The Legendre transformation is one of the ways to rewrite the theory by small number of selected variables. It leads naturally to the generalized action functional called the effective action and everything starts from this generating functional. It is applied to various problems of physics ranging from the relativistic field theory to the condensed matter physics. The main techniques used here are the on-shell expansion and the inversion method. The scheme presented in this article is thus a generating functional formalism of the many particle systems.
The Legendre transformation is one of the ways to rewrite the theory by small number of selected variables. It leads naturally to the generalized action functional called the effective action and everything starts from this generating functional. It is applied to various problems of physics ranging from the relativistic field theory to the condensed matter physics. The main techniques used here are the on-shell expansion and the inversion method. The scheme presented in this article is thus a generating functional formalism of the many particle systems.
We study the response of generating functionals to a variation of parameters (couplings) in equilibrium systems i.e. in quantum field theory (QFT) and equilibrium statistical mechanics. These parameters can be either physical ones such as coupling constants or artificial ones which are intentionally introduced such as the renormalization scale in field theories. We first derive general functional flow equations for the generating functional (grand-canonical potential) W[J] of the connected diagrams. Then, we obtain functional flow equations for the one-particle irreducible (1PI) vertex functional (canonical potential) Γ[ϕ] by performing the Legendre transformation. By taking the functional derivatives of the flow equations, we can obtain an infinite hierarchical equations for the 1PI vertices. We also point out that a Callan-Symanzik type equation holds among the vertices when partition function is invariant under some changes of the parameters. After discussing general aspects of parameter response, we apply our formalism to several examples and reproduce the well-known functional flow equations. Our response theory provides us a systematic and general way to obtain various functional flow equations in equilibrium systems.
Theoretical foundation of density functional theory in terms of field theoretical Legendre transformation is presented. The ground state energy is first written as a functional of local probe coupled to the density operator. The density functional is then defined by the functional Legendre transformation, which leads to a systematic formulation of density functional theory. Excitation spectrum is also determined within the same formalism in a unified way. The diagrammatic evaluation is most conveniently done by using auxiliary field method. Several generalizations of the formalism and extension to the case other than the density operator are also discussed.
The methods of quantum field theory underpin many conceptual advances in contemporary condensed matter physics and neighbouring fields. This book provides a praxis-oriented and pedagogical introduction to quantum field theory in many-particle physics, emphasizing the application of theory to real physical systems. This third edition is organized into two parts: the first half of the text presents a streamlined introduction, elevating readers to a level where they can engage with contemporary research literature, from the introduction of many-body techniques and functional integration to renormalization group methods, and the second half addresses a range of advanced topics including modern aspects of gauge theory, topological and relativistic quantum matter, and condensed matter physics out of thermal equilibrium. At all stages, the text seeks a balance between methodological aspects of quantum field theory and practical applications. Extended problems with worked solutions provide a bridge between formal theory and a research-oriented approach.
Special relativity implies that mass can be created from energy — it implies particle production. A relativistic quantum theory is of necessity a theory of many particles. We shall therefore first describe systems of many identical particles but in the simpler context of non-relativistic theory. As we shall soon see, many-particle systems can be described by field operators that create and destroy particles, with these operators acting in a vastly enlarged vector space that spans states with an arbitrary number of particles, including a state of no particles — the vacuum state. Both Bose and Fermi statistics are handled in this way, and the resulting formalism is compact, convenient, and powerful. For the Bose case, the functional integral representation of the quantum many-particle system is direct generalization of the coherent-state functional integral developed in the previous chapter. To describe the analogous formulation for the Fermi case, we must first develop the theory of anticommuting variables. The power and utility of field operator and functional integral methods will be illustrated by applying it to systems in thermal equilibrium, first to free Fermi and Bose gases, and then to a model of liquid helium which exhibits the phenomenon of spontaneous symmetry breaking, a phenomenon of great importance in condensed matter physics as well as in elementary particle theory.
In the previous chapter we introduced the one-particle irreducible effective action by collecting the one-particle irreducible vertex functions into a generator whose argument is the field, the one-state amplitude in the presence of the source. The effective action thus generates the one-particle irreducible amputated Green's functions. We shall now enhance the usability of the non-equilibrium effective action by establishing its relationship to the sum of all one-particle irreducible vacuum diagrams. To facilitate this it is convenient to add the final mathematical tool to the arsenal of functional methods, viz. functional integration or path integrals over field configurations. We are then following Feynman and instead of describing the field theory in terms of differential equations, we get its corresponding representation in terms of functional or path integrals. This analytical condensed technique shall prove powerful when unraveling the content of a field theory. The loop expansion of the non-equilibrium effective action is developed, and taken one step further as we introduce the two-particle irreducible effective action valid for non-equilibrium states. As an application of the effective action approach, we consider a dilute Bose gas and a trapped Bose–Einstein condensate. Functional integration Functional differentiation has its integral counterpart in functional integration. We shall construct an integration over functions and not just numbers as in elementary integration of a function. We approach this infinite-dimensional kind of integration with care (or, from a mathematical point of view, carelessly), i.e. we base it on our usual integration with respect to a single variable and take it to a limit.
The basic concern of density functional theory is the groundstate energy of an inhomogeneous interacting many-particle system as a function of the external potential acting on its density (and possibly as a function of the particle number, allowing for the consideration of ionization energies, affinities, and excitation gaps). This groundstate energy function is replaced by its Legendre transform as a function of the particle density, and the Legendre back transformation leads via insertion of an interaction-free reference system to an effective one-particle equation whose solution finds the ground-state energy and density to any given external potential. As was seen in Part I, this theory provides rather a reformulation than a solution of the many-body problem for the groundstate, because it does not explicitly provide the Legendre transform H [ n ] to E [ v, N ] (cf. (6.35, 6.31, and 6.36). The original many-body problem is now contained in H [ n ] which only can be guessed, although not without success. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.