A New Method of Quantization of Classical Solutions

Recently new equations for vacuum correlators in the φ theory and in gluodynamics were derived via stochastic quantization method [2]. These equations, alternative to the Schwinger–Dyson and Makeenko–Migdal equations, allow one to connect correlators with different number of fields, but, in contrast to the Schwinger–Dyson equations, they are gauge–invariant in the case of gauge theories, and their mathematical structure is simpler than the structure of Makeenko–Migdal equations. In this letter we shall apply this approach to derivation of a set of equations for correlators of quantum fluctuations around the classical solution (lipaton) [3] in the massless λφ theory. To this end we solve the Langevin equation for quantum fluctuation through Feynman– Schwinger path integral representation, and, introducing the generating functional Z[J ], defined by the formula (4), apply to it cumulant expansion [4,5]. The latter in this case has pure perturbative meaning and corresponds to the usual semiclassical method of quantization of classical solutions, suggested in [6] and developed in [7], providing expansion in the powers of coupling constant (or in the powers of Planck constant). The generating functional Z[0], where one neglects for simplicity the second term in the exponent in (3), has a meaning of a one–loop expression for the effective potential, generated by quantum fluctuations. This analogy becomes clear after applying to Z[0] cumulant expansion: an n–th term of cumulant expansion is an n–point Green function with n lipatonic insertions. After that we expand the obtained system of bilocal approximation in the two lowest orders of perturbation theory and solve the obtained equations in the case, when the lipaton size is large enough (the particular meanings of this approximation for each of the equations