A phase cell approach to Yang-Mills theory

Abstract

Variables are chosen to describe the continuum Yang-Mills fields, a discrete set of group valued variables. These are group elements associated to the sequence of lattice field theory configurations realizing the continuum field: The field is "laid down" inductively: At each inductive step one of three types of "field excitations" makes its contribution to the total field. These are either "pure modes", "averaging correction modes", or "chunks". The pure modes are small field excitations, as studied in previous papers in this series [2, 3]. The averaging correction modes are small excitations added to make sure the block spin transformation is satisfied at each edge: The chunks, encompassing most of our difficulties, are large field excitations, Topological obstructions in n3(G) must be dealt with in defining a gauge choice for each chunk. The laying down process is complex, but fiendishly clever, ensuring a principle of "gauge invariant coupling": Each group valued variable is either the "amplitude" of a pure mode or an "internal variable" in a chunk, The amplitude of an averaging correction mode is a dependent variable, a function of the (independent) variables used to describe the field. The (independent) variables herein defined are those whose mutual interaction will later be inductively decoupled in defining the phase celt cluster expansion (of course treating the variables of each chunk as a unit).

The leading term of the Yang–Mills free energy

“…Ultraviolet stability of three-and four-dimensional non-Abelian lattice gauge theories by phase cell renormalization, as outlined above, was famously established by Ba laban (1983, 1984a,b,c,d, 1985a,b,c,d,e, 1987, 1988, 1989a,b) in a long series of papers spanning six years. A somewhat different approach, again using phase cell renormalization, was pursued by Federbush (1986Federbush ( , 1987aFederbush ( ,b, 1988Federbush ( , 1990 and Federbush and Williamson (1987).…”

Scale-Dependent Functions, Stochastic Quantization and Renormalization

“…As it is seen from (23), the local interaction term φ n becomes nonlocal after the application of wavelet transform (21). Of course, since the wavelet transform provides the partition of a unity, the integration over all scale arguments in the theory (22) drives it back to the original field theory with polynomial interactions, which is UV divergent.…”

“…It have been understood then that different subgrids involved in the wavelet expansion (25) represent different independent degrees of freedom [21], rather than being just different approximation of the same field like that in Kadanoff block expansion. The corresponding diagram technique, called "wavelet diagrams" and originally proposed by Federbush, can be found in Battle's book [19].…”

Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

“…Divergence free wavelets were first developed in the 1980s for fourdimensional spacetime [11,12,13]. In the 1990s, several compactly supported, divergence-free wavelets for Euclidean space were proposed [20,2,3,30].…”