems presented here.
This paper calls attention to the current state of the probability (P) domain which presents weak points at the mathematical level and more significant flaws at the application level. Popper notices how fundamental issues raised in quantum mechanics (QM) directly derive from unresolved probabilistic questions. Endless philosophical debates create more problems than solutions, so the author of this research suggests going directly to the root of the issues and searching for the probability theory which formalizes the multifold nature of P. This paper offers a brief overview of the structural theory of probability, recently published in a book, and applies it to QM in order to show its completeness. The whole probability-based interpretation of QM goes beyond the limits of a paper and these pages condense a few aspects of this theoretical scheme. The double slit experiment is used to corroborate the theorems presented here.
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
Probabilities may be subjective or objective; we are concerned with both kinds of probability, and the relationship between them. The fundamental theory of objective probability is quantum mechanics: it is argued that neither Bohr's Copenhagen interpretation, nor the pilot-wave theory, nor stochastic state-reduction theories, give a satisfactory answer to the question of what objective probabilities are in quantum mechanics, or why they should satisfy the Born rule; nor do they give any reason why subjective probabilities should track objective ones. But it is shown that if probability only arises with decoherence, then they must be given by the Born rule. That further, on the Everett interpretation, we have a clear statement of what probabilities are, in terms of purely categorical physical properties; and finally, along lines laid out by Deutsch and Wallace, that there is a clear basis in the axioms of decision theory as to why subjective probabilities should track these objective ones. These results hinge critically on the absence of hidden-variables or any other mechanism (such as state-reduction) from the physical interpretation of the theory. The account of probability has traditionally been considered the principal weakness of the Everett interpretation; on the contrary it emerges as one of its principal strengths.