Search Results for Quantum

Biographies

1. Bell John biography
   
   • John Bell's great achievement was that during the 1960s he was able to breathe new and exciting life into the foundations of quantum theory, a topic seemingly exhausted by the outcome of the Bohr-Einstein debate thirty years earlier, and ignored by virtually all those who used quantum theory in the intervening period.
     
   • Indeed, almost wholly due to Bell's pioneering efforts, the subject of quantum foundations, experimental as well as theoretical and conceptual, has became a focus of major interest for scientists from many countries, and has taught us much of fundamental importance, not just about quantum theory, but about the nature of the physical universe.
     
   • In addition, and this could scarcely have been predicted even as recently as the mid-1990s, several years after Bell's death, many of the concepts studied by Bell and those who developed his work have formed the basis of the new subject area of quantum information theory, which includes such topics as quantum computing and quantum cryptography.
     
   • Attention to quantum information theory has increased enormously over the last few years, and the subject seems certain to be one of the most important growth areas of science in the twenty-first century.
     
   • Bell was already thinking deeply about quantum theory, not just how to use it, but its conceptual meaning.
     
   • In an interview with Jeremy Bernstein, given towards the end of his life and quoted in Bernstein's book [Quantum Profiles (Princeton, 1991).
     
   • He would also have liked to study the conceptual basis of quantum theory more thoroughly.
     
   • Economic considerations, though, meant that he had to forget about quantum theory, at least for the moment, and get a job, and in 1949 he joined the UK Atomic Research Establishment at Harwell, though he soon moved to the accelerator design group at Malvern.
     
   • Through his career he gained much from discussions with Mary, and when, in 1987, his papers on quantum theory were collected [Speakable and Unspeakable in Quantum Mechanics (Cambridge, 1987).
     
   • During his time in Birmingham, Bell did work of great importance, producing his version of the celebrated CPT theorem of quantum field theory.
     
   • Bell published around 80 papers in the area of high-energy physics and quantum field theory.
     
   • The most important work was that of 1969 leading to the Adler-Bell-Jackiw (ABJ) anomaly in quantum field theory.
     
   • Reinhold Bertlmann, who himself did important work with Bell, has written a book titled Anomalies in Quantum Field Theory [Anomalies in Quantum Field Theory (Oxford, 2000).',10)">10], and the two surviving members of ABJ, Adler and Jackiw shared the 1988 Dirac Medal of the International Centre for Theoretical Physics in Trieste for their work.
     
   • While particle physics and quantum field theory was the work Bell was paid to do, and he made excellent contributions, his great love was for quantum theory, and it is for his work here that he will be remembered.
     
   • They were therefore pleased when John von Neumann proved a theorem claiming to show rigorously that it is impossible to add hidden variables to the structure of quantum theory.
     
   • This was his Copenhagen interpretation of quantum theory.
     
   • (3) Quantum theory is not exactly true in these rather special experiments.
     
   • He therefore concluded that if quantum theory was correct, so one ruled out possibility (3), then (2) must be true.
     
   • In Einstein's terms, quantum theory was not complete but needed to be supplemented by hidden variables.
     
   • He told Bernstein [Quantum Profiles (Princeton, 1991).
     
   • He was delighted by the creation in 1952 by David Bohm of a version of quantum theory which included hidden variables, seemingly in defiance of von Neumann's result.
     
   • Bell wrote [Speakable and Unspeakable in Quantum Mechanics (Cambridge, 1987).
     
   • In 1964, Bell made his own great contributions to quantum theory.
     
   • Von Neumann had illegitimately extended to his putative hidden variables a result from the variables of quantum theory that the expectation value of A + B is equal to the sum of the expectation values of A and of B.
     
   • (The expectation value of a variable is the mean of the possible experimental results weighted by their probability of occurrence.) Once this mistake was realised, it was clear that hidden variables theories of quantum theory were possible.
     
   • Bell had showed rigorously that one could not have local realistic theories of quantum theory.
     
   • So it has been assumed that quantum theory is exactly true, but of course this can never be known.
     
   • In EPR-type experiments, this inequality is obeyed by local hidden variables, but may be violated by other theories, including quantum theory.
     
   • While at least one loophole still remains to be closed [in August 2002], it seems virtually certain that local realism is violated, and that quantum theory can predict the results of all the experiments.
     
   • For the rest of his life, Bell continued to criticise the usual theories of measurement in quantum theory.
     
   • Gradually it became at least a little more acceptable to question Bohr and von Neumann, and study of the meaning of quantum theory has become a respectable activity.
     
   • Since that date, the amount of interest in his work, and in its application to quantum information theory has been steadily increasing.
     

2. Dirac biography
   
   • Fowler was then the leading theoretician in Cambridge, well versed in the quantum theory of atoms; his own research was mostly on statistical mechanics.
     
   • No doubt Fowler aroused his interest in the quantum theory, and in May 1924 Dirac completed his first paper dealing with quantum problems.
     
   • He realised that Heisenberg's uncertainty principle was a statement of the noncommutativity of the quantum mechanical observables.
     
   • This similarity provided the clue which led him to formulate for the first time a mathematically consistent general theory of quantum mechanics in correspondence with Hamiltonian mechanics.
     
   • The ideas were laid out in Dirac's doctoral thesis Quantum mechanics for which he was awarded a Ph.D.
     
   • Also in 1928 he found a connection between relativity and quantum mechanics, his famous spin-1/2 Dirac equation.
     
   • In 1930 Dirac published The principles of Quantum Mechanics and for this work he was awarded the Nobel Prize for Physics in 1933.
     
   • His lectures at Cambridge were closely modelled on [The principles of Quantum Mechanics], and they conveyed to generations of students a powerful impression of the coherence and elegance of quantum theory.
     
   • In 1933 he published a pioneering paper on Lagrangian quantum mechanics which became the foundation on which Feynman later built his ideas of the path integral.
     
   • in recognition of his remarkable contributions to relativistic dynamics of a particle in quantum mechanics.
     
   • Dirac unified the theories of quantum mechanics and relativity theory, but he also is remembered for his outstanding work on the magnetic monopole, fundamental length, antimatter, the d-function, bra-kets, etc.
     
   • we vividly see everywhere the brilliant imprints of Dirac, unifier of quantum mechanics and relativity theory.
     

3. Pauli biography
   
   • I was not spared the shock which every physicist accustomed to the classical way of thinking experienced when he came to know Bohr's basic postulate of quantum theory for the first time.
     
   • He wrote his first paper on quantum physics in June 1920, a work on the magnetic properties of matter.
     
   • Pauli received his doctorate, which had been supervised by Sommerfeld, in July 1921 for a thesis on the quantum theory of ionised molecular hydrogen.
     
   • Looking at it now one can see that it showed that quantum theory, as then formulated, was not in itself going to provide the necessary structure on which to build a logical theory of atomic structure which agreed with experimental evidence.
     
   • In 1924 Pauli proposed a quantum spin number for electrons.
     
   • He is best known for the Pauli exclusion principle , proposed in 1925, which states that no two electrons in an atom can have the same four quantum numbers.
     
   • Less than a year after this Heisenberg submitted his article on quantum mechanics which was to change the whole approach to the topic.
     
   • He found that four quantum numbers are in general needed in order to define the energy state of an electron.
     
   • Three quantum numbers only can be related to the revolution of the electron round the nucleus.
     
   • The necessity of a fourth quantum number proved the existence of interesting properties of the electron.
     
   • In 1925 and 1926 essential progress of another kind was made in the quantum theory, which is the foundation of atomic physics.
     
   • The spin proposal, which gave meaning to Pauli's fourth quantum number, was first suggested by Uhlenbeck in 1925.
     
   • His starting point was the philosophy of quantum mechanics, but this led him to psychology, the history of ideas and many other fields, not least the relation of religion to natural science.
     
   • History Topics: The quantum age begins .
     

4. Heisenberg biography
   
   • Heisenberg invented matrix mechanics, the first version of quantum mechanics, in 1925.
     
   • The creation of quantum mechanics, the application of which has led, among other things, to the discovery of the allotropic forms of hydrogen.
     
   • These different kinds of motion for atoms and molecules produce different systems in Heisenberg's quantum mechanics.
     
   • As the fundamental factor of Heisenberg's theory can be put forward the rule set out by him with reference to the relationship between the position coordinate and the velocity of an electron, by which rule Planck's constant is introduced into the quantum-mechanics calculations as a determining factor.
     
   • Heisenberg's quantum mechanics has been applied by himself and others to the study of the properties of the spectra of atoms and molecules, and has yielded results which agree with experimental research.
     
   • It can be said that Heisenberg's quantum mechanics has made possible a systemization of spectra of atoms.
     
   • Heisenberg is perhaps best known for the Uncertainty Principle, discovered in 1927, which states that determining the position and momentum of a particle necessarily contains errors the product of which cannot be less than the quantum constant h.
     
   • The great obstacles that had occupied all our efforts in the preceding years had been cleared out of the way, the gate to an entirely new field, the quantum mechanics of the atomic shells stood wide open, and fresh fruits seemed ready for the picking.
     
   • Heisenberg published The Physical Principles of Quantum Theory in 1928.
     
   • These papers opened the way for others to apply quantum theory to the atomic nucleus.
     
   • Relativity and quantum theory were classed as "Jewish" and as a consequence Heisenberg's appointment to Munich was blocked.
     
   • History Topics: Light through the ages: Relativity and quantum era .
     
   • History Topics: The quantum age begins .
     

5. Wigner biography
   
   • He then began the work for which he is famous, namely applying group theory to quantum mechanics.
     
   • Hilbert, already interested in quantum mechanics, felt that he needed a physicist as an assistant to complement his own expertise.
     
   • This was an important time for Wigner who produced papers of great depth and significance, introducing in his paper On the conservation laws of quantum mechanics (1927) the new concept of parity.
     
   • Wigner returned to Berlin after the year in Gottingen where he lectured on quantum mechanics, worked on writing his famous text Group theory and its application to the quantum mechanics of atomic spectra and continued his research.
     
   • In fact Wigner's book on the applications of group theory to quantum mechanics was not the first to appear, since Weyl published his a little before Wigner.
     
   • Weyl's ideas differed from those of Wigner in that he wanted to apply group representations to get a better understanding of the foundations of quantum mechanics in general and not so much to gain insight into particular problems.
     
   • epoch-making work on how symmetry is implemented in quantum mechanics, the determination of all the irreducible unitary representations of the Poincare group, and his work with Bargmann on realizing those irreducible unitary representations as the Hilbert spaces of solutions of relativistic wave equations, ..
     
   • discrete symmetries and superselection rules in quantum mechanics, symmetry implications for atomic and molecular spectra, natural line-width theory, contrast of microscopic and macroscopic physics and of general relativity and quantum mechanics, explanation of why symmetry yields more information for quantum than for classical mechanics, philosophical questions such as what nature laws should be, limits on causality, and whether quantum mechanics could in principle explain life.
     
   • course by Wigner on advanced quantum mechanics which I had the good fortune to attend at Princeton in 1940.
     

6. Polkinghorne biography
   
   • Polkinghorne took Part II of the Mathematical tripos in his second year, then took Part III in his third year, specialising in quantum mechanics.
     
   • In 1954 he published An identity for the S matrix for a finite time interval, Renormalization of the transformation operators of quantum electrodynamics, and Normal products of Heisenberg operators.
     
   • A new definition is given for the "normal product" of a set of field operators in the Heisenberg representation of quantum field theory.
     
   • Dyson, reviewed Temporally ordered graphs in quantum field theory:- .
     
   • A graphical representation of the perturbation-theory expansions of quantum field theory is defined, in which the vertices (points at which the interaction operates) are given a definite order in time.
     
   • The equivalence of the Feynman method of setting up a quantum field theory with the usual canonical formalism is here proved.
     
   • Then he published General dispersion relations in 1956 and Causal products in quantum field theory in the following year.
     
   • Also in a joint paper in 1957 he published Cauchy's problem in quantum field theory which explores the relation between the classical and quantum versions of field theories.
     
   • There are also a number of popular books on mathematical physics such as The Particle Play (1979), The Quantum World (1984) and Rochester Roundabout (1989) and Quantum Theory : A very short introduction (2002).
     
   • Such works include One World : The interaction of science and theology (1986), Science and Creation : The search for understanding (1988), Science and Providence : God's interaction with the world (1989), Reason and Reality : The relationship between science and religion (1991), Science and Christian Belief (published in North America as The Faith of a Physicist) (1994), Searching for truth : A scientist looks at the Bible (1997), Belief in God in an Age of Science (1998), Faith, Science and Understanding (2000), Science and Trinity (2004), and Quantum Physics and Theology (2007).
     

7. Planck biography
   
   • I described it as the elementary quantum of action.
     
   • Either the quantum of action was a fictional quantity, then the whole deduction of the radiation law was essentially an illusion representing only an empty play on formulas of no significance, or the derivation of the radiation law was based on a sound physical conception.
     
   • In this case the quantum of action must play a fundamental role in physics, and here was something completely new, never heard of before, which seemed to require us to basically revise all our physical thinking, built as this was, from the time of the establishment of the infinitesimal calculus by Leibniz and Newton, on accepting the continuity of all causative connections.
     
   • Planck himself in [Scientific Autobiography, and Other Papers (1949).',7)">7] explains how despite having invented quantum theory he did not understand it himself at first:- .
     
   • I tried immediately to weld the elementary quantum of action somehow in the framework of classical theory.
     
   • My futile attempts to put the elementary quantum of action into the classical theory continued for a number of years and they cost me a great deal of effort.
     
   • Planck who was 42 years old when he made his historic quantum announcement, took only a minor part in the further development of quantum theory.
     
   • Sadly his life was filled with tragedy in the years following his remarkable initiation of the study of quantum mechanics.
     
   • Max Planck: Quantum Theory .
     
   • History Topics: The quantum age begins .
     
   • History Topics: Light through the ages: Relativity and quantum era .
     

8. Bohr Niels biography
   
   • Using quantum ideas due to Planck and Einstein, Bohr conjectured that an atom could exist only in a discrete set of stable energy states.
     
   • set out his startling attempt to combine aspects of classical physics with the concept of Planck's quantum of action.
     
   • He talked of atomic stability and electrodynamic theory giving an account of the origins of quantum theory, the hydrogen spectrum, explaining the relationships between the elements.
     
   • His explanation covered the absorption and excitation of spectral lines and the correspondence principle which he had set out in three papers On the quantum theory of spectra between 1918 and 1922.
     
   • Quantum mechanics may be said to have arrived in 1925 and two years later Heisenberg stated his uncertainty principle.
     
   • He proposed complementarity of perceptions and pictures, particle-wave, conjugate variables, quantum evolution - classical measurements etc.
     
   • as a fundamentally new interpretation of the foundations of quantum theory.
     
   • Bohr thought that his idea of complementarity could play an important role in fields other than quantum physics and he worked on these ideas throughout the rest of his life.
     
   • It was Bohr's view of quantum theory which was eventually to become accepted.
     
   • Bohr's other major contributions, in addition to quantum theory, include his theoretical description of the periodic table of elements around 1920, his theory of the atomic nucleus being a compound structure in 1936, and his understanding of uranium fission in terms of the isotope 235 in 1939.
     
   • History Topics: Light through the ages: Relativity and quantum era .
     
   • History Topics: The quantum age begins .
     

9. Klein Oskar biography
   
   • Klein chose to solve the problem by essentially extending his work to a fifth dimension, though his early unification ideas centred around quantum physics as the catalyst.
     
   • After a time Klein argued less and less that quantum physics could lead to a unified picture, in fact he later abandoned the idea entirely.
     
   • As Kragh [Quantum Generations: A History of Physics in the Twentieth Century (Princeton, 1999)',4)">4] explains, Klein attempted to explain the atomicity of electricity as a quantum law.
     
   • It is interesting to note that this equation appeared exactly as it has been written in David Bohm's 1951 book Quantum Theory but was not called the Klein-Gordon equation.
     
   • However, Bethe and Jackiw's Intermediate Quantum Mechanics, originally written in 1964, does refer to the same equation as the Klein-Gordon equation.
     
   • But, nonetheless, it was an important point in quantum theory and, along with his unification theory, was to ensure a lasting legacy for Klein and cemented 1926 as a pivotal year in his life.
     
   • In 1927, Klein was appointed Lektor in Copenhagen but nonetheless continued his research working with Pascual Jordan on the second quantization in quantum mechanics.
     
   • In his work with Jordan, he demonstrated the close connection between quantum fields and quantum statistics.
     
   • His continued work included the quantum mechanics of the second law of thermodynamics and Klein's lemma.
     

10. Ehrenfest biography
    
    • An important paper was published by Ehrenfest in 1911 in Annalen der Physik on the essential features of quantum theory.
      
    • On his travels he learnt that Poincare had written a paper on quantum theory which gave similar results to those in his Annalen der Physik paper.
      
    • Among Ehrenfest's contributions to quantum statistics was an understanding of the nature of photons, and their properties which were implied by Planck's radiation law.
      
    • He worked on quantum theory applying it to rotating bodies.
      
    • Ehrenfest's graduate lectures consisted of a two-year course: Maxwell theory, ending with the theory of electrons and some relativity, one year; and statistical mechanics, ending with atomic structure and quantum theory the other.
      
    • In 1925 when quantum mechanics began to dominate work in theoretical physics, Ehrenfest felt he had problems [The Niels Bohr Archives (Copenhagen, unpublished).',9)">9]:- .
      
    • When they met in Leiden [The lessons of the quantum theory (Amsterdam, 1986), 325.',7)">7]:- .
      
    • [Ehrenfest] and Bohr had much to talk about together -- from the current problems of quantum theory to the Icelandic sagas, from the stages of a child's development to the difference between genuine physicists and the other.
      
    • Ehrenfest was unhappy at the disagreement between Bohr and Einstein over quantum theory.
      

11. Feynman biography
    
    • There was no course on quantum mechanics, a topic that Feynman was very keen to study, so together with a fellow undergraduate, T A Welton, he began to read the available texts in the spring of 1936.
      
    • By 1937 Feynman was reading Dirac's The principles of quantum mechanics and seeing how his highly original ideas fitted into Dirac's approach.
      
    • However, he then went on to develop a new approach to quantum mechanics using the principle of least action.
      
    • Suddenly the desire to undertake research hit him again and he returned to the quantum theory of electrodynamics that he was working on before World War II.
      
    • Feynman's main contribution was to quantum mechanics, following on from the work of his doctoral thesis.
      
    • for fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.
      
    • For example Quantum Electrodynamics (1961) and The Theory of Fundamental Processes (1961), The Feynman Lectures on Physics (1963-65) (3 volumes), The Character of Physical Law (1965) and QED: The Strange Theory of Light and Matter (1985).
      
    • History Topics: Light through the ages: Relativity and quantum era .
      

12. Born biography
    
    • Beginning in 1926, Born collaborated with Pauli and Heisenberg, who was a student of Born's, on quantum mechanics (the term "quantum mechanics" is due to Born).
      
    • He recognised Heisenberg's approach to quantum mechanics as being matrix algebra.
      
    • Born produced work of fundamental importance in quantum mechanics beginning with this collaboration.
      
    • His treatment replaced the original quantum theory, which regarded electrons as particles, with a mathematical description representing their observed behaviour more accurately.
      
    • for his contributions to theoretical physics an general and to the development of quantum mechanics in particular.
      
    • History Topics: Light through the ages: Relativity and quantum era .
      
    • History Topics: The quantum age begins .
      

13. Schrodinger biography
    
    • He was able to continue research and it was at this time that he published his first results on quantum theory.
      
    • From 1921 he studied atomic structure, then in 1924 he began to study quantum statistics.
      
    • Wave mechanics, as proposed by Schrodinger in these papers, was the second formulation of quantum theory, the first being matrix mechanics due to Heisenberg.
      
    • I am convinced that you have made a decisive advance with your formulation of the quantum condition..
      
    • In 1935 Schrodinger published a three-part essay on The present situation in quantum mechanics in which his famous Schrodinger's cat paradox appears.
      
    • This was a thought experiment where a cat in a closed box either lived or died according to whether a quantum event occurred.
      
    • History Topics: Light through the ages: Relativity and quantum era .
      
    • History Topics: The quantum age begins .
      

14. Dyson biography
    
    • It was from this time that Dyson's work focused on quantum electrodynamics.
      
    • Something happened at this time that greatly pleased Dyson; Tomanaga in Japan had developed significant work in relativistic quantum field theory.
      
    • The Tomonaga-Schwinger quantum electrodynamics is discussed with due emphasis on the physical ideas involved and the equivalence with a mainly unpublished theory by Feynman is established.
      
    • Dyson's famous paper on renormalisation of the S-matrix The S matrix in quantum electrodynamics in 1949 became a very highly regarded and influential work in quantum electrodynamics.
      
    • The application of quantum electrodynamics to scattering problems is discussed in terms of the calculation of the S matrix, an operator which converts the ingoing waves of the initial state into the outgoing waves of the final state.
      
    • It has a wealth of topics and of seminal contributions: the famous QED [quantum electrodynamics] papers, the stability of matter, the invention of the hierarchical Ising models, the disordered linear chain, random matrices, spin wave theory, etc.; Dyson has made his mark in all these varied subjects.
      

15. Weyl biography
    
    • With his application of group theory to quantum mechanics he set up the modern subject.
      
    • It was his lecture course on group theory and quantum mechanics in Zurich in session 1927-28 which led to his third major text Gruppentheorie und Quantenmechanik published in 1928.
      
    • Wheeler's theory, like Weyl's, lacks the connection with quantum phenomena that is so important for interactions other than gravitation.
      
    • [In the fourth lecture he] shows how the special theory of relativity is essentially the study of the inherent symmetry of the four-dimensional space-time continuum, where the symmetry operations are the Lorentz transformations; and how the symmetry operations of an atom, according to quantum mechanics, include the permutations of its peripheral electrons.
      
    • Preface to H Weyl's Theory of groups and quantum mechanics - First Edition .
      
    • Preface to H Weyl's Theory of groups and quantum mechanics - Second Edition .
      
    • Introduction to H Weyl's theory of groups and quantum mechanics .
      

16. Schwinger biography
    
    • Still more important were his contributions to the development of the modern form of quantum electrodynamics, through introduction of the "renormalization" technique.
      
    • In 1951 he proposed, what is today called the Schwinger effect in quantum electrodynamics, where electron-positron pairs are sucked out of a vacuum by an electric field.
      
    • Schwinger was joint winner of the Nobel Prize for Physics (1965) for his work in formulating quantum electrodynamics and thus reconciling quantum mechanics with Einstein's special theory of relativity.
      
    • The electrons of an atom move according to the laws of quantum mechanics established in 1925 and the next following years.
      
    • Schwinger had developed the formalism of the new quantum electrodynamics in several fundamental papers ..
      
    • The list of his contributions is staggering, from his early work leading to the Schwinger action principle, Euclidean quantum field theory, and the genesis of the standard model, to later valuable work on magnetic charge and the Casimir effect.
      

17. Stueckelberg biography
    
    • The Michigan summer physics programme was held every year and, in 1928, Kramers lectured on quantum theory and Ehrenfest on statistical physics.
      
    • In fact, these meals turned into a series of tutorials on the quantum mechanics.
      
    • We saw a chance to get in on the ground floor of research in quantum mechanics.
      
    • They were helped by Edward U Condon, a pioneer in quantum mechanics, and Howard P Robertson, who was appointed as professor of physics at Princeton in 1929.
      
    • A big advance in theoretical physics was the renormalization programme in quantum field theory.
      
    • This was not Stueckelberg's only contribution to the renormalization programme, however, for in the early 1940s he wrote a long paper outlining a complete and correct description of the renormalization procedure for quantum electrodynamics.
      
    • for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.
      

18. Uhlenbeck biography
    
    • Ehrenfest's graduate lectures consisted of a two-year course: Maxwell theory, ending with the theory of electrons and some relativity, one year; and statistical mechanics, ending with atomic structure and quantum theory the other.
      
    • In January 1925 Pauli had proposed that the electron should be given an additional fourth quantum number which was a half integer.
      
    • it occurred to me that , since (I had learned) each quantum number corresponds to a degree of freedom of the electron, Pauli's fourth quantum number must mean that the electron had an additional degree of freedom -- in other words the electron must be rotating.
      
    • The concept immediately excited Niels Bohr, Pauli, Einstein, Heisenberg and others interested in quantum theory.
      
    • It was of fundamental importance in quantum mechanics, systematising statistical notions and expanding on the electron spin ideas which had announced two years earlier.
      
    • As well as fundamental work on quantum mechanics, Uhlenbeck worked on atomic structure and the kinetic theory of matter.
      

19. Von Neumann biography
    
    • Veblen invited von Neumann to Princeton to lecture on quantum theory in 1929.
      
    • It was in this period also that he began his classical work on quantum theory, the mathematical foundation of the theory of measurement in quantum theory and the new statistical mechanics.
      
    • His text Mathematische Grundlagen der Quantenmechanik (1932) built a solid framework for the new quantum mechanics.
      
    • Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann's stature.
      
    • His interest in ergodic theory, group representations and quantum mechanics contributed significantly to von Neumann's realisation that a theory of operator algebras was the next important stage in the development of this area of mathematics.
      
    • History Topics: The quantum age begins .
      

20. Landau Lev biography
    
    • In fact his first publication appeared in print in the year he graduated, being a paper on quantum theory.
      
    • He worked on atomic collisions, astrophysics, low-temperature physics, atomic and nuclear physics, thermodynamics, quantum electrodynamics, kinetic theory of gases, quantum field theory, and plasma physics.
      
    • These include Statistical physics (1938), Mechanics, Field theory, Quantum mechanics, and Theory of elasticity.
      
    • The chapters of the book indicates the main topics of their joint research: Mechanics, theory of fields, quantum mechanics, quantum electrodynamics, classical statistical physics, quantum statistical physics, fluid mechanics, theory of elasticity, electrodynamics of continuous media, and physical kinetics.
      

21. Broglie biography
    
    • the mystery in which the structure of matter and of radiation was becoming more and more enveloped as the strange concept of the quantum, introduced by Planck in 1900 in his researches into black-body radiation, daily penetrated further into the whole of physics.
      
    • De Broglie's doctoral thesis Recherches sur la theorie des quanta (Researches on the quantum theory) of 1924 put forward this theory of electron waves, based on the work of Einstein and Planck.
      
    • And I realised that, on the one hand, the Hamilton-Jacobi theory pointed somewhat in that direction, for it can be applied to particles and, in addition, it represents a geometrical optics; on the other hand, in quantum phenomena one obtains quantum numbers, which are rarely found in mechanics but occur very frequently in wave phenomena and in all problems dealing with wave motion.
      
    • History Topics: Light through the ages: Relativity and quantum era .
      
    • History Topics: The quantum age begins .
      

22. Witten biography
    
    • It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics.
      
    • Of course, quantum physics had from the beginning a marked influence in many areas of mathematics - functional analysis and representation theory, to mention just two.
      
    • obligatory reading for geometers interested in understanding modern quantum field theory.
      
    • Witten explains that "supersymmetric quantum mechanics" is just Hodge-de Rham theory.
      
    • The real aim of the paper is however to prepare the ground for supersymmetric quantum field theory as the Hodge-de Rham theory of infinite dimensional manifolds.
      
    • In recent years, Witten focused his attention on topological quantum field theories.
      

23. Penrose biography
    
    • Another was a course by Paul Dirac on quantum mechanics which was beautiful in a completely different way ..
      
    • Penrose looked for a unified theory combining relativity and quantum theory since quantum effects become dominant at the singularity.
      
    • One of Penrose's major breakthroughs was his introduction of twistor theory in an attempt to unite relativity and quantum theory.
      
    • In the process of the argument elegant expositions, at a level suitable for the unlearned but reasonably sophisticated reader, are given of a wide variety of topics ranging from the nature of algorithms and abstract computability, through results on undecidability and incompleteness, the basic structures of classical physics, the basic structures and philosophical puzzles in quantum mechanics, the basic features of entropic asymmetry and its relation to cosmological structure, the search for an adequate quantum theory of gravity, to some of the results of neuro-anatomy and research into the functioning of the brain.
      

24. Bose biography
    
    • This university was a research institution for postgraduate studies and here Bose was able to study recent European texts on quantum theory and relativity which, before the opening of the new institution, had not been readily available in India.
      
    • He did important work in quantum theory, in particular on Planck's black body radiation law.
      
    • While he was in Berlin Bose attended a course on quantum theory given by Born.
      
    • History Topics: The quantum age begins .
      
    • History Topics: Light through the ages: Relativity and quantum era .
      

25. Temple biography
    
    • Chapman obtained a scholarship for Temple to undertake further research and he spent a year at Imperial working on quantum theory before going to Cambridge where he worked with Eddington.
      
    • Relativity theory, aerodynamics and quantum mechanics have been mentioned above but he also worked on analysis contributing to the study of the Lebesgue integral.
      
    • He wrote seven books, two on quantum theory An introduction to quantum theory (1931) and The general principles of quantum theory (1934).
      

26. Einstein biography
    
    • Einstein used Planck's quantum hypothesis to describe the electromagnetic radiation of light.
      
    • He made important contributions to quantum theory, but he sought to extend the special theory of relativity to phenomena involving acceleration.
      
    • Niels Bohr and Einstein were to carry on a debate on quantum theory which began at the Solvay Conference in 1927.
      
    • History Topics: Light through the ages: Relativity and quantum era .
      
    • History Topics: The quantum age begins .
      

27. Hau biography
    
    • But after a while I discovered quantum mechanics, and that got me interested in physics again, and I've been hooked ever since.
      
    • For her doctoral studies in quantum theory Hau worked on ideas similar to those involved in fibre optic cables carrying light, but her work involved strings of atoms in a silicon crystal carrying electrons.
      
    • Denmark has a long scientific tradition that included the great Niels Bohr, one of the founders of quantum theory.
      
    • For instance, research in quantum mechanics has been supported in Denmark by the makers of Carlsberg beer since the 1920's.
      
    • Hau produced slow light by inducing quantum interference in the condensate.
      
    • In addition to these and other awards, in 2007 Hau and her team were selected by Nature as the "Favourite of 2007 in Quantum Physics" and her team was also selected by the American Institute of Physics as "Top Ten Physics News Stories of 2007".
      

28. Bethe biography
    
    • This was a major contribution not only to solid state physics but also to the foundations of quantum mechanics as it further demonstrated the unusual consequences of the wave-like properties of matter.
      
    • While in Cambridge he coauthored a paper on the quantum theory of absolute zero with fellow visitors Guido Beck and Wolfgang Riezler that turned out to be a hoax concocted to embarrass another Cambridge physicist (and good friend of Fowler's), Arthur Eddington.
      
    • With his work on quantum mechanics in solid state crystal lattices and his work on stellar nucleosynthesis, he was already a highly accomplished physicist in more than one subfield.
      
    • And yet, he was still able to make significant contributions to yet another distinct subfield: quantum electrodynamics (QED).
      
    • It was just after this that he published a review for the Handbuch der Physik dealing heavily with the quantum theory of hydrogen and helium.
      

29. Mackey biography
    
    • His interest in physics continued, however, and he published A theorem of Stone and von Neumann (1949) in which he generalised a theorem about quantum mechanics proved by Stone and von Neumann in 1930.
      
    • In the spring of 1960 he gave a lecture course at Harvard University on the mathematical foundations of quantum mechanics.
      
    • An edited version of these lectures became his famous classic text The mathematical foundations of quantum mechanics published in 1963:- .
      
    • the aim of the book is to explain, or at least to illuminate, the essential aspects of classical and quantum mechanics from a point of view more congenial to pure mathematicians than that encountered in physics texts.
      
    • But the author's theory is surprisingly versatile, with applications in number theory, harmonic analysis, ergodic theory, quantum mechanics, and statistical mechanics, and these applications are worked out in detail..
      

30. Kramers biography
    
    • Bohr agreed to have Kramers as a student and under his supervision he wrote his doctoral thesis Intensities of spectral lines: On the application of the quantum theory to the problem of the relative intensities of the components of the fine structure and of the Stark effect of the lines of the hydrogen spectrum which he submitted to the University of Leiden.
      
    • It was translated into English by Dirk ter Haar, a former student of Kramers, and published as Quantum mechanics in 1957.
      
    • Kramers divided the subject into two parts, one dealing with the foundations of quantum theory and one presenting the quantum theory of the electron and of radiation.
      
    • The whole volume, however, has great unity and it constitutes one of the very best presentations of quantum theory (nuclear phenomena excluded) every published.
      

31. Fock biography
    
    • Fock had already published two papers, one on quantum mechanics and one on mathematical physics, before he graduated from Petrograd University in 1922.
      
    • Schrodinger published his two fundamental papers on quantum theory in the spring of 1926 and Fock immediately started to develop the ideas and by the end of the year two of his own important papers on the Schrodinger equation had been published.
      
    • Further in 1932 Fock published an important paper with Podolsky and Dirac on quantum electrodynamics in which the concept of multiple time formalism was introduced, and in the same year Fock introduced the concept of the Fock space in another classical paper.
      
    • The articles presented also possess a great historical value, most of them representing important steps in the development of quantum mechanics and quantum field theory during the first half of last century, and should be subject to careful and detailed analysis from historians of science.
      

32. Sommerfeld biography
    
    • From 1911 his main area of interest became quantum theory.
      
    • Sommerfeld's work led him to replace the circular orbits of the Niels Bohr atom with elliptical orbits; he also introduced the magnetic quantum number in 1916 and, four years later, the inner quantum number.
      
    • It was theoretical work attempting to explain the inner quantum number that led to the discovery of electron spin.
      
    • [He] was at the forefront of the work in electromagnetic theory, relativity and quantum theory and he was the great systematizer and teacher who inspired many of the most creative physicists in the first thirty years of this century.
      

33. Reichenbach biography
    
    • However, in the United States he also wrote major works on the philosophical foundations of quantum mechanics and on time.
      
    • On the first of these topics he published the book Philosophic foundations of quantum mechanics (1944), and the two papers Uber die erkenntnistheoretische Problemlage und den Gebrauch einer dreiwertigen Logik in der Quantenmechanik (1951) and Les fondements logiques de la theorie des quanta.
      
    • there is not any exhaustive interpretation of quantum mechanics which is free from causal anomalies.
      
    • In the two papers be examines using Lukasiewicz's three-valued logic in quantum mechanics.
      

34. Daubechies biography
    
    • in physics for a thesis entitled Representation of quantum mechanical operators by kernels on Hilbert spaces of analytic functions.
      
    • In 1978 An application of hyperdifferential operators to holomorphic quantization appeared, then a number of papers written jointly with Dirk Aerts: A characterization of subsystems in physics; Physical justification for using the tensor product to describe two quantum systems as one joint system; A mathematical condition for a sublattice of a propositional system to represent a physical subsystem, with a physical interpretation; and A connection between propositional systems in Hilbert spaces and von Neumann algebras.
      
    • One of the basic rules of Hilbert space quantum mechanics is that when two physical systems, say S1 and S, are viewed as the pieces of a compound system S, then the Hilbert space to be associated to S is the tensor product of the Hilbert spaces H1 and H associated to S1 and S.
      
    • This rule found little justification in the traditional logico-algebraic approach to quantum mechanics.
      

35. Drinfeld biography
    
    • Entitled Quantum groups, the talk reviewed the results obtained by Drinfeld and M Jimbo on Hopf algebras (quantum groups).
      
    • He discussed the concepts of quantum groups and quantization, and also talked about Poisson groups, Lie bi-algebras and the classical Yang-Baxter equation.
      
    • for his work on quantum groups and for his work in number theory.
      
    • Not only do they span work in algebraic geometry and number theory, but his most recent ideas have taken a strikingly different direction: he has been doing significant work on mathematical questions motivated by physics, including the relatively new theory of quantum groups.
      
    • Drinfeld's main achievements are his proof of the Langlands conjecture for GL(2) over a functional field; and his work in quantum group theory.
      
    • The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
      
    • Drinfeld's work on Langlands conjectures, quantum groups, p-adic uniformizations etc.
      
    • Drinfeld's work deeply influenced the world of mathematics of the last two decades, Several research monographs, Seminar Notes and hundreds of papers were dedicated to the two new chapters of mathematics created by him - the so-called Drinfeld modules and quantum groups.
      

36. Nikodym biography
    
    • the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics.
      
    • In 1946 Nikodym and his wife Stanislawa left for Belgium and France where he began his work on mathematical foundations of quantum mechanics.
      
    • His last book The Mathematical Apparatus for Quantum-Theories, based on the Theory of Boolean Lattices published in 1966 by Springer-Verlag contains, on almost thousand pages, the mathematical formalism for quantum mechanics or more precisely a detailed study of the Boolean subalgebras of the logic of closed subspaces of a complex Hilbert space.
      

37. Lorentz biography
    
    • This conference looked at the problems of having two approaches, namely that of classical physics and of quantum theory.
      
    • However Lorentz never fully accepted quantum theory and always hoped that it would be possible to incorporate it back into the classical approach.
      
    • History Topics: The quantum age begins .
      
    • History Topics: Light through the ages: Relativity and quantum era .
      

38. Manin biography
    
    • He has written papers on: algebraic geometry including ones on the Mordell conjecture for function fields and a joint paper with V Iskovskikh on the counter-example to the Luroth problem; number theory including ones about torsion points on elliptic curves, p-adic modular forms, and on rational points on Fano varieties; and differential equations and mathematical physics including ones on string theory and quantum groups.
      
    • This book presents a kind of philosophical assessment along with an introductory description of developments and the present status of quantum field theory as a basis for the theory of elementary particles.
      
    • Other books by Manin include Cubic forms: algebra, geometry, arithmetic (Russian) (1972), A course in mathematical logic (1977), Computable and noncomputable (Russian) (1980), Quantum groups and noncommutative geometry (1988), Topics in noncommutative geometry (1991), Frobenius manifolds, quantum cohomology, and moduli spaces (1999).
      

39. Jeans biography
    
    • Of course Jeans' paper can be seen as a mathematical "proof" that classical physics does not suffice, but it is interesting to note that his pre-quantum ideas concerning the very long time required for systems to come into equilibrium and the observed breakdown of equipartition in specific heat measurements on molecular gases have been used again in relatively recent times more than 80 years after Jeans introduced them.
      
    • Certainly Jeans continued to produce a remarkable output, and he wrote an excellent report on Radiation and Quantum Theory for the Physical Society in 1914.
      
    • Although World War I prevented Jeans' report from being widely read in Britain until after 1918, it then had a major impact on having quantum theory and the Bohr theory of the atom accepted by the British scientific community.
      
    • Although Jeans never published original contributions to quantum theory, he showed in such popular books that he had kept up with the developments in this area.
      

40. Khinchin biography
    
    • In 1951 he extended the work of this 1943 book when he published Mathematical foundations of quantum statistics.
      
    • Topics covered included: local limit theorems for sums of identically distributed random variables; the foundations of quantum mechanics; general principles of quantum statistics; the foundations of the statistics of photons; entropy; and the second law of thermodynamics.
      
    • The book has been rated as being equal in quality to von Neumann's masterpiece Mathematical foundations of quantum mechanics.
      

41. Peierls biography
    
    • It was Sommerfeld who introduced Peierls to quantum mechanics during these two years and this proved highly significant for Peierls' career.
      
    • The topics covered are mostly from quantum theory and its applications; statistical physics and even relativity are touched upon.
      
    • this book can be warmly recommended to physics students and to their teachers as a valid auxiliary tool for courses in quantum mechanics, structure of matter and statistical physics.
      

42. Jeffreys Bertha biography
    
    • There she studied quantum theory and other topics under Max Born and Werner Heisenberg at a really exciting time with Heisenberg's theory of quantum mechanics published in 1925 and Schrodinger's wave mechanics in 1926.
      
    • At Girton, Bertha Jeffreys continued to undertake research on quantum theory publishing papers such as The classification of multipole radiation (1952), The use of the Airy functions in a potential barrier problem (1956), and The asymptotic approximation (AA) method (1961).
      

43. Hawking biography
    
    • Using quantum theory and general relativity he was able to show that black holes can emit radiation.
      
    • His success with proving this made him work from that time on combining the theory of general relativity with quantum theory.
      
    • These mini black holes have large gravitational attraction governed by general relativity, while the laws of quantum mechanics would apply to objects that small.
      
    • While many prominent physicists, cosmologists and astronomers have made important contributions to the study of quantum gravity and cosmology, the impact of Stephen Hawking's contributions to the field truly stand out.
      
    • Although his work on black hole thermodynamics is perhaps the most well known, Hawking has also made major contributions to the study of singularity theorems in general relativity, black hole uniqueness, quantum fields in curved spacetimes, Euclidean quantum gravity, the wave function of the universe and many other areas as well.
      
    • Furthermore, it would be hard to imagine assembling any list of researchers working in quantum cosmology without including a large number of Hawking's students and close colleagues.
      

44. Friedrichs biography
    
    • During the 1950s Friedrichs wrote five articles on Mathematical aspects of the quantum theory of fields which eventually became part of his book with the same title.
      
    • mathematicians who are familiar with the fundamental concepts of quantum theory of single particles and who would like to learn which mathematical concepts are involved in the simplest problems of field quantum theory.
      

45. Fuchs Klaus biography
    
    • During his years at Bristol, Fuchs published A Quantum Mechanical Investigation of Cohesive Forces of Metallic Copper (1935), A Quantum Mechanical Calculation of the Elastic Contants of Monovalent Metals (1936), and The Elastic Contants and Specific Heats of the Alkali Metals (1936).
      
    • Born had been very impressed with Fuch's paper A Quantum Mechanical Calculation of the Elastic Contants of Monovalent Metals, and this made him keen to employ the brilliant young man.
      

46. Schwarzschild biography
    
    • While in Russia he wrote two papers on Einstein's relativity theory and one on Planck's quantum theory.
      
    • The quantum theory paper explained that the Stark effect, namely the splitting of the spectral lines of hydrogen by an electric field (the amount being proportional to the field strength), could be proved from the postulates of quantum theory.
      

47. Lifshitz biography
    
    • The chapters of the book indicates the main topics of Lifshitz's research: Mechanics, theory of fields, quantum mechanics, quantum electrodynamics, classical statistical physics, quantum statistical physics, fluid mechanics, theory of elasticity, electrodynamics of continuous media, physical kinetics.
      

48. Turing biography
    
    • He read Einstein's papers on relativity and he also read about quantum mechanics in Eddington's The nature of the physical world.
      
    • At about the same time he read von Neumann's 1932 text on quantum mechanics, a subject he returned to a number of times throughout his life.
      
    • Not only did he press forward with further study of morphogenesis, but he also worked on new ideas in quantum theory, on the representation of elementary particles by spinors, and on relativity theory.
      

49. Uhlenbeck Karen biography
    
    • Witten, who gave the next talk on Geometry and quantum field theory at the symposium said:- .
      
    • Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
      
    • Two years later, in 1990, Witten received a Fields Medal for his work on topological quantum field theories.
      

50. Eddington biography
    
    • In [Eddington\'s search for a fundamental theory : a key to the universe (Cambridge, 1994).',9)">9] Kilmister delves deeply into the ideas which led Eddington to the theories he put forward in Fundamental Theory in attempting to unite quantum mechanics and general relativity.
      
    • It was Dirac's 1928 paper on the wave equation of the electron which had first set Eddington on the path of seeking ways to unify quantum mechanics and general relativity.
      
    • History Topics: Light through the ages: Relativity and quantum era .
      

51. Wien biography
    
    • Max Planck, who was a colleague of Wien's when he was carrying out this work, later, in 1900, based quantum theory on the fact that Wien's law, while valid at high frequencies, broke down completely at low frequencies.
      
    • History Topics: Light through the ages: Relativity and quantum era .
      
    • History Topics: The quantum age begins .
      

52. Fowler biography
    
    • 1926 marked the publication of his most seminal individual paper which linked the gaseous degenerate state (obeying quantum statistics, co-discovered by P A M Dirac, who was introduced to quantum theory by Fowler himself) to white dwarf stars.
      
    • It was Fowler who ultimately introduced Paul Dirac to the burgeoning field of quantum theory in 1923 leading Dirac to the forefront of its ultimate discovery in 1925.
      

53. Kirchhoff biography
    
    • Fundamental work by Kirchhoff on black body radiation (a term he introduced in 1862) was important in the development of quantum theory.
      
    • History Topics: The quantum age begins .
      

54. Atiyah biography
    
    • The index theorem could be interpreted in terms of quantum theory and has proved a useful tool for theoretical physicists.
      
    • More recently Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists, notably E Witten, to the attention of the mathematical community.
      

55. Lemaitre biography
    
    • If the world has begun with a single quantum, the notions of space and time would altogether fail to have any meaning at the beginning; they would only begin to have a sensible meaning when the original quantum had been divided into a sufficient number of quanta.
      

56. Cayley biography
    
    • His work on matrices served as a foundation for quantum mechanics, which was developed by Werner Heisenberg in 1925.
      
    • History Topics: quantum theory .
      

57. Robertson biography
    
    • However he did make outstanding contributions to differential geometry, quantum theory, the theory of general relativity, and cosmology [H P Robertson : January 27, 1903-August 26, 1961.
      
    • In fact his association with Weyl went much further and in 1931 he published an English translation of the second edition of Weyl's classic text The theory of groups and quantum mechanics.
      

58. Feigenbaum biography
    
    • His official courses were on quantum mechanics, classical mechanics, and complex function theory.
      
    • During his two years at Cornell he taught courses on variational techniques and on quantum mechanics.
      

59. Aleksandrov Aleksandr biography
    
    • These first three works were all as a result of his mathematical work with Delone but also in 1934 he published two physics papers on quantum mechanics On the calculation of the energy of a bivalent atom by Fok's method and Remark on the commutation rule in Schrodinger's equation.
      
    • He published on optics, quantum mechanics, and relativity.
      

60. Levi-Civita biography
    
    • In 1933 Levi-Civita contributed to Dirac's equations of quantum theory.
      
    • History Topics: The quantum age begins .
      

61. Hoyle biography
    
    • For example Born taught him quantum mechanics, Eddington taught him general relativity, and he was also taught by Dirac.
      
    • In 1939 Hoyle published a major paper on Quantum electrodynamics in the Proceedings of the Cambridge Philosophical Society.
      

62. Wiener Norbert biography
    
    • Especially important was his contacts with Paul Levy and with Gottingen where his work was seen to have important connections with quantum mechanics.
      
    • Wiener had an extraordinarily wide range of interests and contributed to many areas in addition to those we have mentioned above including communication theory, cybernetics (a term he coined), quantum theory and during World War II he worked on gunfire control.
      

63. Albert Abraham biography
    
    • These algebras had been introduced by Pascual Jordan as being related to quantum theory.
      
    • Albert was able to use his expertise in structural questions regarding algebras to solve some of the problems in his 1934 paper On certain algebras of quantum mechanics.
      

64. Murnaghan biography
    
    • Over the period up to 1936, in addition to the major texts we have already mentioned, Murnaghan undertook research and published papers on a wide variety of topics such as electrodynamics, relativity, tensor analysis, elasticity, dynamics, aerodynamics, quantum mechanics, and celestial mechanics.
      
    • elementary and self-contained account of the theory of group representations with special reference to those groups which have turned out to be of fundamental significance for quantum mechanics, especially nuclear physics.
      

65. Cartan biography
    
    • These are complex vectors that are used to transform three-dimensional rotations into two-dimensional representations and they later played a fundamental role in quantum mechanics.
      
    • History Topics: The quantum age begins .
      

66. Boltzmann biography
    
    • History Topics: The quantum age begins .
      
    • History Topics: Light through the ages: Relativity and quantum era .
      

67. Riesz biography
    
    • This is of fundamental importance in early quantum theory.
      
    • History Topics: Quantum theory .
      

68. Green Sandy biography
    
    • More recently, he has made substantial contributions to the study of representations of quantum groups via a relationship with the Hall algebras that he had studied earlier in his 1955 paper.
      
    • Other published lectures include Classical invariants and Classical groups both delivered at the University of Coimbra in 1993, and Hall algebras and quantum groups delivered at the University of Coimbra in March 1994.
      

69. Kato biography
    
    • However he had published many papers by the time the doctorate was awarded including work on pair creation by gamma rays, the motion of an object through a fluid and results on the spectral theory of operators arising in quantum mechanics.
      
    • Also, resulting from his visit to New York University, Kato's notes On the eigenfunctions of many-particle systems in quantum mechanics were published by the Mathematical Sciences research Institute of New York University in 1956.
      

70. Kneser Hellmuth biography
    
    • His doctoral studies there were directed by Hilbert and he submitted a dissertation on the mathematics of quantum mechanics in 1921 Untersuchungen zur Quantentheorie.
      
    • After his doctoral work on quantum theory he turned toward topology and the theory of analytic functions in several indeterminates.
      

71. Julia biography
    
    • This book is the sixteenth of the well known series, 'Cahiers Scientifiques,' and is the first of a series which proposes to give the mathematical foundation of quantum mechanics.
      
    • In this first volume the essential difficulties of quantum mechanics (some of which concern the fact that Hubert space is not finite dimensional) are merely foreshadowed, the attention being directed in the main to vector analysis in a space of finite dimensions.
      

72. Larmor biography
    
    • Larmor's contributions came at a time when there were major revolutions in physics with the passing of classical physics to be replaced by quantum theory and relativity.
      
    • It was difficult to ascertain how much he appreciated the new developments (especially quantum theory), because he was accustomed to adopt a pose which exaggerated his aloofness.
      

73. Hertz Gustav biography
    
    • Their work confirmed experimentally quantum theory as proposed by Bohr by showing that when an electron strikes an atom of mercury vapour, it must possess a certain minimum energy before that energy is absorbed by the atom.
      
    • He applied Planck's quantum theory to the problem of atomic structure and light emission, and thereby greatly extended this theory.
      

74. Cooper biography
    
    • He studied the unbounded operators which arose from quantum theory publishing The characterization of quantum-mechanical operators (1950).
      
    • He corresponded with Einstein on logical inconsistency in quantum theory in 1949 and this led to his paper The paradox of separated systems in quantum theory (1950) in which he discussed the paradox put forward by Einstein, Podolsky and Rosen in 1935.
      
    • These problems originate in questions in the quantum mechanics of many particle systems posed to me by Professor Heinz Post; my thanks are due to him for drawing my attention to the problem and to discussions of the physics involved.
      

75. Todd John biography
    
    • He became ill while giving a lecture series on group theory and quantum mechanics.
      
    • I knew a bit of quantum mechanics because my first introduction to matrix theory was Heisenberg's matrix mechanics, based on infinite matrices.
      

76. Sneddon biography
    
    • The book discussed applications of quantum mechanics rather than studying the theoretical foundations of the topic.
      
    • Also considered are interatomic forces and valence, the theory of solids, collision problems, radiation theory and relativistic quantum theory.
      

77. McShane biography
    
    • an outgrowth of his search for a mathematically correct setting in which to treat the divergent integrals in quantum physics.
      
    • In 1974, the year he retired and was made Professor Emeritus at Virginia, McShane published Stochastic calculus and stochastic models which again reflected his work on the mathematical setting for quantum mechanics.
      

78. Hilbert biography
    
    • This work also established the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics.
      

79. Enskog biography
    
    • Although the advent of quantum theory was to lessen the impact of this theory, it was later seen to be still important in the new context.
      

80. Stone biography
    
    • In 1932 he proved results on spectral theory, arising from group theoretical methods in quantum mechanics, which had been conjectured by Weyl.
      

81. Whittaker John biography
    
    • His first four papers in the late 1920s were on quantum theory.
      

82. Krylov Nikolai biography
    
    • This paper began their work which established the theory of perturbations and transitions of state on a new and uniform basis, both in classical mechanics and quantum mechanics.
      

83. De Rham biography
    
    • The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years.
      

84. Kostrikin biography
    
    • This chapter includes symmetry and applications to quantum mechanics.
      

85. Stefan Josef biography
    
    • History Topics: The quantum age begins .
      

86. Young Alfred biography
    
    • Weyl also began to make use of Young's ideas and Young tableau appear in his famous book Theory of groups and quantum mechanics.
      

87. Petit biography
    
    • The law has exceptions and was not fully understood until quantum theory was used.
      

88. Frobenius biography
    
    • Frobenius's representation theory for finite groups was later to find important applications in quantum mechanics and theoretical physics which may not have entirely pleased the man who had such "pure" views about mathematics.
      

89. Grassmann biography
    
    • Clifford algebras are used today in the theory of quadratic forms and in relativistic quantum mechanics.
      

90. Wintner biography
    
    • In fact at this time the development of Hilbert spaces had become particularly important for the study of quantum theory since this mathematics underlay the theory.
      

91. Weil biography
    
    • In fact Weil's work in this area was basic to work by mathematicians such as Yau who was awarded a Fields Medal in 1982 for work in three dimensional algebraic geometry which has major applications to quantum field theory.
      

92. Poincare biography
    
    • In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology.
      

93. Connes biography
    
    • To place Alain Connes's fundamental and pioneering contributions to operator algebras in context, recall that von Neumann and Murray in the 1930s and 1940s were led by, among other things, the spectral theory of operators on Hilbert space, and by considerations of constructing mathematical models for quantum mechanical systems, to introduce what they called rings of operators - since renamed von Neumann algebras.
      
    • In naming Connes the 2004 Gold Medallist, the CNRS called him "one of the greatest mathematicians of our time." Throughout his career, Connes has applied himself to solving mathematical problems arising from quantum physics and the theory of relativity.
      

94. Friedmann biography
    
    • This group discussed quantum theory, relativity and statistical mechanics.
      

95. Fermi biography
    
    • Fermi gave lectures on quantum theory.
      

96. Dynkin biography
    
    • Around 1980 Dynkin interpreted and vastly generalized an identity which had first come up in the context of quantum field theory.
      
    • Around 1980 Dynkin interpreted and vastly generalized an identity which had first come up in the context of quantum field theory.
      

97. Clifford Alfred biography
    
    • He enjoyed playing bridge, but also decided that he wanted to teach himself quantum mechanics.
      

98. Ince biography
    
    • These tables were useful not only in the problems originally envisaged but also in more recent investigations such as quantum-mechanical problems leading to Mathieu's equation.
      

99. Bronowski biography
    
    • Quantum physics was transformed by Dirac and the others.
      

100. Birkhoff Garrett biography
     
     • A course by E C Kemble on quantum mechanics as well as courses on Lebesgue integration and topology gave him a broad education in mathematics.
       

101. Jones Vaughan biography
     
     • These included (in addition to knots and links) that part of statistical mechanics having to do with exactly solvable models, the very new area of quantum groups, and also Dynkin diagrams and the representation theory of simple Lie algebras.
       

102. Rennie biography
     
     • Among the courses he attended at Cambridge during session 1940-41 was Quantum mechanics by Dirac.
       

103. Helmholtz biography
     
     • History Topics: Light through the ages: Relativity and quantum era .
       

104. Fomin biography
     
     • The reason for his interest in this subject came from another of his many interests, namely quantum physics.
       

105. Van Vleck biography
     
     • He produced the first quantum mechanical theory of magnetism.
       

106. Littlewood Dudley biography
     
     • He also studied quantum mechanics and some of the problems in representation theory he considered were motivated by this.
       

107. Penney biography
     
     • After two years at the University of Wisconsin he returned to England and obtained a doctorate from the University of Cambridge in 1935 on the application of quantum mechanics to the physics of crystals.
       

108. Bremermann biography
     
     • The year 1957 saw Bremermann move into a new area of research when he collaborated with the physicists R Oehme and J G Taylor applying his expertise in complex analysis to work on quantum field theory.
       

109. Gibbs biography
     
     • His work on statistical mechanics was also important, providing a mathematical framework for quantum theory and for Maxwell's theories.
       

110. Loewner biography
     
     • The functions which Loewner called n-monotonic turned out to be of importance for electrical engineering and for quantum physics ..
       

111. Libermann biography
     
     • For example she gave a survey of various geometric concepts and results used in analytical mechanics in her lecture Liouville forms, parallelisms and Cartan connections to the Jean Leray '99 Conference, and reviewed and summarized the theory of Cartan connections in her lecture Cartan connections and momentum maps given at the Classical and Quantum Integrability conference held in Warsaw in 2001.
       

112. Morse biography
     
     • He also wanted to produce a topological version of quantum theory, but this largely remained a dream which he never achieved.
       

113. Frenkel biography
     
     • His book were seen to oppose the deterministic Socialist philosophy, particularly his work on quantum mechanics.
       

114. Wolf Frantisek biography
     
     • Such perturbation results are important, for example, in quantum mechanics where physical phenomena are interpreted through linear operators on Hilbert space.
       

115. Maxwell biography
     
     • History Topics: Light through the ages: Relativity and quantum era .
       

116. Koopmans biography
     
     • However his first two publications, while still an undergraduate at Utrecht, were on quantum mechanics.
       

117. Rosanes biography
     
     • He later said these lectures were particularly important in his development as a mathematician and led him to one of his greatest ideas namely the realisation that Heisenberg's quantum mechanics was represented by matrices.
       

118. Harish-Chandra biography
     
     • Here he studied theoretical physics, this direction being the result of reading Principles of Quantum Mechanics by Dirac which he found himself in the university library.
       

119. FitzGerald biography
     
     • History Topics: Light through the ages: Relativity and quantum era .
       

120. Birkhoff biography
     
     • The foundations of relativity and quantum mechanics were also topics which Birkhoff studied.
       

121. Kaluza biography
     
     • The theory, initially a popular topic of research, quickly lost favour with the introduction of quantum mechanics.
       

122. Chandrasekhar biography
     
     • Next he looked at the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950, followed by hydrodynamic and hydromagnetic stability from 1950 to 1961.
       

123. Schubert Hans biography
     
     • He had previously been with Niels Bohr in Copenhagen and was already a world leader having published his theory of quantum mechanics two years earlier; he received the Nobel prize for this work in 1932.
       

124. Taylor Geoffrey biography
     
     • He also undertook experimental work, following a suggestion by J J Thomson, to test quantum theory.
       

125. Darwin C G biography
     
     • The Royal Medal of the Royal Society has been awarded to Professor C G Darwin, F.R.S., of the Tate Chair of Natural Philosophy in Edinburgh University, for his researches in mathematical physics, especially in quantum mechanics.
       

126. Rayleigh biography
     
     • History Topics: Light through the ages: Relativity and quantum era .
       

127. Rey Pastor biography
     
     • Amongst them were: Alberto Gonzalez Dominguez, who became an important quantum physicist; Alberto Calderon, who would become the chairman of the Department of Mathematics at Chicago University; and many more who considerably enhanced the collective teaching capacity of the mathematical community.
       

128. Van der Waerden biography
     
     • Van der Waerden worked on algebraic geometry, abstract algebra, groups, topology, number theory, geometry, combinatorics, analysis, probability theory, mathematical statistics, quantum mechanics, the history of mathematics, the history of modern physics, the history of astronomy and the history of ancient science.
       

129. Brauer biography
     
     • This work was to provide a background for the work of Paul Dirac in his exposition of the theory of the spinning electron within the framework of quantum mechanics.
       

130. Conway Arthur biography
     
     • He continued to publish on electromagnetic theory, quantum theory, and on electron spin after it was proposed by Uhlenbeck in 1926.
       

131. Titchmarsh biography
     
     • From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics.
       

132. Ricci-Curbastro biography
     
     • History Topics: The quantum age begins .
       

133. Hellinger biography
     
     • Hellinger would keep in touch with Born and developments in quantum mechanics for much of his life.
       

134. Livsic biography
     
     • He remained there until 1957, publishing results on applications of his functional analysis results to quantum theory.
       

135. Bruno Giordano biography
     
     • Other insights amaze physicists today who can see ideas of quantum theory in Bruno's writings.
       

136. Grauert biography
     
     • Other areas on which Grauert wrote papers, but are not included in [Hans Grauert : Selected papers (Springer-Verlag, Berlin, 1994).',2)">2], are hyperbolicity, non-Archimedean function theory and quantum physics.
       

137. Specker biography
     
     • Geneve, Geneva, 1982), 11-24.',5)">5] where his 32 publications up to 1979 are divided into 10 categories: topology, recursive analysis, combinatorial set theory, type theory, axiomatic set theory, Ramsey's theorem, arithmetic, logic of quantum mechanics, algorithms, and miscellaneous.
       

138. Novikov Sergi biography
     
     • He studied a wide variety of applications of mathematics such as dynamical systems in the theory of homogeneous cosmological models, the theory of solitons, the spectral theory of linear operators, quantum field theory and string theory.
       
     • He constructed a global version of Morse theory on manifolds and loop spaces that had novel applications to quantum field theory (multivalued action functionals).
       

139. Mathisson biography
     
     • The subject was of particular interest at that time, as it had become clear that quantum mechanics cannot solve the difficulties that had arisen in connection with the interaction of point particles with fields, and a deeper classical analysis of the problem was needed.
       

140. McCrea biography
     
     • His first research topic involved applying advanced mathematical methods to the study of quantum theory and relativity.
       

141. Drach biography
     
     • The work by Drach in 1919 is the subject of the paper [Classical quantum models and arithmetic problems, Lecture Notes in Pure and Appl.
       

142. Faraday biography
     
     • History Topics: Light through the ages: Relativity and quantum era .
       

143. Singer biography
     
     • Singer is justifiably famous among mathematicians for his deep and spectacular work in geometry, analysis, and topology, culminating in the Atiyah-Singer Index theorem and its many ramifications in modern mathematics and quantum physics.
       

144. Barkla biography
     
     • He has also shown both the applicability and the limitation of the quantum theory in relation to Rontgen radiation.
       

History Topics

1. Quantum mechanics history references
   
   • References for: A history of Quantum Mechanics .
     
   • D Aschman, The legacy of Erwin Schrodinger : quantum mechanics, Trans.
     
   • M Beller, The conceptual and the anecdotal history of quantum mechanics, Found.
     
   • L M Brown, Quantum mechanics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 1252-1260.
     
   • B L Cline, Men who made a new physics : physicists and the quantum theory (Chicago, 1987).
     
   • A B Datsev, The role of Erwin Schrodinger (1887-1961) in the creation and interpretation of quantum mechanics (Bulgarian), Fiz.-Mat.
     
   • C P Enz, Heisenberg's applications of quantum mechanics (1926-33) or the settling of the new land, Helv.
     
   • A Fine, Einstein's interpretations of the quantum theory, in Einstein in context (Cambridge, 1993), 257-273.
     
   • P A Hanle, Erwin Schrodinger's reaction to Louis de Broglie's thesis on the quantum theory, Isis 68 (244) (1977), 606-609.
     
   • J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
     
   • G S Im, Experimental constraints on formal quantum mechanics : the emergence of Born's quantum theory of collision processes in Gottingen, 1924-1927, Archive for History of Exact Sciences 50 (1) (1996), 73-101.
     
   • M Jammer, The philosophy of quantum mechanics : the interpretations of quantum mechanics in historical perspective (New York, 1974).
     
   • C W Kilmister, Quantum mechanics 1899-1925 : a survey of concept formation, Bull.
     
   • P T Matthews, Dirac and the foundation of quantum mechanics, Reminiscences about a great physicist : Paul Adrien Maurice Dirac (Cambridge, 1987), 199-224.
     
   • J Mehra, Dirac's contribution to the early development of quantum mechanics, in Tributes to Paul Dirac (Bristol, 1987), 63-75.
     
   • J Mehra and H Rechenberg, The historical development of quantum theory (5 volumes) (New York-Berlin, 1982-1987).
     
   • K von Meyenn, Pauli, Schrodinger and the conflict about the interpretation of quantum mechanics, in Symposium on the foundations of modern physics (Singapore, 1985), 289-302.
     
   • A I Miller (ed.), Sixty-two years of uncertainty : historical, philosophical, and physical inquiries into the foundations of quantum mechanics (New York, 1990).
     
   • N Mukunda, The mathematics and physics of quantum mechanics, Math.
     
   • L Navarro, On Einstein's statistical-mechanical approach to the early quantum theory (1904-1916), Historia Sci.
     
   • R M Nugayev, The history of quantum mechanics as a decisive argument favoring Einstein over Lorentz, Philos.
     
   • A Pais, Einstein and the quantum theory, Rev.
     
   • A Pais, Max Born's statistical interpretation of quantum mechanics, Science 218 (4578) (1982), 1193-1198.
     
   • M Paty, The nature of Einstein's objections to the Copenhagen interpretation of quantum mechanics, Found.
     
   • J C Polkinghorne, Dirac and the interpretation of quantum mechanics, in Tributes to Paul Dirac (Bristol, 1987), 76-83.
     
   • K Popper, A critical note on the greatest days of quantum theory, Found.
     
   • H Reichenbach, The space problem in the new quantum mechanics, Erkenntnis 35 (1-3) (1991), 29-47.
     
   • F Rohrlich, Schrodinger and the interpretation of quantum mechanics, Found.
     
   • F Rohrlich, Schroedinger's criticism of quantum mechanics-fifty years later, in Symposium on the foundations of modern physics (Singapore, 1985), 555-572.
     
   • B L van der Waerden, From matrix mechanics and wave mechanics to unified quantum mechanics, Notices Amer.
     
   • D Wick, The infamous boundary : seven decades of controversy in quantum physics (Boston, 1995).
     
   • I : Schrodinger and his path to quantum mechanics (Czech), Pokroky Mat.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_Quantum_age_begins.html] .
     

2. Quantum mechanics history references
   
   • References for: A history of Quantum Mechanics .
     
   • D Aschman, The legacy of Erwin Schrodinger : quantum mechanics, Trans.
     
   • M Beller, The conceptual and the anecdotal history of quantum mechanics, Found.
     
   • L M Brown, Quantum mechanics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 1252-1260.
     
   • B L Cline, Men who made a new physics : physicists and the quantum theory (Chicago, 1987).
     
   • A B Datsev, The role of Erwin Schrodinger (1887-1961) in the creation and interpretation of quantum mechanics (Bulgarian), Fiz.-Mat.
     
   • C P Enz, Heisenberg's applications of quantum mechanics (1926-33) or the settling of the new land, Helv.
     
   • A Fine, Einstein's interpretations of the quantum theory, in Einstein in context (Cambridge, 1993), 257-273.
     
   • P A Hanle, Erwin Schrodinger's reaction to Louis de Broglie's thesis on the quantum theory, Isis 68 (244) (1977), 606-609.
     
   • J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
     
   • G S Im, Experimental constraints on formal quantum mechanics : the emergence of Born's quantum theory of collision processes in Gottingen, 1924-1927, Archive for History of Exact Sciences 50 (1) (1996), 73-101.
     
   • M Jammer, The philosophy of quantum mechanics : the interpretations of quantum mechanics in historical perspective (New York, 1974).
     
   • C W Kilmister, Quantum mechanics 1899-1925 : a survey of concept formation, Bull.
     
   • P T Matthews, Dirac and the foundation of quantum mechanics, Reminiscences about a great physicist : Paul Adrien Maurice Dirac (Cambridge, 1987), 199-224.
     
   • J Mehra, Dirac's contribution to the early development of quantum mechanics, in Tributes to Paul Dirac (Bristol, 1987), 63-75.
     
   • J Mehra and H Rechenberg, The historical development of quantum theory (5 volumes) (New York-Berlin, 1982-1987).
     
   • K von Meyenn, Pauli, Schrodinger and the conflict about the interpretation of quantum mechanics, in Symposium on the foundations of modern physics (Singapore, 1985), 289-302.
     
   • A I Miller (ed.), Sixty-two years of uncertainty : historical, philosophical, and physical inquiries into the foundations of quantum mechanics (New York, 1990).
     
   • N Mukunda, The mathematics and physics of quantum mechanics, Math.
     
   • L Navarro, On Einstein's statistical-mechanical approach to the early quantum theory (1904-1916), Historia Sci.
     
   • R M Nugayev, The history of quantum mechanics as a decisive argument favoring Einstein over Lorentz, Philos.
     
   • A Pais, Einstein and the quantum theory, Rev.
     
   • A Pais, Max Born's statistical interpretation of quantum mechanics, Science 218 (4578) (1982), 1193-1198.
     
   • M Paty, The nature of Einstein's objections to the Copenhagen interpretation of quantum mechanics, Found.
     
   • J C Polkinghorne, Dirac and the interpretation of quantum mechanics, in Tributes to Paul Dirac (Bristol, 1987), 76-83.
     
   • K Popper, A critical note on the greatest days of quantum theory, Found.
     
   • H Reichenbach, The space problem in the new quantum mechanics, Erkenntnis 35 (1-3) (1991), 29-47.
     
   • F Rohrlich, Schrodinger and the interpretation of quantum mechanics, Found.
     
   • F Rohrlich, Schroedinger's criticism of quantum mechanics-fifty years later, in Symposium on the foundations of modern physics (Singapore, 1985), 555-572.
     
   • B L van der Waerden, From matrix mechanics and wave mechanics to unified quantum mechanics, Notices Amer.
     
   • D Wick, The infamous boundary : seven decades of controversy in quantum physics (Boston, 1995).
     
   • I : Schrodinger and his path to quantum mechanics (Czech), Pokroky Mat.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_Quantum_age_begins.html .
     

3. Quantum mechanics history
   
   • A history of Quantum Mechanics .
     
   • The neutron was not discovered until 1932 so it is against this background that we trace the beginnings of quantum theory back to 1859.
     
   • This work was not done with quantum theory in mind but, as so often happens, the mathematics necessary to embody a physical theory had appeared at precisely the right moment.
     Go directly to this paragraph
     
   • Einstein proposed a quantum theory of light to solve the difficulty and then he realised that Planck's theory made implicit use of the light quantum hypothesis.
     Go directly to this paragraph
     
   • By 1906 Einstein had correctly guessed that energy changes occur in a quantum material oscillator in changes in jumps which are multiples of planck v where planck is Planck's reduced constant and v is the frequency.
     Go directly to this paragraph
     
   • Arthur Compton derived relativistic kinematics for the scattering of a photon (a light quantum) off an electron at rest in 1923.
     Go directly to this paragraph
     
   • However there were concepts in the new quantum theory which gave major worries to many leading physicists.
     Go directly to this paragraph
     
   • change by quantum amounts.
     
   • Up to this stage quantum theory was set up in Euclidean space and used Cartesian tensors of linear and angular momentum.
     
   • However quantum theory was about to enter a new era.
     
   • One does not get an answer to the question, What is the state after collision? but only to the question, How probable is a given effect of the collision? From the standpoint of our quantum mechanics, there is no quantity which causally fixes the effect of a collision in an individual event.
     
   • Heisenberg wrote his first paper on quantum mechanics in 1925 and 2 years later stated his uncertainty principle.
     Go directly to this paragraph
     
   • Dirac, in 1928, gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity.
     Go directly to this paragraph
     
   • In 1932 von Neumann put quantum theory on a firm theoretical basis.
     Go directly to this paragraph
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_Quantum_age_begins.html .
     

4. 20th century time
   
   • Let us now look at another revolution in time which took place in the 20th century with the discovery of quantum mechanics, see [Nuncius Ann.
     
   • It is really impossible in an article such as this to cover all aspects of time in relation to quantum theory but we will look at one or two issues to gain a feeling for their relation.
     
   • The first point to note is that quantum theory was developed within the absolute time scenario of Newton.
     
   • Even in this form it has a direct consequence for aspects of time we have already discussed, for it means that Laplace's realisation that Newton's laws meant that the future was completely determined by the present would not extend to quantum theory.
     
   • In the "clock in the box" thought experiment we have seen how relativity and quantum theory begin to interact.
     
   • Milne developed a complex theory of cosmology, attempting to unify relativity and quantum theory, that included a non-constant value for G, which we know as the gravitational constant.
     
   • There are ways that quantum theory time appears to contradict relativity time, and this is worrying.
     
   • It relies on the fact that a quantum event sometimes creates a pair of particles with complementary properties - for example they must have opposite spins.
     
   • In quantum theory the particle will have the properties of both possible states until we measure it when it collapses into one of the two states.
     
   • Einstein firmly believed that no information could be transmitted faster than the speed of light, and saw this as an objection to quantum theory.
     
   • He showed that Bell's inequalities were violated and so the quantum interpretation held rather than the classical one.
     
   • An interpretation of quantum theory put forward by Hugh Everett in 1957 is the many worlds interpretation.
     
   • In this the universe splits into two every time a quantum event is forced to choose between two states.
     

5. Modern light references
   
   • References for: Light through the ages: Relativity and quantum era .
     
   • J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
     
   • H Konno, Bohr's search for the quantum theory of dispersion : the number of dispersion electrons, absorption and emission of light and the oscillator model.
     
   • Discovery of energy quanta and development of early quantum theory, Historia Sci.
     
   • J Stachel, Einstein's light-quantum hypothesis, or why didn't Einstein propose a quantum gas a decade-and-a-half earlier?, in Einstein: the formative years, 1879-1909 (Boston, MA, 2000), 231-251.
     

6. Modern light references
   
   • References for: Light through the ages: Relativity and quantum era .
     
   • J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
     
   • H Konno, Bohr's search for the quantum theory of dispersion : the number of dispersion electrons, absorption and emission of light and the oscillator model.
     
   • Discovery of energy quanta and development of early quantum theory, Historia Sci.
     
   • J Stachel, Einstein's light-quantum hypothesis, or why didn't Einstein propose a quantum gas a decade-and-a-half earlier?, in Einstein: the formative years, 1879-1909 (Boston, MA, 2000), 231-251.
     

7. Modern light
   
   • Light through the ages: Relativity and quantum era .
     
   • Another important development in the understanding of light, namely the development of quantum theory, had taken place over this same period of time, roughly 1880 to 1926.
     
   • A second problem also led to a quantum theory of light, and this time to a belief in the physical reality of the quanta.
     
   • Two mathematical models of quantum mechanics were presented, that of matrix mechanics, proposed by Werner Heisenberg, Max Born, and Pascual Jordan, and that of wave mechanics proposed by Erwin Schrodinger.
     
   • By 1930 the interpretation of quantum theory called the Copenhagen interpretation, mainly due to Bohr and his co-workers, was essentially complete.
     
   • On the other hand, I think I can safely say that nobody understands quantum mechanics.
     

8. Wave versus matrix
   
   • We want to examine here the different models for the atom provided by wave mechanics and by quantum mechanics.
     
   • The theory by Heisenberg to which Schrodinger refers is quantum mechanics which he put forward in 1925.
     
   • Don't take it as an unfriendliness to you but look on the expression as my objective conviction that quantum phenomena naturally display aspects that cannot be expressed by the concepts of continuum physics.
     
   • Investigation of the type of physical reality which is proper to electrons and atoms is precisely the subject of atomic physics and thus also of quantum mechanics.
     
   • You surely must understand, Bohr, that the whole idea of quantum jumps necessarily leads to nonsense.
     
   • But that doesn't prove that there are no quantum jumps.
     
   • It only proves that we can't visualise them, that means that the pictorial concepts we use to describe the events of everyday life and the experiments of the old physics do not suffice also to represent the process of a quantum jump.
     

9. Classical light
   
   • Light in the quantum era .
     
   • By 'classical' here we meant pre-relativity and pre-quantum theory.
     
   • We will study the developments in relativity-quantum theory era in a separate article; see Light through the ages: Relativity and quantum era.
     
   • Planck, who made one of the next major breakthoughts described in Light through the ages: Relativity and quantum era, said on the occasion of the centenary of Maxwell's birth in 1931, that this theory:- .
     
   • Light in the quantum era .
     

10. Classical light references
    
    • J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
      
    • J Stachel, Einstein's light-quantum hypothesis, or why didn't Einstein propose a quantum gas a decade-and-a-half earlier?, in Einstein : the formative years, 1879-1909 (Boston, MA, 2000), 231-251.
      

11. Classical light references
    
    • J Hendry, The development of attitudes to the wave-particle duality of light and quantum theory, 1900-1920, Ann.
      
    • J Stachel, Einstein's light-quantum hypothesis, or why didn't Einstein propose a quantum gas a decade-and-a-half earlier?, in Einstein : the formative years, 1879-1909 (Boston, MA, 2000), 231-251.
      

12. Classical time
    
    • Quantum mechanics and relativity theory in the 20th century have shown the complexities, and sometime apparent paradoxes, in the notion of time.
      
    • This requires time to be continuous and a time interval to always be divisible, yet quantum theory tells us that time is quantised and quite unlike mathematical time which forms the basis of applied mathematics.
      

13. General relativity references
    
    • A J Kox, Einstein, Lorentz, Leiden and general relativity, Les Journees Relativistes, Classical Quantum Gravity 10 (1993), S187-S191.
      

14. Babylonian mathematics references
    
    • I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995), 23-27.
      

15. Mayan mathematics references
    
    • I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995), 23-27.
      

16. Egyptian mathematics references
    
    • I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995), 23-27.
      

17. references
    
    • (1948), Space-time approach to non-relativistic quantum mechanics, Review of Modern Physics, 20, 367-387.
      

18. Mayan mathematics references
    
    • I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995), 23-27.
      

19. Special relativity references
    
    • R M Nugayev, Special relativity as a step in the development of the quantum programme : revolution in a revolution, Centaurus 29 (2) (1986), 100-109.
      

20. General relativity references
    
    • A J Kox, Einstein, Lorentz, Leiden and general relativity, Les Journees Relativistes, Classical Quantum Gravity 10 (1993), S187-S191.
      

21. Mayan mathematics references
    
    • I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995), 23-27.
      

22. Special relativity references
    
    • R M Nugayev, Special relativity as a step in the development of the quantum programme : revolution in a revolution, Centaurus 29 (2) (1986), 100-109.
      

23. Pi history
    
    • How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
      Go directly to this paragraph
      

24. Babylonian and Egyptian references
    
    • I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995), 23-27.
      

Famous Curves

Societies etc

1. AMS Steele Prize
   
   • for the following two papers in mathematical physics characterized by leaders of the field as extremely innovative "A quartic interaction in two dimensions in Mathematical Theory of Elementary Particles", and "Construction of quantum fields from Markoff fields in Journal of Functional Analysis".
     
   • In these papers he showed for the first time how to use the powerful tools of probability theory to attack the hard analytic questions of constructive quantum field theory, controlling renormalizations with Lp estimates in the first paper, and in the second turning Euclidean quantum field theory into a subset of the theory of stochastic processes.
     
   • for his paper "Pseudo-holomorphic curves in symplectic manifolds", which revolutionized the subject of symplectic geometry and topology and is central to much current research activity, including quantum cohomology and mirror symmetry.
     

2. BMC 1995
   
   • Carter, R W New developments in the representation theory of Lie algebras, algebraic groups and quantum groups .
     
   • Lenagan, T HCatenarity in quantum algebras .
     
   • Ringel, C The Hall algebra approach to quantum groups .
     
   • Lenagan, T HCatenarity in quantum algebras .
     

3. BMC 2002
   
   • Veselov, A Deformed root systems and quantum integrability .
     
   • Special session: Quantum phenomeman Organisers: G Friesecke and L Mason .
     
   • Dowker, FThe deep quantum structure of spacetime .
     
   • Josza, R Quantum computation - some theoretical challenges .
     

4. International Congress Speakers
   
   • George David Birkhoff, On the Foundations of Quantum Mechanics.
     
   • Nikolay Nikolaevich Bogolyubov and V S Vladimirov, On Some Mathematical Problems of Quantum Field Theory.
     
   • James Glimm, Analysis over Infinite-Dimensional Spaces and, Applications to Quantum Field Theory.
     
   • Mikio Sato, Monodromy Theory and Holonomic Quantum Fields - a New Link between Mathematics and Theoretical Physics.
     
   • Jurg Frohlich, The Fractional Quantum Hall Effect, ChernSimons Theory and Integral Lattices.
     
   • Tetsuji Miwa, Solvable Lattice Models and Representation Theory of Quantum Affine Algebras.
     
   • Peter Williston Shor, Quantum Computing.
     

5. BMC 2000
   
   • Jones, V The quantum Platonic solids .
     
   • Lenagan, T HThe algebra of quantum matrices .
     
   • Rieffel, M String theory and quantum tori .
     
   • Veselov, A Quantum integrability, configurations of hyperplanes and Hadamard's problem .
     

6. European Mathematical Society Prizes
   
   • Recent work of Gaitsgory relates finite quantum groups and chiral Hecke algebras.
     
   • In further works, Seidel constructed a natural representation of the fundamental group of the group of Hamiltonian symplectomorphisms into the quantum cohomology ring.
     
   • Although he has worked widely in ergodic theory, his recent proof of the quantum unique ergodicity conjecture for arithmetic hyperbolic surfaces breaks fertile new ground, with great promise for future applications to number theory.
     

7. Bakerian Lecturers
   
   • The physical interpretation of quantum mechanics (Proc.
     
   • The semiclassical chaology of quantum eigenvalues (Proc.
     

8. AMS Wiener Prize
   
   • for his distinguished work in the perturbation theory of quantum mechanics.
     
   • for his outstanding contribution of original and non-perturbative mathematical methods in statistical mechanics by means of which he was able to solve several long open important problems concerning critical phenomena, phase transitions, and quantum field theory; and to Jerrold E Marsden for his outstanding contributions to the study of differential equations in mechanics: he proved the existence of chaos in specific classical differential equations; his work on the momentum map, from abstract foundations to detailed applications, has had great impact.
     

9. BMC 1994
   
   • Jordan, D Skew polynomial rings from Hilbert to quantum groups .
     
   • Lance, E C Compact quantum groups .
     

10. BMC 2008
    

11. Kharkov Mathematical Society
    
    • Despite the difficulties, mathematics flourished in Kharkov and they achieved their greatest honour when, on 21 August 1990, Vladimir Gershonovich Drinfeld was awarded a Fields medal at the International Congress of Mathematicians in Kyoto, Japan, for his work on quantum groups and on number theory.
      

12. BMC 1981
    
    • Gibbons, G WThe Euclidean approach to quantum gravity .
      

13. Minutes for 2000
    
    • There were satellite conferences on harmonic maps; model theory; regular dynamics; rings and quantum groups; there was also the British Topology Meeting.
      

14. BMC 1985
    

15. BMC 1993
    
    • Effros, E G What is a discrete quantum group? .
      

16. BMC 1986
    
    • Streater, R F Classical and quantum probability .
      

References

1. References for Bell John
   
   • Bernstein, Quantum Profiles (Princeton, 1991).
     
   • Jammer, The Philosophy of Quantum Mechanics (New York, 1974).
     
   • Rae, Quantum Physics: Illusion or Reality (Cambridge, 1986).
     
   • Redhead, Incompleteness, Nonlocality and Realism, a Prolegomenon to the Philosophy of Quantum Mechanics (Oxford, 1987).
     
   • Squires, The Mystery of the Quantum World (Bristol, 1994) .
     
   • Whitaker, Einstein, Bohr and the Quantum Dilemma (Cambridge, 1996).
     
   • Braunstein, Quantum Computing: Where do we Want to Go Tomorrow? (Chichester, 1999).
     
   • Bertlmann, Anomalies in Quantum Field Theory (Oxford, 2000).
     
   • Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).
     
   • Whitaker, Theory and experiment in the foundations of quantum theory, Progress in Quantum Electronics 24, 1-106 (2000).
     
   • [This review contains many references to the sizeable literature concerning the applications of Bell's work in quantum theory.] .
     
   • Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge, 1987).
     
   • [Contains nearly all Bell's papers on quantum theory.] .
     
   • Bell, Quantum Mechanics, High Energy Physics and Accelerators (Singapore, 1995) (edited by M.
     
   • Bell on the Foundations of Quantum Mechanics (Singapore, 2001).
     
   • McMullin (eds.), Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem (Notre Dame, 1989).
     
   • Amati (eds.), Quantum Reflections (Cambridge, 2000).
     
   • Bertlmann and A.Zeilinger (eds.), Quantum (Un)speakables: from Bell to Quantum Information (Berlin, 2002).
     

2. References for Einstein
   
   • A Whitaker, Einstein, Bohr and the quantum dilemma (Cambridge, 1996).
     
   • The background, in Microphysical reality and quantum formalism (Dordrecht, 1988), 299-308.
     
   • B Carazza, Historical considerations on the conceptual experiment by Einstein, Podolsky and Rosen, in The nature of quantum paradoxes (Dordrecht, 1988), 355-369.
     
   • Einstein and quantum theory : Excerpts from the correspondence of A Einstein with M Besso (Russian), Voprosy Istor.
     
   • M A Elyashevich, Einstein's part in the development of quantum concepts, Soviet Phys.
     
   • M A Elyashevich, Einstein's part in the development of quantum concepts (Russian), Uspekhi Fiz.
     
   • H Ezawa, Einstein's contribution to statistical mechanics, classical and quantum, Japan.
     
   • A Fine, Einstein's interpretations of the quantum theory, in Einstein in context (Cambridge, 1993), 257-273.
     
   • M Jammer, Einstein and quantum physics, in Albert Einstein, Jerusalem, 1979 (Princeton, NJ, 1982), 59-76.
     
   • N V Karlov and A M Prokhorov, Quantum electronics and Einstein's theory of radiation, Soviet Phys.
     
   • N V Karlov and A M Prokhorov, Quantum electronics and Einstein's theory of radiation (Russian), Uspekhi Fiz.
     
   • A J Kox, Einstein, Lorentz, Leiden and general relativity, Classical Quantum Gravity 10 (Suppl.) (1993) S187-S191.
     
   • L Navarro, On Einstein's statistical-mechanical approach to the early quantum theory (1904-1916), Historia Sci.
     
   • R M Nugayev, The history of quantum mechanics as a decisive argument favoring Einstein over Lorentz, Philos.
     
   • A Pais, Einstein and the quantum theory, Rev.
     
   • M Paty, The nature of Einstein's objections to the Copenhagen interpretation of quantum mechanics, Found.
     
   • J Stachel, Einstein and quantum mechanics, in Conceptual problems of quantum gravity (Boston, MA, 1991), 13-42.
     
   • J Stachel, Einstein and the quantum : fifty years of struggle, in From quarks to quasars (Pittsburgh, PA, 1986), 349-385.
     
   • V Tonini, Continuity and discontinuity : the Einstein-Bohr conflict of ideas and the Bohr-Fock discussion, in The nature of quantum paradoxes (Dordrecht, 1988), 371-384.
     

3. References for Bohr Niels
   
   • J Hendry, The creation of quantum mechanics and the Bohr-Pauli dialogue (Dordrecht, 1984).
     
   • J Honner, The description of nature : Niels Bohr and the philosophy of quantum physics (Oxford, 1988).
     
   • A Whitaker, Einstein, Bohr and the quantum dilemma (Cambridge, 1996).
     
   • Microphysical reality and quantum formalism, in Fund.
     
   • H R Holcomb, Latency versus complementarity : Margenau and Bohr on quantum mechanics, British J.
     
   • A Ott, Didactic aspects on the controversy between Niels Bohr and Albert Einstein, in Problems in quantum physics, Gda'nsk '87 (Teaneck, NJ, 1988), 702-705.
     
   • S Petruccioli, Classical physics and quantum concepts in the works of Niels Bohr : 1913-1927 (Italian), Science and philosophy (Milan, 1985), 738-759.
     
   • J Rayski, Reconcilement of Einstein's realistic with Bohr's positivistic attitude, in Problems in quantum physics, Gda'nsk '87 (Teaneck, NJ, 1988), 247-255.
     
   • M Sachs, Bohr versus Einstein : A possible shaping of 21st century physics, in Problems in quantum physics, Gda'nsk '87 (Teaneck, NJ, 1988), 745-768.
     
   • V Tonini, Continuity and discontinuity : the Einstein-Bohr conflict of ideas and the Bohr-Fock discussion, in The nature of quantum paradoxes, Cesena, 1985, (Dordrecht, 1988), 371-384.
     
   • On the path to quantum mechanics (Czech), Pokroky Mat.
     

4. References for Fock
   
   • Quantum mechanics and quantum field theory (Chapman & Hall/CRC, Boca Raton, FL, 2004).
     
   • Functional methods in quantum field theory and statistical physics.
     
   • Yu V Novozhilov and V Yu Novozhilov, The works of Vladimir Aleksandrovich Fok on quantum theory (on the centenary of his birth) (Russian), Teoret.
     
   • Yu V Novozhilov and V Yu Novozhilov, The works of Vladimir Aleksandrovich Fok on quantum theory (on the centenary of his birth), Theoret.
     
   • O I Zav'yalov and A M Malokostov, V A Fok and N N Bogolyubov and their role in establishing modern quantum field theory (Russian), Teoret.
     
   • O I Zav'yalov and A M Malokostov, V A Fok and N N Bogolyubov and their role in establishing modern quantum field theory, Theoret.
     

5. References for Klein Oskar
   
   • Bethe and R Jackiw, Intermediate Quantum Mechanics (Reading, 1997).
     
   • Bohm, Quantum Theory (New York, 1979).
     
   • Gasiorowicz, Quantum Physics (New York, 1996).
     
   • Kragh, Quantum Generations: A History of Physics in the Twentieth Century (Princeton, 1999) .
     

6. References for Dirac
   
   • B L Cline, The Questioners: Physicists and the Quantum Theory (1965).
     
   • P T Matthews, Dirac and the foundation of quantum mechanics, in Reminiscences about a great physicist : Paul Adrien Maurice Dirac (Cambridge, 1987), 199-224.
     
   • B V Medvedev and D V Shirkov, P A M Dirac and the formation of the basic ideas of quantum field theory, Soviet Phys.
     
   • J Mehra, Dirac's contribution to the early development of quantum mechanics, in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 63-75.
     

7. References for Pauli
   
   • J Hendry, The creation of quantum mechanics and the Bohr-Pauli dialogue (Dordrecht, 1984).
     
   • E A Giannetto and F Pozzi, Non-separability and synchronicity : Pauli, Jung and a new historical, philosophical perspective on quantum physics, in The foundations of quantum mechanics, Lecce, 1998 (River Edge, NJ, 2000), 251-259.
     
   • K von Meyenn, Pauli, Schrodinger and the conflict about the interpretation of quantum mechanics, in Symposium on the foundations of modern physics, Joensuu, 1985 (Singapore, 1985), 289-302.
     

8. References for Schrodinger
   
   • J Mehra and H Rechenberg, The historical development of quantum theory Vol.5: Erwin Schrodinger and the rise of wave mechanics (New York, 1987).
     
   • P A Hanle, The Coming of Age of Erwin Schrodinger : His Quantum Statistics of Ideal Gases, Archive for History of Exact Science 17 (1977), 165-.
     

9. References for Ehrenfest
   
   • M J Klein, in The lessons of the quantum theory (Amsterdam, 1986), 325.
     
   • M J Klein, Ehrenfest's contributions to the development of quantum statistics.
     

10. References for Wigner
    
    • Foundations of quantum mechanics (Berlin, 1997).
      
    • G G Emch, The philosophy of Eugene P Wigner, Classical and quantum systems (River Edge, NJ, 1993), 2-8.
      

11. References for Feynman
    
    • O S Razumovskii and V A Firsov, Feynman's formulation in quantum mechanics, and path integration : The historical-methodological aspect (Russian), Studies in the history of physics and mechanics, 1988 'Nauka' (Moscow, 1988), 37-51.
      
    • J Schwinger, A path to quantum electrodynamics, in 'Most of the good stuff' (New York, 1993), 59-73.
      

12. References for Von Neumann
    
    • H Araki, Some of the legacy of John von Neumann in physics: theory of measurement, quantum logic, and von Neumann algebras in physics, The legacy of John von Neumann (Providence, R.I., 1990), 119-136.
      
    • L van Hove, Von Neumann's contributions to quantum theory, Bull.
      

13. References for Heisenberg
    
    • M Jammer, The Philosophy of Quantum Mechanics: The Interpretation of Quantum Mechanics in Historical Perspective (1974).
      

14. References for Lemaitre
    
    • M Heller, Lemaitre, big bang, and the quantum universe, Pachart History of Astronomy Series 10 (Pachart Publishing House, Tucson, AZ, 1996).
      

15. References for Witten
    
    • R Schmid, Strings, knots, and quantum groups: a glimpse at three 1990 Fields medalists, SIAM Rev.
      

16. References for Planck
    
    • M J Klein, Max Planck and the Beginnings of Quantum Theory, Archive for History of Exact Sciences 1 (1962), 459-479.
      

17. References for Drach
    
    • D V Chudnovsky and G V Chudnovsky, Travaux de J Drach (1919), in Classical quantum models and arithmetic problems, Lecture Notes in Pure and Appl.
      

18. References for Bose
    
    • W A Blanpied, Satyendranath Bose: Co-Founder of Quantum Statistics, Amer.
      

19. References for Poisson
    
    • B Geller and Y Bruk, A portrait of Poisson, Quantum (1991), 21-25.
      

20. References for Jones Vaughan
    
    • R Schmid, Strings, knots, and quantum groups: a glimpse at three 1990 Fields medalists, SIAM Rev.
      

21. References for Poincare
    
    • G Ciccotti and G Ferrari, Was Poincare a herald of quantum theory?, European J.
      

22. References for Chandrasekhar
    
    • N Panchapakesan, Seeing beauty in the simple and the complex : Chandrasekhar and general relativity, in Classical and quantum aspects of gravitation and cosmology, Madras, 1996 (Madras, 1998), 1-10.
      

23. References for Broglie
    
    • B L Cline, The Questioners: Physicists and the Quantum Theory (1965).
      

24. References for Black Fischer
    
    • (1948), Space-time approach to non-relativistic quantum mechanics, Review of Modern Physics, 20, 367-387.
      

Additional material

1. Max Planck: 'Quantum Theory
   
   • Max Planck: Quantum Theory .
     
   • Max Planck lectured on The Origin and Development of the Quantum Theory in German and an English translation was published by Methuen & Co in 1925.
     
   • The Origin and Development of the Quantum Theory .
     
   • In this lecture I will endeavour to give a general account of the origin of the quantum theory, to sketch concisely its development up to the present, and to point out its immediate significance in physics.
     
   • Looking back over the last twenty years to the time when the conception and magnitude of the physical quantum of action first emerged from the mass of experimental facts, and looking back at the long and complicated path which finally led to an appreciation of its importance, the whole history of its development reminds me of the well-proved adage that "to err is human." And all the hard intellectual work of an industrious thinker must often appear vain and fruitless, but that striking occurrences sometimes provide him with an irrefutable proof of the fact that at the end of all his attempts, he does ultimately get one step nearer the truth.
     
   • I called it the elementary quantum of action, since it is a product of energy and time, and was calculated to be 6.55 × 10-27 erg sec.
     
   • Either the quantum of action was a fictitious quantity, in which case all the deductions from the radiation theory were largely illusory and were nothing more than mathematical juggling.
     
   • Or the radiation theory is founded on actual physical ideas, and then the quantum of action must play a fundamental role in physics, and proclaims itself as something quite new and hitherto unheard of, forcing us to recast our physical ideas, which, since the foundation of the infinitesimal calculus by Leibniz and Newton, were built on the assumption of continuity of all causal relations.
     
   • That this decision should be made so soon and so certainly is not due to the verification of the law of distribution of energy in heat radiation, much less to my special derivation of this law, but to the restless, ever-advancing labour of those workers who have made use of the quantum of action in their investigations.
     
   • The first advance in this work was made by A Einstein, who proved, on the one hand, that the introduction of the energy quanta, required by the quantum of action, appeared suitable for deriving a simple explanation for a series of remarkable observations of light effects, such as Stokes's rule, emission of electrons, and ionization of gases.
     
   • Further, quantum effects are very noticeable when considering the specific heat of gases.
     
   • W Nernst had shown at an early stage that the quantum of energy of an oscillation must correspond to the quantum of energy of a rotation, and accordingly expected that the energy of rotation of a gas molecule would decrease with temperature.
     
   • The work of N Bjerrum, E von Bahr, H Rubens, and G Hettner, etc., on absorption bands in the infrared rays, shows that there can be no doubt that the rotations of the gas molecules indicated by the quantum conditions do actually exist.
     
   • The determination by James Franck and Gustav Hertz of the so-called resonance potential, or that critical velocity, the minimum velocity which an electron must have to bring about the emission of a quantum of light by collision with a neutral atom, is as direct a method of measuring the quantum of action as can be desired.
     
   • The liberation of quanta of light by electronic impulses is the converse of the emission of electrons by projection of light, Rontgen or Gamma rays, and here, again, the quanta of energy determined from the quantum of action and the frequency of oscillations play a characteristic part in the same way as we have seen above, in that the velocity of the electrons emitted does not depend on the intensity of the radiation, but on the wavelength of the light emitted.
     
   • From a quantitative point of view, also, Einstein's relations for light quanta mentioned above have been verified in every way, particularly by R A Millikan, who determined the initial velocities of the emitted electrons, while the significance of the light quantum in causing photo-chemical reactions has been made clear by E Warburg.
     
   • The results quoted above, collected from the most varied branches of physics, present an overwhelming case for the existence of the quantum of action, and the quantum hypothesis was put on a very firm foundation by Niels Bohr's theory of the atom.
     
   • This theory was destined, by means of the quantum of action, to open a door into the wonderland of spectroscopy, which had obstinately defied all investigators since the discovery of spectral analysis.
     
   • In view of all these results - a complete explanation would involve the inclusion of many more well-known names - an unbiased critic must recognize that the quantum of action is a universal physical constant, the value of which has been found from several very different phenomena to be 6.54 × 10-27 ergs secs.
     
   • Yet no actual quantum theory has been formed by the introduction of the quantum of action.
     
   • The difficulties in the way of introducing the quantum of action into classical theory from the beginning have been mentioned above.
     
   • But that the quite sharply defined frequency of an emitted light quantum should be different from the frequency of the emitted electrons must seem, at first sight, to a physicist educated in the classical school, an almost unreasonable demand on his imagination.
     
   • If a surmise be allowed as to the probable outcome of this struggle, everything seems to indicate that the great principles of thermo-dynamics, derived from the classical theory, will not only maintain their central position in the quantum theory, but will be greatly extended.
     
   • The adiabatic hypothesis of P Ehrenfest plays the same part in the quantum theory as the original experiments played in the founding of classical thermodynamics.
     
   • A question, from the complete answer to which we may expect far-reaching explanations, is what becomes of the energy of a light quantum after perfect emission? Does it spread out, as it progresses, in all directions, as in Huygens's wave theory, and while covering an ever-larger amount of space, diminish without limit? Or does it travel along as in Newton's emanation theory like a projectile in one direction? In the first case the quantum could never concentrate its energy in a particular spot to enable it to liberate an electron from the atomic influences; in the second case we would have the complete triumph of Maxwell's theory, and the continuity between static and dynamic fields must be sacrificed, and with it the present complete explanation of interference phenomena, which have been investigated in all details.
     
   • Until this goal is attained the problem of the quantum of action will not cease to stimulate research and to yield results, and the greater the difficulties opposed to its solution, the greater will be its significance for the extension and deepening of all our knowledge of physics.
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Planck_quantum_theory.html .
     

2. H Weyl: 'Theory of groups and quantum mechanics' Introduction
   
   • H Weyl: Theory of groups and quantum mechanics Introduction .
     
   • In 1929 Hermann Weyl's The theory of groups and quantum mechanics was published in German.
     
   • The quantum theory of atomic processes was proposed by Niels Bohr in the year 1913, and was based on the atomic model proposed earlier by Rutherford.
     
   • But about this time it began to become more and more apparent that the Bohr theory was a compromise between the old "classical" physics and a new quantum physics which has been in the process of development since Planck's introduction of energy quanta in 1900.
     
   • From these results it seems to follow that, in the general problem of the quantum theory, one is faced not with a modification of the mechanical and electrodynamical theories describable in terms of the usual physical concepts, but with an essential failure of the pictures in space and time on which the description of natural phenomena has hitherto been based.
     
   • The foundations of the new quantum physics, or at least its more important theoretical aspects, are to be treated in this book.
     
   • The spectroscopic data, presented in accordance with the new quantum theory, together with complete references to the literature, are given in the following three volumes of the series Struktur der Materie, edited by Born and Franck:- .
     
   • The development of quantum theory has only been made possible by the enormous refinement of experimental technique, which has given us an almost direct insight into atomic processes.
     
   • While the quantum theory can be traced back only as far as 1900, the origin of the theory of groups is lost in a past scarcely accessible to history; the earliest works of art show that the symmetry groups of plane figures were even then already known, although the theory of these was only given definite form in the latter part of the eighteenth and in the nineteenth centuries.
     
   • Until the present, its most important application to natural science lay in the description of the symmetry of crystals, but it has recently been recognized that group theory is of fundamental importance for quantum physics; it here reveals the essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved.
     
   • We may well expect that it is just this part of quantum physics which is most certain of a lasting place.
     
   • The investigation of groups first becomes a connected and complete theory in the theory of the representation of groups by linear transformations and it is exactly this mathematically most important part which is necessary for an adequate description of the quantum mechanical relations.
     
   • All quantum numbers, with the exception of the so-called principal quantum number, are indices characterising representations of groups.
     
   • Chapter II is devoted to preparation on the physical side; only that has been given which seemed to me indispensable for an understanding of the meaning and methods of quantum theory.
     
   • A multitude of physical phenomena, which have already been dealt with by quantum theory, have been omitted.
     
   • Chapter III develops the elementary portions of the theory of representations of groups and Chapter IV applies them to quantum physics.
     
   • Thus mathematics and physics alternate in the first four chapters, but in Chapter V the two are fused together, showing how completely the mathematical theory is adapted to the requirements of quantum physics.
     

3. H Weyl: 'Theory of groups and quantum mechanics'Preface to Second Edition
   
   • H Weyl: Theory of groups and quantum mechanicsPreface to Second Edition .
     
   • In 1929 Hermann Weyl's The theory of groups and quantum mechanics was published in German.
     
   • I may mention in this connection the derivation of the Clebsch-Gordan series, which is of fundamental importance for the whole of spectroscopy and for the applications of quantum theory to chemistry, the section on the Jordan-Holder theorem and its analogues, and above all the careful investigation of the connection between the algebra of symmetric transformations and the symmetric permutation group.
     
   • But above all several sections have been added which deal with the energy-momentum theorem of quantum physics and with the quantization of the wave equation in accordance with the recent work of Heisenberg and Pauli.
     
   • This extension already leads so far away from the fundamental purpose of the book that I felt forced to omit the formulation of the quantum laws in accordance with the general theory of relativity, as developed by V Fock and myself, in spite of its desirability for the deduction of the energy-momentum tensor.
     
   • The fundamental problem of the proton and the electron has been discussed in its relation to the symmetry properties of the quantum laws with respect to the interchange of right and left, past and future, and positive and negative electricity.
     
   • At present no solution of the problem seems in sight; I fear that the clouds hanging over this part of the subject will roll together to form a new crisis in quantum physics.
     
   • I have intentionally presented the more difficult portions of these problems of spin and second quantization in considerable detail, as they have been for the most part either entirely ignored or but hastily indicated in the large number of texts which have now appeared on quantum mechanics.
     
   • It has been rumoured that the "group pest" is gradually being cut out of quantum physics.
     
   • The constants c and h, the velocity of light and the quantum of action, have caused some trouble.
     
   • The insight into the significance of these constants, obtained by the theory of relativity on the one hand and quantum theory on the other, is most forcibly expressed by the fact that they do not occur in the laws of Nature in a thoroughly systematic development of these theories.
     

4. H Weyl: 'Theory of groups and quantum mechanics'Preface to First Edition
   
   • H Weyl: Theory of groups and quantum mechanicsPreface to First Edition .
     
   • In 1929 Hermann Weyl's The theory of groups and quantum mechanics was published in German.
     
   • The importance of the standpoint afforded by the theory of groups for the discovery of the general laws of quantum theory has of late become more and more apparent.
     
   • Since I have for some years been deeply concerned with the theory of the representation of continuous groups, it has seemed to me appropriate and important to give an account of the knowledge won by mathematicians working in this field in a form suitable to the requirements of quantum physics.
     
   • My desire to show how the concepts arising in the theory of groups find their application in physics by discussing certain of the more important examples has necessitated the inclusion of a short account of the foundations of quantum physics, for at the time the manuscript was written there existed no treatment of the subject to which I could refer the reader.
     
   • In brief this book, if it fulfils its purpose, should enable the reader to learn the essentials of the theory of groups and of quantum mechanics as well as the relationships existing between these two subjects; the mathematical portions have been written with the physicist in mind, and vice versa.
     
   • I have particularly emphasized the "reciprocity" between the representations of the symmetric permutation group and those of the complete linear group; this reciprocity has as yet been unduly neglected in the physical literature, in spite of the fact that it follows most naturally from the conceptual structure of quantum mechanics.
     
   • The continuum of real numbers has retained its ancient prerogative in physics for the expression of physical measurements, but it can justly be maintained that the essence of the new Heisenberg-Schrodinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense, the elements of which are the physical quantities themselves.
     

5. Born Inaugural
   
   • The new conception of the quantum of action which helped to elucidate them was first put forward by Planck in 1900.
     
   • The most important consequences of this conception were deduced by Einstein, who laid the foundations of the quantum theory of light in 1905, the year in which he published his relativity theory, and by Niels Bohr in 1913, when he applied the idea of the quantum to the structure of atoms.
     
   • Traces of it can be seen in fundamental papers of Heisenberg on quantum theory; but it has also met with strenuous opposition, for instance, from Planck.
     
   • The second revolution of physics, called quantum theory, is, however, built on an enormous accumulation of experience, which is still growing from day to day.
     
   • In spite of this difficulty, I shall try to outline the problem and its solution, called quantum mechanics.
     
   • I cannot follow out here the historical development of the quantum idea which led step by step to the recognition that we have here to do with a much more general conception.
     
   • But when it became known, theoretical physics was already prepared to treat it by proper mathematical methods, the so-called quantum mechanics, initiated by Heisenberg, worked out in collaboration with Jordan and myself, and quite independently by Dirac; and another form of the same theory, the wave-mechanics, worked out by Schrodinger in close connection with de Broglie's suggestion.
     
   • To understand how the quantum idea and causality are connected, we must explain the second fundamental law relating particles and waves.
     
   • Now we can analyse the connection between the quantum laws and causality.
     
   • The corresponding probability wave must also be restricted to this small part of space, according to our second quantum law.
     
   • Using now the first quantum law stating the proportionality of frequency and energy, we see that this geometrically well-defined state must contain a wide range of energies.
     
   • The quantitative law found by Heisenberg states that for each direction in space the product of the uncertainty interval of space and that of momentum (equal to mass times velocity) is always the same, being given by Planck's quantum constant h.
     
   • The quantum laws contradict this supposition, and this means the break-down of causality and determinism.
     
   • The result that the discovery of the quantum laws puts an end to the strict determinism which was unavoidable in the classical period is of great philosophical importance by itself.
     
   • The mathematical theory called quantum mechanics which expresses these ideas in a precise form is a most wonderful structure, not only comparable with, but superior to, classical mechanics.
     
   • As I said before, this standpoint has proved itself productive by inducing physicists to adopt a critical attitude towards traditional assumptions, and has helped in the building of relativity and quantum theory.
     
   • In quantum theory we are only at the beginning of this process.
     
   • Therefore I cannot tell you in a few words of ordinary language what the reality is which quantum mechanics deals with.
     
   • Then there is another dimensionless number 137, connecting the elementary charge, Planck's quantum constant, and the velocity of light.
     

6. Wolfgang Pauli and the Exclusion Principle
   
   • I was not spared the shock which every physicist, accustomed to the classical way of thinking, experienced when he came to know of Bohr's "basic postulate of quantum theory" for the first time.
     
   • At that time there were two approaches to the difficult problems connected with the quantum of action.
     
   • One was an effort to bring abstract order to the new ideas by looking for a key to translate classical mechanics and electrodynamics into quantum language which would form a logical generalization of these.
     
   • The most fundamental of his results thereby was the use of half-integers as magnetic quantum numbers for the doublet-spectra of the alkali metals.
     
   • On the other hand, the anomalous splitting was hardly understandable from the standpoint of the mechanical model of the atom, since very general assumptions concerning the electron, using classical theory as well as quantum theory, always led to the same triplet.
     
   • One was the absence of a general key to translate a given mechanical model into quantum theory which one tried in vain by using classical mechanics to describe the stationary quantum states themselves.
     
   • In the autumn of 1924 I published some arguments against this point of view, which I definitely rejected as incorrect and proposed instead of it the assumption of a new quantum theoretic property of the electron, which I called a two-valuedness not describable classically.
     
   • For a given value of the principal quantum number is the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field the same as the number of electrons in the closed shell of the rare gases which corresponds to this principal quantum number.
     
   • The fundamental idea can be stated in the following way: The complicated numbers of electrons in closed subgroups are reduced to the simple number one if the division of the groups by giving the values of the four quantum numbers of an electron is carried so far that every degeneracy is removed.
     
   • The gap was filled by Uhlenbeck and Goudsmit's idea of electron spin, which made it possible to understand the anomalous Zeeman effect simply by assuming that the spin quantum number of one electron is equal to 1/2 and that the quotient of the magnetic moment to the mechanical angular moment has for the spin a value twice as large as for the ordinary orbit of the electron.
     
   • On the other hand, my earlier doubts as well as the cautious expression classically non-describable two-valuedness experienced a certain verification during later developments, since Bohr was able to show on the basis of wave mechanics that the electron spin cannot be measured by classically describable experiments (as, for instance, deflection of molecular beams in external electromagnetic fields) and must therefore be considered as an essentially quantum-mechanical property of the electron .
     
   • The subsequent developments were determined by the occurrence of the new quantum mechanics.
     
   • Nor does time permit me to illustrate in detail the general epistemological significance of the new discipline of quantum mechanics, which has been done, among others, in a number of articles by Bohr, using hereby the idea of complementarity as a new central concept.
     
   • I shall only recall that the statements of quantum mechanics are dealing only with possibilities, not with actualities.
     
   • Only this renouncement concerning the old claims for an objective description of the physical phenomena, independent of the way in which they are observed, made it possible to reach again the self-consistency of quantum theory, which actually had been lost since Planck's discovery of the quantum of action.
     

7. Max Planck: 'The Nature of Light
   
   • The address was given at an interesting time in the development of ideas on the nature of light at just the time when quantum theory was being proposed and the lecture considers both the traditional and quantum-mechanical view.
     
   • Such a light-quantum, striking the metal, communicates its energy to an electron, and the energy always remains the same, however great the distance from the source of light.
     
   • But interference, which was a bar to the further development of Newton's emanation theory, is also an enormous difficulty in the quantum theory of light, for it is difficult at present to see how two exactly similar light quanta, moving independently in space, and meeting on a common path, can neutralize each other, without violating the principle of energy.
     
   • It is the surplus amount of energy liberated by the atom which travels out into space as a light quantum.
     
   • It is definitely determined by the amount of energy emitted, since the more rapid the oscillations, the greater is the light quantum.
     
   • It follows that a short wave-length corresponds to a large amount of energy, considered as a light quantum.
     
   • What becomes of them later as light disperses - whether the energy of a quantum remains concentrated as in Newton's emanation theory or whether, as in Huygens's wave theory, it spreads out in all directions and gets less dense indefinitely - is another question of a very fundamental character, to which I have referred above.
     
   • In fact, this question, whether light rays themselves consist of quanta, or whether the quanta exist only in matter, is the chief and most difficult dilemma before which the whole quantum theory halts, and the answer to this question will be the first step towards further development.
     

8. A I Khinchin: 'Statistical Mechanics' Introduction
   
   • In closing this brief sketch we should mention that the development of atomic mechanics during the last decades has changed the face of physical statistics to such a degree that, naturally, statistical mechanics had to extend its mathematical apparatus in order to include also quantum phenomena.
     
   • Quantum statistics also presents some new mathematical problems.
     
   • Nevertheless, it could be stated that the transition from the classical systems to the quantum systems did not introduce any essentially new mathematical difficulties.
     
   • Precisely for these reasons in the present book we have restricted ourselves to the discussion of the classical systems, leaving completely out of consideration everything concerning quantum physics, although all the methods which we develop after suitable modifications could be applied without any difficulties to the quantum systems.
     
   • We have chosen the classical systems mainly because our book is designed, in the first place, for a mathematical reader, who cannot always be assumed to have a sufficient knowledge of the foundations of quantum mechanics.
     
   • Such an inclusion would have considerably increased the size of the book, and would not attain the desired purpose since quantum mechanics with its novel ideas, often contradicting the classical representations, could not be substantially assimilated by studying such a brief exposition.
     

9. Wave versus matrix mechanics
   
   • We want to examine here the different models for the atom provided by wave mechanics and by quantum mechanics.
     
   • The theory by Heisenberg to which Schrodinger refers is quantum mechanics which he put forward in 1925.
     
   • Don't take it as an unfriendliness to you but look on the expression as my objective conviction that quantum phenomena naturally display aspects that cannot be expressed by the concepts of continuum physics.
     
   • Investigation of the type of physical reality which is proper to electrons and atoms is precisely the subject of atomic physics and thus also of quantum mechanics.
     
   • You surely must understand, Bohr, that the whole idea of quantum jumps necessarily leads to nonsense.
     
   • But that doesn't prove that there are no quantum jumps.
     
   • It only proves that we can't visualise them, that means that the pictorial concepts we use to describe the events of everyday life and the experiments of the old physics do not suffice also to represent the process of a quantum jump.
     

10. Edmund Whittaker: 'Physics and Philosophy
    
    • Among the qualitative assertions are included the postulates of impotence, e.g., the uncertainty principle in quantum mechanics, the impossibility of constructing a perpetual motion machine.
      
    • Freewill poses to metaphysicians the question: If God knows what we shall do in the future, how can we be said to act freely?, The problem of freewill has also engaged the attention of both classical and quantum physicists and a searching analysis of the problem was given by Whittaker in the twenty-seventh Guthrie lecture entitled Chance, Freewill and Necessity in the Scientific Conception of the Universe.
      
    • The advent of quantum mechanics, and in particular the uncertainty relation, has compelled scientists and philosophers to recast some of the notions about cause and effect which were previously regarded as axiomatic.
      
    • In quantum theory the diffraction of electrons by a crystal, the passage of a plane-polarised light say through a Nicol prism, and radioactivity exhibit features which are certainly not deterministic.
      
    • The question as to which class does a theory belong has, with the advent of relativity and quantum theory, ceased to have a meaning.
      
    • Moreover the absence of determinism and of crypto-determinism in quantum mechanics suggests that there is a continual succession of divine intrusions.
      

11. What do mathematicians do?
    
    • In the 1920's, for example, the discovery of quantum mechanics went a very long way toward reducing chemistry to the solution of well-defined mathematical problems.
      
    • While the physicists were struggling with contradictions and anomalies in the so-called 'old quantum theory', two quite distinct branches of pure mathematics were being developed by two different sets of mathematicians with no thought for one another or for physics.
      
    • Then the discoveries of Schrodinger and Heisenberg in 1924-25 provided the key to the mystery, and physics found its way to that subtle refinement of Newtonian mechanics known as quantum mechanics.
      
    • Almost immediately it was found that these two separate new branches of pure mathematics were not only what quantum mechanics needed for its precise formulation and further development, but they could be regarded moreover as two facets of a bigger and better unified new branch which was even more adapted to the needs of quantum physics.
      

12. Collected Papers of Paul Ehrenfest' Preface
    
    • Ehrenfest's own contributions are mainly in the field of statistical mechanics and its relations with quantum mechanics.
      
    • His work on quantum statistics led to the formulation of his theorem of adiabatic invariance which played an important role throughout the further development of quantum mechanics.
      
    • Now that atoms have become almost tangible realities and the theory of atomic structure can be built up from first principles, it is instructive to realize to what extent the development of quantum theory was initially due to statistical and thermodynamical considerations.
      

13. Max Planck and the quanta of energy
    
    • When I look back to the time, already twenty years ago, when the concept and magnitude of the physical quantum of action began, for the first time, to unfold from the mass of experimental facts, and again, to the long and ever tortuous path which led, finally, to its disclosure, the whole development seems to me to provide a fresh illustration of the long-since proved saying of Goethe's that man errs as long as he strives.
      
    • Because it represents the product of energy and time, I described it as the elementary quantum of action.
      
    • Either the quantum of action was a fictional quantity, then the whole deduction of the radiation law was in the main illusory and represented nothing more than an empty non-significant play on formulae, or the derivation of the radiation law was based on a sound physical conception.
      
    • In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was, since the establishment of the infinitesimal calculus by Leibniz and Newton, upon the acceptance of the continuity of all causative connections.
      

14. Planck's quanta.html
    
    • When I look back to the time, already twenty years ago, when the concept and magnitude of the physical quantum of action began, for the first time, to unfold from the mass of experimental facts, and again, to the long and ever tortuous path which led, finally, to its disclosure, the whole development seems to me to provide a fresh illustration of the long-since proved saying of Goethe's that man errs as long as he strives.
      
    • Because it represents the product of energy and time, I described it as the elementary quantum of action.
      
    • Either the quantum of action was a fictional quantity, then the whole deduction of the radiation law was in the main illusory and represented nothing more than an empty non-significant play on formulae, or the derivation of the radiation law was based on a sound physical conception.
      
    • In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was, since the establishment of the infinitesimal calculus by Leibniz and Newton, upon the acceptance of the continuity of all causative connections.
      

15. Arthur Eddington's 1927 Gifford Lectures
    
    • The theory of relativity and the quantum theory have led to strange new conceptions of the physical world; the progress of the principles of thermodynamics has wrought more gradual but no less profound change.
      
    • It would not serve my purpose to give an easy introduction to the rudiments of the relativity and quantum theories; it was essential to reach the later and more recondite developments in which the conceptions of greatest philosophical significance are to be found.
      
    • There is a familiar table parallel to the scientific table, but there is no familiar electron, quantum or potential parallel to the scientific electron, quantum or potential.
      

16. Coulson: 'Electricity
    
    • The theory of the earth's magnetism has been only lightly touched upon, and electrolysis has been completely omitted, for this, like the theory of electrons, belongs more properly to the field of quantum theory, and is outside the range of this book.
      
    • In the microscopic point of view we deal with individual atoms and electrons, which are the field of atomic physics and quantum theory.
      
    • Such studies have been made, but they introduce us to two essentially new ideas - the quantum theory and the wave nature of matter.
      

17. Max Born's matrices
    
    • In the book Born, in an extremely modest way, explains how he came to realise that Heisenberg's quantum mechanics was represented by matrices:- .
      
    • This form of quantum mechanics, which was also brought to a high degree of perfection by Dirac quite independently, is not only the earliest form of quantum mechanics, but perhaps the most fundamental; but it is so mathematical and abstract that it cannot be made intelligible without the use of mathematics.
      

18. Whittaker EMS Obituary.html
    
    • the superbly beautiful theory which springs from Hamilton's equations and which has turned out to be of such fundamental importance for the development of quantum mechanics.
      
    • The second volume, which appeared in 1953, contains an account of the developments which took place between 1900 and 1926 and is mainly concerned with Relativity and the Quantum Theory.
      
    • I think he would have had great difficulty in shaping it into final form, for some of the recent theories are in anything but a finished state and the "bad" mathematics so successfully used in quantum field theory was revolting to him.
      

19. E P Adams
    
    • He was, therefore, commissioned by the National Research Council to write a report on the existing state of the quantum theory, which had had a considerable development using classical ideas with superimposed quantum conditions, as in Bohr's atomic theory.
      
    • In a letter written in October 1956, he remarked about the teaching of physics: "I still think there must be a solid foundation of Newtonian and Maxwellian physics, but how to make the transition to quantum and relativistic physics is what puzzles me.
      

20. Born's matrices.html
    
    • In the book Born, in an extremely modest way, explains how he came to realise that Heisenberg's quantum mechanics was represented by matrices:- .
      
    • This form of quantum mechanics, which was also brought to a high degree of perfection by Dirac quite independently, is not only the earliest form of quantum mechanics, but perhaps the most fundamental; but it is so mathematical and abstract that it cannot be made intelligible without the use of mathematics.
      

21. G H Hardy's schedule of lectures in the USA
    
    • Representations of groups, applications and representations of groups for quantum mechanics .
      
    • Dirac lectured on quantum mechanics at the University of Wisconsin in April and May, 1929.
      
    • He and Milne were two of the four lecturers giving courses at a symposium on theoretical physics at the University of Michigan, 24 June-16 August, 1929: Milne spoke on problems in astrophysics and vector and tensor methods in statics and dynamics; Dirac gave an introduction to quantum mechanics.
      

22. Levi-Civita.html
    
    • There are few modern branches of mathematical physics to which he did not at one time or another contribute - classical mechanics, hydromechanics, thermodynamics, elasticity, the strength of materials, astronomy, electromagnetism, optics, relativity and quantum mechanics - yet some of his greatest work was in pure mathematics.
      
    • He made numerous other contributions to relativity, and in 1933 interested himself in the problem of reconciling Dirac's equations in quantum mechanics with the relativistic principle of covariance.
      

23. Von Neumann: 'The Mathematician' Part 2
    
    • Michelson's experiment leading to special relativity, the difficulties of certain ionization potentials and of certain spectroscopic structures leading to quantum mechanics exemplify the first case; the conflict between special relativity and Newtonian gravitational theory leading to general relativity exemplifies the second, rarer, case.
      
    • Accordingly, the subject of theoretical physics was at almost all times enormously concentrated; at almost all times most of the effort of all theoretical physicists was concentrated on no more than one or two very sharply circumscribed fields-quantum theory in the 1920's and early 1930's and elementary particles and structure of nuclei since the mid-1930's are examples.
      

24. Tullio Levi-Civita
    

25. EMS obituary
    
    • There are few modern branches of mathematical physics to which he did not at one time or another contribute - classical mechanics, hydromechanics, thermodynamics, elasticity, the strength of materials, astronomy, electromagnetism, optics, relativity and quantum mechanics - yet some of his greatest work was in pure mathematics.
      
    • He made numerous other contributions to relativity, and in 1933 interested himself in the problem of reconciling Dirac's equations in quantum mechanics with the relativistic principle of covariance.
      

26. Finlay Freundlich's Inaugural Address, Part 2
    
    • Although followers of Einstein's theory of relativity are reluctant, to say the least, even to consider such a possibility, it may not be overlooked that also in other respects a deep gap is dividing the two branches of physics the one approaching the phenomena from a macroscopic point of view; this is the theory of relativity; the other, the quantum theory, based on fundamentally different conceptions, which takes account of the atomic structure of matter and the quantum structure of energy.
      

27. N S Krylov's monograph - Introduction
    
    • Closely connected with the first type of difficulties are the problem of the mechanical interpretation of irreversibility and, among other things, all the well-known objections to Boltzmann's treatment of the H-theorem, and all the attempts still being made at achieving a quantum-mechanical solution of this problem.
      
    • The second chapter contains a similar analysis in quantum mechanics.
      

28. Jenó Wigner's student years
    
    • Jeno (Eugene) P Wigner was born in Budapest in 1902, and received the Nobel Prize in Physics as a professor at Princeton University in 1963 for understanding the role of symmetries in quantum mechanics, for the discovery of parity, and for applying quantum mechanics to atomic nuclei.
      

29. Schrödinger: 'Statistical Thermodynamics
    
    • The object of this seminar is to develop briefly one simple, unified standard method, capable of dealing, without changing the fundamental attitude, with all cases (classical, quantum, Bose-Einstein, Fermi-Dirac, etc.) and with every new problem that may arise.
      
    • 6 have exchanged their roles; but the latter is merely a question of enumerating correctly the states of the single system, of describing correctly its quantum-mechanical nature.
      

30. Sneddon: 'Special functions
    
    • Throughout the text an attempt is made to show how these functions may be used in the discussion of problems in classical physics and in quantum theory.
      

31. EMS obituary
    
    • In Berlin and some other continental universities the situation was very different; nevertheless this was by no means uniformly so as is evidenced by the story that Heisenberg did not recognise the relationship between his quantum mechanical operators and matrices.
      

32. A D Aleksandrov's view of Mathematics
    
    • Similarly, in the present-day theory of atomic phenomena, in the so-called quantum mechanics, essential use is made of many extremely abstract mathematical concepts and theories, as for example the concept of infinite-dimensional space.
      

33. Gordon Preston on semigroups
    
    • an old friend from undergraduate days who was a theoretical chemist at Cambridge, that they might be of use in the quantum theory of the molecule.
      

34. James Jeans: 'Physics and Philosophy' II
    
    • But the quantum theory finds, as we shall see later, that the fundamental activities of nature cannot be represented as occurring in space and time; they cannot, then, be mechanical in the ordinary sense of the word.
      

35. János Neumann's student years
    
    • Born in Budapest in 1903, he laid the mathematical foundation of quantum mechanics in his twenties, in Gottingen.
      

36. EMS 1934 Colloquium
    
    • The second, given by Professor G Temple (London), on the General Principles of the Quantum Theory and Eddington's theory of the fine-structure constant, was remarkable not only for the breadth of knowledge the lecturer revealed but also for the fascinating manner in which he presented his subject.
      

37. Rota's lecture on 'Mathematical Snapshots
    
    • New theories are sprouting up: quantum groups, the Yang-Baxter equations, monoidal categories, and what not.
      

38. Raoul Bott on John Nash
    
    • We were reading von Neumann's book on quantum mechanics, which developed Hilbert spaces at the same time.
      

39. Mark Kac on education, physics and mathematics
    
    • I would attempt, I wouldn't be very good at it, but I would attempt to teach a first semester course in quantum mechanics, and I would probably teach it reasonably well.
      

40. Einstein: 'Ether and Relativity
    
    • Further, in contemplating the immediate future of theoretical physics we ought not unconditionally to reject the possibility that the facts comprised in the quantum theory may set bounds to the field theory beyond which it cannot pass.
      

41. EMS 1914 Colloquium 3.html
    
    • Mr Cunningham's third lecture began with a short reference to the famous quantum hypothesis, and suggested one or two difficulties in reference to it.
      

42. Mathematical and Physical Journal for Secondary Schools
    
    • (Professor Rudolf Ortvay introduced the most active school teachers in this way to quantum mechanics already in 1930.) .
      

43. EMS 1934 Colloquium 2.html
    
    • The second, given by Professor G Temple (London), on the General Principles of the Quantum Theory and Eddington's theory of the fine-structure constant, was remarkable not only for the breadth of knowledge the lecturer revealed but also for the fascinating manner in which he presented his subject.
      

44. EMS 1938 Colloquium 4.html
    
    • Professor Whittaker covered this immensity, and described the stages in which the transformation took place, through relativity, the early quantum theory and its developments up to the latest speculations on heavy electrons and cosmic rays.
      

45. Eddington: 'Mathematical Theory of Relativity' Introduction
    
    • Indeed it has been suspected that the perplexities of quantum phenomena may arise from the tacit assumption that the notions of length and duration acquired primarily from experiences in which the average effects of large numbers of quanta are involved, are applicable in the study of individual quanta.
      

46. EMS honours Maxwell and Tait
    
    • This result, when modified by the requirements of the quantum theory, had important consequences in astrophysics and also in the theory of metals.
      

47. Malcev: 'Foundations of Linear Algebra' Introduction
    
    • Applications to quantum mechanics stimulated a still more rapid development of the theory of these spaces, which has become one of the most important parts of contemporary functional analysis.
      

48. EMS 1938 Colloquium
    
    • Professor Whittaker covered this immensity, and described the stages in which the transformation took place, through relativity, the early quantum theory and its developments up to the latest speculations on heavy electrons and cosmic rays.
      

49. James Jeans addresses the British Association in 1934, Part 2
    
    • The parable tells us that light consists of discrete light-particles, called photons, each carrying a single quantum of energy.
      

50. Sneddon: 'Special functions
    
    • Throughout the text an attempt is made to show how these functions may be used in the discussion of problems in classical physics and in quantum theory.
      

51. EMS obituary
    
    • Yet these were the years, immediately subsequent to 1926, that saw legions of researchers and a spate of papers on quantum theory.
      

52. Harold Jeffreys on Logic and Scientific Inference
    
    • At present we are faced with the inaccuracy of Euclid's parallel axiom, which for millennia was considered intuitively obvious; with the inaccuracy of Newton's law of gravitation, which had been well established by experience and had been believed for centuries to be exact; with the failure in stars of the law of the indestructibility of matter; and with the discordance of the classical undulatory theory of light with the group of facts known as quantum phenomena.
      

53. Eddington on the Expanding Universe
    
    • But one knows the sort of effect that curvature can have; and the way it will appear in the equation is pretty well dictated by the quantum laws which make a speciality of the properties of "closed circuits" such as are introduced by curved space.
      

54. Eddington on the Expanding Universe
    

55. R A Fisher: 'Statistical Methods' Introduction
    
    • In Quantum Theory this is now clearly recognised.
      

56. Tietze: 'Famous Problems of Mathematics
    
    • If a work by Albert Einstein can now be quoted without further ado, at that time the book's chances of appearing were endangered by mere mention in it of the theory of relativity, which had the effect of a red flag on many in those days when, blinded by the universal propaganda, some scientists really believed that this theory was pure humbug and when, worse still, one of our greatest theoretical physicists was replaced by a man, foisted upon us from on high, who tossed Planck's quantum theory into the same kettle of damnation with the theory of relativity, so that our theoretical physicists had great difficulty in interceding on behalf of continued use of the theory of relativity in research and in teaching.
      

57. Sommerfeld: 'Atomic Structure
    
    • All integral laws of spectral lines and of atomic theory spring originally from the quantum theory.
      

58. Von Neumann Silliman lectures
    
    • His publications included works on quantum theory, mathematical logic, ergodic theory, continuous geometry, problems dealing with rings of operators, and many other areas of pure mathematics.
      

59. EMS 1914 Colloquium
    
    • Mr Cunningham's third lecture began with a short reference to the famous quantum hypothesis, and suggested one or two difficulties in reference to it.
      

60. James Jeans addresses the British Association in 1934, Part 3
    
    • The parable tells us that light consists of discrete light-particles, called photons, each carrying a single quantum of energy.
      

Quotations

1. Quotations by Bohr Niels
   
   • Quoted in L I Ponomarev, The Quantum Dice .
     
   • Anyone who is not shocked by quantum theory has not understood it.
     

2. Quotations by Weyl
   
   • In this very radical sense, quantum physics supports the doctrine that the whole is more than the combination of its parts.
     

3. Quotations by Schrodinger
   
   • [On quantum mechanics ] .
     

4. Quotations by Planck
   
   • If anybody says he can think about quantum problems without getting giddy, that only shows he has not understood the first thing about them.
     

5. Quotations by Napier
   
   • Te, quantum magnis mille voluminibus .
     

6. Quotations by Feynman
   
   • I think that I can safely say that nobody understands quantum mechanics.
     

7. Quotations by Dirac
   
   • Preface to The principles of Quantum Mechanics (Oxford, 1930) .
     

8. Quotations by Wiener Norbert
   
   • The modern physicist is a quantum theorist on Monday, Wednesday, and Friday and a student of gravitational relativity theory on Tuesday, Thursday, and Saturday.
     

Chronology

1. Mathematical Chronology
   
   • Planck proposes quantum theory.
     
   • Von Neumann publishes Grundlagen der Quantenmechanik on quantum mechanics.
     
   • Drinfeld is awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan for his work on quantum groups and for his work in number theory.
     

2. Chronology for 1980 to 1990
   
   • Drinfeld is awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan for his work on quantum groups and for his work in number theory.
     

3. Chronology for 1900 to 1910
   
   • Planck proposes quantum theory.
     

4. Chronology for 1990 to 2000
   
   • Drinfeld is awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan for his work on quantum groups and for his work in number theory.
     

5. Chronology for 1930 to 1940
   
   • Von Neumann publishes Grundlagen der Quantenmechanik on quantum mechanics.