Decomposition of State Spaces into Subobjects in Quantum Field Theory

TL;DR

The paper develops a comprehensive framework to decompose a quantum field theory's state space into entangled subobjects and to analyze how projecting some subobjects onto background backgrounds yields parameterized effective fields. It provides two parallel formalisms—one based on saddle-point projections of action functionals and another based on projecting onto lowest eigenstates of operators—both yielding slices of the state space labeled by degeneracy eigenvalues. By averaging transitions between slices and incorporating cloud-like parameter spaces, the work derives a dynamics-like structure on an effectively background-independent theory, with a geometry that depends on both the remaining fields and the projected background. The approach culminates in a multi-part, scalable formalism that generalizes to many subobjects and decompositions, culminating in a field-theory-on-a-fibred-state-space picture where transitions between theories correspond to variations in global states. The framework also connects to more conventional field-theory language via a fields formulation, operator perspectives, and a projective-invariance generalization, illustrating potential paths to dynamical descriptions emerging from static parameterized geometry.

Abstract

This paper introduces a comprehensive formalism for decomposing the state space of a quantum field into several entangled subobjects, i.e., fields generating a subspace of states. Projecting some of the subobjects onto degenerate background states reduces the system to an effective field theory depending on parameters representing the degeneracies. Notably, these parameters are not exogenous. The entanglement among subobjects in the initial system manifests as an interrelation between parameters and non-projected subobjects. Untangling this dependency necessitates imposing linear first-order equations on the effective field. The geometric characteristics of the parameter spaces depend on both the effective field and the background of the projected subobjects. The system, governed by arbitrary variables, has no dynamics, but the projection of some subobjects can be interpreted as slicing the original state space according to the lowest eigenvalues of a parameter-dependent family of operators. The slices can be endowed with amplitudes similar to some transitions between each other, contingent upon these eigenvalues. Averaging over all possible transitions shows that the amplitudes are higher for maps with increased eigenvalue than for maps with decreasing eigenvalue.