Application of Many-Body Techniques to Canonically Quantized

Andrei Yu. Khrennikov, Sergei Vladimirovich Kozyrev

The contextual probabilistic quantization procedure is formulated. This approach to quantization has much broader field of applications, compared with the canonical quantization. The contextual probabilistic quantization procedure is based on the notions of probability context and the principle of complementarity of probabilities. The general definition of probability context is given. The principle of complementarity of probabilities, which combines the ideas of the Bohr complementarity principle and the technique of noncommutative probability, is introduced. The principle of complementarity of probabilities is the criterion of possibility of the contextual quantization.

Laurent Sanchez-Palencia

So Hirata

A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.

K. T. Hecht

Luca Salasnich

In this chapter we review the main ideas of the canonical quantization in nonrelativistic quantum mechanics and optics, which we suppose the reader learned in previous courses. We first consider the least action principle and the action functional for classical many-body systems and also for the Schrödinger wavefunction, interpreted as the complex field of a classical field theory. Then, after having analyzed the Legendre transformation from the Lagrangian to the Hamiltonian formulation, we illustrate the canonical quantization of particles, and finally we extend the approach to the canonical quantization of bosonic and fermionic fields, considering also the role of temperature.

George Rosensteel