Relative normalizers of automorphism groups, infravacua and the problem of velocity superselection in QED

Abstract

We advance superselection theory of pure states on a $C^{\ast }$-algebra $\mathfrak{A}$ outside of the conventional (DHR) setting. First, we canonically define conjugate and second conjugate classes of such states with respect to a given reference state ${\omega }_{\mathrm{vac}}$ and background $a\in \mathrm{Aut}(\mathfrak{A})$. Next, for some subgroups $RSG\subset \mathrm{Aut}(\mathfrak{A})$ we study the family $\{{\omega }_{\mathrm{vac}}\circ s|s\in S\}$ of infrared singular states whose superselection sectors may be disjoint for different $s$. We show that their conjugate and second conjugate classes always coincide provided that $R$ leaves the sector of ${\omega }_{\mathrm{vac}}$ invariant and $a$ belongs to the relative normalizer $N_G(R,S):=\{g\in G|g\cdot S\cdot g^{-1}\subset R\}$. We study the basic properties of this apparently new group theoretic concept and show that the Kraus-Polley-Reents infravacuum automorphisms belong to the relative normalizers of the automorphism group of a suitable CCR algebra. Following up on this observation we show that the problem of velocity superselection in non-relativistic QED disappears at the level of conjugate and second conjugate classes, if they are computed with respect to an infravacuum background. We also demonstrate that for more regular backgrounds such merging effect does not occur.

arXiv comment31 pages

References in corpus (14)

• Soft Hair on Black Holes
• Infraparticle Scattering States in Non-Relativistic QED: I. The Bloch-Nordsieck Paradigm
• Infraparticle Scattering States in Non-Relativistic QED: II. Mass Shell Properties
• New Light on Infrared Problems: Sectors, Statistics, Symmetries and Spectrum
• Renormalized Electron Mass in Nonrelativistic QED
• Absence of Ground States for a Class of Translation Invariant Models of Non-relativistic QED
• Effective dynamics of an electron coupled to an external potential in non-relativistic QED
• The mass shell in the semi-relativistic Pauli-Fierz model
• A Note on Polarization Vectors in Quantum Electrodynamics
• On Infravacua and the Localisation of Sectors
• Compton scattering in the Buchholz-Roberts framework of relativistic QED
• Coulomb scattering in the massless Nelson model III. Ground state wave functions and non-commutative recurrence relations
• Infraparticles with superselected direction of motion in two-dimensional conformal field theory
• On Infravacua and Superselection Theory

Cited by in corpus (3)

• Curing velocity superselection in non-relativistic QED by restriction to a lightcone
• Local normality of infravacua and relative normalizers for relativistic systems
• Ground States for Infrared Renormalized Translation-Invariant Non-Relativistic QED