Elements of Advanced Quantum Theory [ペーパーバック]

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.