Imaginary-time method in quantum mechanics and field theory

Providing a new perspective on quantum field theory, this book gives a pedagogical and up-to-date exposition of non-perturbative methods in relativistic quantum field theory and introduces the reader to modern research work in theoretical physics. It describes in detail non-perturbative methods in quantum field theory, and explores two- dimensional and four- dimensional gauge dynamics using those methods. The book concludes with a summary emphasizing the interplay between two- and four- dimensional gauge theories. Aimed at graduate students and researchers, this book covers topics from two-dimensional conformal symmetry, affine Lie algebras, solitons, integrable models, bosonization, and 't Hooft model, to four-dimensional conformal invariance, integrability, large N expansion, Skyrme model, monopoles and instantons. Applications, first to simple field theories and gauge dynamics in two dimensions, and then to gauge theories in four dimensions and quantum chromodynamics (QCD) in particular, are thoroughly described.

This is a concise graduate level introduction to analytical functional methods in quantum field theory. Functional integral methods provide relatively simple solutions to a wide range of problems in quantum field theory. After introducing the basic mathematical background, this book goes on to study applications and consequences of the formalism to the study of series expansions, measure, phase transitions, physics on spaces with nontrivial topologies, stochastic quantisation, fermions, QED, non-abelian gauge theories, symmetry breaking, the effective potential, finite temperature field theory, instantons and compositeness. Serious attention is paid to the shortcomings of the conventional formalism (e.g. problems of measure) as well as detailed appraisal of the ambiguities of series summation. This book will be of great use to graduate students in theoretical physics wishing to learn the use of functional integrals in quantum field theory. It will also be a useful reference for researchers in theoretical physics, especially those with an interest in experimental and theoretical particle physics and quantum field theory.

Stochastic methods in quantum field theory and hydrodynamics

A different method in quantum field theory: Strong coupling and the infrared structure of φ 4 theory

If the concepts and methods of modern physics research are reasonably integrated into the teaching of basic physics courses, students can better understand the application value of what they have learnt, and thus be prepared for higher level teaching and research. In this paper, we take the calculation of the potential of a uniformly charged infinite-length self-conducting wire as an example, and discuss how to introduce and penetrate some concepts and methods of modern quantum field theory in the teaching of electrodynamics in the second year of undergraduate courses. Specific contents include good mathematical definitions and pathological mathematical definitions, regularisation of divergence integrals, physical quantities and non-physical quantities, and etc. This paper attempts to explain these physics and mathematical concepts that are too abstract for thousands of undergraduate students in a relatively simple way, so as to help students understand and use the ideas and methods more easily and naturally when they study modern physics such as quantum field theory in the future, and to have a better understanding of some general problems related to mathematics and physics.

In the paper we begin a description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and are the q-generalization of the colored particles which appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we prove the q-functional analogues of Hori's formulation of Wick's theorems for the different ordered q-particle creation and annihilation operators. The formulae have the same formal expressions as fermionic and bosonic ones but differ by a nature of fields. This allows us to derive the perturbation series for the theory and develop analogues of standard quantum field theory constructions in q-functional form.

The self-consistent method in quantum field theory is applied to the itinerant-electron ferromagnet. The magnon as a bound state of electrons is studied by using Bethe-Salpeter equations in the pair approximation. The spin rotation of electrons is induced by anE2-symmetry transformation of magnons and quasi-electrons which is different from a rotation. A simple derivation of the low-energy theorem is discussed.

The long-debated problem of the corrections proposed to the Glauber eikonal approximation in potential scattering theory is here studied in a general and elementary framework. A number of expansions is described and the Fourier-Bessel representation of Wallace is easily obtained in the high-energy limit. Our approach has meaningful analogies with similar methods in quantum field theory. A newly proposed approximation for the scattering amplitude has some optimal features.

The renormalisation of Feynman diagrams is considered in the context of perturbative algebraic quantum field theory. The construction of Wick polynomials is related to the renormalisation of diagrams with short loops (tadpoles). Microlocal techniques are used to prove a smoothness condition for these renormalised diagrams. The renormalisation of diagrams without tadpoles is treated from the perspective of causal perturbation theory. A locally covariant construction of renormalisation maps is given for Feynman diagrams with edges labelled by conormal distributions.

Julian Schwinger's development of his Green's functions methods in quantum field theory is placed in historical context. The relation of Schwinger's quantum action principle to Richard Feynman's path-integral formulation of quantum mechanics is reviewed. The nonperturbative character of Schwinger's approach is stressed as well as the ease with which it can be extended to finite temperature situations.

Discusses some difficulties which arise in proposals to extend the inverse scattering transform method to nonlinear quantum field theory. In particular, for the nonlinear Schrodinger equation, the author shows that the usual classical methods for obtaining the Poisson brackets of the scattering data reach an impasse if they are extended to find the commutators of the corresponding quantum operators.

Book reviews

Over the past few decades, in concert with ground-breaking experimental advances, condensed matter theory has drawn increasingly from the language of low-energy quantum field theory. This primer is aimed at elevating graduate students of condensed matter theory to a level where they can engage in independent research. It emphasizes the development of modern methods of classical and quantum field theory with applications oriented around condensed matter physics. Topics covered include second quantization, path and functional field integration, mean-field theory and collective phenomena, the renormalization group, and topology. Conceptual aspects and formal methodology are emphasized, but the discussion is rooted firmly in practical experimental application. As well as routine exercises, the text includes extended and challenging problems, with fully worked solutions, designed to provide a bridge between formal manipulations and research-oriented thinking. This book will complement graduate level courses on theoretical quantum condensed matter physics.

We consider the easy-plane anisotropic spin-$\frac{1}{2}$ Heisenberg chain in combined uniform longitudinal and transverse staggered magnetic fields. The low-energy limit of his model is described by the sine-Gordon quantum field theory. Using methods of integrable quantum field theory we determine the various components of the dynamical structure factor. To do so, we derive explicit expressions for all matrix elements of the low-energy projections of the spin operators involving at most two particles. We discuss applications of our results to experiments on one-dimensional quantum magnets.

In this paper we investigate the massive Sine-Gordon model in the ultraviolet finite regime in thermal states over the two-dimensional Minkowski spacetime. We combine recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive quantum field theory over Euclidean spacetimes to construct the correlation functions of the equilibrium state of the Sine-Gordon theory in the adiabatic limit. First of all, the observables of the Sine-Gordon theory are seen as functionals over the free configurations and are obtained as a suitable combination of the S−matrices of the interaction Lagrangian restricted to compact spacetime regions over the free massive theory. These S−matrices are given as power series in the coupling constant with values in the algebra of fields over the free massive theory. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian signature, we prove that these power series converge when evaluated on a generic field configuration. The latter observation implies convergence in the strong operator topology in the GNS representations of the considered states. In the second part of the paper, adapting the cluster expansion technique to the Lorentzian case, we prove that the correlation functions of the interacting equilibrium state at finite temperature (KMS state) can be constructed also in the adiabatic limit, where the interaction Lagrangian is supported everywhere in space.