A brief survey of the mathematics of quantum physics

The mathematics of quantum physics started from matrices and from differential operators. It inspired the theory of linear operators in Hilbert space and of unitary representation for symmetry groups and spectrum generating groups. The Dirac bra-ket formalism led first to Schwartz's theory of distributions and then to its generalization, the Rigged Hilbert Space (RHS) or Gelfand triplet. This Schwartz-RHS provided the mathematical justification for Dirac's continuous basis vector expansion and for the algebra of continuous observables of quantum theory. To obtain also a mathematical theory of scattering, resonance and decay phenomena one needed to make a mathematical distinction between prepared in-states and detected observables (“out-states”). This leads to a pair of Hardy RHS's and using the Paley-Wiener theorem, to solutions of the dynamical equations (Schrodinger or Heisenberg) given by time-asymmetric semi-groups, expressing Einstein causality.

[1]  Jean-Pierre Antoine,et al.  Dirac Formalism and Symmetry Problems in Quantum Mechanics. I. General Dirac Formalism , 1969 .

[2]  S. Weinberg.  The Quantum Theory of Fields: THE CLUSTER DECOMPOSITION PRINCIPLE , 1995 .

[3]  W. PauliJr..  Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik , 1926 .

[4]  O. Melsheimer.  Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. II. Transformation theory in nonrelativistic quantum mechanics , 1974 .

[5]  On the mass and width of the Z-boson and other relativistic quasistable particles , 2000, hep-ph/0001206.

[6]  G. Gamow,et al.  Zur Quantentheorie des Atomkernes , 1928 .

[7]  Murray Gell-Mann,et al.  The formal theory of scattering , 1953 .

[8]  Erwin Schrödinger,et al.  Quantisierung als Eigenwertproblem , 1925 .

[9]  Arno R Bohm,et al.  Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics , 1981 .

[10]  W. Heisenberg,et al.  Zur Quantenmechanik. II. , 1926 .

[11]  Arno R Bohm,et al.  Quantum Mechanics: Foundations and Applications , 1993 .

[12]  Arno R Bohm.  Associative Algebra in the Problem of Mass Formulas , 1967 .

[13]  Schmitt,et al.  Precision lifetime measurements on NaI3p 2P1/2 and 3p 2P 3/2 by beam-gas-laser spectroscopy. , 1996, Physical review letters.

[14]  W. Heisenberg.  A quantum-theoretical reinterpretation of kinematic and mechanical relations , 1925 .

[15]  Carl Eckart,et al.  Operator Calculus and the Solution of the Equations of Quantum Dynamics , 1926 .

[16]  H. Kaldass,et al.  Time asymmetric quantum theory - II. Relativistic resonances from S-matrix poles , 2003 .

[17]  Pamela W. Jordan.  Über eine neue Begründung der Quantenmechanik , 1927 .

[18]  Hall,et al.  High precision linewidth measurement of laser-cooled atoms: Resolution of the Na 3p 2P3/2 lifetime discrepancy. , 1996, Physical review letters.

[19]  A. Böhm.  Rigged Hilbert space and mathematical description of physical systems , 1967 .

[20]  John von Neumann,et al.  Uber Einen Satz Von Herrn M. H. Stone , 1932 .

[21]  Paul Adrien Maurice Dirac,et al.  The fundamental equations of quantum mechanics , 1925 .

[22]  Paul Adrien Maurice Dirac,et al.  The physical interpretation of the quantum dynamics , 1927 .

[23]  Donald Babbitt,et al.  Rigged Hilbert spaces and one-particle Schrödinger operators , 1972 .

[24]  L. Schwartz.  Théorie des distributions , 1966 .

[25]  A. Bohm,et al.  Quantum mechanical irrebersibility , 1997 .

[26]  Über Fundamentalbereiche diskontinuierlicher Gruppen , 1959 .

[27]  F. Riesz,et al.  Leçons d,analyse fonctionnelle , 1953 .

[28]  F. London.  Winkelvariable und kanonische Transformationen in der Undulationsmechanik , 1927 .

[29]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[30]  D. Hilbert,et al.  Über die Grundlagen der Quantenmechanik , 1928 .

[31]  George W. Kattawar,et al.  Scattering theory of waves and particles (2nd ed.) , 1983 .

[32]  S. L. Sobolev,et al.  Some Applications of Functional Analysis in Mathematical Physics , 1991 .

[33]  B. Lippmann.  Variational Principles for Scattering Processes. II. Scattering of Slow Neutrons by Para-Hydrogen , 1950 .

[34]  R. Newton.  Scattering theory of waves and particles , 1966 .

[35]  J. E. Roberts.  The Dirac Bra and Ket Formalism , 1966 .

[36]  J. Roberts,et al.  Rigged Hilbert spaces in quantum mechanics , 1966 .

[37]  P. Kristensen,et al.  Tempered distributions in infinitely many dimensions , 1965 .

[38]  Resonances of Perturbed Selfadjoint Operators and their Eigenfunctionals , 1976 .

[39]  J. Schwinger,et al.  Variational Principles for Scattering Processes. I , 1950 .

[40]  Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution , 2004, hep-ph/0412106.

[41]  Roland Omnès.  The Interpretation of Quantum Mechanics , 1987 .

[42]  Manuel Gadella,et al.  A rigged Hilbert space of Hardy‐class functions: Applications to resonances , 1983 .

[43]  M. Stone.  On One-Parameter Unitary Groups in Hilbert Space , 1932 .

[44]  P. Jordan.  Über kanonische Transformationen in der Quantenmechanik , 1926 .

[45]  D. Fredricks,et al.  Tight riggings for a complete set of commuting observables , 1975 .

[46]  Erwin Schrödinger,et al.  Über das Verhältnis der Heisenberg‐Born‐Jordanschen Quantenmechanik zu der meinem , 1926 .

[47]  Paul Roman,et al.  The Analytic S-Matrix , 1967 .

[48]  Max Born,et al.  Eine neue Formulierung der Quantengesetze für periodische und nicht periodische Vorgänge , 1926 .

[49]  R. Morrow,et al.  Foundations of Quantum Mechanics , 1968 .

[50]  Paul Adrien Maurice Dirac,et al.  Quantum Mechanics and a Preliminary Investigation of the Hydrogen Atom , 1926 .

[51]  Semigroup representations of the Poincaré group and relativistic Gamow vectors , 1999, hep-th/9911059.

[52]  P. Jordan.  Über eine neue Begründung der Quantenmechanik. II. , 1927 .

[53]  Olaf Melsheimer.  Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory , 1974 .

[54]  Arno R Bohm,et al.  Symmetry representations in the rigged Hilbert space formulation of quantum mechanics , 2002 .

[55]  E. Schrödinger.  Der stetige Übergang von der Mikro- zur Makromechanik , 1926, Naturwissenschaften.

[56]  W. Brenig,et al.  Allgemeine Quantentheorie der Stoßprozesse , 1959 .

[57]  Arno R Bohm,et al.  Gamow state vectors as functionals over subspaces of the nuclear space , 1979 .

[58]  W. Jr. Pauli,et al.  Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik , 1926 .

[59]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .

[60]  Günther Ludwig.  Foundations of quantum mechanics , 1983 .

[61]  F. London.  Über die Jacobischen Transformationen der Quantenmechanik , 1926 .

[62]  Jean-Pierre Antoine,et al.  Dirac Formalism and Symmetry Problems in Quantum Mechanics. II. Symmetry Problems , 1969 .

[63]  B. Guo,et al.  First measurement of the 2H(6He,7Li)n angular distribution and proton spectroscopic factor in 7Li , 2010 .