Abstract
Gauge theories possess nonlocal features that, in the presence of boundaries, inevitably lead to subtleties. We employ geometric methods rooted in the functional geometry of the phase space of Yang-Mills theories to: (1) characterize a basis for quasilocal degrees of freedom (dof) that is manifestly gauge-covariant also at the boundary; (2) tame the non-additivity of the regional symplectic forms upon the gluing of regions; and to (3) discuss gauge and global charges in both Abelian and non-Abelian theories from a geometric perspective. Naturally, our analysis leads to splitting the Yang-Mills dof into Coulombic and radiative. Coulombic dof enter the Gauss constraint and are dependent on extra boundary data (the electric flux); radiative dof are unconstrained and independent. The inevitable non-locality of this split is identified as the source of the symplectic non-additivity, i.e. of the appearance of new dof upon the gluing of regions. Remarkably, these new dof are fully determined by the regional radiative dof only. Finally, a direct link is drawn between this split and Dirac's dressed electron.
Highlights
In this article we limit ourselves to laying down some general considerations on the non-Abelian case and leave the detailed analysis of the symplectic geometry associated to these charges to future work
As in the previous section, vectors fields of this form are called vertical. Through their span they locally define an integral distribution V ⊂ TT∗A, and a foliation Fof T∗A, which identifies the pure-gauge directions in phase space
In [55] this circumstance is interpreted—and we agree—as meaning that a gauge symmetry that can be “neutralized” in this way is non-substantial. This is the case e.g. when the gauge symmetry is introduced through a so-called Stückelberg trick, but it is the case for the Lorentz gauge symmetry in tetrad gravity and, with certain subtleties [19, Sect. 9], in the presence of spontaneous symmetry breaking
Summary
Physical degrees of freedom in gauge theories cannot be completely localized, since gaugeinvariant quantities have a certain degree of nonlocality; the prototypical example being a Wilson line. This means that the physical relevance of global charges in the non-Abelian theories is less clear (fluctuations that are not fine-tuned generically break the global symmetry under study), and that an extension of our geometric formalism that encompasses non-Abelian reducible configurations would require substantially more work For these reasons, in this article we limit ourselves to laying down some general considerations on the non-Abelian case and leave the detailed analysis of the symplectic geometry associated to these charges to future work. At reducible configurations, and in the presence of matter, gluing is ambiguous due to the presence of the non-trivial global symmetries analyzed in (2) This is relevant in the Abelian case, where the ambiguity is related to the total regional electric charge. A list of symbols can be found in appendix C
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