Quantum mechanics and the theory of operator algebras have been intertwined since their origin.In the 1930s [20] von Neumann laid the foundations for the theory of (what are now known as) von Neumann algebras with the explicit goal of establishing Heisenberg's matrix mechanics on a rigorous footing (quoting from the preface, in the translation by Beyer: "The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous").Following the initial explorations of Murray and von Neumann, the new theory took on a life of its own, eventually leading to multiple applications unrelated to quantum mechanics, such as to free probability or noncommutative geometry.In his 1976 paper completing the classification of injective von Neumann algebras [6] Connes made a casual
Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.
Using techniques from the theory of von Neumann algebras, we propose a framework for addressing questions of controllability of bilinear systems on infinite dimensional Hilbert spaces. In the setup, we assume only that the drift and control terms arising in a bilinear control system are affiliated with a von Neumann algebra of finite type acting on the same Hilbert space. When the control terms satisfy basic norm bound conditions, we prove existence of time-optimal controls. In the more general setting where all operators may be unbounded, we show how the dynamical Lie algebra for the system is still well-defined and may be used to check approximate controllability of the system in question. We discuss how this approach can be applied to classical dynamical systems through the Koopman operator formalism, and investigate potential candidates for the von Neumann algebra which may guide the choice of controls. We illustrate how an affiliation relation naturally arises in both classical and quantum control systems with a few examples.
The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, within one or the other of the infinitely many {\it reduced power algebras} which can replace the usual real numbers
R, or complex numbers
C. A major advantage of such a re-formulation is, among others, the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics". The use of reduced power algebras also opens up a foundational question about the role, and in fact, about the very meaning and existence, of fundamental constants in Physics, such as Planck's constant
h. A role, meaning, and existence which may, or on the contrary, may not be so objective as to be independent of the scalars used, be they the usual real numbers
R, complex numbers
C, or scalars given by any of the infinitely many reduced power algebras, algebras which can so easily be constructed and used.
One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.