Abelian Monopoles inSU(2)Lattice Gauge Theory as Physical Objects

Abstract

By numerical calculations we show that the Abelian monopole currents are locally correlated with the density of the SU͑2͒ lattice action. This fact is established for the maximal Abelian projection. Thus, in the maximal Abelian projection, the monopoles are physical objects; they carry the SU͑2͒ action. Calculations on the asymmetric lattice show that the correlation between monopole currents and the density of the SU͑2͒ lattice action also exists in the deconfinement phase of gluodynamics. [S0031-9007(97) The monopoles in the maximal Abelian projection (MaA projection) of SU͑2͒ lattice gluodynamics [1] seem to be responsible for the formation of the flux tube between the test quark-antiquark pair. The SU͑2͒ string tension is well described by the contribution of the Abelian monopole currents [2-4] which satisfy the London equation for a superconductor [5]. The study of monopole creation operators shows that the Abelian monopoles are condensed [6][7][8] in the confinement phase of gluodynamics.On the other hand, the Abelian monopoles arise in the continuum theory [9] from the singular gauge transformation, and it is not clear whether these monopoles are "real" objects. A physical object is something which carriers action, and in the present publication we only study the question of if there are any correlations between Abelian monopole currents and SU͑2͒ action. In Ref.[10] it has found that the total action of SU͑2͒ fields is correlated with the total length of the monopole currents, so there exists a global correlation. Below, we discuss the local correlations between the action density and the monopole currents.Correlators of monopole currents and density of SU͑2͒ action-The simplest quantity which reflects the correlation of the local action density and the monopole current is the relative excess of SU͑2͒ action density in the region near the monopole current. It can be defined as follows. Consider the average action S m on the plaquettes closest to the monopole current j m ͑x͒. Then the relative excess of the action iswhere S is the standard expectation value of the lattice section, S ͗͑1 2 1 2 Tr U P ͒͘. S m is defined as follows:

The Berry Phase and Monopoles in Non-Abelian Gauge Theories

“…[10]. Moreover, the excess of the action associated with monopoles (46) is numerically close to the action excess around the Abelian monopoles in the Maximal Abelian gauge [12].…”

“…Note that the asymmetry needs not to be unity at β < β crit on the asymmetric lattice since the Lorenz invariance is explicitly broken. In order to clarify the physical relevance of the monopoles constructed consider the correlation of the local SU(2) action density and the monopole currents [12]. In particular, we define the relative excess of the action density associated with the monopole as follows:…”

“…Finally, it is shown that these monopoles are locally correlated with the non-Abelian action density, like the monopoles in the Maximal Abelian gauge, Ref. [12].…”

Anatomy of the lattice magnetic monopoles

“…Actually, this kind of measurements have been reported earlier [6]. Namely, it was shown that the non-Abelian action on the plaquettes close to the monopole trajectory is larger than the average plaquette action, S. It is easy to realize, however, that if this were true at arbitrary small distances the monopoles would be strongly suppressed by the action factor and could not condense, see, e.g., [7].…”

Monopoles and Confining Strings in QCD

“…The total action of SU(2) fields is correlated with the total length of the monopole currents [35], so there exists a global correlation. The Abelian monopoles in the MaA projection are also locally correlated with the non-Abelian action density [36]. This fact shows that monopoles are some physical objects (not the artifacts of the singular gauge transformation), since by definition we call the object physical if it carries the action.…”