A Novel Variational Approach for Quantum Field Theory: Example of Study of the Ground State and Phase Transition in Nonlinear Sigma Model

We discuss a novel form of the variational approach in Quantum Field Theory in which the trial quantum configuration is represented directly in terms of relevant expectation values rather than, e.g., increasingly complicated structure from Fock space. The quantum algebra imposes constraints on such expectation values so that the variational problem is formulated here as an optimization under constraints. As an example of application of such approach we consider the study of ground state and critical properties in a variant of nonlinear sigma model. The variational approach is one of the cornerstones of nonperturbative methods in Quantum Mechanics and Quantum Field Theory. In this approach, the expectation value of the Hamiltonian is analyzed on a set of quantum configurations of specific form and its minimum is sought. Variational Method takes roots in Ritz Theorem [1] which states that for a hermitian Hamiltonian operator with the spectrum bounded from below H = Φ|H|Φ Φ|Φ ≥ E 0 , (1) for any quantum state Φ. Here E 0 is the lowest eigenvalue of H. Ritz theorem can be transparently motivated using the fact that hermitian operator should have a complete set of eigenstates, i.e. an arbitrary quantum state can be represented as a linear superposition of the Hamiltonian eigenstates. Then, after doing a simple algebra one can get