Quantum Mechanics from Symmetry and Statistical Modeling

Inge Svein Helland

The Basic Elements Statistical Theory and Practice Statistical Inference Under Symmetry The Transition from Statistics to Quantum Theory Quantum Mechanics from a Statistical Basis Further Development of Quantum Mechanics Decisions in Statistics Multivariate Data Analysis and Statistics Quantum Mechanics and the Diversity of Concepts.

Carlos Osácar, Amalio F. Pacheco

The symmetry of self-duality (Savit 1980 Rev. Mod. Phys. 52 453) of some models of statistical mechanics and quantum field theory is discussed for finite spin blocks of the Ising chain in a transverse magnetic field. The existence of this symmetry in a specific type of these blocks, and not in others, is manifest by the degeneracy of their energy levels. This illustrates the fact that in quantum mechanics, more symmetry in the Hamiltonian induces more degeneracy in the energy spectrum.

B. Freedman. Physics Department

Michael J. Creutz, Barry A. Freedman

Inge Svein Helland

Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The quantum Hilbert space is constructed from the parameters of the various experiments using group representation theory. It is shown under natural assumptions that a subset of the set of unit vectors of this space, the generalized coherent state vectors, can be put in correspondence with questions of the kind: What is the value of the (complete) parameter? - together with a crisp answer to that question. Links are made to statistical models in general, to model reduction of overparametrized models and to the design of experiments. It turns out to be essential that the range of the statistical parameter is an invariant set under the relevant symmetry group.