Abstract
Recently, based on a new procedure to quantize the theory in the continuum, it was argued that Singer's theorem points towards the existence of a Yang-Mills ensemble. In the new approach, the gauge fields are mapped into an auxiliary field space used to separately fix the gauge on different sectors labeled by center vortices. In this work, we study this procedure in more detail. We provide examples of configurations belonging to sectors labeled by center vortices and discuss the existence of nonabelian degrees of freedom. Then, we discuss the importance of the mapping injectivity, and show that this property holds infinitesimally for typical configurations of the vortex-free sector and for the simplest example in the one-vortex sector. Finally, we show that these configurations are free from Gribov copies.
Highlights
After Singer’s theorem [1], it became clear that the usual Faddeev-Popov procedure to quantize non-Abelian YangMills theories must be somehow modified in the nonperturbative regime
The gauge fields are mapped into an auxiliary field space used to initially determine sectors labeled by center vortices, and separately fix the gauge on them
A condition can be obtained by first considering the minimization with respect to G 1⁄4 eiθ, with infinitesimal θ, and fixed Σ: 1⁄2∂μ þ aΣμ ; Aμ − ∂μaΣμ 1⁄4 0: ð13Þ. If this step were free from Gribov copies, we would have a unique gauge field AΣ that satisfies Eq (13), and the continuum maximal center gauge would be completed by determining the best Σ: Z
Summary
After Singer’s theorem [1], it became clear that the usual Faddeev-Popov procedure to quantize non-Abelian YangMills theories must be somehow modified in the nonperturbative regime. Because of a topological obstruction, there is no condition gðAÞ 1⁄4 0 that can globally fix the gauge on the whole configuration space fAμg. When such condition is imposed, the path integral still contains redundant degrees of freedom (d.o.f) associated with gauge fields obeying gðAÞ 1⁄4 0 and related by nontrivial gauge transformations. Such spurious configurations are typically called Gribov copies. [17], a different procedure to deal with Singer’s obstruction was introduced, by splitting the configuration space into domains θα ⊂ fAμg where local sections are well defined. The important point is that, in order for these regions to serve as a basis to implement the new proposal, they must form a partition fAμg 1⁄4 ∪αθα; θα ∩ θβ 1⁄4 ∅ if α ≠ β: ð1Þ
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