Related papers: Embeddable Quantum Homogeneous Spaces
Quantum embedding theories are powerful tools for approximately solving large-scale strongly correlated quantum many-body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale…
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles (`quantum bundles') over the spacetime manifold yield a quantum ``formalism'' along…
We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms.…
Spatially homogeneous universes can be described in (loop) quantum gravity as condensates of elementary excitations of space. Their treatment is easiest in the second-quantised group field theory formalism which allows the adaptation of…
Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).
We determine the quantum automorphism groups of finite spaces and find they are all compact quantum groups in the sense of Woronowicz. This solves a problem of Connes for finite spaces.
We extend the definition of Lagrangian quantum homology to monotone Lagrangian cobordism and establish its general algebraic properties. In particular we develop a relative version of Lagrangian quantum homology associated to a cobordism…
We apply the Tannaka-Krein duality theory for quantum homogeneous spaces, developed in the first part of this series of papers, to the case of the quantum SU(2) groups. We obtain a classification of their quantum homogeneous spaces in terms…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
We introduce an abstract framework of Cartesian squares beyond the context of fiber products, and use it to extend the notion of pullback from classical to compact quantum principal bundles. Based only on our abstract notion of a Cartesian…
By exploring a possible physical realisation of the geometric concept of noncommutative tangent bundle, we outline an axiomatic quantum picture of space as topological manifold and time as a count of its reconfiguration events.
In this work, we make new developments in generic cotangent bundle geometries, depending on all phase-space variables. In particular, we will focus on the so-called generalized Hamilton spaces, discussing how the main ingredients of this…
Quantum embedding theories are promising approaches to investigate strongly-correlated electronic states of active regions of large-scale molecular or condensed systems. Notable examples are spin defects in semiconductors and insulators. We…
Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within…
Associated to the standard $SU_{q}(n)$ R-matrices, we introduce quantum spheres $S_{q}^{2n-1}$, projective quantum spaces $CP_{q}^{n-1}$, and quantum Grassmann manifolds $G_{k}(C_{q}^{n})$. These algebras are shown to be homogeneous quantum…
In the years 1952 and 1965, H.-C. Wang and U. Hirzebruch showed that the two-point homogeneous compact spaces with convex metrics are isometric to the spheres, the real, complex, octonion projective spaces and the Moufang plane and as well…
We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf-Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
We construct a topos of quantum sets and embed into it the classical topos of sets. We show that the internal logic of the topos of sets, when interpreted in the topos of quantum sets, provides the Birkhoff-von Neumann quantum propositional…
In physics, experiments ultimately inform us as to what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the…