Suborbits of (m,k)-isotropic subspaces under finite singular classical groups

AbstractLet Fq2ν+δ+l be one of the (2ν+δ+l)-dimensional singular classical spaces and let G2ν+δ+l,2ν+δ be the corresponding singular classical group of degree 2ν+δ+l. All the (m,k)-isotropic subspaces form an orbit under G2ν+δ+l,2ν+δ, denoted by M(m,k;2ν+δ+l,2ν+δ). Let Λ be the set of all the orbitals of (G2ν+δ+l,2ν+δ,M(m,k;2ν+δ+l,2ν+δ)). Then (M(m,k;2ν+δ+l,2ν+δ),Λ) is a symmetric association scheme. First, we determine all the orbitals and the rank of (G2ν+δ+l,2ν+δ,M(m,k;2ν+δ+l,2ν+δ)), calculate the length of each suborbit. Next, we compute all the intersection numbers of the symmetric association scheme (M(ν+k,k;2ν+δ+l,2ν+δ),Λ), where k=1 or k=l−1. Finally, we construct a family of symmetric graphs with diameter 2 based on M(2,0;4+δ+l,4+δ)

Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I (The two antagonistic localizations and their asymptotic compatibility)

It is shown that there are significant conceptual differences between QM and QFT which make it difficult to view the latter as just a relativistic extension of the principles of QM. At the root of this is a fundamental distiction between Born-localization in QM (which in the relativistic context changes its name to Newton-Wigner localization) and modular localization which is the localization underlying QFT, after one separates it from its standard presentation in terms of field coordinates. The first comes with a probability notion and projection operators, whereas the latter describes causal propagation in QFT and leads to thermal aspects of locally reduced finite energy states. The Born-Newton-Wigner localization in QFT is only applicable asymptotically and the covariant correlation between asymptotic in and out localization projectors is the basis of the existence of an invariant scattering matrix. In this first part of a two part essay the modular localization (the intrinsic content of field localization) and its philosophical consequences take the center stage. Important physical consequences of vacuum polarization will be the main topic of part II. Both parts together form a rather comprehensive presentation of known consequences of the two antagonistic localization concepts, including the those of its misunderstandings in string theory.Comment: 63 pages corrections, reformulations, references adde