the Sine-Gordon field theory as a model for the Yukawa gas and the neutral Coulomb gas

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ETH Library

Euclidean constructive quantum

field theory and classical statistical mechanics

the Sine-Gordon field theory as a model for the Yukawa gas and the neutral Coulomb gas

Doctoral Thesis Author(s):

Steinmann, Adrian Walter Publication date:

1988

Permanent link:

https://doi.org/10.3929/ethz-a-000473190 Rights / license:

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Diss. ETH No. 8376

Euclidean

Constructive Quantum ^Field Theory

and

Classical Statistical Mechanics

The Sine

^-

Gordon Field Theory

as a

Model for the Yukawa Gas

and the

Neutral Coulomb Gas

A dissertation submittedtothe

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the

degree

^of

DoctorofMathematics

presented by

ADRIAN WALTER STEINMANN

dipl.

^{Math. ETH} bornonOctober24,¹⁹⁵⁸ citizen of Altishofen

(LU)

accepted

^on^therecommendationof Prof. Dr. Konrad

Osterwalder,

examiner Prof. Dr. Giovanni

Gallavotti,

co-examiner

Prof. Dr.

Jiirg Frohlich,

co-examiner

1988

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Abstract

Theproperty^ofUltraviolet

Stability

^{for the}^twodimensional Sine-Gordon interac¬

tion

A

/

: cosa<px: dx J\

infinite volumeA CIR2isstudiedas a

problem

ⁱⁿEuclidean Constructive

Quantum

Field

Theory.

InClassical Statistical

Mechanics,

this model hasanatural Yukawa Gas inter¬

pretation

^under

application

of the so-called Sine-Gordon^or

Siegert

transformation.

Furthermore, by applying

^a^second

transformation,

the interaction_may also_rep¬

resent ^aclassicalneutral Coulomb Gas.

Preliminary chapters

^are ^devoted ^to

describing

^the

underlying phenomena

ⁱⁿ classical statistical mechanics in additionto

introducing

^the mathematical tools needed in

subsequent chapters. Throughout

the technicalpart^{of the}

work,

^{the Eu¬}

clidean

perspective

of Constructive

Quantum

^Field

Theory

^is

adopted.

^A

powerful technique recently developed

^to

study

Ultraviolet

Stability

and Renormalization Group^is^reviewed^and

applied extensively.

It isshown that the

(trivial)

^additiverenormalization

gives

^{rise to}

(non-trivial) large

fluctuationswhich ^are overcome

by applying

^two ^related yet

conceptually independent

methods. The

properties

^of

Asymptotic

Freedom and

Superrenormal¬

izability

of the Sine-Gordon model for alla2 G

[0,87r[

^follows^{as a}

corollary

^{of the}

preliminary analysis. Thereupon

^{it is}further shown that both the

positivity

^{of the} action inthe

large fluctuating regions

^as^well^as their small

probability

^lead ^to^an

upperbound forUltraviolet

Stability.

^As^aconsequence,the

Mayer

series for the pressureof theneutral Coulomb_gasinthe

Thermodynamic

^{Limit is}

asymptotic

^to

anyorderfora2 > 8ir,^while^for a2< 8nitis shown that the

partial

^sums^{of the}

(divergent) Mayer expansion

^is

asymptotic

up to^anorder

depending

^on^a2.

In additionto asummaryand

outlook,

^abrief review of related work is_pre¬

sented.

Keywords:

Asymptotic Freedom,

^Cosine

Interaction,

^Coulomb

Gas, Mayer Expansion,

^Renor¬

malization

Group, Sine-Gordon, Superrenormalizability, Thermodynamic Limit,

Ultraviolet

Stability,

^Yukawa^Gas

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Zusammenfassung

Das Problem der Ultravioletten Stabilitat wird im Rahmen der Konstruktiven Euklidischen

Quantenfeldtheorie

^am Sinus-GordonModell

A

/

:cosaifx: dx Jh

in zwei Dimensionen undimendlichenVolumen ACIR2studiert.

Unter

Anwendung

der Sinus-Gordon oder

Siegert Transformation,

^kann ^man dieses Modell als ein GasvonYukawa in der klassischen statistischen Mechanik

interpretieren.

Das Modell kann aber auch ein klassisches neutrales Coulomb Gas

darstellen,

^indem^maneine zweite Transformation anwendet.

Dieersten

Kapitel

beschreiben die

physikalischen

Phanomene dieausder Sicht der klassischen statistischen Mechanikerwartetwerden. In diesen

Kapiteln

^werden auch die mathematischen Hilfsmittel

eingefiihrt,

^als

Vorbereitung

^{auf ihre}

Anwendung

ⁱⁿ^den

nachfolgenden Kapiteln,

^wo^dertechnische Teil derArbeitunter dem

Gesichtspunkt

^derEuklidischen

Quantenfeldtheorie ausgefuhrt

^wird. ^Insbe-

sondere wird eineneu

entwickelte, anwendungskraftige

^Technik^zum^Studieren^der

Renormierungsgruppe

^und^derUltraviolettenStabilitat

eingehend

beschrieben und

weitlaufig angewendet.

Es wird

gezeigt

^wie ^die

(triviale)

^additive Renormierung ^zu einem Problem

(nicht-trivialer)

grosserFluktuationenfiihrt. Unter

Anwendung

^zweier^zwar^ahn- lichen

jedoch konzeptionell unabhangigen

Methoden werden dann hierfur zwei verschiedene

Losungen

^erzielt. ^Die

Eigenschaften

^der

Asymptotischen

^Freiheit und der

Super-Renormierbarkeit

des Sinus-Gordon Modells werdenfuraiieWerte o? e

[0, 8tt[

^bewiesen. Insbesondere wird

gezeigt,

wie sowohl die Positivitatder

Wirkung

^im^Gebiet^dergrossenFluktuationensowieauch deren kleine Wahrschein- lichkeit _genugen,umdie Ultraviolette Stabilitat zubeweisen. Danach

folgen

^Ko- rollare uber den

Thermodynamischen

^{Limes der}

Mayer-Entwicklung

fiir den Druck desneutralen Coulomb Gases. Fiir Werteo?^>8irist diese

Entwicklung

^zur^be-

liebigen Ordnung asymptotisch,

andererseits

zeigt sich,

^dass^fiir ^Werte ^a2 ^< ^8ft

nur die Partialsummen bis zu einer a2 ^-

abhangigen Ordnung

^der

(divergenten) Entwicklung asymptotisch

^zumDruck sind.

Inder

Einfuhrung

^wirdfiber andereWerkekurz berichtet undzumAbschluss sind die

Hauptzuge

^der

Beweisfuhrung

^imHinblick aufweitere Problemezudiesem Thema

zusammengefasst.

Stichworte:

Asymptotische Freiheit,

^Coulomb

Gas, Kosinus-Wechselwirkung, Mayer-Entwick-

lung, Renormierungsgruppe, Sinus-Gordon, Super-Renormierbarkeit, Thermody-

namischer Limes, Ultraviolette

Stabilitat,

^{Yukawa Gas}

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Riassunto

II

problema

della Stabilita Ultraviolettaestudiatonell'ambito della Teoria Euclidea Costruttiva dei

Campi Quantistici

sul modello Seno-Gordon

A

/

:cosa^,: dx J\

inunvolume A CIR2 finito di dimensionedue.

Nella Meccanica Statistica

Classica,

questo^{modello ha}

un'interpretazione

^natu- ralecome ungasdiYukawasottolatrasformazione di

Seno-Gordon,

^anche^chiamata latrasformazione di

Siegert. Inoltre,

^via^una^seconda

trasformazione,

l'interazione descrive ancheunGasdi Coulomb classicocon carica

complessiva

^zero.

I

capitoli preliminari

descrivono il fenomeno fisico

previsto

dalla meccanicasta¬

tistica classicae introducono anche i metodi matematici che verranno

applicati

nei

capitoli seguenti.

^{Tutta la}parte tecnica del lavoro viene

eseguita

nell'ambito dellateoriaeuclidea dei

campi,

ⁱⁿ

particolare

^si ^introduce^un metodo molto _po- tente

sviluppato

recentemente perstudiare laStabilitaUltravioletta ed il

Gruppo

di Rinormalizzazione.

Si dimostra che la rinormalizzazione additiva

(banale)

^da

luogo

^afluttuazioni

grandi (non-banali)

^chepossono ^essere controllate tramite due metodi similima

concettualmente

indipendenti.

^Le

proprieta

^di^Liberta^Asintotica^e^di

Super-rinor-

malizzabilita del modello Seno-Gordon_per tutti i valori a2

[0,87r[

segue^come corollario dalla nostra analisi. Si dimostra anche che sia la

positivita

^nelle ^re-

gioni

^delle

grandi

fluttuazioni sia la loro

probabilita piccola

portano^ad^una^stima

superiore

perla Stabilita Ultravioletta. Comeconseguenza, nel Limite Termodi-

namico,

^la^serie ^di

Mayer

perla

pressione

^delgas diCoulombneutroe asintotica ad

ogni

^ordinepera2 >

8w,

mentre pera2< 8nsidimostrache lesomme

parziali dell'espansione

^di

Mayer

^sonoasintotichefino adunordine che

dipende

^da^a2.

Oltre adundiscorso sulle

prospettive,

^unriassunto breve di altri lavorisuquesto temaepresentato.

Parole Chiave:

Espansione

^di

Mayer,

^Gas ^di

Coulomb,

^{Gas di}

Yukawa, Gruppo

di Rinorma¬

lizzazione,

Interazione

Coseno,

^Liberta

Asintotica,

^Limite

Termodinamico,

^Seno-

Gordon,

^Stabilita