Interacting Electron–Hole–Phonon System

Chapter 11 employs variational differential techniques and the Schwinger Action Principle to derive coupled-field Green’s function equations for a multi-component system, modeled as an interacting electron-hole-phonon system. The coupled Fermion Green’s function equations involve five interactions (electron-electron, hole-hole, electron-hole, electron-phonon, and hole-phonon). Starting with quantum Hamilton equations of motion for the various electron/hole creation/annihilation operators and their nonequilibrium average/expectation values, variational differentiation with respect to particle sources leads to a chain of coupled Green’s function equations involving differing species of Green’s functions. For example, the 1-electron Green’s function equation is coupled to the 2-electron Green’s function (as earlier), also to the 1-electron/1-hole Green’s function, and to the Green’s function for 1-electron propagation influenced by a nontrivial phonon field. Similar remarks apply to the 1-hole Green’s function equation, and all others. Higher order Green’s function equations are derived by further variational differentiation with respect to sources, yielding additional couplings. Chapter 11 also introduces the 1-phonon Green’s function, emphasizing the role of electron coupling in phonon propagation, leading to dynamic, nonlocal electron screening of the phonon spectrum and hybridization of the ion and electron plasmons, a Bohm-Staver phonon mode, and the Kohn anomaly. Furthermore, the single-electron Green’s function with only phonon coupling can be rewritten, as usual, coupled to the 2-electron Green’s function with an effective time-dependent electron-electron interaction potential mediated by the 1-phonon Green’s function, leading to the polaron as an electron propagating jointly with its induced lattice polarization. An alternative formulation of the coupled Green’s function equations for the electron-hole-phonon model is applied in the development of a generalized shielded potential approximation, analysing its inverse dielectric screening response function and associated hybridized collective modes. A brief discussion of the (theoretical) origin of the exciton-plasmon interaction follows.

Schwinger Action Principle and Variational Calculus

Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.

RENORMALIZATION AND THE ACTION PRINCIPLE

An action principle technique provides an unusual perspective on the infrared problem in the effective action for gauge field theories. It is shown by means of a dynamical analogy that the renormalization group and the ansatz of nonlocal sources can be simultaneously presented through generalized variations of an action supplemented by sources in the manner of the Schwinger action principle. Indiscriminate resummations of the effective potential often lead to erroneous conclusions about phase transitions in a gauge theory if they resum matter self-energies at the expense of the gauge sector. The action principle method illuminates the reason for this and shows a way of proceeding, without having to go to nonzero momentum. Some examples are computed to the lowest order, reproducing results previously obtained through the renormalization group, and a new example is computed in Yang–Mills theory. The dynamical analogy leads to a comparison with a class of phenomenological nonequilibrium Lagrangians in which time-dependent couplings are used to model the influence of an external system. These models and some of their implications are discussed in terms of the action principle.

BRST–BFV QUANTIZATION AND THE SCHWINGER ACTION PRINCIPLE

We introduce an operator version of the BRST–BFV effective action for arbitrary systems with first class constraints. Using the Schwinger action principle we calculate the propagators corresponding to: (i) the parametrized nonrelativistic free particle, (ii) the relativistic free particle and (iii) the spinning relativistic free particle. Our calculation correctly imposes the BRST invariance at the end points. The precise use of the additional boundary terms required in the description of fermionic variables is incorporated.