Large-Nexpansion as a semiclassical approximation to the third-quantized theory

Abstract

The semiclassical theory for the large-N field models is developed from an unusual point of view. Analogously to the procedure of the second quantization in quantum mechanics, the functional Schr\odinger large-N equation is presented in a third-quantized form. The third-quantized creation and annihilation operators depend on the field \ensuremath{\varphi}(x). If the coefficient of the ${\ensuremath{\varphi}}^{4}$ term is of order $1/N$ (this is the usual condition of applicability of the $1/N$ expansion), one can rescale the third-quantized operators in such a way that their commutator will be small, while the Heisenberg equations will not contain large or small parameters. This means that the classical equation of motion is an equation on the functional \ensuremath{\Phi}[\ensuremath{\varphi}(\ensuremath{\cdot})]. This equation, being a nonlinear analogue of the functional Schr\odinger equation for the one-field theory, is investigated. The exact solutions are constructed and the renormalization problem is analyzed. We also perform a quantization procedure about found classical solutions. The corresponding semiclassical theory is a theory of a variable number of fields. The developed third-quantized semiclassical approach is applied to the problem of finding the large-N spectrum. The results are compared with formulas obtained by known methods. We show that not only the known but also new energy levels can be found.