Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · hep-th/9712153

Implementation of an iterative map in the construction of (quasi)periodic instantons: chaotic aspects and discontinuous rotation numbers

A. Chakrabarti
View original source ↗Read source document ↗Export metadata ↓

Abstract

An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits. The iterations do not correspond directly to a discretized time evolution of the gauge fields. They are implemented in an indirect fashion. First, (t,r,θ,ϕ) being the standard coordinates, the (r,t) half plane is mapped on the unit disc in an appropriate way. This provides an (r,t) parametrization (Sec.1) of Z_0, the starting point of the iterations and makes the iterates increasingly complex functions of r and t. These are then incorporated as building blocks in the generating function of the fields (Sec.2). We explain (starting in Sec.1 and at different stages) in what sense and to what extent some remarkable features of our map (indicated in the title) are thus carried over into the continuous time development of the fields. Special features for quasiperiodicity are studied (Sec.3). Spinor solutions (Sec.4) and propagators (Sec.5) are discussed from the point of view of the mapping. Several possible generalizations are indicated (Sec.6). Some broader topics are discussd in conclusion (Sec.7).

Explore related subjects

hep-th

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. Chakrabarti. 1998-01-05. Implementation of an iterative map in the construction of (quasi)periodic instantons: chaotic aspects and discontinuous rotation numbers. https://doi.org/10.1063/1.532680

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Scalar, vector and tensor fields on $dS_3$ with arbitrary sources: harmonic analysis and antipodal maps

The scalar, vector and tensor spherical harmonics on three-dimensional de Sitter spacetime are defined and analyzed. Each harmonic defines two sets of asymptotic data on the two-sphere in the asymptotic expansion close to both the past and the future of de Sitter spacetime. For each case, we make explicit the antipodal relationship of both sets of asymptotic data between past and future infinity, which can be non-local. A procedure is defined to extract these asymptotic data in the presence of sources. This provides for each class of propagating field on de Sitter the relationship between two independent sets of data defined on the sphere in the asymptotic future with the corresponding data defined in the asymptotic past. We also provide several theorems on the decomposition of vectors and tensors on de Sitter such as one proving that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse traceless tensor. These results are instrumental in the description of interacting four-dimensional asymptotically flat fields at spatial infinity.

hep-th↗
arXiv2026

Physical Predictions in Closed Quantum Gravity

Recent developments in gravitational path integrals indicate that the nonperturbative physical Hilbert space of a closed universe is one-dimensional within each superselection sector. This raises a basic puzzle: how can a unique quantum-gravity state give rise to semiclassical physics, measurement outcomes, and classical probabilities? In this paper, we develop a framework in which nontrivial and statistically stable predictions emerge despite the one-dimensionality of the fully constrained Hilbert space. The key idea is to extract physical predictions in an enlarged, unconstrained Hilbert space by conditioning on observational data. We show that partial observability -- reflecting the limited access of observers to the degrees of freedom of the universe -- suppresses ensemble fluctuations associated with microscopic structure in the gravitational path integral, thereby restoring semiclassical predictability with exponential accuracy. We formulate the construction explicitly including contributions from the Hartle--Hawking no-boundary state, define a gauge-invariant Hilbert space for observations via a density operator, and generalize the formalism to conditioning on histories, clarifying the emergence of classical probabilities and an effective arrow of time. Finally, we explore whether this framework can support a realistic cosmology and identify assumptions that the underlying theory of quantum gravity must satisfy.

hep-th↗
arXiv2026

de Sitter extremal surfaces, time contours, complexifications and pseudo-entropies

We study no-boundary de Sitter extremal surfaces and their pseudo-entropy areas for generic subregions at the future boundary, building on previous work. For large subregions, timelike+Euclidean extremal surfaces exist with transparent geometric interpretations, as do complex ones. The situation for small subregions is analogous to Poincare $dS$ and only complex extremal surfaces exist. In general, the extremal surface area integrals are defined via time contours in the complex time plane. We find multiple extremal surfaces with indistinguishable areas whose time contours can be deformed into each other in the complex time plane without obstruction, which are equivalent for these purposes. This also suggests equivalences between complex $dS$ replica geometries. We discuss $dS_3$ as a simple example at length. This suggests a picture for multiple subregions and entropy inequalities in de Sitter, as encoding $AdS$ ones via analytic continuation. We also discuss mapping future boundary subregions and those on constant time slices in the static patch via lightrays.

hep-th↗