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achine learning scientists aharonov bell bethe blackett bloch bohm bohr born bose de broglie compton dirac davisson debye ehrenfest einstein everett fock fermi feynman glauber gutzwiller heisenberg hilbert jordan kramers pauli lamb landau laue moseley millikan onnes planck rabi raman rydberg schrödinger simmons sommerfeld von neumann weyl wien wigner zeeman zeilinger v t e von neumann was the first to establish a rigorous mathematical framework for quantum mechanics known as the dirac von neumann axioms in his widely influential 1932 work mathematical foundations of quantum mechanics 225 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 226 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 226 when heisenberg was informed von neumann had clarified the difference between an unbounded operator that was a self adjoint operator and one that was merely symmetric heisenberg replied eh what is the difference 227 von neumann s abstract treatment permitted him also 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 228 hermann s work was largely ignored until after john s bell made essentially the same argument in 1966 229 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 230 the validity of bub s argument is in turn disputed 231 in any case gleason s theorem of 1957 fills the gaps in von neumann s approach 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 232 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 although this view was accepted by eugene wigner 233 the von neumann wigner interpretation never gained acceptance among the majority of physicists 234 the von neumann wigner interpretation has been summarized as follows 235 the rules of quantum mechanics are correct but there is only one system which may be treated with quantum mechanics namely the entire material world there exist external observers which cannot be treated within quantum mechanics namely human and perhaps animal minds which perform measurements on the brain causing wave function collapse 235 though theories of quantum mechanics continue to evolve there is a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches that can be traced back to the mathematical formalisms and techniques first used by von neumann in other words discussions about interpretation of the theory and extensions to it are now mostly conducted on the basis of shared assumptions about the mathematical foundations 225 viewing von neumann s work on quantum mechanics as a part of the fulfilment of hilbert s sixth problem noted mathematical physicist a s wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date in the publication of 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 236 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 operator theory von neumann algebras continuous geometries etc he did not find a satisfactory mathematical framework for quantum theory as a whole including quantum field theory 237 238 von neumann entropy edit main article von neumann entropy statistical mechanics thermodynamics kinetic theory particle statistics spin statistics theorem identical particles maxwell boltzmann bose einstein fermi dirac parastatistics anyonic statistics braid statistics thermodynamic ensembles nve microcanonical nvt canonical µvt grand canonical nph isoenthalpic isobaric npt isothermal isobaric models debye einstein ising potts potentials internal energy enthalpy helmholtz free energy gibbs free energy grand potential landau free energy scientists maxwell boltzmann gibbs einstein ehrenfest von neumann tolman debye fermi bose v t e von neumann entropy is extensively used in different forms conditional entropy relative entropy etc in the framework of quantum information theory 239 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 ρ displaystyle rho it is given by s ρ tr ρ ln ρ displaystyle s rho operatorname tr rho ln rho many of the same entropy measures in classical information theory can also be generalized to the quantum case such as holevo entropy and conditional quantum entropy quantum mutual information edit quantum information theory is largely concerned with the interpretation and uses of von neumann entropy the von neumann entropy is the cornerstone in the development of quantum information theory while the shannon entropy applies to classical information theory this is considered a historical anomaly as shannon entropy might have been expected to be discovered before von neumann entropy given the latter s more widespread application to quantum information theory but von neumann discovered von neumann entropy first and applied it to questions of statistical physics decades later shannon developed an information theoretic formula for use in classical information theory and asked von neumann what to call it von neumann said to call it shannon entropy as it was a special case of von neumann entropy 240 density matrix edit main article density matrix the formalism of density operators and matrices was introduced by von neumann 241 in 1927 and independently but less systematically by lev landau 242 and felix bloch 243 in 1927 and 1946 respectively the density matrix is an alternative way to represent the state of a quantum system which could otherwise be represented using the wavefunction the density matrix allows the solution of certain time dependent problems in quantum mechanics von neumann measurement scheme edit the von neumann measurement scheme the ancestor of quantum decoherence theory represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object the projective measurement scheme introduced by von neumann led to the development of quantum decoherence theories 244 245 quantum logic edit main article quantum logic von neumann first proposed a quantum logic in his 1932 treatise mathematical foundations of quantum mechanics where he noted that projections on a hilbert space can be viewed as propositions about physical observables the field of quantum logic was subsequently inaugurated in a famous paper of 1936 by von neumann and garrett birkhoff the first work ever to introduce quantum logics 246 wherein von neumann and birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics the concept of creating a propositional calculus for quantum logic was first outlined in a short section in von neumann s 1932 work but in 1936 the need for the new propositional calculus was demonstrated through several proofs for example photons cannot pass through two successive filters that are polarized perpendicularly e g horizontally and vertically and therefore a fortiori it cannot pass if a third filter polarized diagonally is added to the other two either before or after them in the succession but if the third filter is added between the other two the photons will indeed pass through this experimental fact is translatable into logic as the non commutativity of conjunction a b b a displaystyle a land b neq b land a it was also demonstrated that the laws of distribution of classical logic p q r p q p r displaystyle p lor q land r p lor q land p lor r and p q r p q p r displaystyle p land q lor r p land q lor p land r are not valid for quantum theory 247 the reason for this is that a quantum disjunction unlike the case for classical disjunction can be true even when both of the disjuncts are false and this is in turn attributable to the fact that it is frequently the case in quantum mechanics that a pair of alternatives are semantically determinate while each of its members is necessarily indeterminate this latter property can be illustrated by a simple example suppose we are dealing with particles such as electrons of semi integral spin spin angular momentum for which there are only two possible values positive or negative then a principle of indetermination establishes that the spin relative to two different directions e g x and y results in a pair of incompatible quantities suppose that the state ɸ of a certain electron verifies the proposition the spin of the electron in the x direction is positive by the principle of indeterminacy the value of the spin in the direction y will be completely indeterminate for ɸ hence ɸ can verify neither the proposition the spin in the direction of y is positive nor the proposition the spin in the direction of y is negative nevertheless the disjunction of the propositions the spin in the direction of y is positive or the spin in the direction of y is negative must be true for ɸ in the case of distribution it is therefore possible to have a situation in which a b c a 1 a displaystyle a land b lor c a land 1 a while a b a c 0 0 0 displaystyle a land b lor a land c 0 lor 0 0 247 as hilary putnam writes von neumann replaced classical logic with a logic constructed in orthomodular lattices isomorphic to the lattices of subspaces of the hilbert space of a given physical system 248 nevertheless he was never satifisied with his work on quantum logic he intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the henry joseph lecture he gave at the washington philosophical society in 1945 he found that he could not especially given that he was busy with war work at the time he just could not make himself write something he did not fully understand to his satisfaction during his address at the 1954 international congress of mathematicians he gave this issue as one of the unsolved problems that future mathematicians could work on 249 250 fluid dynamics edit von neumann made fundamental contributions in the field of fluid dynamics von neumann s contributions to fluid dynamics included his discovery of the classic flow solution to blast waves 251 and the co discovery independently of yakov borisovich zel dovich and werner döring of the znd detonation model of explosives 252 during the 1930s von neumann became an authority on the mathematics of shaped charges 253 later with robert d richtmyer von neumann developed an algorithm defining artificial viscosity that improved the understanding of shock waves when computers solved hydrodynamic or aerodynamic problems they tried to put too many computational grid points at regions of sharp discontinuity shock waves the mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics 254 von neumann soon applied computer modelling to the field developing software for his ballistics research during ww2 he arrived one day at the office of r h kent the director of the us army s ballistic research laboratory with a computer program he had created for calculating a one dimensional model of 100 molecules to simulate a shock wave von neumann then gave a seminar on his computer program to an audience which included his friend theodore von kármán after von neumann had finished von kármán said well johnny that s very interesting of course you realize lagrange also used digital models to simulate continuum mechanics it was evident from von neumann s face that he had been unaware of lagrange s mécanique analytique 255 other work in physics edit while not as prolific in physics as he was in mathematics he nevertheless made several other notable contributions to it his pioneering papers with subrahmanyan chandrasekhar on the statistics of a fluctuating gravitational field generated by randomly distributed stars were considered a tour de force 256 in this paper they developed a theory of two body relaxation 257 and used the holtsmark distribution to model 258 the dynamics of stellar systems 259 he wrote several other unpublished manuscripts on topics in stellar structure some of which were included in chandresekhar s other works 260 261 in some earlier work led by oswald veblen von neumann helped develop basic ideas involving spinors that would lead to roger penrose s twistor theory 262 263 much of this was done in seminars conducted at the ias during the 1930s 264 from this work he wrote a paper with a h taub and veblen extending the dirac equation to projec...
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