New observables in topological instantonic field theories

TL;DR

This paper introduces novel geometric observables in topological instantonic quantum mechanics that enable trajectory jumps and can be used to compute linking numbers, expanding the toolkit for topological invariants.

Contribution

It presents new 'arbitrary jump' observables that differ from standard evaluation observables, with explicit examples and potential applications in calculating linking numbers.

Findings

New observables allow trajectory jumps at insertion points.

These observables depend on linking of points and can compute linking numbers.

Explicit examples demonstrate their geometric and operator formalism.

Abstract

Instantonic theories are quantum field theories where all correlators are determined by integrals over the finite-dimensional space (space of generalized instantons). We consider novel geometrical observables in instantonic topological quantum mechanics that are strikingly different from standard evaluation observables. These observables allow jumps of special type of the trajectory (at the point of insertion of such observables). They do not (anti)commute with evaluation observables and raise the dimension of the space of allowed configurations, while the evaluation observables lower this dimension. We study these observables in geometric and operator formalisms. Simple examples are explicitly computed; they depend on linking of the points. The new "arbitrary jump" observables may be used to construct correlation functions computing e.g. the linking numbers of cycles, as we illustrate on Hopf fibration.