To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

This review discusses instabilities of the Fermi-liquid state of conduction electrons in metals with particular emphasis on magnetic quantum critical points. Both the existing theoretical concepts and experimental data on selected materials are presented; with the aim of assessing the validity of presently available theory. After briefly recalling the fundamentals of Fermi-liquid theory, the local Fermi-liquid state in quantum impurity models and their lattice versions is described. Next, the scaling concepts applicable to quantum phase transitions are presented. The Hertz-Millis-Moriya theory of quantum phase transitions is described in detail. The breakdown of the latter is analyzed in several examples. In the final part experimental data on heavy-fermion materials and transition-metal alloys are reviewed and confronted with existing theory.

http://export.arxiv.org/pdf/cond-mat/0606317

Fermi-liquid instabilities at magnetic quantum phase transitions

The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A $d=1$ warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in $d=2 \mathrm{and} 3$, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function $F$, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a $\frac{1}{N}$ expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in $d=2$ using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.

Renormalization-group approach to interacting fermions

The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. The evidence from symmetry truncations and from the truncated flow of the effective average action is presented in detail. A dimensional reduction phenomenon for the residual interactions in the extreme ultraviolet links both results. For practical reasons the background effective action is used as the central object in the quantum theory. In terms of it criteria for a continuum limit are formulated and the notion of a background geometry self-consistently determined by the quantum dynamics is presented. Self-contained appendices provide prerequisites on the background effective action, the effective average action, and their respective renormalization flows.

/pdf/the-asymptotic-safety-scenario-in-quantum-gravity-a2o8tsfgoi.pdf

The Asymptotic Safety Scenario in Quantum Gravity

When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Ei…

Bose-Einstein condensation

The Euclidean massive Gross-Neveu model in two dimensions is just renormalizable and asymptotically free. Thanks to the Pauli principle, bare perturbation theory with an ultra-violet cut-off (and the correct ansatz for the bare mass) is convergent in a disk, whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant. Therefore, the theory can be fully constructed in a perturbative way. It satisfies the O.S. axioms and is the Borel sum of the renormalized perturbation expansion of the model

/pdf/a-renormalizable-field-theory-the-massive-gross-neveu-model-nty6tjkw02.pdf

A renormalizable field theory: The massive Gross-Neveu model in two dimensions

We construct the thermodynamic limit of the critical (massless) φ4 model in 4 dimensions with an ultraviolet cutoff by means of a “partly renormalized” phase space expansion. This expansion requires in a natural way the introduction of effective or “running” constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial role. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.

/pdf/construction-and-borel-summability-of-infrared-ph-4-4-by-a-2bib9229vx.pdf

Construction and Borel summability of infrared Φ 4 4 by a phase space expansion

We prove the convergence of a simplified cluster/Mayer expansion at one energy scale for three space-time dimensional many Fermion systems. The bounds are uniform in the scale. We iterate them to show that the sum of all diagrams that contain no two or four-legged subdiagrams converges. Our results are suited to a multiscale construction of the full system

/pdf/an-infinite-volume-expansion-for-many-fermion-green-s-1vn8d9q08z.pdf

AN infinite volume expansion for many Fermion Green's functions

Perturbation theory for many fermion systems

In a series of ten papers (see the flow chart at the end of §I), of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many–fermion models in two space dimensions have nonzero radius of convergence. The models have ‘‘asymmetric’’ Fermi surfaces and short range interactions. One consequence of the convergence of the perturbation expansions is the existence of a discontinuity in the particle number density at the Fermi surface. Here, we present a self contained formulation of our main results and give an overview of the methods used to prove them.

/pdf/a-two-dimensional-fermi-liquid-part-1-overview-3ptrml8c50.pdf

Joel Feldman

Papers

A renormalizable field theory: The massive Gross-Neveu model in two dimensions

Construction and Borel summability of infrared Φ 4 4 by a phase space expansion

AN infinite volume expansion for many Fermion Green's functions

Perturbation theory for many fermion systems

A Two Dimensional Fermi Liquid. Part 1: Overview