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1

Shishmarev, I. A. (Ilʹi͡a︡ Andreevich)., ed. Nonlinear nonlocal equations in the theory of waves. Providence, R.I: American Mathematical Society, 1994.

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2

E, Zorumski W., Watson Willie R, and Langley Research Center, eds. Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1995.

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3

E, Zorumski W., Watson Willie R, and Langley Research Center, eds. Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1995.

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4

Nonlocal quantum field theory and stochastic quantum mechanics. Dordrecht: D. Reidel pub. Co., 1986.

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5

Nonlocal diffusion problems. Providence, R.I: American Mathematical Society, 2010.

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6

Andreu-Vaillo, Fuensanta. Nonlocal diffusion problems. Providence, R.I: American Mathematical Society, 2010.

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7

Nonlocal and abstract parabolic equations and their applications. Warszawa: Institute of Mathematics, Polish Academy of Sciences, 2009.

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8

Naumkin, P. I. Nonlinear nonlocal equations in the theory of waves. Providence, R.I: American Mathematical Society, 1994.

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9

E, Zorumski William, and Langley Research Center, eds. Periodic time-domain nonlocal nonreflecting boundary conditions for duct acoustics. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1996.

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10

E, Zorumski William, and Langley Research Center, eds. Periodic time-domain nonlocal nonreflecting boundary conditions for duct acoustics. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1996.

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11

E, Zorumski William, and Langley Research Center, eds. Periodic time-domain nonlocal nonreflecting boundary conditions for duct acoustics. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1996.

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12

Kamenskiĭ, G. A. Extrema of nonlocal functionals and boundary value problems for functional differential equations. Hauppauge, N.Y: Nova Science Publishers, 2007.

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13

1958-, Biler Piotr, Karch Grzegorz, and Nadzieja Tadeusz 1951-, eds. Nonlocal elliptic and parabolic problems: Proceedings of the conference held at Będlewo , September 12-15, 2003. Warszawa: Institute of Mathematics, Polish Academy of Sciences, 2004.

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14

Karch, Grzegorz, and Tadeusz Nadzieja. Nonlocal elliptic and parabolic problems: Proceedings of the conference held at Bedlewo, September 12-15, 2003. Warszawa, Poland: Institute of Mathematics, Polish Academy of Sciences, 2004.

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15

Ordinary differential equations: Qualitative theory. Providence, R.I: American Mathematical Society, 2010.

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16

Mashhoon, Bahram. Field Equation of Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0006.

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In extended general relativity (GR), Einstein’s field equation of GR can be expressed in terms of torsion and this leads to the teleparallel equivalent of GR, namely, GR||, which turns out to be the gauge theory of the Abelian group of spacetime translations. The structure of this theory resembles Maxwell’s electrodynamics. We use this analogy and the world function to develop a nonlocal GR|| via the introduction of a causal scalar constitutive kernel. It is possible to express the nonlocal gravitational field equation as modified Einstein’s equation. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. The field equation of NLG can be expressed as Einstein’s field equation with an extra source that has the interpretation of the effective dark matter. It is possible that the kernel of NLG, which is largely undetermined, could be derived from a more general future theory.
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17

Morawetz, Klaus. Nonlocal Collision Integral. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0013.

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The kinetic equation with the nonlocal shifts is presented as the final result on the way to derive the kinetic equation with nonlocal corrections. The exclusive dependence of the nonlocal and non-instant corrections on the scattering phase shift confirms the results from the theory of gases. With the approximation on the level of the Brueckner reaction matrix, the corresponding non-instant and nonlocal scattering integral in parallel with the classical Enskog’s equation, can be treated with Monte-Carlo simulation techniques. Neglecting the shifts, the Landau theory of quasiparticle transport appears. In this sense the presented kinetic theory unifies both approaches. An intrinsic symmetry is found from the optical theorem which allows for representing the collision integral equivalently either in particle-hole symmetric or space-time symmetric form.
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18

Mashhoon, Bahram. Linearized Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0007.

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The only known exact solution of the field equation of nonlocal gravity (NLG) is the trivial solution involving Minkowski spacetime that indicates the absence of a gravitational field. Therefore, this chapter is devoted to a thorough examination of NLG in the linear approximation beyond Minkowski spacetime. Moreover, the solutions of the linearized field equation of NLG are discussed in detail. We adopt the view that the kernel of the theory must be determined from observation. In the Newtonian regime of NLG, we recover the phenomenological Tohline-Kuhn approach to modified gravity. A simple generalization of the Kuhn kernel leads to a three-parameter modified Newtonian force law that is always attractive. Gravitational lensing is discussed. It is shown that nonlocal gravity (NLG), with a characteristic galactic lengthscale of order 1 kpc, simulates dark matter in the linear regime while preserving causality.
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19

Mashhoon, Bahram. Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.001.0001.

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A postulate of locality permeates through the special and general theories of relativity. First, Lorentz invariance is extended in a pointwise manner to actual, namely, accelerated observers in Minkowski spacetime. This hypothesis of locality is then employed crucially in Einstein’s local principle of equivalence to render observers pointwise inertial in a gravitational field. Field measurements are intrinsically nonlocal, however. To go beyond the locality postulate in Minkowski spacetime, the past history of the accelerated observer must be taken into account in accordance with the Bohr-Rosenfeld principle. The observer in general carries the memory of its past acceleration. The deep connection between inertia and gravitation suggests that gravity could be nonlocal as well and in nonlocal gravity the fading gravitational memory of past events must then be taken into account. Along this line of thought, a classical nonlocal generalization of Einstein’s theory of gravitation has recently been developed. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. A significant observational consequence of this theory is that the nonlocal aspect of gravity appears to simulate dark matter. The implications of NLG are explored in this book for gravitational lensing, gravitational radiation, the gravitational physics of the Solar System and the internal dynamics of nearby galaxies as well as clusters of galaxies. This approach is extended to nonlocal Newtonian cosmology, where the attraction of gravity fades with the expansion of the universe. Thus far only some of the consequences of NLG have been compared with observation.
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20

Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1995.

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21

National Aeronautics and Space Administration (NASA) Staff. Solution of the Three-Dimensional Helmholtz Equation with Nonlocal Boundary Conditions. Independently Published, 2018.

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22

Morawetz, Klaus. Simulations of Heavy-Ion Reactions with Nonlocal Collisions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0023.

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The scenario of heavy-ion reactions around the Fermi energy is explored. The quantum BUU equation is solved numerically with and without nonlocal corrections and the effect of nonlocal corrections on experimental values is calculated. A practical recipe is presented which allows reproducing the correct asymptotes of scattering by acting on the point of closest approach. The better description of dynamical correlations by the nonlocal kinetic equation is demonstrated by an enhancement of the high-energy part of the particle spectra and the enhancement of mid-rapidity charge distributions. The time-resolved solution shows the enhancement of neck formation. It is shown that the dissipated energy increases due to the nonlocal collision scenario which is responsible for the observed effects and not due to the enhancement of collisions. As final result, a method is presented how to incorporate the effective mass and quasiparticle renormalisation with the help of the nonlocal simulation scenario.
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23

Morawetz, Klaus. Nonequilibrium Quantum Hydrodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0015.

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The balance equations resulting from the nonlocal kinetic equation are derived. They show besides the Landau-like quasiparticle contributions explicit two-particle correlated parts which can be interpreted as molecular contributions. It looks like as if two particles form a short-living molecule. All observables like density, momentum and energy are found as a conserving system of balance equations where the correlated parts are in agreement with the forms obtained when calculating the reduced density matrix with the extended quasiparticle functional. Therefore the nonlocal kinetic equation for the quasiparticle distribution forms a consistent theory. The entropy is shown to consist also of a quasiparticle part and a correlated part. The explicit entropy gain is proved to complete the H-theorem even for nonlocal collision events. The limit of Landau theory is explored when neglecting the delay time. The rearrangement energy is found to mediate between the spectral quasiparticle energy and the Landau variational quasiparticle energy.
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24

Morawetz, Klaus. Properties of Non-Instant and Nonlocal Corrections. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0014.

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The derived nonlocal and non-instant shifts are discussed with respect to various symmetries and gauges. The classical counterparts are derived and found in agreement with the expected phenomenological ones from chapter 3. The explicit forms of the hard-sphere like offsets and the delay time in terms of the scattering phase shifts are calculated and discussed on the example of nuclear collision. The numerical results reveal an interesting inside into the microscopic correlations developed in dependence on the scattering angle and scattering energy. The just-accomplished derivation of the nonlocal scattering integrals is far from being intuitive. We have reached our task, the kinetic equation, being guided by nothing but systematic implementation of the quasiclassical approximation and the limit of small scattering rates.
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25

Morawetz, Klaus. Classical Kinetic Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0003.

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The classical non-ideal gas shows that the two original concepts of the pressure based of the motion and the forces have eventually developed into drift and dissipation contributions. Collisions of realistic particles are nonlocal and non-instant. A collision delay characterizes the effective duration of collisions, and three displacements, describe its effective non-locality. Consequently, the scattering integral of kinetic equation is nonlocal and non-instant. The non-instant and nonlocal corrections to the scattering integral directly result in the virial corrections to the equation of state. The interaction of particles via long-range potential tails is approximated by a mean field which acts as an external field. The effect of the mean field on free particles is covered by the momentum drift. The effect of the mean field on the colliding pairs causes the momentum and the energy gains which enter the scattering integral and lead to an internal mechanism of energy conversion. The entropy production is shown and the nonequilibrium hydrodynamic equations are derived. Two concepts of quasiparticle, the spectral and the variational one, are explored with the help of the virial of forces.
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26

Cardaliaguet, Pierre, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions. The Master Equation and the Convergence Problem in Mean Field Games. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691190716.001.0001.

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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.
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27

Horing, Norman J. Morgenstern. Interacting Electron–Hole–Phonon System. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0011.

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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.
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28

Morawetz, Klaus. Deep Impurities with Collision Delay. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0017.

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The linearised nonlocal kinetic equation is solved analytically for impurity scattering. The resulting response function provides the conductivity, plasma oscillation and Fermi momentum. It is found that virial corrections nearly compensate the wave-function renormalizations rendering the conductivity and plasma mode unchanged. Due to the appearance of the correlated density, the Luttinger theorem does not hold and the screening length is influenced. Explicit results are given for a typical semiconductor. Elastic scattering of electrons by impurities is the simplest but still very interesting dissipative mechanism in semiconductors. Its simplicity follows from the absence of the impurity dynamics, so that individual collisions are described by the motion of an electron in a fixed potential.
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29

Morawetz, Klaus. Elementary Principles. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0002.

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The many-body theory combines ideas of thermodynamics with ideas of mechanics. In this introductory chapter, the symbiosis of these two different fields of physics is demonstrated on overly simplified models. We explore the principles of finite-range forces to show the twofold nature of virial corrections. Infrequent collisions with a large deflection angle lead to collision integrals and rather frequent encounters with deflections on small angles act as a mean field. The (mean-field) corrections to drift result in the internal pressure and the nonlocal correction to the collisions results in the effect of the molecular volumes. The concept of distribution functions is introduced and the measure of information as entropy. The binary correlation allows one to distinguish tails and cores of the interaction potential. The concept of binary correlation is thus behind the intuitive picture of the kinetic equation.
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30

Necula, Mihai, Ioan I. Vrabie, Monica-Dana Burlică, and Daniela Roșu. Delay Differential Evolutions Subjected to Nonlocal Initial Conditions. Taylor & Francis Group, 2016.

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31

Necula, Mihai, Ioan I. Vrabie, Monica-Dana Burlică, and Daniela Roșu. Delay Differential Evolutions Subjected to Nonlocal Initial Conditions. Taylor & Francis Group, 2018.

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32

Necula, Mihai, Ioan I. Vrabie, Monica-Dana Burlică, and Daniela Roșu. Delay Differential Evolutions Subjected to Nonlocal Initial Conditions. Taylor & Francis Group, 2018.

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33

Mingione, Giuseppe, Alessio Figalli, José Antonio Carrillo, Matteo Bonforte, Gabriele Grillo, Manuel del Pino, and Juan Luis Vázquez. Nonlocal and Nonlinear Diffusions and Interactions : New Methods and Directions: Cetraro, Italy 2016. Springer, 2017.

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34

Periodic time-domain nonlocal nonreflecting boundary conditions for duct acoustics. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1996.

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35

Extrema of Nonlocal Functionals and Boundary Value Problems for Functional Differential Equations. Nova Science Pub Inc, 2007.

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36

Petrosyan, Arshak, Camelia A. Pop, and Donatella Danielli. New Developments in the Analysis of Nonlocal Operators: AMS Special Session on New Developments in the Analysis of Nonlocal Operators, October 28-30, 2016, University of St. Thomas, Minneapolis, Minnesota. American Mathematical Society, 2019.

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37

Morawetz, Klaus. Systems with Condensates and Pairing. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0012.

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The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown to correct the T-matrix from unphysical multiple-scattering events. The resulting generalised Soven scheme provides the Beliaev equations for Boson’s and the Nambu–Gorkov equations for fermions without the usage of anomalous and non-conserving propagators. This systematic theory allows calculating the fluctuations above and below the critical parameters. Gap equations and Bogoliubov–DeGennes equations are derived from this theory. Interacting Bose systems with finite temperatures are discussed with successively better approximations ranging from Bogoliubov and Popov up to corrected T-matrices. For superconductivity, the asymmetric theory leading to the corrected T-matrix allows for establishing the stability of the condensate and decides correctly about the pair-breaking mechanisms in contrast to conventional approaches. The relation between the correlated density from nonlocal kinetic theory and the density of Cooper pairs is shown.
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38

Morawetz, Klaus. Scattering on a Single Impurity. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0004.

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Evolution of a many-body system consists of permanent collisions among particles. Looking at the motion of a single particle, one can identify encounters by which a particle abruptly changes the direction of flight, these are seen as true collisions, and small-angle encounters, which in sum act as an applied force rather than randomising collisions. The scattering on impurities is used to introduce the mentioned mechanisms and, in particular, to show how they affect each other. Point impurities are assumed, i.e. impurities the potential of which is restricted to a single atomic site of the crystal lattice. In this case interaction potentials never overlap and many-body effects are due to nonlocal character of the quantum particle. To introduce elementary components of the formalism, in this chapter we first describe the interaction of an electron with a single impurity. Lippman–Schwinger equations are derived and the physics behind the collision delay, dissipativeness and optical theorems is explored.
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39

Horing, Norman J. Morgenstern. Quantum Statistical Field Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.001.0001.

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The methods of coupled quantum field theory, which had great initial success in relativistic elementary particle physics and have subsequently played a major role in the extensive development of non-relativistic quantum many-particle theory and condensed matter physics, are at the core of this book. As an introduction to the subject, this presentation is intended to facilitate delivery of the material in an easily digestible form to students at a relatively early stage of their scientific development, specifically advanced undergraduates (rather than second or third year graduate students), who are mathematically strong physics majors. The mechanism to accomplish this is the early introduction of variational calculus with particle sources and the Schwinger Action Principle, accompanied by Green’s functions, and, in addition, a brief derivation of quantum mechanical ensemble theory introducing statistical thermodynamics. Important achievements of the theory in condensed matter and quantum statistical physics are reviewed in detail to help develop research capability. These include the derivation of coupled field Green’s function equations of motion for a model electron-hole-phonon system, extensive discussions of retarded, thermodynamic and non-equilibrium Green’s functions, and their associated spectral representations and approximation procedures. Phenomenology emerging in these discussions includes quantum plasma dynamic, nonlocal screening, plasmons, polaritons, linear electromagnetic response, excitons, polarons, phonons, magnetic Landau quantization, van der Waals interactions, chemisorption, etc. Considerable attention is also given to low-dimensional and nanostructured systems, including quantum wells, wires, dots and superlattices, as well as materials having exceptional conduction properties such as superconductors, superfluids and graphene.
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