Ations of motion for a Legendre transform arbitrary order

Васильев, А. Н.; Kazanskiǐ, A. K.

doi:10.1007/bf01029302

Cited by 29 publications

(22 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since W V is composed of all connected graphs, including those containing higher order cumulants of W 0 , we see that the terms · · · Γ (2) 0 W V Γ (2) 0 · · · appearing in the iteration produce those unusual terms of the form W An alternative approach is the analog of the Dyson Schwinger equation for the effective action [35,52,53], reviewed in [5, sec. 1.8, 6.2].…”

Section: Discussionmentioning

confidence: 99%

Expansion of the effective action around non-Gaussian theories

Kühn

Helias

2018

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well known and composed of one-line irreducible diagrams only. For the expansions around an arbitrary, non-Gaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the Thouless-Anderson-Palmer mean-field theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the high-temperature expansion are special cases of the general formalism presented here.

show abstract

Section: Discussionmentioning

confidence: 99%

Expansion of the effective action around non-Gaussian theories

Kühn

Helias

2018

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…We mention here that the diagrams of φ (n) is obtained by attaching a black dot, in all possible ways, to one of the lines in the graphs of the n-th order of W . For example, the 31 diagrams of φ (4) is obtained through the fourth order of W ;…”

Section: Rule For J (N)mentioning

confidence: 99%

“…Note here that 1VI can not be replaced by 2PI (2-particle-irreducible) as in the case of the effective action for the non-local operator ϕ(x)ϕ(y) [2,4,6]. This is clear from the second and third (1VI) graphs from the last on the left-hand side of (2.40), which are 2PR (2-particlereducible).…”

Section: Rule For J (N)mentioning

confidence: 99%

“…In spite of its wide-spread use, the precise rule of constructing the effective action for a local composite field has not been derived up to now although the graphical rules for an elementary field and for non-local composite fields up to four-body operators are already known [2,3,4,5,6]. The study of the effective action for local composite operator amounts to rewrite the theory in terms of physical variables such as the expectation values of the number density operator, spin density operator, local gauge invariant operator etc.…”

Section: Introductionmentioning

confidence: 99%

“…Last example of QED is given in Sec. 4 which may be a one-step toward a gauge-invariant study of the gauge field theory. Appendix A explains the reason why φ is to be considered as independent of λ in the process of inversion in a way different from the one given in the literature.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Okumura

1996

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

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.

show abstract

Vacuum Energy‐Density in Quantum Electrodynamics

Manoukian

1976

Fortschr. Phys.

View full text Add to dashboard Cite

The vacuum energy‐density in quantum electrodynamics is studied by renormalization group techniques as well as a diagrammatic analysis is carried out to investigate the dependence of on an ultraviolet cut‐off as the latter is led to become large. The study corresponds to the situation of finite electrodynamics with the renormalized fine‐structure constant α fixed in the sens of the renormalization group. In this case explicit statements about the problem at hand may be made. In passing a study of the so‐called second Legendre transform method for electrodynamics is given in an appendix.

show abstract

Ations of motion for a Legendre transform arbitrary order

Cited by 29 publications

References 7 publications

Expansion of the effective action around non-Gaussian theories

Expansion of the effective action around non-Gaussian theories

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

Vacuum Energy‐Density in Quantum Electrodynamics

Contact Info

Product

Resources

About

Ations of motion for a Legendre transform arbitrary order

Cited by 29 publications

References 7 publications

Expansion of the effective action around non-Gaussian theories

Expansion of the effective action around non-Gaussian theories

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

Vacuum Energy‐Density in Quantum Electrodynamics

Contact Info

Product

Resources

About

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