Large-N Expansion as Semiclassical Approximation to the Third-Quantized Theory

TL;DR

This paper develops a third-quantized semiclassical framework for large-N field models, enabling the analysis of the spectrum and solutions, including new energy levels, by extending the analogy of second quantization.

Contribution

It introduces a novel third-quantized semiclassical approach for large-N theories, allowing for the construction of solutions and spectrum analysis beyond traditional methods.

Findings

Exact solutions for the third-quantized equations are constructed.

The approach reveals both known and new energy levels in the spectrum.

The method provides a new perspective on the large-N spectrum analysis.

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 Schrodinger large-N equation is presented in a third-quantized form. The third-quantized creation and annihilation operators depend on the field $\phi({\bf x})$. If the coefficient of the $\phi^4$-term is of order 1/N (this is a 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 classical equation of motion is an equation on the functional $\Phi[\phi(\cdot)]$. This equation being a nonlinear analog of the functional Schrodinger equation for the one-field theory is investigated. The exact solutions are constructed and the renormalization problem is analysed. 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 obtained by known methods. We show that not only known but also new energy levels can be found.