Renormalization and Statistical Mechanics in Many-Particle Systems. I. Hamiltonian Perturbation Method

A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\theta }_i}^{†}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\theta }_i}^{†}]={\omega }_i{{\theta }_i}^{†}$, where $\mathcal{H}$ is the Hamiltonian and ${\omega }_i$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order $m$, and also to satisfy the appropriate boson commutators and fermion anticommutators to order $m$, then the problem is solved to order $m$. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of $\mathcal{H}$ and are orthonormal, all to order $m$. A procedure is otained for finding the ($m+1\mathrm{}$)-order contributions to the particle creation energies, in terms of the $m$-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schrödinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.