A Note on Invariant Measure on the Local Gauge Group

In this paper we investigate the problem of the existence of invariant measures on the local gauge group. We prove that it is impossible to define a finite translationally invariant measure on the local gauge group C∞(Rn, G)(where G is an arbitrary matrix Lie group). Functional integral over the local gauge group is a commonly employed technique in the study of gauge field theories. Its earliest appearance in the literature is Faddeev and Popov’s paper in 1967 [1], which investigated the quantization of non-Abelian gauge fields. In [2] this technique was used to construct a projection operator which implements the Gauss law, and in [3] it was used to construct a gauge invariant ground state wave-functional. In all these applications a key point is the assumption that we can define a translationally invariant measure on the local gauge group. It is known that on locally compact groups this is possible [4]. On non-locally-compact groups we cannot guarantee such a measure exists. In the case of vector spaces a standard result [5] says that an invariant measure exists only when the vector space is finite dimensional, in which case it is locally compact. The local gauge group is infinite dimensional and not locally compact. Since functional integral over the local gauge group is an important technique the existence of an invariant measure deserves a careful study. In [6] it is proved that on the local gauge group C(R, G) there exists no finite translationally invariant Borel measure(Haar measure). In this paper we study the case of the local gauge group C(R, G) (where G is an arbitary matrix Lie group) and prove that it is impossible to define a finite translationally invariant Borel measure on it (provided that it is endowed with a suitable topology). The local gauge group G0 = C (R, G) is the group of all smooth mappings from R to G. For a generic element ω(x, · · ·x) of G0, the quantity ω (0)∂1ω(0)(here by ∂1ω(0) we certainly mean ∂ ∂x ω(x)|x=0) belongs to the Lie algebra g of G. For ξ = ∑m i=1 uiξi ∈ g(where {ξ1 · · · ξm} is a basis of g) we define f(ξ) = e − ∑ m i=1 u i and consider the following bounded continuous function of ω (in some suitable topology of the group G0): F [ω] = f(ω(0)∂1ω(0)) (1) e-mail address: [email protected] e-mail address: [email protected]