EFFECTIVE ACTIONS OF LOCAL COMPOSITE OPERATORS: THE CASE OF φ⁴ THEORY, THE ITINERANT ELECTRON MODEL, AND QED

Abstract

The effective action Γ[φ] is defined from the generating function(al) W [J] through Legendre transformation and plays the role of action functional in the zero-temperature field theory and of generalized thermodynamical function(al) in equilibrium statistical physics. A compact graph rule for the effective action Γ[φ] of a local composite operator is given in this paper. This long-standing problem of obtaining Γ[φ] in this case is solved directly without using the auxiliary field. The rule is first deduced with help of the inversion method, which is a technique for making the Legendre transformation perturbatively. It is then proved by using a topological relation and also by the sum-up rule. The latter is a technique for making the Legendre transformation in a graphical language. In the course of proof a special role is played by J (0) [φ], which is a function(al) of the variable φ and defined through the lowest inversion formula. Here J (0) [φ] has the meaning of the source J for the noninteracting system expressed by φ. Explicitly derived are the rules for the effective action of ϕ(x) 2 in the ϕ 4 theory, of the number density n rσ in the itinerant electron model, and of the gauge invariant operator ψ γ µ ψ in QED.

Landau gauge gluon and ghost propagators in the refined Gribov-Zwanziger framework in 3 dimensions

“…Let us proceed thus by explicitly performing the Legendre transformation, along the lines of [51,52]. The generating functional W(J) will be obtained as a series in the coupling constant, W(J) = W 0 (J) + g 2 W 1 (J) + .…”

Thermal effective potential of the O(N) linear σ model

“…They were first encountered in the study of certain condensed-matter problems, but even for what concerns those problems a general consensus on the procedures to be followed has not yet been reached (see the recent Ref. [10] and references therein). In the context of the relativistic quantum field theories of interest to particle physics the same, essentially infrared, problems associated to the description of thermal effects are accompanied by the familiar "pathology" of ultraviolet divergencies.…”

“…[1][2][3]), are best estimated within the framework of non-equilibrium (classical or quantum) field theory [3][4][5][6][7][8], there is also considerable interest (see, e.g., the recent Refs. [9][10][11]) in the quasi-equilibrium/thermal description of the chiral phase transition. Moreover, some of the scenarios of phenomenological interest consist [5,6] of a first stage in which thermal equilibrium is reached, followed by a stage in which the system evolves out of equilibrium.…”

Unified time-path approach to the effect of anharmonicity on the molecular vibrational spectroscopy in solution

“…[47][48][49] They are originally applied to the problems in high energy physics and have proved their wide applicability in various fields. [50][51][52][53][54][55] By this method one can rewrite the theory in terms of the physical observable in question by using the Legendre transformation. The unperturbative contributions are systematically taken into account by perturbative calculations in this method ͑though it may sound paradoxical͒.…”