We develop a comprehensive many-body theory for systems of particles obeying quantum unified statistics, or quons. After exploring the properties of the Fock space of this system, we formulate a systematic S-matrix expansion and a generalized Wick's theorem. A consistent Green-function formalism is constructed at both zero and finite temperatures, accompanied by a generalized Wick's theorem appropriate for infinite-statistics operator algebras. Within this framework, we establish diagrammatic rules for interacting quon systems. Employing the random phase approximation, we derive the dielectric function and reveal the emergence of anomalous plasmon modes that have no direct counterpart in conventional Bose or Fermi systems. We further analyze the ground-state energy, energy-loss function, generalized Thomas-Fermi screening wave vectors, and Friedel oscillations, elucidating how infinite statistics qualitatively modifies collective behavior and screening properties.
We develop a comprehensive many-body theory for systems of particles obeying quantum unified statistics, or quons. After exploring the properties of the Fock space of this system, we formulate a systematic S-matrix expansion and a generalized Wick's theorem. A consistent Green-function formalism is constructed at both zero and finite temperatures, accompanied by a generalized Wick's theorem appropriate for infinite-statistics operator algebras. Within this framework, we establish diagrammatic rules for interacting quon systems. Employing the random phase approximation, we derive the dielectric function and reveal the emergence of anomalous plasmon modes that have no direct counterpart in conventional Bose or Fermi systems. We further analyze the ground-state energy, energy-loss function, generalized Thomas-Fermi screening wave vectors, and Friedel oscillations, elucidating how infinite statistics qualitatively modifies collective behavior and screening properties.
We present a comprehensive quantum many-body theory for q-deformed fermions and bosons, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters k and q, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick theorem and extended Feynman diagrammatics tailored to kq particles, allowing us to calculate two types of Green's functions. Explicit expressions for these Green's functions are derived in both direct and momentum spaces, providing key insights into the collective properties of q-fermionic and bosonic systems. Using a random-phase approximation, we estimate the dielectric function for q-fermion gas ($k=\ensuremath{-}1$) and analyze the Friedel oscillations, the plasmon excitations, and the energy-loss function. Our results demonstrate that the effective interaction is tuned by the value of q so that a noninteracting limit is obtained as $q\ensuremath{\rightarrow}0$, where the Friedel and the plasma oscillations disappear. An optimal value of q, namely, q∗, the plasma frequency, and the energy-loss function show an absolute maximum and the effective interaction changes behavior.
We present a comprehensive quantum many body theory for kq deformed particles, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters k and q, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick's theorem and extended Feynman diagrammatic tailored to kq-particles, allowing us to calculate two types of Green functions. Explicit expressions for these Green functions are derived in both direct and momentum spaces, providing key insights into the collective properties of kq-deformed systems. Using a random phase approximation (RPA), we estimate the dielectric function for q-fermion gas, and analyze the Friedel oscillations, the plasmon excitations, and the energy loss function. Our results demonstrate that the effective interaction is tuned by the value of q, so that a non interacting limit is obtained as q goes to zero, where the Friedel as well as the plasma oscillations disappear. There is an optimal value of q, the plasma frequency, as well as the energy loss function show an absolute maximum, and the effective interaction changes behavior.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.
Develops the many-body one-particle Green function, and explains its physical interpretation in terms of the spectral function, self-energy, and quasiparticle. lifetime. Its application in angle-resolved photoemission spectroscopy is presented in detail. The time-evolution operator in the interaction picture is derived, and time-ordering and adiabatic switching-on are introduced as precursor tools to construct the Feynman–Dyson many-body perturbation theory. A detailed account of Wick’s theorem, normal ordering, and contractions is outlined. Feynman diagrams are constructed, and the emergence of the infinite Dyson series from irreducible diagrams is outlined. Two-particle Green function and the particle-hole excitation spectra are developed. Diagramatic application of RPA for interacting systems is described. The finite-temperature Matsubara Green function is introduced and developed, together with its Fourier series representation and the evaluation of Matsubara sums.
A general fermion system interacting with a time-periodic external field is discussed in the second quantization formalism transposed in the Fock-Floquet space. The method simulates the evolution of the system with the help of an auxiliary parameter which is analog to the time and defines the vectors associated to the states of the system, such that the matrix elements in the Hilbert space have a physical correspondence. Assuming that only the principal quasi-energies have physical relevance, the ground state of the free system is constructed in the occupation-numbers representation and an analog of Wick's theorem is derived. The perturbation series of the Green's functions is subsequently obtained, and the linear response of a given observable is expressed in terms of the correlation functions. The results have formal similarities to those of the standard theory of the many-particle conservative systems.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .