Are ZN-Bubbles Really There?

Abstract

We argue that different Z N thermal vacua of hot pure Yang-Mills theory distinguished in the standard approach by different values of Polyakov loop average 〈P〉 T correspond actually to one and the same physical state. A critical discussion of the arguments, which are usually put forward in favor of the opposite conclusion (that, in pure continuum Yang-Mills theory, distinct Z N -phases may coexist in the physical space, being separated by the domain walls with finite surface energy), is given. In particular, we note that the same arguments can be applied with equal ease to abelian theories and would lead to the existence of the walls in high- T four-dimensional QED and to the appearance of the queer high- T solitons with the mass ∝ T 2/ e in the Schwinger model. We emphasize that these configurations may be relevant for the Euclidean path integral but do not correspond to real Minkowski space objects. We also discuss the lattice theories and confront the usual SU( N) version thereof with the version involving adjoint matrices ∈ SU( N)/ Z N . These two theories should coincide in the continuum limit, but the latter (in contrast to the former) has no trace of Z N -symmetry so that nothing is broken (this is especially clear in the strong coupling limit). The recent numerical lattice calculations of the wall surface energy are done too close to the strong coupling regime and are not conclusive. We note also that the phase of 〈P〉 T is not a physically measurable quantity and that the proper order parameter associated with the deconfinement phase transition is not 〈P〉 T but only the correlator 〈 P( x) P*(0)〉 T at large distances.