Field theory of strings as a collective field theory of $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ gauge fields

We develop a collective field theory of a $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ gauge field, which involves gauge-invariant operators only. We treat gauge-invariant path-ordered phase factors as collective fields on string space. The theory is formulated in such a way that in the $N\to \infty$ limit the collective field theory exactly approaches the original $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ gauge field theory. Even for finite $N$ it is expected that it provides correct excitation energy and degeneracy for low-lying states. The final form of the collective field theory is a field theory of closed strings. Although our formalism is Lorentz noncovariant, it is manifestly gauge invariant. The present paper is devoted mainly to the formalism, although a few remarks for the actual calculations are given in the final section.