Abstract
Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.
Highlights
The interplay between quantum field theory (QFT) and geometry has a long and rich history
A new and promising instance has emerged, which concerns geometrization of state preparation or representations of, primarily, the time evolution operator in systems described by quantum fields. In this approach one seeks to find a way of preparing a state or an operator of interest minimizing the use of what one regards as more primitive operators or states by relating it to a geodesic problem in an auxiliary geometry
While in general integro-differential equation (IDE) do find uses in mathematical physics [61] and, what might be interesting in light of the currently ongoing SARS-CoV-2 pandemic, in modeling the spread of diseases [62], they may not feel very natural to most general relativists and quantum field theorists
Summary
The interplay between quantum field theory (QFT) and geometry has a long and rich history. The key guiding principle behind our approach to this problem, as elucidated already in our earlier paper [21], is the requirement that the considered cost functions allow for a well-posed initial value problem between two arbitrary local conformal transformations In this way we encapsulate both the state and circuit complexity associated with local.
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