# Diagrammar in classical scalar field theory

## Abstract

In this paper we analyze perturbatively a g{phi}{sup 4}classical field theory with and without temperature. In order to do that, we make use of a path-integral approach developed some time ago for classical theories. It turns out that the diagrams appearing at the classical level are many more than at the quantum level due to the presence of extra auxiliary fields in the classical formalism. We shall show that a universal supersymmetry present in the classical path-integral mentioned above is responsible for the cancelation of various diagrams. The same supersymmetry allows the introduction of super-fields and super-diagrams which considerably simplify the calculations and make the classical perturbative calculations almost 'identical' formally to the quantum ones. Using the super-diagrams technique, we develop the classical perturbation theory up to third order. We conclude the paper with a perturbative check of the fluctuation-dissipation theorem. - Highlights: > We provide the Feynman diagrams of perturbation theory for a classical field theory. > We give a super-formalism which links the quantum diagrams to the classical ones. > We check perturbatively the fluctuation-dissipation theorem.

- Authors:

- Department of Physics (Miramare Campus), University of Trieste, Strada Costiera 11, Miramare-Grignano 34014, Trieste (Italy)
- (Italy)
- Departamento de Engenharia de Producao, Administracao e Economia, Escola de Minas, Campus Morro do Cruzeiro, UFOP, 35400-000 Ouro Preto MG (Brazil)

- Publication Date:

- OSTI Identifier:
- 21583337

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 326; Journal Issue: 9; Other Information: DOI: 10.1016/j.aop.2011.05.009; PII: S0003-4916(11)00100-X; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FEYNMAN DIAGRAM; FIELD THEORIES; FLUCTUATIONS; PATH INTEGRALS; PERTURBATION THEORY; SCALAR FIELDS; SUPERSYMMETRY; DIAGRAMS; INFORMATION; INTEGRALS; SYMMETRY; VARIATIONS

### Citation Formats

```
Cattaruzza, E., E-mail: Enrico.Cattaruzza@gmail.com, Gozzi, E., E-mail: gozzi@ts.infn.it, INFN, Sezione di Trieste, and Francisco Neto, A., E-mail: antfrannet@gmail.com.
```*Diagrammar in classical scalar field theory*. United States: N. p., 2011.
Web. doi:10.1016/j.aop.2011.05.009.

```
Cattaruzza, E., E-mail: Enrico.Cattaruzza@gmail.com, Gozzi, E., E-mail: gozzi@ts.infn.it, INFN, Sezione di Trieste, & Francisco Neto, A., E-mail: antfrannet@gmail.com.
```*Diagrammar in classical scalar field theory*. United States. doi:10.1016/j.aop.2011.05.009.

```
Cattaruzza, E., E-mail: Enrico.Cattaruzza@gmail.com, Gozzi, E., E-mail: gozzi@ts.infn.it, INFN, Sezione di Trieste, and Francisco Neto, A., E-mail: antfrannet@gmail.com. Thu .
"Diagrammar in classical scalar field theory". United States. doi:10.1016/j.aop.2011.05.009.
```

```
@article{osti_21583337,
```

title = {Diagrammar in classical scalar field theory},

author = {Cattaruzza, E., E-mail: Enrico.Cattaruzza@gmail.com and Gozzi, E., E-mail: gozzi@ts.infn.it and INFN, Sezione di Trieste and Francisco Neto, A., E-mail: antfrannet@gmail.com},

abstractNote = {In this paper we analyze perturbatively a g{phi}{sup 4}classical field theory with and without temperature. In order to do that, we make use of a path-integral approach developed some time ago for classical theories. It turns out that the diagrams appearing at the classical level are many more than at the quantum level due to the presence of extra auxiliary fields in the classical formalism. We shall show that a universal supersymmetry present in the classical path-integral mentioned above is responsible for the cancelation of various diagrams. The same supersymmetry allows the introduction of super-fields and super-diagrams which considerably simplify the calculations and make the classical perturbative calculations almost 'identical' formally to the quantum ones. Using the super-diagrams technique, we develop the classical perturbation theory up to third order. We conclude the paper with a perturbative check of the fluctuation-dissipation theorem. - Highlights: > We provide the Feynman diagrams of perturbation theory for a classical field theory. > We give a super-formalism which links the quantum diagrams to the classical ones. > We check perturbatively the fluctuation-dissipation theorem.},

doi = {10.1016/j.aop.2011.05.009},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 9,

volume = 326,

place = {United States},

year = {2011},

month = {9}

}