Semiclassical quantization of kinks on compact spaces [Elektronische Ressource] / vorgelegt von Michael Pawellek

Semiclassical quantizationof kinks on compact spacesDer Naturwissenschaftlichen Fakult atder Friedrich-Alexander-Universit at Erlangen-Nurn? bergzurErlangung des Doktorgradesvorgelegt vonMichael Pawellekaus Furth?Als Dissertation genehmigt von der Naturwissen-schaftlichen Fakult at der Universit at Erlangen-Nurn? bergTag der mundlic? hen Prufung:? 10.April 2008Vorsitzender derPromotionskommission: Prof. Dr. Eberhard B anschErstberichterstatter: Prof. Dr. Frieder LenzZweitberichterstatter: Prof. Dr. Andreas WipfWir sollten einfach herumgehen und Ideen einfangen, irgendwann verlieben wir unsin eine davon, und die zeigt uns dann den richtigen Weg.David LynchZusammenfassungIn dieser Arbeit wird die semiklassische Quantisierung von Kinks in 1+1 dimen-sionalen Modellfeldtheorien auf kompaktem Raum untersucht. Fur? die Bestim-mung der 1-loop Quantenkorrekturen zur Masse/Energie der Kink- bzw. Solitonenwird fur? gew ohnlich das Eigenwertspektrum des zugeh origen Fluktuationsoperators4ben otigt. Im Fall des sine-Gordon- und ` -Modells ist dies die n = 1 bzw. n = 2Lam´egleichung, deren periodisches bzw. antiperiodisches Eigenwertspektrum nichtvollst andig bekannt ist. Trotz der Unkenntnis des vollen Spektrums wird mit Hilfeder Integraldarstellung der Spektralzetafunktionen die 1-loop Massenkorrekturen fur?4das ` - und sine-Gordon-Modell analytisch bestimmt.
Publié le : mardi 1 janvier 2008
Lecture(s) : 24
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Source : WWW.OPUS.UB.UNI-ERLANGEN.DE/OPUS/VOLLTEXTE/2008/907/PDF/MICHAELPAWELLEKDISSERTATION.PDF
Nombre de pages : 89
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Semiclassical quantization
of kinks on compact spaces
Der Naturwissenschaftlichen Fakult at
der Friedrich-Alexander-Universit at Erlangen-Nurn? berg
zur
Erlangung des Doktorgrades
vorgelegt von
Michael Pawellek
aus Furth?Als Dissertation genehmigt von der Naturwissen-
schaftlichen Fakult at der Universit at Erlangen-Nurn? berg
Tag der mundlic? hen Prufung:? 10.April 2008
Vorsitzender der
Promotionskommission: Prof. Dr. Eberhard B ansch
Erstberichterstatter: Prof. Dr. Frieder Lenz
Zweitberichterstatter: Prof. Dr. Andreas WipfWir sollten einfach herumgehen und Ideen einfangen, irgendwann verlieben wir uns
in eine davon, und die zeigt uns dann den richtigen Weg.
David LynchZusammenfassung
In dieser Arbeit wird die semiklassische Quantisierung von Kinks in 1+1 dimen-
sionalen Modellfeldtheorien auf kompaktem Raum untersucht. Fur? die Bestim-
mung der 1-loop Quantenkorrekturen zur Masse/Energie der Kink- bzw. Solitonen
wird fur? gew ohnlich das Eigenwertspektrum des zugeh origen Fluktuationsoperators
4ben otigt. Im Fall des sine-Gordon- und ` -Modells ist dies die n = 1 bzw. n = 2
Lam´egleichung, deren periodisches bzw. antiperiodisches Eigenwertspektrum nicht
vollst andig bekannt ist. Trotz der Unkenntnis des vollen Spektrums wird mit Hilfe
der Integraldarstellung der Spektralzetafunktionen die 1-loop Massenkorrekturen fur?
4das ` - und sine-Gordon-Modell analytisch bestimmt. Das Eigenwertspektrum wird
dabei nur implizit in Form der Spektraldiskriminante ben otigt, welche fur? die n = 2
Lam´egleichung konstruiert wird. Fur? antiperiodische Randbedingungen existiert
dabei ein Radius mit minimaler Energie.
Im zweiten Teil der Arbeit wird gezeigt, dass die Fluktuationsgleichungen fur?
6das ` -, Doppel-Sine-Gordon- und Jacobimodell jeweils zu einer verallgemeinerten
Lam´egleichung fuhrt.? Sie ist aquiv? alent zu einer gew ohnlichen Differentialgleichung
zweiter Ordnung mit funf? regul ar singul aren Punkten. Um Eigenschaften und
L osungen dieser Gleichung in einer kanonischen Form wie im Fall der Lam´egleichung
zu diskutieren, werden die sog. verallgemeinerten Jacobi Elliptischen Funktio-
nen eingefuhrt.? Wichtige Eigenschaften und Identit aten dieser Funktionen werden
hergeleitet und diskutiert.
Ein zentrales Resultat dieses zweiten Teils der Arbeit ist, dass die verallgemeinerte
Lam´e-gleichung nur 15 Polynoml osungen besitzt, ausdruc? kbar in verallgemeinerten
JacobiElliptischeFunktionen. DiePolynoml osungwirdphysikalischalsNullmodeim
Fluktuationsspektrum der Kinks in der jeweiligen Feldtheorie identifiziert.Abstract
In this thesis the semiclassical quantization of kinks in 1+1 dimensional field theories
on compact spaces is considered. Usually the spectrum of the corresponding fluctu-
ation operator is needed for the determination of the 1-loop quantum corrections to
4the mass/energy of kinks or solitons. In the case of the sine-Gordon and ` model it
is the n = 1 and n = 2 Lam´e equation, respectively. Their (anti-) periodic spectrum
is not completely known. Despite this lack of knowledge the 1-loop mass corrections
4for the ` and sine-Gordon model are analytically determined with the help of the
integral representation of the spectral zeta function. The corresponding spectrum is
only implicitly needed by the sp discriminant, which for the n = 2 Lam´e equa-
tion is constructed. For antiperiodic boundary conditions a radius of minimal energy
is obtained.
In the second part of this thesis it is shown, that the fluctuation equations of the
6` , double-sine-Gordon and Jacobi model correspond to a generalized Lam´e equa-
tion. Thisequationisequivalenttoasecondorderordinarydifferentialequationwith
five regular singular points. In order to discuss the properties and solutions of this
equation in a canonical manner as in the case of the ordinary Lam´e equation the gen-
eralized Jacobi elliptic functions are introduced. Important properties and identities
of these functions are derived and discussed.
A central result of this second part is, that the generalized Lam´e equation possesses
exactly15polynomialsolutionsexpressibleintermsofgeneralizedJacobiellipticfunc-
tions. The p solution is physically identified as zero mode in the fluctuation
spectrum of the mentioned field theories.Contents
1 Introduction 3
2 Twisted scalar field 6
2.1 Classical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 1-Loop fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Spectral Zeta functions 11
3.1 Zeta function regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Sample calculation: Free scalar field with periodic boundary conditions . . . . . . 14
3.2.1 Casimir energy of a free massive scalar field . . . . . . . . . . . . . . . . . . 14
3.2.2 of a free massless scalar field . . . . . . . . . . . . . . . . . 16
4 The quantized twisted kink 18
4.1 Construction of the spectral discriminant . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 The 1-loop contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.1 Regularization in the sector R<R . . . . . . . . . . . . . . . . . . . . . . 200
4.2.2 and renormalization in the sector R>R . . . . . . . . . . 220
4.2.3 Renormalization in the sector R<R . . . . . . . . . . . . . . . . . . . . . 240
4.2.4 The limit R!1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Discussion of numerical evaluations and physical implications . . . . . . . . . . . . 26
15 Sine-Gordon solitons on S 28
5.1 Classical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.1 Periodic boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.2 Anti-periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 1-loop Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2.1 Spectral discriminant for n=1 Lam´e equation . . . . . . . . . . . . . . . . 31
5.2.2 Periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.3 Anti-periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 The generalized Jacobi functions 39
6.1 Towards a generalization of the Lam´e equation . . . . . . . . . . . . . . . . . . . . 39
6.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 The integrals of generalized Jacobi functions. . . . . . . . . . . . . . . . . . . . . . 44
6.4 Relation to theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.5 Addition Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.6 Special Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
12 CONTENTS
7 The generalized Lam´e equation 48
7.1 Relation to the generalized Ince equation . . . . . . . . . . . . . . . . . . . . . . . 48
7.1.1 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 Quasi-doubly periodic solutions to the generalized Lam´e equation . . . . . . . . . . 49
7.2.1 Even functions with period … . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2.2 Odd with period … . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.2.3 Odd functions with period 2… . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2.4 Even with period 2… . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2.5 Further polynomial solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2.6 Expansion in s(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Hermite-like ansatz for the generalized Lam´e equation . . . . . . . . . . . . . . . . 54
7.3.1 The case n=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3.2 The case n=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8 Further models with periodic kink solutions 59
68.1 ` model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.2 The Double Sine-Gordon model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.3 The Jacobi model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9 Summary and outlook 64
A Elliptic Functions 66
A.1 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.1.1 Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.2 Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.2.1 Fundamental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.2.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2.3 Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2.4 Addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2.5 Glaisher’s notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2.6 Jacobi Zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B Miscellaneous calculational details 68
B.1 First derivative of the n=2 quasi-momentum . . . . . . . . . . . . . . . . . . . . . 68
B.2 n=2 Spectral discriminant in the limit k!0 . . . . . . . . . . . . . . . . . . . . . 69
C RecurrencerelationsforpowerseriesexpansionsingeneralizedJacobifunctions 70
D Expansions 74Chapter 1
Introduction
Since the pioneering work of Casimir [1], it is known that physical quantities, such as the ground
state energy or the mass [2, 3], in quantum field theories depend on the geometry and topology
which are imposed on these fields. There are also field theories, especially scalar fields in 1+1
dimensions, admitting classical soliton and kink solutions, which become quantum states of new
heavy particles, stabilized by their topological charge [4]. It is of interest, what happens e.g. to
1a sine-Gordon soliton when placed on a compact S space, which is realized by imposing periodic
boundaryconditionsonthebackgroundfieldconfiguration. Canitexistatall? Howdoesthemass
depend on the radius or circumference R of the compact dimension? It has also been known for a
long time, that on non-simply connected spaces there can exist besides the standard scalar fields
topological twisted fields [5, 6] with anti-periodic boundary conditions.
Recently there is renewed interest in phenomenological and theoretical aspects of this kind
4of field theories. The twisted version of the ` -model with kink [8, 9] possess some interesting
features, e.g. there appears a critical radius R for the compactified dimension, so that for R<R0 0
a twisted kink solution does not exist. Even more, for R > R the twisted kink is energetically0
preferred, compared to the constant field configuration, which can be interpreted as spontaneous
breakdown of translational invariance in the compact dimension [9]. In [10] this was proposed as
a new mechanism for spontaneous breaking of supersymmetry.
Compact spaces are also important in superstring theories, which are consistent only in ten
space-time dimensions. If these theories describe the observed physical world, one has to explain
why six space dimensions remain compactified and unobservable small. There are proposals that
a Casimir energy with a nontrivial behaviour with respect to the size of the compact dimensions
may play a significant role in their stabilization [11].
Assuming that in a (1+1)-dim. quantum field theory the radius R and a mass scale 1=m are
the only parameters with dimensions of length (c =~ = 1) then from dimensional considerations
the ground state energy has the general property E(R) = f(r)=R, where the scaling function
f(r) only depends on the dimensionless parameter r = Rm. These scaling functions contain
information about the conformal field theory in the UV-fixpoint reached for r ! 0, e.g. the
anomalous dimensions of the operators corresponding to the ground state [7, 8].
4To understand the quantum properties of the twisted ` -model in the semiclassical regime,
one has to consider the 1-loop corrections to the ground state. For R < R the spectrum of the0
fluctuation equation can be found in [8]. For R>R the ground state is the twisted kink and one0
has to quantize the small fluctuations in a spatial non-constant background, where the fluctuation
equationisthen=2Lameequation,whichisaquasi-exactlysolvabledifferentialequation[12,13].
This means that only a finite subset of the (anti-)periodic spectrum is exactly known. Therefore
only approximate expressions for the mass of the kink for R… R and R!1 were obtained so0
far [8].
InthisworkIusethecontourintegralrepresentationofspectralzetafunctions[14,15]toobtain
34 Introduction
4analytic expressions for the 1-loop quantum corrections to the mass of the twisted ` kink and the
1sine-Gordon soliton on S valid for all values of R, without explicit knowledge of the fluctuation
spectrum. This is possible since for the integral representation only an implicit knowledge of the
spectrum in terms of the spectral discriminant suffices to determine the 1-loop energy of a smooth
background field configuration (for an early attempt in this direction see [16, 17]). This method
was successfully applied to Casimir energy calculations (for a review see [15] or [21]) or to the
evaluation of functional determinants [32, 33].
I construct the spectral discriminant of the n = 2 Lam´e equation in terms of Jacobi’s elliptic
functions [19] by solving a corresponding set of transcendental equations. These equations have
been known for a long time [19] and are the Bethe ansatz equations for the n = 2 Lame equation
[53], since the problem of solving a differential equation is shifted to the equivalent problem of
solving certain algebraic or transcendental equations. This is possible because the Lam´e equation
is an example of a Schr odinger equation with finite gap potential [36], which have the special
property, that the spectrum has only a finite number of forbidden bands. Although the case
n = 2 was considered in [17] an explicit construction of the spectral discriminant appropriate for
numerical evaluations was missing there.
The renormalized expression for the 1-loop quantum mass of the twisted kink in the sector
R > R obtained by this method interpolates continuously between the well known result for the0
4ordinary kink of the ` -model [14, 4] for R!1 and the ground state energy in the sector R<R0
[8]. The physical energy which is the sum of the classical and renormalized 1-loop contributions
develops a minimum as function of R, which indicates the existence of an energetically preferred
radius R <R .min 0
With this method I also worked out the renormalized 1-loop contribution for the sine-Gordon
1soliton on S [8, 28], where the corresponding fluctuation equation is the n = 1 Lam´e equation.
The spectral discriminant for n = 1 was found in [16]. The sine-Gordon theory is an example
of an integrable theory also in the quantized version [37], where at least numerically all physical
quantities can be obtained. For the sine-Gordon theory therefore exist exact numerical results for
1the soliton mass on S [40]. Comparing our semiclassical results with [40], it seems that for the
1sine-Gordon model on S the semiclassical approximation gives valuable results also beyond the
semiclassical regime.
The Lam´e equation also appears when considering periodic soliton solutions of the Korteweg-
de-Vries equation. These soliton solutions can be obtained by solving an auxiliary pair of linear
equations (Lax pair) [36], one of which is the Schr odinger equation with a periodic finite gap
1potential , where the Lam´e equation is an example of. In this thesis the Lam´e equation appears as
fluctuation equation of the classical soliton of sine-Gordon theory (known as integrable) and the
4 1classical kink of ` theory (known as non-integrable) on S .
In the second part of this thesis I show that the semiclassical quantum fluctuations around
6kink solutions of the double sine-Gordon model [27], ` -theory [8] and the Jacobi model [29] on
1S are uniformly described by a generalization of the Lam´e equation. While the Lam´e equation
is equivalent to a second order ordinary differential equation of Fuchsian type with four regular
singular points and one parameter n, the generalization of the Lam´e equation has five points and five free numerical parameters.
IintroducetheconceptofgeneralizedJacobiellipticfunctions,whichmakethediscussionofthe
properties of the generalized Lam´e equation very illuminative. In [26] I had presented for the first
time some mathematical results for this type of differential equation for which up to now nearly
nothing was known [8, 42]. The main result is that the possess exactly 15 polynomial
solutions in term of generalized Jacobi functions.
4In Chapter 2 I present the ` model with anti-periodic boundary conditions and the corre-
sponding twisted kink solution. I discuss some well known properties of the Lame equation in the
context of its appearance as 1-loop fluctuation equation of the twisted kink.
1non-periodic solitons of KdV-equations are related to Schr odinger equations with reflectionless potentials

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