Quantum geometry from commutators: a Heisenberg-picture framework and a toy application to early structure

We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth” of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of M_N -spaces needed to represent (up to a constant C>1 ) the M_N -version of the n -dimensional operator Hilbert space OH_n as a direct sum of copies of M_N . We show that, when C is close to 1, this multiplicity grows as \exp{\beta n N^2} for some constant \beta>0 . The main idea is to relate quantum expanders with “smooth” points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on n -dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.

Self-Adjoint Realizations in Another Hilbert Space

We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

Several conceptual aspects of quantum gravity (QG) are studied on the example of the homogeneous isotropic loop quantum cosmology (LQC) model. In particular: (i) the proper time of the comoving observers is shown to be a quantum operator and a quantum spacetime metric tensor operator is derived. (ii) Solutions of the quantum scalar constraint for two different choices of the lapse function are compared and contrasted. In particular it is shown that in the case of a model with massless scalar field and cosmological constant Λ, the physical Hilbert spaces constructed for two choices of lapse are the same for Λ < 0 while they are significantly different for Λ > 0. (iii) The mechanism of the singularity avoidance is analyzed via detailed studies of an energy density operator, whose essential spectrum was shown to be an interval , where . (iv) The relation between the kinematical and the physical quantum geometry is discussed on the level of relation between observables.

In the search of fractional quantum anomalous Hall (FQAH) effect, the conventional wisdom is to start from a flat Chern band isolated from the rest of the Hilbert space by band gaps, so that many-body interaction can be projected to a landscape that mimics a Landau level. Singular flat bands (SFB), which share protected touching points with other dispersive bands, represent another type of flat landscapes differing from Landau levels and Chern bands in topological and geometric properties. Here we report the finding of FQAH phases in a SFB, which emerges in the bipartite limit of the nearest-neighbor tight-binding model of twisted bilayer MoTe_{2}. At 1/3 and 2/3 filling of the SFB, FQAH effects are demonstrated using density matrix renormalization group calculations with all bands, as well as exact diagonalization calculations with the two touching bands. Gapping the band touching can turn the SFB into a nearly flat Chern band, but counterintuitively this suppresses the FQAH effect, as the gap opening introduces strong inhomogeneity to the quantum geometry. An optical scheme to realize such SFB for cold atoms is provided. Our findings uncover a new arena for the exploration of fractional quantum Hall physics beyond the Landau level and Chern insulator paradigms.

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.

We analyze the energy-momentum properties of relativistic short-lived particles with the result that they are characterized by two 4-vectors: in addition to the familiar energy-momentum vector (timelike) there is an energy-momentum `spread vector' (spacelike). The wave functions in space and time for unstable particles are constructed. For the relativistic properties of unstable states we refer to Wigner's method of Poincare group representations that are induced by representations of the space-time translation and rotation groups. If stable particles, unstable particles and resonances are treated as elementary objects that are not fundamentally different one has to take into account that they will not generally be orthogonal to each other in their state space. The scalar product between a stable and an unstable state with otherwise identical properties is calculated in a particular Lorentz frame. The spin of an unstable particle is not infinitely sharp but has a `spin spread' giving rise to `spin neighbors'. This opens the possibility of a non-zero scalar product between states with unequal spin. - A first practical application of non-orthogonal states is seen in diffraction dissociation reactions whose large cross-sections are attributed to interference of states that are `partially identical'.