Elements of Advanced Quantum Theory by J. M. Ziman | (PDF) Free Download

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This is a textbook of advanced quantum theory for graduate students and research workers which gives a connected mathematical derivation of the important results, concentrating on the central ideas without burdening the exposition with elaborate detail or unnecessary rigour, and explains, in the simplest possible terms, the symbols and concepts which frequently confront the active research worker in solid state, nuclear and high-energy physics, and in theoretical chemistry. Professor Ziman brings to his task the sympathetic guidance of a lecturer who has not forgotten the difficulties that he himself had to surmount in mastering his subject.

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⭐Before I commence, a few words from the text of Zagoskin, Quantum Theory of Many-Body Systems (Springer, 1998): ” Excellent introduction into the very heart of the method of Green’s functions.” (Page 220, his reference list). Thus, the book of Ziman (first printing 1969) is still recommended, as Zagoskin has thoughtfully done ! In fact, Ziman makes excellent prelude to the more advanced textbook of Zagoskin.A few things which should be kept uppermost in mind: this is a more heuristic book, lacking exercises for the reader. As compensation, many derivations are left for the reader ( One example: “…the corresponding anticommutation relations for the Dirac matrices may be laboriously worked out…” see Page 195, or, “…this can be constructed by contour integration.”)With that caveat, this text provides firm physical foundation between elementary and advanced accounts: straddling the fence between Griffith’s Introduction to Quantum Mechanics and Sakurai, Advanced Quantum Mechanics. As author writes, Dirac’s Principles of Quantum Mechanics (third edition) remains primary prerequisite to this offering.Let us read a few words from the author:(1) “…although one can write down elementary perturbation expressions based upon the interaction between the fields, these often lead to infinite, uninterpretable results.” (Page 48). This, chapter two–a fine one detailing Fermions.(2) “…a propagator also has a deeper significance as a Green’s function of the corresponding field.” (Page 83) and, “…it must be emphasized that a Feynman diagram represents only a succession of virtual processes” (Page 85); “We can often arrive at many of the results of diagrammatic analysis by direct self-consistency requirements”(Page 93). It is here, Third Chapter, where Diagrammatic techniques are presented and discussed from a qualitative perspective. If familiar with the book of Mattuck, you will most likely enjoy this presentation. Physical processes take precedence over mathematical formalism. No equation herein presented without a masterful elucidation of its physical construct.(3) “The virtue of the density matrix lies in its symbolic mathematical invariance, not in any practical arithmetical simplicity.” Reading, as we learn of ‘Green’ Functions and density matrix, in fourth chapter: “… allows one to unify many apparently diverse aspects, scattering,ensemble averages, bound states, excitation spectra…” Bethe-Salpeter Equation here, too, justified. Green and Poisson–their connections elaborated upon in fine form (Pages 117-120). Mathematical companions along this route: contour integration, series summations, chain rule, Fourier transforms.(4) Chapter Five–Physics at the fore–placing the previous four chapters into perspective, pressed into service of physical systems:Dielectric screening, Fermi Liquids, Bose gas, superconductivity–to name a few of those physical systems.(Compare to the second chapter of Zagoskin; Ziman is elementary, Zagoskin is advanced; Zagoskin is terse where Ziman is verbose.)(5) Relativistic considerations: Chapter Six. And, what a pleasant surprise. First, note: no pesky “ict” ! Also, note: the vector cross product is symbolized by a wedge (not an ‘x’); our author utilizes metric tensor with diagonal components (1,1,1,-1). Equation # 6.46 ( Page 184), boson propagator, makes its entrance. Reading,” the reader may verify for himself, as an instructive exercise, that equation # 6.46 is exactly equivalent to # 3.115 ” ( Page 185). Spinors and Dirac Equations, via Pauli, are introduced. We read: “Dirac matrices, being fourth roots of unity, appear in pure mathematics as generators of a hypercomplex algebra.”(6) The final chapter introduces (in elementary, rudimentary fashion) the methods and language of Group Theory. Read: “…the full power of group theory is seen when we go from simple systems to the study of composite systems.” (Page 230). The chapter makes excellent preliminary to Wigner’s book, Group Theory and Applications to Quantum Mechanics.(7) And, thus culminates an interesting foray into advanced quantum theory. While not a full-fledged textbook, it best serves as collateral to the more technical offerings. Also, one needs to be aware of typographic slips. Zimon writes with pleasant discursiveness: providing physical elaboration of mathematical formulations. Nothing introduced mathematically which remains unaccompanied by sufficient physical prose.Physics is the primary focus. An excellent preliminary to advanced presentations.

⭐For the reader interested in a modern introduction to quantum field theory using the latest mathematical tools and one that will take one to the frontiers of research, this would not be an appropriate book to begin from. One might describe it as “the old quantum field theory”, as it approaches the subject from the standpoint of what was being done in the sixties and seventies. That is not to say however that it could not be used by someone interested in going into the field of condensed matter physics for example. The many-body quantum physics used in that field is detailed very effectively in this book. Readers who are interesting in high energy physics though should perhaps select another book. Some of the more unique and interesting discussions in this book that are still relevant today include: 1. The quantization of continuous fields and the treatment of the Rayleigh scattering of phonons. Here one is introducing a point mass into a continuous medium and asking for its effect on the phonon field. The familiar Rayleigh scattering formula is derived, and the author points out that for scattering between modes containing many particles, the transition rate also depends on the state of occupation of the mode into which a phonon is going, which is the familiar stimulated emission. Replacing the point mass by an extended object, such as a grain boundary, and attempting to solve for the phonon scattering is non-trivial and has been the subject of much research. 2. The fermion-boson interaction and the origin of the concept of a “polaron”. This arises in the consideration of the interaction of an electron with the optical modes in a polar crystal. The author calculates the self-energy of the fermion in the boson field, and shows it leads to a correction of the relationship between the energy and momentum of the electron, giving the electron an “effective mass”. The effective mass is dependent on the mass of the electron and the effective dielectric constant. A polaron is then this “dressed” electron which is “more massive” than the electron because of the electron’s interaction with the optical modes. Also, in the context of perturbation theory and the S-matrix, the author eliminates the term in the fermion-boson interaction in order to study purely the properties of the fermion field. This means that the interaction Hamiltonian operates only on the vacuum state for bosons, and thus only excitations of single bosons into and out of the vacuum are considered. This results in an effective interaction between the fermions, due to the exchange of bosons, and this interaction can be attractive or repulsive, depending on the range of momenta. This effective interaction between electrons due to the exchange of virtual phonons is the explanation for superconductivity. The fermion-boson interaction is still of considerable interest in the context of explanations for high-temperature superconductivity. 3. The derivation of the Kubo formula as a first crack at the formulation of transport theory in the quantum realm. The author explains the formula as one that shows that conductivity is an intrinsic property of quantum-mechanical systems, in that the application of a weak electron field will make apparent the time-correlations of the electric current fluctuations in equilibrium. He cautions the reader though that practical calculations may make the use of the Kubo formula problematic. The author returns to the Kubo formula later in his treatment of the spectral representation of the dielectric function, and proves a case of the famous fluctuation-dissipation theorem. A comparison between the Kubo formula shows that dissipation has been expressed in terms of Fourier transform of a two-body time-correlation function which describes the fluctuations in the many-body system. The Kubo formula and its generalizations are still discussed widely in the context of nonequilibrium statistical mechanics, quantum transport theory, and the theory of mesoscopic systems. 4. An illustration of the properties of the time-independent Green’s function via the consideration of impurity states in a medal. The author introduces a single impurity atom with delta function potential at a fixed point in the metal, and calculates the Green function of the perturbed system in terms of the unperturbed one. The resulting singularities in the Green function motivate the author to consider the role of the strength of the potential, and he shows that for a certain range of this strength, one obtains a bound state or “localized” level. 5. The treatment of the random phase approximation. The author writes the Hamiltonian for an interacting system of fermions in a way that makes the density fluctuations of various wavelengths manifest. Noting the the commutator of the density part with the Hamiltonian results in an intractable problem, he replaces the operator products by expectation values (or ensemble averages for finite temperature). This results in the off-diagonal terms cancelling one another, due to them being randomly out of phase with each other. He then proceeds to solve for the equations of motion of the system, obtaining a dispersion formula for the frequency of a self-consistent excited mode of the system, which he then views as a pole of an approximation to the inverse dielectric function. He mentions, but does not discuss in detail, what this implies for the theory of an electron gas in a metal, namely the phenomenon of dielectric screening and the existence of plasmons. 6. The brief but informative discussion of (zero-temperature) superconductivity. He accounts for the phenomenon by the use of an effective electron-electron interaction which is attractive when the energy difference of the two electron states is small. This interaction is modeled by a small negative constant for momentum transfers between these types of electrons, and zero otherwise. A perturbation calculation then shows that the effect of this interaction is infinite for any pair of electrons with exactly opposite momenta, and thus one obtains a bound state, the famous Cooper pair. The author then goes on to show the existence of an energy gap for the system, thus showing that a superconducting system does not have excitations of vanishingly small energy.

⭐Ziman’s book addresses a number of topics associated with quantum field theory, including Feynman diagrams. It is not these topics that interested me in this book, however, as there are other books that address the same topics in QFT. My favorite part of this book was Ziman’s discussion of the density matrix (sections 4.1-4.3, pp.94-101)–especially that he placed it in terms of the integral for the expectation value rather than solely in terms of bra/ket notation. I also liked the section (6.1) on relativistic kinematics that included Mandelstam diagrams (pp.203-207).

⭐多体系の量子論の本です。この訳者が訳している本は良書が多く，本書も旧装版と，誤字が多かった記憶があるのでこの新装版と持っています。良い点は記述が，原著者自身の言葉によって行われていることだと思います。ただし，その代償として，項目によって，記述の細かさに若干ムラが見られます。最初この本1冊で読破しようとしましたが，基礎知識が追いつかず一度戦略的撤退をした記憶があります。一冊根気よく読めば理解できる系統の本ではないのですが，教科書的な本と並行利用するには良い本です。

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Elements of Advanced Quantum Theory PDF Free Download
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Elements of Advanced Quantum Theory PDF Free Download
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