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2 results

  • 8 - Representations of the Orthogonal and the Lorentz Group

  • from Part II - Spin
  • Michel Talagrand
  • Book:
    What Is a Quantum Field Theory?
    Published online:
    22 February 2022
    Print publication:
    17 March 2022, pp 183-207

  • Summary

    The orthogonal group admits projective unitary representations which do not derive from true representations, and we describe a fundamental family of such representations. As a consequence there exist quantum systems that change state under a full turn rotation along a given axis (although a second full turn rotation brings them back to the original state). Amazingly, Nature has made essential use of this structure. In order to study the projective representations of the orthogonal and Lorentz groups, it is convenient to replace them by “better versions“; the groups SU(2) and SL(2,C), which are groups of 2 by 2 matrices, and for which projective representations are simply related to true representations. The orthogonal and Lorentz groups are then images of these groups under two-to-one group homomorphisms, and it is these isomorphisms that concentrate the behavior of their projective representations. Finally we describe how the introduction of parity in our theory leads to the discovery of the Dirac matrices.

  • 11 - The ⋆-Function in Complex Analysis

  • Albert Baernstein II, Washington University, St Louis
  • Book:
    Symmetrization in Analysis
    Published online:
    22 February 2019
    Print publication:
    14 March 2019, pp 398-453

  • Summary

    The star function was originally developed to prove the spread theorem, a problem dealing with meromorphic functions in the complex plane. The first sections prove the spread theorem, along with other applications to the study of these functions. Later sections center on analytic functions in the unit disk. The star function technique yields to sharp estimates for integral means of univalent functions and the (harmonic) conjugate function, along with the behavior of the Green function and harmonic measure under symmetrization. The final section extends some results to domains of arbitrary connectivity. The chapter includes the necessary background in Nevanlinna theory and the Poincaré metric on hyperbolic plane domains, and in almost all cases, the mappings which exhibitextremal behavior are identified.