A Framework for Systematic Study of QCD Vacuum Structure I: Kolmogorov Entropy and the Principle of Chiral Ordering

In this series of articles we describe a systematic approach for studying QCD vacuum structure using the methods of lattice gauge theory. Our framework incorporates four major components. (i) The recently established existence of space–time order at all scales (fundamental structure) observed directly in typical configurations of regularized path–integral ensembles. (ii) The notion of scale–dependent vacuum structure (effective structure) providing the means for representing and quantifying the influence of fluctuations at various scales on physical observables (phenomena). (iii) The unified description of gauge and fermionic aspects of the theory which facilitates a high level of space–time order in the path–integral ensembles. (iv) The strict “Bottom–Up” approach wherein the process of identifying the vacuum structure proceeds inductively, using the information from valid lattice QCD ensembles as the only input. In this work we first elaborate on the meaning of the notion of space–time order in a given configuration which is conceptually at the heart of the path–integral approach to vacuum structure. It is argued that the algorithmic complexity of binary strings associated with coarse-grained descriptions of the configuration provides a relevant quantitative measure. The corresponding ensemble averages define the ranking of different lattice theories at given cutoff by the degree of space–time order generated via their dynamics. We then introduce the set of local transformations of a configuration, chiral orderings, in which the transformed gauge connection represents an effective matrix phase acquired by chiral fermion when hopping over a given link. It is proposed that chiral orderings facilitate the evolution in the set of actions which increases the degree of space–time order while preserving the physical content of the theory, and should thus be used in the search for the fundamental QCD vacuum structure. The relation to renormalization group ideas is discussed, and the first step in general formulation of effective lattice QCD realizing the notion of scale–dependent vacuum structure is given.