A 1.5V class A 5/sup th/ order log domain filter in SiGe technology

A new kind of geometric phases in open quantum systems and higher gauge theory

“…These three types of curvature satisfy some gauge transformation formulae (studied in Appendix C) which are also characteristic of a higher gauge theory [26,27,28,29,30,31]). The C * -geometric phases of an open quantum system are not related to a principal bundle over M (as would occur for the geometric phases of closed quantum systems) because the family of local principal G-bundles {P α G } α cannot be lift to a single global principal bundle (this is due to the failure of the cocycle relation with g αβ measured by h αβγ ).…”

“…The gauge transformations of the gauge potential A α and of the potentialtransformation η αβ are characteristic of a higher gauge theory, as described with several different notations in refs. [26,27,28,29,30,31]. In accordance with these previous works, we introduce here three types of curvature.…”

“…This identification provides a new interpretation of the geometric phases of open quantum systems. The geometric structure in which this new kind of geometric phase is sited is not a principal bundle (as used for the geometric phases of a closed quantum system) but a higher gauge structure, a nonabelian bundle gerbes [26,27,28] or equivalently a principal 2-bundle [29,30,31], i.e. a categorical generalization of a principal bundle.…”

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

“…(2.32) 13 Strictly speaking, they are n-term L ∞ -algebras, but for all intents and purposes, they can be regarded as (categorically) equivalent to Lie n-algebras. The categorical equivalence has been proven for Lie 2algebras and 2-term L ∞ -algebras [40] ; the extension to Lie n-algebras and n-term L ∞ -algebras should be very involved, but ultimately a mere technicality. 14 The isomorphism is the shift isomorphism s • defined in (2.10).…”

“…Clearly, L ∞ -morphisms concentrated in degree 0 are of this type, and for those the relation (2.52) reduces to φ 1 (μ 2 ( 1 , 2 ))) = μ 2 (φ 1 ( 1 ), φ 1 ( 2 )), (2.53) that is, the expected relation for a morphism of Lie algebras. The notion of a weak morphism between 2-term L ∞ -algebras was derived in [40], where also many more details on 2-term L ∞algebras can be found. Morphisms of L ∞ -algebras are composable, and the formulas for the composition map can be derived using the coalgebra picture in Appendix A in which composition is evident.…”