John von Neumann

John von Neumann (/vɒn ˈnɔɪmən/ von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He had perhaps the widest coverage of any mathematician of his time, [9] integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer. His analysis of the structure of selfreplication preceded the discovery of the structure of DNA. During World War II, von Neumann worked on the Manhattan Project. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon. [10] Before and after the war, he consulted for many organizations including the Office of Scientific Research and Development, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project and the Oak Ridge National Laboratory. [11] At the peak of his influence in the 1950s, he chaired a number of Defense Department committees including the Strategic Missile Evaluation Committee and the ICBM Scientific Advisory Committee. He was also a member of the influential Atomic Energy Commission in charge of all atomic energy development in the country. He played a key role alongside Bernard Schriever and Trevor Gardner in the design and development of the United States' first ICBM programs. [12] At that time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the Pentagon. Von Neumann's contributions and intellectual ability drew praise from colleagues in physics, mathematics, and beyond. Accolades he received range from the Medal of Freedom to a crater on the Moon named in his honor. 

Life and education

Von Neumann was born in Budapest, Kingdom of Hungary (then part of the Austro-Hungarian Empire), [13][14][15] on December 28, 1903, to a wealthy, non-observant Jewish family. His birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names

are equivalent to John Louis in English.[16] He was the eldest of three brothers; his two younger siblings were Mihály (Michael) and Miklós (Nicholas). [17] His father Neumann Miksa (Max von Neumann) was a banker and held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. [18] Miksa's father and grandfather were born in Ond (now part of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (English: Margaret Kann); [19] her parents were Jakab Kann and Katalin Meisels of the Meisels family. [20] Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor. [21] On February 20, 1913, Emperor Franz Joseph elevated John's father to the Hungarian nobility for his service to the AustroHungarian Empire. [22] The Neumann family thus acquired the hereditary appellation Margittai, meaning "of Margitta" (today Marghita, Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen coat of arms depicting three marguerites. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann.[23] Von Neumann was a child prodigy who at six years old could divide two eight-digit numbers in his head [24][25] and converse in Ancient Greek. [26] He, his brothers and his cousins were instructed by governesses. Von Neumann's father believed that knowledge of languages other than their native Hungarian was essential, so the children were tutored in English, French, German and Italian. [27] By age eight, von Neumann was familiar with differential and integral calculus, and by twelve he had read Borel's La Théorie des Fonctions. [28] He was also interested in history, reading Wilhelm Oncken's 46-volume world history series Allgemeine Geschichte in Einzeldarstellungen (General History in Monographs). [29] One of the rooms in the apartment was converted into a library and reading room. [30]

 Child prodigy 

Hamburg, where the prospects of becoming a tenured professor were better, [48] then in October of that year moved to Princeton University as a visiting lecturer in mathematical physics. [49] Von Neumann was baptized a Catholic in 1930.[50] Shortly afterward, he married Marietta Kövesi, who had studied economics at Budapest University. [49] Von Neumann and Marietta had a daughter, Marina, born in 1935; she would become a professor. [51] The couple divorced on November 2, 1937.[52] On November 17, 1938, von Neumann married Klara Dan. [53][54] 

In 1933 Von Neumann accepted a tenured professorship at the Institute for Advanced Study in New Jersey, when that institution's plan to appoint Hermann Weyl appeared to have failed.[55] His mother, brothers and inlaws followed von Neumann to the United States in 1939.[56] Von Neumann anglicized his name to John, keeping the Germanaristocratic surname von Neumann.[23] Von Neumann became a naturalized U.S. citizen in 1937, and immediately tried to become a lieutenant in the U.S. Army's Officers Reserve Corps. He passed the exams but was rejected because of his age. [57] Klara and John von Neumann were socially active within the local academic community. [58] His white clapboard house on Westcott Road was one of Princeton's largest private residences. [59] He always wore formal suits. [60] He enjoyed Yiddish and "off-color" humor. [28] In Princeton, he received complaints for playing extremely loud German march music; [61] Von Neumann did some of his best work in noisy, chaotic environments. [62] Per Churchill Eisenhart, von Neumann could attend parties until the early hours of the morning and then deliver a lecture at 8:30.[63] He was known for always being happy to provide others of all ability levels with scientific and mathematical advice. [4][64][65] Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician.[66] His daughter wrote that he was very concerned with his legacy in two aspects: her life and the durability of his intellectual contributions to the world.[67] Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing on technical ones. Herbert York described the many "Von Neumann Committees" that he participated in as "remarkable in style as well as output". The way the committees von Neumann chaired worked directly and intimately with the necessary military or corporate entities became a blueprint for all Air Force long-range missile programs. [68] Many people who had known von Neumann were puzzled by his relationship to the military and to power structures in general. [69] Stanisław Ulam suspected that he had a hidden admiration for people or organizations that could influence the thoughts and decision making of others. [70] Von Neumann's gravestone He also maintained his knowledge of languages learnt in his youth. He knew Hungarian, French,  

German and English fluently, and maintained a conversational level of Italian, Yiddish, Latin and Ancient Greek. His Spanish was less perfect. [71] He had a passion for and encyclopedic knowledge of ancient history, [72][73] and he enjoyed reading Ancient Greek historians in the original Greek. Ulam suspected they may have shaped his views on how future events could play out and how human nature and society worked in general. [74] Von Neumann's closest friend in the United States was the mathematician Stanisław Ulam. [75] Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up.[62] Ulam noted that von Neumann's way of thinking might not be visual, but more aural. [76] Ulam recalled, "Quite independently of his liking for abstract wit, he had a strong appreciation (one might say almost a hunger) for the more earthy type of comedy and humor". [77]

Von Neumann paradox

Building on the Hausdorff paradox of Felix Hausdorff (1914), Stefan Banach and Alfred Tarski in 1924 showed how to subdivide a three-dimensional ball into disjoint sets, then translate and rotate these sets to form two identical copies of the same ball; this is the Banach–Tarski paradox. They also proved that a twodimensional disk has no such paradoxical decomposition. But in 1929,[92] von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving affine transformations instead of translations and rotations. The result depended on finding free groups of affine transformations, an important technique extended later by von Neumann in his work on measure theory. [93

Proof theory

With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems. [94] By 1927, von Neumann was involving himself in discussions in Göttingen on whether elementary arithmetic followed from Peano axioms. [95] Building on the work of Ackermann, he began attempting to prove (using the finistic methods of Hilbert's school) the consistency of first-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on induction). [96] He continued looking for a more general proof of the consistency of classical mathematics using methods from proof theory. [97] Von Neumann paradox Proof theory A strongly negative answer to whether it was definitive arrived in September 1930 at the Second Conference on the Epistemology of the Exact Sciences, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete. [98] At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers. [99] Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency. [98] Gödel replied that he had already discovered this consequence, now known as his second incompleteness theorem, and that he would send a preprint of his article containing both results, which never appeared.[100][101][102] Von Neumann acknowledged Gödel's priority in his next letter. [103] However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did.[104][105] With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in the foundations of mathematics and metamathematics and instead spent time on problems connected with applications. [106

Ergodic theory 

In a series of papers published in 1932, von Neumann made foundational contributions to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure. [107] Of the 1932 papers on ergodic theory, Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". [108] By then von Neumann had already written his articles on operator theory, and the application of this work was instrumental in his mean ergodic theorem. [109] The theorem is about arbitrary one-parameter unitary groups and states that for every vector in the Hilbert space, exists in the sense of the metric defined by the Hilbert norm and is a vector which is such that for all . This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to Boltzmann's ergodic hypothesis. He also pointed out that ergodicity had not yet been achieved and isolated this for future work.[110] Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic measure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with Paul Halmos have significant applications in other areas of mathematics. [110][111]

Measure theory 

In measure theory, the "problem of measure" for an n-dimensional Euclidean space R n may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of R n ?" [112] The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, Ergodic theory Measure theory and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."[113] Around 1942 he told Dorothy Maharam how to prove that every complete σ-finite measure space has a multiplicative lifting; he did not publish this proof and she later came up with a new one. [114] In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. [115] A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". [116] He proved this in the positive, and in later papers with Stone discussed various generalizations and algebraic aspects of this problem. [117] He also proved by new methods the existence of disintegrations for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for compact groups. [116] He had to create entirely new techniques to apply this to locally compact groups. [118] He also gave a new, ingenious proof for the Radon–Nikodym theorem. [119] His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.[120][121][122] 

Topological groups

Using his previous work on measure theory, von Neumann made several contributions to the theory of topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups. [123] He continued this work with another paper in conjunction with Bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers. [124] In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers. [125][126] In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact groups. [127] The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of linear transformations and found that closed subgroups of a general linear group are Lie groups. [128] This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem. [129][116

Functional analysis

Von Neumann was the first to axiomatically define an abstract Hilbert space. He defined it as a complex vector space with a Hermitian scalar product, with the corresponding norm being both separable and complete. In the same papers he also proved the general form of the Cauchy–Schwarz inequality that had previously been known only in specific examples. [130] He continued with the development of the spectral theory of operators in Hilbert space in three seminal papers between 1929 and 1932.[131] This work cumulated in his Mathematical Foundations of Quantum Mechanics which alongside two other books by Stone and Banach in the same year were the first monographs on Hilbert space theory. [132] Previous work by others showed that a theory of weak topologies could not be obtained by using sequences. Von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining locally convex spaces and topological vector spaces for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new Topological groups Functional analysis topological ideas from Hausdorff from Euclidean to Hilbert spaces) [133] such as boundness and total boundness are still used today. [134] For twenty years von Neumann was considered the 'undisputed master' of this area. [116] These developments were primarily prompted by needs in quantum mechanics where von Neumann realized the need to extend the spectral theory of Hermitian operators from the bounded to the unbounded case. [135] Other major achievements in these papers include a complete elucidation of spectral theory for normal operators, the first abstract presentation of the trace of a positive operator, [136][137] a generalisation of Riesz's presentation of Hilbert's spectral theorems at the time, and the discovery of Hermitian operators in a Hilbert space, as distinct from self-adjoint operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage of infinite matrices, common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of operator algebras. [138] His later work on rings of operators lead to him revisiting his work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces. [135] Like in his work on measure theory he proved several theorems that he did not find time to publish. He told Nachman Aronszajn and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the invariant subspace problem. [139] With I. J. Schoenberg he wrote several items investigating translation invariant Hilbertian metrics on the real number line which resulted in their complete classification. Their motivation lie in various questions related to embedding metric spaces into Hilbert spaces. [140][141] With Pascual Jordan he wrote a short paper giving the first derivation of a given norm from an inner product by means of the parallelogram identity. [142] His trace inequality is a key result of matrix theory used in matrix approximation problems. [143] He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as symmetric absolute norms). [144][145][146] This paper leads naturally to the study of symmetric operator ideals and is the beginning point for modern studies of symmetric operator spaces. [147] Later with Robert Schatten he initiated the study of nuclear operators on Hilbert spaces, [148][149] tensor products of Banach spaces, [150] introduced and studied trace class operators, [151] their ideals, and their duality with compact operators, and preduality with bounded operators. [152] The generalization of this topic to the study of nuclear operators on Banach spaces was among the first achievements of Alexander Grothendieck. [153][154] Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on and proving several other results on what are now known as Schatten–von Neumann ideals. [155]

Operator algebras

Von Neumann founded the study of rings of operators, through the von Neumann algebras (originally called W*-algebras). While his original ideas for rings of operators existed already in 1930, he did not begin studying them in depth until he met F. J. Murray several years later. [156][157] A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. [158] The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant. [159] After elucidating the study of the commutative algebra case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the noncommutative case, the general study of factors classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the Operator algebras masterpieces of analysis in the twentieth century"; [160] they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of factors. [161] In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949.[162][163] Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out. [164][165] Another important result on polar decomposition was published in 1932.[166

Lattice theory

Between 1935 and 1937, von Neumann worked on lattice theory, the theory of partially ordered sets in which every two elements have a greatest lower bound and a least upper bound. As Garrett Birkhoff wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". [167] Von Neumann combined traditional projective geometry with modern algebra (linear algebra, ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of general modules over rings. His work laid the foundations for some of the modern work in projective geometry. [168] His biggest contribution was founding the field of continuous geometry. [169] It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry, where instead of the dimension of a subspace being in a discrete set it can be an element of the unit interval . Earlier, Menger and Birkhoff had axiomatized complex projective geometry in terms of the properties of its lattice of linear subspaces. Von Neumann, following his work on rings of operators, weakened those axioms to describe a broader class of lattices, the continuous geometries. While the dimensions of the subspaces of projective geometries are a discrete set (the non-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval . Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor. [170][171] In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of (continuous-dimensional projective geometry over an arbitrary division ring ) in abstract language of lattice theory. [172] Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices (properties that arise in the lattices of subspaces of inner product spaces): Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity. For any integer every -dimensional abstract projective geometry is isomorphic to the subspacelattice of an -dimensional vector space over a (unique) corresponding division ring . This is known as the Veblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case. [173] This coordinatization theorem stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques. [168][174] Birkhoff described this theorem as follows: Lattice theory Any complemented modular lattice L having a "basis" of n ≥ 4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself— realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe. [175] This work required the creation of regular rings. [176] A von Neumann regular ring is a ring where for every , an element exists such that . [175] These rings came from and have connections to his work on von Neumann algebras, as well as AW*-algebras and various kinds of C*-algebras. [177] Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding distributivity (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory of metric lattices. [178] Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication.[179]

Mathematical statistics

Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. [180] This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic [181] for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression. [181] Subsequently, Denis Sargan and Alok Bhargava extended the results for testing whether the errors on a regression model follow a Gaussian random walk (i.e., possess a unit root) against the alternative that they are a stationary first order autoregression.[182]

Other work 

In his early years, von Neumann published several papers related to set-theoretical real analysis and number theory. [183] In a paper from 1925, he proved that for any dense sequence of points in , there existed a rearrangement of those points that is uniformly distributed. [184][185][186] In 1926 his sole publication was on Prüfer's theory of ideal algebraic numbers where he found a new way of constructing them, thus extending Prüfer's theory to the field of all algebraic numbers, and clarified their relation to p-adic numbers. [187][188][189][190][191] In 1928 he published two additional papers continuing with these themes. The first dealt with partitioning an interval into countably many congruent subsets. It solved a problem of Hugo Steinhaus asking whether an interval is -divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are -divisible by translations (i.e. that these intervals can be decomposed into subsets that are congruent by translation). [192][193][194][195] His next paper dealt with giving a constructive proof without the axiom of choice that algebraically independent reals exist. He Mathematical statistics Other work proved that are algebraically independent for . Consequently, there exists a perfect algebraically independent set of reals the size of the continuum. [196][197][198][199] Other minor results from his early career include a proof of a maximum principle for the gradient of a minimizing function in the field of calculus of variations, [200][201][202][203] and a small simplification of Hermann Minkowski's theorem for linear forms in geometric number theory. [204][205][206] Later in his career together with Pascual Jordan and Eugene Wigner he wrote a foundational paper classifying all finite-dimensional formally real Jordan algebras and discovering the Albert algebras while attempting to look for a better mathematical formalism for quantum theory. [207][208] In 1936 he attempted to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras [209] in a paper investigating the infinite-dimensional case; he planned to write at least one further paper on the topic but never did.[210] Nevertheless, these axioms formed the basis for further investigations of algebraic quantum mechanics started by Irving Segal. [211][212] Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, know

Physics 

Quantum mechanics 

Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, in his influential 1932 work Mathematical Foundations of Quantum Mechanics. [213] After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system. [214] The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. [214] Von Neumann's abstract treatment permitted him to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables", as in classical statistical mechanics. In 1935, Grete Hermann published a paper arguing that the proof contained a conceptual error and was therefore invalid.[215] Hermann's work was largely ignored until after John S. Bell made essentially the same argument in 1966.[216] In 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories. [217] The validity of Bub's argument is, in turn, disputed. Gleason's theorem of 1957 provided an argument against hidden variables along the lines of von Neumann's, but founded on assumptions seen as better motivated and more physically meaningful. [218][219] Von Neumann's proof inaugurated a line of research that ultimately led, through Bell's theorem and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires a notion of reality substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity. [220] Physics Quantum mechanics In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. In other words, while the line between observer and observed could be drawn in different places, the theory only makes sense if an observer exists somewhere. [221] Although the idea of consciousness causing collapse was accepted by Eugene Wigner, [222] the Von Neumann–Wigner interpretation never gained acceptance among the majority of physicists. [223] Though theories of quantum mechanics continue to evolve, a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches can be traced back to the mathematical formalisms and techniques first used by von Neumann. Discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations. [213] Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of Hilbert's sixth problem, mathematical physicist Arthur Wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. With his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems. [224] Nevertheless, von Neumann in his later years felt he had failed in this aspect of his scientific work as despite all the mathematics he developed, he did not find a satisfactory mathematical framework for quantum theory as a whole. [225][226]

Von Neumann entropy

Von Neumann entropy is extensively used in different forms (conditional entropy, relative entropy, etc.) in the framework of quantum information theory. [227] Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix , it is given by Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy[228] and conditional quantum entropy. Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy, a cornerstone in the former's development; the Shannon entropy applies to classical information theory. [229] The formalism of density operators and matrices was introduced by von Neumann[230] in 1927 and independently, but less systematically by Lev Landau [231] and Felix Bloch [232] in 1927 and 1946 respectively. The density matrix is an alternative way to represent the state of a quantum system including statistical probabilities, which are not easily represented using wavefunctions. V

Density matrix 

The formalism of density operators and matrices was introduced by von Neumann[230] in 1927 and independently, but less systematically by Lev Landau [231] and Felix Bloch [232] in 1927 and 1946 respectively. The density matrix is an alternative way to represent the state of a quantum system including statistical probabilities, which are not easily represented using wavefunctions.

Von Neumann measurement scheme