Abstract
We discuss a Polyakov loop in non-covariant operator formalism which consists of only physical degrees of freedom at finite temperature. It is pointed out that although the Polyakov loop is expressed by a Euclidean time component of gauge fields in a covariant path integral formalism, there is no direct counterpart of the Polyakov loop operator in the operator formalism because the Euclidean time component of gauge fields is not a physical degree of freedom. We show that by starting with an operator which is constructed in terms of only physical operators in the non-covariant operator formalism, the vacuum expectation value of the operator calculated by trace formula can be rewritten into a familiar form of an expectation value of Polyakov loop in a covariant path integral formalism at finite temperature for the cases of axial and Coulomb gauge.
Highlights
Gauge invariance is undoubtedly one of the fundamental principles in particle physics. Both local and non-local gauge-invariant operators play an important role in gauge theory. An example of such non-local operators is the Wilson loop, which will be given by a line integral along a rectangular contour such that one side is taken to be in a space-like direction and another side to be in a time direction
Confinement in non-Abelian gauge theories [1]. Another example of non-local gauge-invariant operators is a Polyakov loop, which is similar to a Wilson loop but is given by a line integral along a Euclidean time axis
One may wonder how one should define the operator, written in terms of only physical degrees of freedom, corresponding to the Polyakov loop operator when one considers the theory in these gauge fixings
Summary
Gauge invariance is undoubtedly one of the fundamental principles in particle physics. We will focus on β the Polyakov loop operator whose explicit form is given by tr P exp ig 0 dτ Aτ , where P denotes a path-ordered symbol, β is an inverse temperature, and Aτ is a Euclidean time component of gauge fields. Another purpose of this paper is to show that an expectation value of the above operator given in the non-covariant operator formalism can be rewritten into a familiar form of an expectation value of a Polyakov loop in a covariant path integral formalism at finite temperature. We shall use the Hamiltonian (2.12) in the discussions given below
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