Micro-Macro Duality for Inductions / Reductions Izumi Ojima

Paradoxical appearance of negative metrics in the processes of emergences will be analyzed from the viewpoint of Morse theory, induced representations and of imprimitivity systems. 1 How to control inclusion relations The essence of the present notes is to discuss the following issues which have been triggered by the requirement for a theoretical framework to treat Dressed Photons: how to fill the gap between Macroscopic Phenomena & Microscopic Theory, on the basis of Micro-Macro Duality in Quadrality Scheme, comined with Saddle-Point Instability, Through the following examples, Lorentz symmetry/ Regge structure/ Dressed photons/ Coulomb modes/ Tomita-Takesaki modular theory for statistical mechnaics, controlling mechanism will be explained on the basis of induced representations. Existence of quantum modes with “indefinite metric” breaks the consistency of theory at Micro level, as is well known by the difficulties caused by longitudinal photons. Therefore, one always tries to avoid longitudinal photons in QED. However, this is in contradiction to the existence of Coulomb modes in Macro world! To understand such contradictory situations, we need first re-examine the concept and phenomena of symmetry breaking. 1 O ff Sh el l: 18 09 O . 0 01 . v 1 2 Symmetry Breaking creates Symmetric Space When symmetry of the system described by a group G is broken up to unbroken subgroup H, a homogeneous space G/H emerges in sector classifying space. In this situation, G/H is shown to be a symmetric space with many nice properties [Hel], according to the following criterion for symmetry breaking. For this reason, induced representation IndH [Mackey] to describe the broken symmetry G on the sector classifying space G/H as a symmetric space has a strong connection with automorphic forms and zeta functions playing important roles in number theory. The mutual relation between the quadrality scheme and the groups to describe symmetries can be depicted as follows: Spec= classifying space = G/H : Visible Macro Emergence ↗ »1⁄4 ↑ (Family of) States ¿ Algebra x G : Micro-Macro boundaries : t-ch »1⁄4 ↗ ↑ Dynamics x H : Invisible Micro: s-ch 2.1 Symmetry Breaking General definition of symmetry breaking [IO03]: Definition (Symmetry Breaking): Let X be a C∗-algebra describing quantum fields with an automorphic action τ :X x τ G of a Lie group G on X and (π, H) be a representation of X . If the spectrum Spec(Zπ(X )) of its center Zπ(X ) = Z(π(X )′′) is pointwise G-invariant (almost everywhere w.r.t. the central measure), the symmetry (G, τ) on X is said to be unbroken in (π, H) and broken otherwise. The reason for complicated situations concerning symmetry breaking in QFT is due to such a contrast between quantum systems with finite vs. infinite degrees of freedom: while the use of a unitary representation U of G leads automatically to the unbroken symmetry (which is always the case for systems with finite degrees of freedom), the very non-existence of U realizable only in those with infinite degrees of freedom characterizes the broken symmetry. This is the reason why we need G-actions both in C∗and W∗versions in the above criterion for symmetry breaking. 2.2 Induced Representation from Unbroken to Broken To streamline the discussion, we define “augmented algebra” [IO03] by a (C∗-)crossed product X o (̂H\G) =: X̂ of X with the dual (̂H\G) of (G/H), which allows unitary implementation of broken G at the expense of non-trivial center Zπ(X̂ ) with