Symmetries and Spectra: The Foundational Link Between Group Theory and Quantum Mechanics

Paul F. McMillan, Anthony C. Hess

Fl. Stancu

Abstract Group theory is a branch of mathematics appropriate for studying symmetries of physical systems. A symmetry is a natural attribute of the physical world and therefore is the starting point of any physical law. The foundation of dynamics is in Lagrangian theory or equivalently in Hamiltonian theory. A symmetry of a physical system is expressed by the invariance of its Lagrangian or Hamiltonian, or ultimately of its equations of motion, under certain transformations. When these transformations form groups it is convenient and useful to study symmetries through group theory. Group theory can explain observed regularities and helps to simplify and unify the physics of apparently different systems. It is a valuable tool that helps in predicting the behaviour of systems for which a symmetry has been found.

Inge Svein Helland

A new foundation of quantum mechanics for systems symmetric under a compact symmetry group is proposed. The foundation is given by a link to classical statistics and coupled to the concept of parameter. A vector ϕ of parameters is called an inaccessible c-variable if experiments can be provided for the single parameter, but no experiment can be provided for ϕ. This is related to the concept of complementarity in quantum mechanics, but more generally to contrafactual parameters. Using these concepts and some weak assumption, the Hilbert space of quantum mechanics is constructed. The complete set of axioms of quantum mechanics is provided by proving Born's formula under weak assumptions.

Bartel Leenert van der Waerden

Zen Revista, 10 PHYSICS

The exactly solvable systems---the harmonic oscillator, the hydrogen atom, and the angular momentum algebra---are quantum mechanics' solved ground: the spectra computed in closed form that calibrated the theory, taught its methods, and revealed the symmetries that make solvability itself a theorem. This article presents a narrative review of the primary literature of that ground, from de Broglie's 1924 thesis---the wave hypothesis the solvable problems confirmed---through Heisenberg's 1925 umdeutung, Schrödinger's 1926 eigenvalue problem, whose oscillator and rotator opened the solvable list, Dirac's 1926 theory of quantum mechanics and Pauli's 1926 hydrogen solution by the new algebra, Darwin's 1927 vector waves, Dirac's 1928 electron---the relativistic hydrogen's fine structure---Wigner's 1931 group theory and its application to the atomic spectra, Condon and Shortley's 1935 atomic spectra canon, Fock's 1935 hydrogen symmetry---the O(4) hidden behind the spectrum---to the factorization method's systematization in Infeld and Hull's 1951 review and its supersymmetric completion in Cooper, Khare, and Sukhatme's 1995 physics reports. The synthesis is organized around three themes: the solved spectra, in which the oscillator's and hydrogen's levels fixed the new mechanics' validity; the hidden symmetries, in which Runge-Lenz and spherical symmetry explained the solvabilities; and the factorization program, in which the ladder operators' algebra became a method---the exactly solvable potentials classified. It is concluded that the solvable systems are quantum mechanics' load-bearing examples---and that their symmetries remain the theory's deepest explanation of why some problems close.