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Reflection Positivity - A Representation Theoretic Perspective (Paperback, 1st ed. 2018)
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Reflection Positivity - A Representation Theoretic Perspective (Paperback, 1st ed. 2018)
Series: SpringerBriefs in Mathematical Physics, 32
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Refection Positivity is a central theme at the crossroads of Lie
group representations, euclidean and abstract harmonic analysis,
constructive quantum field theory, and stochastic processes. This
book provides the first presentation of the representation
theoretic aspects of Refection Positivity and discusses its
connections to those different fields on a level suitable for
doctoral students and researchers in related fields. It starts with
a general introduction to the ideas and methods involving refection
positive Hilbert spaces and the Osterwalder--Schrader transform. It
then turns to Reflection Positivity in Lie group representations.
Already the case of one-dimensional groups is extremely rich. For
the real line it connects naturally with Lax--Phillips scattering
theory and for the circle group it provides a new perspective on
the Kubo--Martin--Schwinger (KMS) condition for states of operator
algebras. For Lie groups Reflection Positivity connects unitary
representations of a symmetric Lie group with unitary
representations of its Cartan dual Lie group. A typical example is
the duality between the Euclidean group E(n) and the Poincare group
P(n) of special relativity. It discusses in particular the curved
context of the duality between spheres and hyperbolic spaces.
Further it presents some new integration techniques for
representations of Lie algebras by unbounded operators which are
needed for the passage to the dual group. Positive definite
functions, kernels and distributions and used throughout as a
central tool.
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