TVE 16 030 maj Examensarbete 15 hp Juni 2016

A summary on Solitons in Quantum field theory

Hugo Laurell Abstract A summary on Solitons in Quantum field theory

Hugo Laurell

Teknisk- naturvetenskaplig fakultet UTH-enheten This text is a brief summary on solitons in quantum field theory. An introduction to quantum Besöksadress: field theory is made through which certain Ångströmlaboratoriet Lägerhyddsvägen 1 notions in the theory of solitons in quantum field Hus 4, Plan 0 theory is possible to understand. The explicit time-independent kink solutions are computed Postadress: for the phi^4- and the sine-Gordon theory. The Box 536 751 21 Uppsala time-dependent kink solutions were then found by Lorentz transformation of the time- Telefon: independent solutions. Interesting quantities 018 – 471 30 03 such as the kink mass, the charge and the

Telefax: energy density is computed in both phi^4- and 018 – 471 30 00 the sine-Gordon theory. An investigation of geometrical methods for finding the integrals of Hemsida: motions for the sine-Gordon field is made. http://www.teknat.uu.se/student Additionally the equivalence between the sine- Gordon model and the massive Thirring model is presented.

Handledare: Luigi Tizzano Ämnesgranskare: Gopi Tummala Examinator: Martin Sjödin ISSN: 1401-5757, TVE 16 030 maj Contents

1Introduction 2

2 Introduction to Quantum field theory 3 2.1 The transition from a discrete system to a continuous 3 2.2 Generalization for higher dimensions 4 2.3 The Euler-Lagrange equations of motion 5 2.4 the Noether theorem: Conservations laws in field theories 6 2.5 Scalar Field Theories and the Klein-Gordon field 7 2.6 Second Quantization 8 2.7 Lorentz invariant normalization 10

3Solitontheoryinscalarfieldtheories 10 3.1 Non-trivial time-independent finite energy solutions for single scalar fields 11 3.1.1 The explicit time independent sine-Gordon solution 12 3.1.2 The explicit time independent 4-field solution 13 3.2 Stability of the kink 14 3.3 Taylor expansion of the sine-Gordon potential around a minimum 16 3.4 Energy density calculation for the sine-Gordon and the 4-model 16 3.5 Non-trivial time-dependent finite energy solutions for single scalar fields 17 3.6 The Topological charge 18 3.7 Derrick’s theorem 19 3.8 Soliton Quantization 20 3.8.1 The weak coupling criterion 20 3.8.2 Perturbation around a minimum of the potential 22 3.8.3 Recapitulation of the diatomic molecule 23 3.8.4 Transition to the Infinitely polyatomic molecule 25 3.8.5 Quantized kink energy levels 25 3.9 Soliton interactions 27 3.9.1 Soliton-soliton interaction 28 3.9.2 Soliton-anstisoliton interaction 29 3.9.3 Breather interaction 30

4Geometricalmethodsinthetheoryofsolitons 31 4.1 Parallel transport and Gauge transformations 32 4.2 An interesting duality for the sine-Gordon model 35

5Summary 37

–1– Populärvetenskaplig sammanfattning

Den här texten sammanfattar och förenklar litteratur kring solitoner i kvantfältteori. Kvant- fältteori är den delen av fysiken där man både behandlar mikroskopiska och makroskopiska fenomen samtidigt. Dessutom betraktar man systemet som ett fält istället för punktvisa generaliserade koordinater. Fördelen med att komprimera oändligt många koordinater i en fält konfiguration är att det blir mycket lättare att räkna med system av högre dimension och komplexitet. En soliton är en ensam vågpuls som inte minskar i styrka med tiden. Om två solitoner kolliderar överlever båda solitonerna kollisionen och fortsätter rörelsen med en fasvridning. Solitonen är inte ett oberoende objekt utan är starkt sammanbunden med sin motpart anti-solitonen. Mellan solitonen och anti-solitonen verkar attraktiv växelverkan samt mellan en soliton och soliton verkan repulsiv växelverkan. Den här texten behandlar främst solitoner i skalära fältteorier, fält där partiklarna som existerar är bosoner. Ett exempel på en boson är fotonen. En speciell egenskap hos bosoner är att man kan placera oändligt många bosoner på varandra utan att de repellerar vandra. Bosoner verkar med svag växelverkan i relation till till exempel fermioner. Ett exempel på en fermion är elektro- nen. Man kan inte placera flera elektroner på varandra. Om man försöker bildas en mycket stor kraft som gör att elektronerna repelleras från varandra. Kraften genereras som en kon- sekvens av Paulis uteslutningsprincip. Det visar sig att sine-Gordon modellen som beskriver ett skalärt fält och massiva Thirring modellen som beskriver ett Dirac fält med fermioner som elementarpartiklar är ekvivalenta under en indentifikation. Solitoner i kvantfältteori modellerar oupptäckta partiklar så som kosmiska strängar. Kosmiska strängar är endimen- sionella topologiska defekter bildade tidigt i universums historia. Om kosmiska strängar skulle detekteras skulle detta vara ett hinder för att förena generell relativitetsteori med kvantfysiken en förening som är en av de största utmaningarna i modern teoretisk fysik.

1Introduction

Finite energy solutions are crucial to understand the interplay between the topology of space-time and physical phenomena. It is very important to deepen our understanding of these kind of solutions because they might be useful in the discovery of new physical phenomena. The study of one (1+1)-dimensional solitons can be thought of as a laboration where a much easier analysis of the properties and consequences of the solitons can be made. For these experiments to say something about the real world the dimension of the solitons can then be inflated. The real-time solitons in quantum field theory model undiscovered particles such as the magnetic monopole and can be extended to model cosmic strings. The hypothetical cosmic string is a one-dimensional topologic defect formed in the early universe. If such a hypothetical particle would be measured it could constrain the unifying of the two major disciplines in modern physics the quantum physics and general relativity. Soliton theory is extremely important in applications such as non-linear optics where self- enforcing solitary waves can be used to increase performance of information transmission in optical fibres. With a continuously increasing demand on the data rate in the optical fiber network the optical pulses are made shorter and shorter. However, with shorter pulses

–2– the spectrum broadens, as understood from the uncertainty principle. The redshifted light of the pulse travels faster than the blue shifted, due to dispersion, which make the pulse stretch out in time. After some propagation distance the pulses in the fiber will become superimposed on each other making the information inaccessible, which ultimately limits the length over which the fiber can be used. However, by increasing the amplitude of the pulse a self-stabilizing pulse can be formed, an optical soliton, where the refractive index of the fiber changes due to non-linear effects such that the speed of the redshifted part of the pulse becomes equal to the blue shifted part. This means that the pulse becomes self enforcing making the information accessible for high frequencies [1]. This is just but one of the applications of solitons in modern science and there is a lot more to unveil in the subject.

2IntroductiontoQuantumfieldtheory

2.1 The transition from a discrete system to a continuous In this section I will briefly introduce the basics of classical field theory and continue with quantum field theory. I will follow closely the method of [2], [3]. To develop a classical field theory first we must look at the transition from a set of discrete generalized coordinates qn, into a continuous field coordinate (x). An example is coupled oscillations in a infinitely long and thin elastic rod. As Goldstein describes the generalized coordinates, qn, are placed such that qn measures the amplitude at x and qn+1 measures the amplitude at x + a and so forth. The total Lagrangian for this system looks as follows.

n n n 1 2 1 2 L = Li = Ti Ui = mq˙i k(qi qi 1) 2 2 Xi=1 Xi=1 Xi=1 Where k is the spring constant and m is the mass of the rod. This expression can also be written as

n 2 1 m 2 1 qi qi 1 L = a q˙i ka 2 " a 2 a # Xi=1 ✓ ◆ The the following five limits describes how the relevant quantities change as the separa- tion between the equally spaced coordinates go to zero and the discrete system becomes continuous.

n qi qi 1 @ m lim a = dx, lim qn = (x), lim = , lim = ⇢, lim ka = Y a 0 a 0 a 0 a @x a 0 a a 0 ! ! ! ! ! ! Xi=1 Z ✓ ◆ Here Y is the Young’s modulus for the material and ⇢ is the density. Using the limits above it is possible to transform the Lagrangian quantity to a Lagrangian density. A continuous

–3– quantity described by the limit below.

n 2 1 m 2 1 qi qi 1 lim L =lim a q˙i ka = a 0 a 0 2 a 2 a ! ! " #! Xi=1 ✓ ◆ 1 @(x) 2 ⇢˙(x)2 Y dx = dx 2 @x L Z Z where is the Lagrangian density. Now the formalism have transformed from n discrete L coordinates to one continuous field coordinate which have compactified the problem sig- nificantly. Equivalently the Hamiltonian can be described as the space integral over the Hamiltonian density. If we consider the example of the continuous rod we get the following expression describing the Hamiltonian.

@L @Li pi = = a @q˙ @q˙i n n @Li H = piq˙i L = a q˙i aLi @q˙i Xi=1 Xi=1 Under the limit a 0 the Hamiltonian transforms to ! @ H = dx L˙ @˙ L Z ✓ ◆ Another interesting quantity is the canonical momentum. It is the momenta of the field in an infinitesimal section of the field. Therefore the canonical momenta is defined as the following limit.

pi @ lim ⇡ = L (2.1) a 0 a ⌘ ˙ ! @ Which gives the following final expression for the continuous Hamiltonian in the example of coupled oscillations in an infinitely long and thin rod.

H = dx ⇡˙ = dx L H Z ⇣ ⌘ Z 2.2 Generalization for higher dimensions To generalize the classical field theory for higher dimensions we simply transform the scalar quantity x to a vector. More specifically a four-vector,

x xµ 7! where xµ is the four-vector and spans µ = 0, 1, 2, 3 . The differentiation operator then { } transforms to @ @ @µ @x 7! ⌘ @xµ

–4– Including Lorentz invariance in our description we can define the metric in Minkowski space with the following convention ( , +, +, +) and defining the one-form. 1000 0100 g = 0 1 µ⌫ 0010 B C B 0001C B C @ µ A @µ = gµ⌫@ This means that we consider time and space as equals with the exception of a minus-sign. Time and space are just extensions of an four-dimensional euclidian geometry. For example if one walks 1 positive unit distance in space the separation is equivalent to standing still and waiting one positive unit time. This is of course only true when all fundamental constants of nature are set to 1. The Lagrangian density is now a function of the four-gradient of the field and the field configuration, (@ ,), so the action over the Lagrangian density L µ becomes S = Ldt= (@ ,) d4x L µ Z Z This action is perhaps the most important quantity in field theories. It has the unit of [Energy time] and as we will se later on using the Hamilton’s principle the variation of the ⇥ action can give us the integrals of motion.

2.3 The Euler-Lagrange equations of motion Hamilton’s principle is a central part of all physics. It states that the variation of the action must be zero. Marpertuis who is credited for the formulation of the principle of least action, the forerunner of the Hamilton’s principle, said the following regarding the principle of least action [4]. “The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements”. -Pierre Louis Maupertuis Applying Hamilton’s principle on the action S will generate the Euler-Lagrange equations of motion for the field. Hence if the variation of the action is zero the Lagrangian will be an extremum, most commonly a minimum. S =0 @ @ S = d4x (@ ,)= d4x L (@ )+ L L µ @(@ ) µ @ Z Z ✓ µ ◆ @ @ @ = d4x L @ L + @ L @ µ @(@ ) µ @(@ ) Z ✓ ✓ µ ◆ ✓ µ ◆◆

–5– The third term vanishes since according to the divergence theorem. @ @ d4x@ L = d3x L =0 µ @(@ ) @(@ ) Z ✓ µ ◆ I µ Since the 3-integral is on the boundary of the 4-volume the variation of the field configura- tion, , is zero by definition on the entire 3-surface. Hence the 3-integral is equal to zero. This gives the following variation of the action. @ @ S = d4x L @ L =0 @ µ @(@ ) Z ✓ µ ◆ Since is arbitrary and non-zero inside of the 3-surface the expression in the brackets must be zero. @ @ L @ L =0 @ µ @(@ ) ✓ µ ◆ Giving the Euler-Lagrange equations for the 4-dimensional Lorentz invariant field.

2.4 the Noether theorem: Conservations laws in field theories Emmy Noether’s theorem generalizes conservation law’s for conformal field theories under special transformations called symmetries. A symmetry is a transform that leaves the equations of motion unchanged, where the equations of motion are given by the Euler- Lagrange equations. Consider the following transformations.

0 = + ↵@ 7! µ µ 0 = + ↵@ L7! L L µJ As we saw earlier a variation of a four-divergence will not change the equations of motions since the four-divergence term will vanish when we consider the variation of the action. The transformations above are therefore symmetries. Consider the variation of the transformed Lagrangian density. @ @ @ @ @ (@ ,)= L + L (@ )= L @ L + @ L L µ @ @(@ ) µ @ µ @(@ ) µ @(@ ) µ ✓ µ ◆ ✓ µ ◆ @ @ @ = L @ L + @ L @ µ @(@ ) µ @(@ ) ✓ ✓ µ ◆◆ ✓ µ ◆ The first term is the Euler-Lagrange equation term and is zero by the formulation. As the transformation is a symmetry up to a four-divergence the variation of the Lagrangian is equal to ↵@ µ, which gives. µJ @ (@ ,)=@ L = ↵@ µ L µ µ @(@ ) µJ ✓ µ ◆ Inserting the variation of the field parameter the expression can be written as. @ @ L @ µ =0 µ @(@ ) µ J ✓ µ ◆ µ @ µ j = L @µ @(@µ) J

–6– These two equations are the Emmy Noether conservation law’s. If the transformation is a translation the resulting conserved current will be the Stress-Energy tensor. Consider the following 4-space translation.

µ µ µ µ x x 0 = x a 7! µ 0 = + a @ 7! µ µ ⌫ µ 0 = + a @ = + a @ ( ) L7!L L µL L µ ⌫ L These transformations are clearly symmetries which means we can consider the variation of the translated Lagrangian density @ @ @ @ @ (@ ,)= L + L (@ )= L @ L + @ L L µ @ @(@ ) µ @ µ @(@ ) µ @(@ ) µ ✓ µ ◆ ✓ µ ◆ @ = @ L = a⌫@ (µ ) µ @(@ ) µ ⌫ L ✓ µ ◆ @ @ L @ µ =0 µ @(@ ) ⌫ ⌫ L ✓ µ ◆ µ @ µ T⌫ = L @⌫ ⌫ @(@µ) L µ Where T⌫ is the Stress-Energy tensor. The conserved charges and the total momentum are given by

H = T 00d3x = d3x, (2.2) H Z Z P i = T 0id3x = ⇡@ d3x (2.3) i Z Z 2.5 Scalar Field Theories and the Klein-Gordon field The Klein-Gordon field is a scalar field described by the Klein-Gordon equation

2 (⇤ + m )(¯x, t)=0

µ The ⇤ operator is the D’Alembertian which can also be written as @µ@ . The m in the equation can be interpreted as the mass of the particle. For simplicity fundamental con- stants of physics are often chosen to 1 in QFT, further on the values of the propagation speed c and the Planck constant ~ will be set equal to 1. Writing the wave function (¯x, t) in momentum space, Fourier-expanded with momentum as the Fourier-variable, the Klein- Gordon equation will transform to a much simpler equation where the solution is a simple harmonic oscillator. 3 d p ip¯ x¯ (¯x, t)= e · (¯p, t) (2⇡)3 Z @2 +¯p2 + m2 (¯p, t)=0 @t2 ✓ ◆ Since the solution to the Klein-Gordon equation in momentum space is a harmonic oscillator we can interpret the result as quantum harmonic oscillator with frequency ! = p¯2 + m2. p

–7– 2.6 Second Quantization For the quantum harmonic oscillator the following commutator relations holds.

[qi,pj]=iij

[qi,qj]=[pi,pj]=0

The Hamiltonian of the harmonic oscillator can be written as 1 1 H = p2 + !2q2 (2.4) 2 2 By inferring creation and annihilation operators the generalized coordinate and the mo- mentum operator can be expressed as follows.

! i ! i a = q + p, a† = q p 2 p 2 p r 2! r 2! 1 ! q = (a + a†),p= i (a a†) p 2 2! r Inserting into the commutator relation between the momentum operator and the generalized coordinate the momentum operator expressed as creation and annihilation operator gives the following commutator relation between the creation and annihilation operator.

[a, a†]=1

Inserting the representation as creation and annihilation operators into the Hamiltonian gives

1 H = ! a†a + 2 ✓ ◆ Now we want to look at a generalization of the commutator relations for the continuous field theory as the generalized coordinate transforms to the wave function and the momentum operator to the canonical momentum operator. It is reasonable that the Kroenecker delta function transforms into the Dirac delta distribution.

[(¯x),⇡(¯y)] = i(3)(¯x y¯) [(¯x),(¯y)] = [⇡(¯x),⇡(¯y)] = 0

Writing the wave functional in momentum space as a linear combination of the creation and annihilation operator gives the form of the wave functional and the canonical momentum in scalar field theories. 3 d p 1 ip¯ x¯ ip¯ x¯ (¯x)= (ape · + a†e · ) (2⇡)3 2! p Z p 3 d p p !p ip¯ x¯ ip¯ x¯ ⇡(¯x)= ( i) (a e · a†e · ) (2⇡)3 2 p p Z r

–8– The commutation relations for the creation and annihilation operators become

3 (3) [a ,a†]=(2⇡) (¯p q¯) p q [ap,aq]=[ap†,aq†]=0

Inserting the quantized wave functional and canonical momentum in the expression of the Hamiltonian 2.4 for the harmonic oscillator gives. 3 3 3 3 2 d pd qd x 1 ipx ipx iqx iqx d x = ape + a†e aqe + a†e (2⇡)6 4! ! p p Z Z p q 3 3 3 ⇣ ⌘⇣ ⌘ d pd qd x p 1 ix(p q) ix(p q) = apa†e + a†aqe (2⇡)6 4! ! q p Z p q 3 3 ⇣ ⌘ d pd q p1 3 (3) = (2⇡) (p q) apa† + a†aq (2⇡)6 4! ! q p Z p q 3 ⇣ ⌘ p d p 1 = 3 apap† + ap†ap (2⇡) 2!p Z ⇣ ⌘

3 3 3 3 2 d pd qd x 2 !p!q ipx ipx iqx iqx d x⇡ = ( i) a e a†e a e a†e (2⇡)6 4 p p q q Z Z r 3 3 3 ⇣ ⌘⇣ ⌘ d pd qd x !p!q ix(p q) ix(p q) = a a†e + a†a e (2⇡)6 4 p q p q Z r 3 3 ⇣ ⌘ d pd q !p!q 3 (3) = (2⇡) (p q) a a† + a†a (2⇡)6 4 p q p q r Z 3 ⇣ ⌘ d p !p = a a† + a†a (2⇡)3 2 p p p p Z ⇣ ⌘

3 3 1 d p d p 1 3 (3) H = ! a a† + a†a = ! a†a + (2⇡) (0) 2 (2⇡)3 p p p p p (2⇡)3 p p p 2 Z h i Z  The last term in the Hamiltonian corresponds to an infinite energy. This infinity will not be a problem since we are interested in energy differences. It can easily be verified that

[H, a†]=! a† [H, a ]= ! a p p p p p p This means that we can construct a particle with momentum p with the following operation

p¯ = a† 0 | i p | i Where 0 is the empty state. The eigenvalue of the Hamiltonian acting on a state p is the | i 2 2 frequency !p = p¯ + m .

p H p¯ = ! p¯ | i p | i The total momentum operator can be defined from the expression from the Noether theorem 3 3 d p P = ⇡@ d x = pa¯ †a i (2⇡)3 p p Z Z

–9– 2.7 Lorentz invariant normalization The vacuum is normalized as.

0 0 =1 h | i The normalization between two particle states are.

p¯ q¯ =(2⇡)3(3)(¯p q¯) h | i To construct a Lorentz invariant normalisation we must find the Lorentz invariant measure. The Lorentz invariant measure is given by.

d3p 2E Z p

Where Ep is given by the relativistic dispersion relation.

2 2 Ep = p¯ + m p This can be shown rather easily. The following measure is obviously Lorentz invariant.

d4p Z Using the relativistic dispersion relation.

p pµ = m2 p2 = E2 =¯p2 + m2 µ ! 0 p Combining these two quantities we can construct a third Lorentz invariant quantity.

3 3 4 2 2 2 d p d p d p(p0 p¯ m ) = = 2p0 2Ep Z p0>0 Z Z This gives us the Lorentz invariant delta function.

2E (3)(¯p q¯) p p Since d3p 2E (3)(¯p q¯)=1 2E p Z p 3Solitontheoryinscalarfieldtheories

A soliton solution to a field equation is a non-dissipative non-trivial finite energy solution. They are a subset of kinks which are also non dissipative non-trivial finite energy solutions what distinguishes the kinks from the solitons is that solitons remain unperturbed in colli- sions with other solitons, while this is not the case for kinks. A necessary condition for the existence of soliton and kink solutions to a field equation is that the potential energy has

–10– at least two degenerate minimas. This is a consequence of the bounded energy of the kink. The standard Lagrangian for scalar field theories is as follows

1 = (@ )(@µ) U() (3.1) L 2 µ Where U() is any potential. In this text two different scalar theories will be examined: the 4-theory and the sine-Gordon theory. The finite energy solutions are called Kinks by Weinberg [5] and lumps by Coleman [6]. Coleman tells us that he will not call them solitons since a soliton is a precisely defined mathematical object which some kinks/lumps does not fulfil. The difference between solitons and kinks as stated above is that solitons are unperturbed by collisions with other solitons while kinks are not [7].

3.1 Non-trivial time-independent finite energy solutions for single scalar fields The kinetic energy is the term in the scalar field Lagrangian density (3.1) associated with the square of the time derivative of the field. The potential energy is defined as the remaining terms, the space derivatives of the field and the field potential. For there to exist non-trivial finite energy solutions the potential U(x) must have at least two degenerate minimas. The time-independent hamiltonian density is given by.

1 = (@ )2 + U() H 2 x For the energy to remain finite the potential U() must go to a minimum as x .Soif !±1 the potential has only one zero this means that the solution (x) must be at the minimum for x = if the energy is to be finite. This is of course a trivial finite energy solution. ±1 However if U() has at least two zeros it is possible for the solution to be at one minima of the potential at x = and another minima of the potential at x = . This means that 1 1 all non-trivial finite energy solutions travel from one minima of the potential at x = 1 to another minima at x = . If the Noether theorem is applied on the field the equation 1 of motion is found.

@ µ @ @U µ @U L = @ , L = , @⌫@ + =0 @(@µ) @ @ ) @

Considering the (1 + 1)-dimensional field where µ = 0, 1 .Forthe(1 + 1)-dimensional { } field the equations of motion becomes.

@U @2 @2 + =0 t x @

First the static solutions will be studied, the solutions where @t =0. After the analy- sis of the time independent solutions through Lorentz transformation the time-dependent solutions will be found. For the time independent scalar field the action becomes.

–11– 1 S = dx (@ )2 + U() 2 x Z ✓ ◆ Under the change of variables x, x t, it is possible to recognise the time-independent 7! 7! problem as the problem in classical mechanics of a particle with unit mass moving in the potential U(x).

1 S = dt (@ x)2 + U(x) =0 2 t Z ✓ ◆ From classical mechanics we know that the particle’s motion has zero total energy which means that. 1 (@ x)2 U(x)=0 2 t Making the inverse change of variables, x , t x, gives the very important relation 7! 7! in soliton theory, the relation from which the explicit time independent solution can be computed.

1 (@ )2 = U() (3.2) 2 x @ = 2U() (3.3) @x ± p By integration equation 3.3 gives the explicit formula for finding the time independent kink solutions. The plus-minus sign on the right hand side has some interesting consequences. If one follows the positive branch when computing the explicit time independent solution the kink solution is found. If the negative branch is followed the anti-kink solution is found.

2 d x2 = dx = x (3.4) 2U() ± ± Z1 Zx1 3.1.1 The explicit time independentp sine-Gordon solution The sine-Gordon scalar field is interesting in many aspects. It is periodic and has infinite degenerate minimas. The potential of the sine-Gordon field is given by.

m2 U()= (1 cos ) 2 This means that the Lagrangian for the sine-Gordon system is defined as follows.

1 1 m2 = ⇡2 (@ )2 (1 cos ) (3.5) L 2 2 x 2

–12– Remember that the Hamiltonian is a function of the Lagrangian density

H = dxT 0 = dx(@ @0 0 ) 0 0 0L Z Z The only aspect of this Hamiltonian that differs from another scalar field hamiltonian is the last term of the integral, the field potential term. The Hamiltonian for the sine-Gordon field is defined as follows.

1 1 m2 H = ⇡2 + (@ )2 + (1 cos ) 2 2 x 2 ZR ✓ ◆ The sine-Gordon potential is periodic and has degenerate minimas at

2n⇡ = ,n Z 2

Inserting the sine-Gordon potential into the explicit integral formula 3.4 gives a way to compute the time independent kink’s for the sine-Gordon field.

2 d 2 d 2 2 1 = = = arcsin t, d = dt 2m p 2 1 m2 1 sin 2 1 t Z 2 2 (1 cos ) Z ⇢ r ⇣ ⌘ 1 t2 1 1 p1 t2 1 s2 1 = dt = s = 1 t2,dt= ds = ds 2 2 m t1 t p1 t ( t ) m s1 s 1 Z p Z 1 s2 1 1 s2 1 1 1 1 s s2 = ds = + ds = ln = x m (s 1)(s + 1) 2m s 1 s +1 2m 1+s ± Zs1 Zs1 ✓ ◆ s1 2 1 cos 2 sin 4 2mx mx 4 mx = = e± tan = e± (x)= arctan e± ) 2 ) 4 ) 1 + cos 2 cos 4 Hence the solution to the time-independent sine-Gordon equation is

4 mx (x)= arctan e± (3.6)

Remember that the plus-minus determines if the solution is a kink or a anti-kink. A plus sign means the solution is a kink and a minus sign an anti-kink.

3.1.2 The explicit time independent 4-field solution For the 4- field the potential is given by.

m2 U()= (2 v2)2,v= 4 r

–13– Where the relationship between the weak-coupling , and v is as follows.

2 2 v = = (3.7) m2 The integral equation becomes.

2 2 d = x 2 v2 ± r Z1 2 2 d 2 1 v = ln = x , 2 v2 2v + v ± r Z1 r 0 m x m x + v e± p2 e⌥ p2 m mp2x = e⌥ = v m x m x = v tanh x v , p2 p2 ±p2 e± + e⌥ This means that the time independent for the 4-field is. m (x)=v tanh ± x p2 The kink is the solution obtained by following the positive branch of the -sign. ± m (x)=v tanh x p2 The anti-kink is found by following the negative branch. m (x)=v tanh x p2 3.2 Stability of the kink To analyse the stability of the kink we study the first order perturbation around a minima of the potential under the assumption that there are at least two degenerate minimas of the potential enabling non dissipative finite energy solutions. The expansion of the potential around the minima min look’s as follows where (x, t) is an arbitrary function of small oscillations.

U( )( )0 U ( )( )1 U ( )( )2 U( )= min min + 0 min min + 00 min min + ( 3) 0! 1! 2! O The definition of a minima of U() is defined as follows.

U 0(min)=0

The Lagrangian for the scalar field becomes.

1 U ( )( )1 U ( )( )2 = (@ )(@µ ) U( ) 0 min min 00 min min ( 3) L 2 µ min 1! 2! O

–14– Through the Noether theorem the equation’s of motion for the Lagrangian is found.

µ 2 @ @ + U 00( ) + ( )=0 v O In first order perturbation theory this becomes. µ @v@ (x, t)+U 00(min) (x, t)=0 (3.8) This equation is invariant under time translation since it is a equation generated from the Noether theorem which means that the solutions to equation (3.8) can be written as.

i!nt (x, t)=Re ane n(x) (3.9) n X and !n and n is determined by. 2 2 @ + U 00( ) = ! (3.10) x n min n n n The solutions (3.9) are stable if and only if the eigenfrequencies are real. This can be showed by the following proof. Consider the real eigenfrequencies

!n R 2 For the solution to be stable means that it does not diverge as t . This can be !±1 written as lim (x, t) < ⌦ t !±1 | | Where ⌦ is a finite constant. For the real eigenfrequencies this becomes.

i!nt lim Re ane n(x) t !±1 n X Using the Triangular inequality this is less or equal to

i!nt i!nt lim ane n(x) lim an n(x) e =lim an n(x) t  t | | t | | !±1 n !±1 n !±1 n X X X an n(x) is bounded by definition. This means that. lim an n(x) < ⌦ t | | !±1 n X This shows that if the eigenfrequencies are real the solitons are stable. lim (x, t) < ⌦ t !±1 | | Now consider the arbitrary complex eigenfrequencies 2 !n = an + ibn, (an,bn) R 2 Inserting this expression in the stability condition gives

(ian bn)t lim (x, t) =lim Re ane n(x) t | | t !±1 !±1 n X The limit above is either zero or infinite, hence the eigenfrequencies must be real for stability.

–15– 3.3 Taylor expansion of the sine-Gordon potential around a minimum One minimum of the periodic sine-Gordon potential is =0. This is a good choice to expand around since the Taylor expansion becomes the simpler MacLaurin expansion. m2 U()= (1 cos ) 2

@U m2 @2U = sin = m2 cos @ @2 @3U @4U = m2 sin = 2m2 cos @3 @4 @5U @6U = 3m2 sin = 4m2 cos @5 @6

m22 m224 m246 m2 1 ( )2n U()= + + (8)= 2! 4! 6! O 2 (2n)! n=1 X The first term in the expansion is the classical harmonic potential.

3.4 Energy density calculation for the sine-Gordon and the 4-model The total energy and the hamiltonian are the same quantities. The energy integral is given by the hamiltonian from the Noether theorem 2.2. 1 1 E = dx = dx (@ )2 + (@ )2 + U() E 2 0 2 1 Z Z  Where is the energy density. Using the identity in 3.2 gives the total energy as. E 2 E = dx(@1) = dx 2U() Z Z For the sine-Gordon model the time-independent energy density becomes 2m2 2m2 =2U()= (1 cos )= sin2 E 2 2 2 The simplest way to compute the total energy of the sine-Gordon field is through the integral of the space derivative of the analytic solution. 2 mx 2 1 16 4m e± 16m 1 1 16m E = dx(@1) = 2 2mx dx = 2 2mx = 2 2 (1 + e± ) ± (1 + e± ) Z Z  1 16m So the total energy for a single soliton in the sine-Gordon model is 2 . This quantity is sometimes referred to as the soliton mass. For the 4-model the energy density is defined as. =2U()= (2 v2)2 E 2

–16– The total energy or the soliton mass for a single soliton becomes.

2v v E = dx2U()= dx (2 v2)2 = ± d 2 v2 2 ⌥p2m Z Z ✓ ◆ Z0 2v 2v3 2p2m3 = = ⌥p2m ⌥ 3 3 ✓ ◆ 3.5 Non-trivial time-dependent finite energy solutions for single scalar fields

To find the time-dependance of the soliton solutions we simply Lorentz transform the time- independent solutions.

x ut x 7! p1 u2 The solution for the sine-Gordon field transforms to

x ut 4 m ± p1 u2 (x, t)= arctan e ✓ ◆

The solution for the 4-field transforms to

m(x ut) (x, t)=v tanh ± " 2(1 u2) # p The soliton masses transforms under the Lorentz transformation as follows.

E E 7! p1 u2 So when the soliton velocity approaches one the mass diverges, as would be expected from special relativity. The solitons are similar to particles. Since they are Lorentz invariant the obey the standard relativistic energy relation.

E = p2 + m2 (3.11) p But there is some differences from particles. If we have one particle solution it is possible to extend it and get a asymptotic many particle solution i.e if all particles are placed at the edge of infinity their interaction goes to zero and superposition is trivial. However the same does not hold for solitons. Consider n equally space points a1 <...

–17– This has interesting consequences to the symmetry of the fields. Since the sine-Gordon 2⇡ potential is -symmetric the only observable functions in the sine-Gordon model are the functions unchanged by the following transformation, i.e are a symmetry with respect to. 2⇡ + 7!

For the 4-field the only observables are the functions symmetric with respect to

7! Figure 1 shows the graph of the sine-Gordon and the 4-potential. The symmetry of the 4-

Sine-Gordon potential 2 U(φ)

1.5 2 β

/ 1 2 m

0.5

0 -6 -4 -2 0 2 4 6 βφ

φ 4 potential 2 U(φ), a = 1

1.5

/2 1 λ

0.5

0 -1.5 -1 -0.5 0 0.5 1 1.5 φ

Figure 1.Graphoverthesine-Gordonandthe4-potential

field has some extraordinary consequences. Solitons and anti-solitons are not independent objects. A soliton must always be followed by an anti-soliton and so forth.

3.6 The Topological charge The topological charge is an integral of motion and is defined as [5].

µ = ✏µ⌫@ Jtop ⌫

–18– Where ✏µ⌫ is anti-symmetric. As we saw in the Noether theorem, if we have a conserved current the conserved charge is the space integral of the conserved current.

✏µ⌫ = ✏⌫µ,✏01 =1 For the sine-Gordon model the topological charge becomes

01 4 ⇡ ⇡ 4⇡ Qtop = dx = dx(@1)=[( ) ( )] = = R J R 1 1 ± 2 ⌥ 2 ± Z Z ⇣ ⇣ ⌘⌘ For the 4-model the topological charge becomes.

01 ⇡ ⇡ Qtop = dx = dx(@1)=[( ) ( )] = v = v⇡ R J R 1 1 ± 2 ⌥ 2 ± Z Z ⇣ ⇣ ⌘⌘ The soliton has positive charge and the anti-soliton has negative charge.

3.7 Derrick’s theorem An important question to ask is whether it is possible to find time-independent soliton solutions in a scalar field and if so how does the solutions change as the spatial dimension increases. As we have seen before it is possible to find time-independent soliton solutions in scalar field theories with one spatial dimension if the potential has at minimum two degen- erate minimas. However we have not investigated what happens if the spatial dimension is grater than one. Derrick’s Theorem says something very important regarding the existence of non-trivial time-independent finite energy soliton solutions in scalar field theories. If the spatial dimension is greater than 1 they cannot exist. I should however emphasise that this hold only for time-independent solutions.

Derricks’s theorem. Let be a set of scalar fields with one time dimension and D spatial dimensions that follows the dynamics determined by the Lagrangian density. 1 = @ @µ U() L 2 µ The potential U is non-negative and zero for the ground states. Then for D 2theonly non-trivial finite energy solutions are the ground states [6].

Proof. Define 1 V = dDx( )2 1 2 r Z D V2 = d xU() Z Where (x) is a time-independent solution. Consider the one-parameter family of field configurations given by

+ (x, ) (x),R ⌘ 2

–19– Since the solution is time-independent the energy for this family becomes

(2 D) D H()= V1 + V2

According to Hamilton’s principle this must be stationary at =1.Thisisequivalentto

@H =(2 D)V DV = ((D 2)V + DV )=0 @ 1 2 1 2 =1 We see that V1 and V2must vanish for D>2.ForD =2, V2 must vanish. The Hamiltonian becomes.

H2 = V1

Applying Hamilton’s principle on the Hamiltonian for D =2implies that V1 is zero.

@H = V =0 @ 1 =1 This is an remarkable result and quite discouraging [6]. It means that we will never find time-independent two or higher dimensional non trivial finite-energy solutions.

3.8 Soliton Quantization

In this section we will quantize the soliton of the scalar field theory. We will follow the methods of [5]. Two things are especially important to think about when quantizing the soliton. Firstly, how does the explicit form of the classical field configuration affect the quantum field configuration and secondly how is the soliton reflected in the states of a quantum theory. When considering a quantum wave-function the uncertainty of the wave- function increases as the scale diminishes according to the uncertainty principle. In fact the fluctuations diverges as the scale goes to zero.

~ x p 2 ~ lim p lim = x 0 x 0 2 1 ! ! x We want to choose a distance scale L such that this limit does not diverge at the same time as the distance scale is small with respect to the classical limit. This is ensured by the weak coupling criterion.

3.8.1 The weak coupling criterion The variation of the classical field is proportional to.

@ ( ) L L classical ⇠ @x

–20– We know the soliton solution so we can differentiate it for both the 4-model and the sine-Gordon model and find that the derivative of the wave-function is proportional to the mass and the weak coupling. @ vm mx = (1 tanh ) vm @x p2 p2 ⇠ @ Lm2 (L)classical L Lmv ⇠ @x ⇠ ⇠ p

If we define a smeared out quantum field L(¯x).

1 (¯y x¯)2 n 2 L(¯x)= n d ye 2L (¯x) (2⇡L2) 2 Z Then we are able to estimate the variation of the quantum fluctuations of the free field around a vacuum with the definition.

y¯2 z¯2 2 2 1 n n ( ) 0 (0) 0 = d yd ze 2L2 0 (¯y)(¯z) 0 L quantum ⌘h | L | i (2⇡L2)n h | | i Z n It is possible to compute 0 (¯y)(¯z) 0 remembering that 0 a aq† 0 =(2⇡) (¯p q¯). h | | i h | p | i n n d pd q 1 ipy ipy iqz iqz 0 (¯y)(¯z) 0 = 0 ape + a†e aqe + a†e 0 h | | i h | (2⇡)2n 4! ! p q | i Z p q n n ⇣ n n ⌘⇣ ⌘ d pd q 1 i(py qz) d pd q 1 i(py qz) = 0 apa†pe 0 = p q e h | (2⇡)2n 4! ! q | i (2⇡)2n 4! ! h | i Z p q Z p q n n ⇣ ⌘ n d pd q 1 (n) i(py qz) d p 1 ip(y z) = p (p q)e = p e (2⇡)n 4! ! (2⇡)n 2 2 Z p q Z 2 p + m This gives. p p

1 1 y¯2 z¯2 2 n n n 2 ip(y z) (L) d yd zd p e 2L e quantum ⌘ (2⇡L)2n 2 2 Z 2 p + m Which is equivalent to. p

n p2L2 d p e ( )2 (3.12) L quantum ⌘ (2⇡)n 2! Z p 1 The variation of the field is primarily in the region of m surrounding the kink for both the sine-Gordon model and the 4-model [5]. Therefore the length scale should be very small in relation to the inverse mass.

1 L m ⌧ mL 1 , ⌧ The behaviour of the integral 3.12 in this interval tells us

1 ln mL if n =1 (L)quantum (n 1) ⇠ 8Lq 2 n>1 < if :

–21– In the case of one dimensional solitons we can now set up the weak coupling criterion. We want the fluctuation of the classical model to be a lot bigger than the fluctuations of the quantum model. 1 Lm2 ln r mL ⌧ p L2m4 1 mLe , ⌧ The weak coupling criterion forces to be very small if the criterion is to hold for L m 1. ⌧ 3.8.2 Perturbation around a minimum of the potential Perturbation around a minimum of the potential is a classic approach in physics. In this section we will see that small perturbations around the ground state of the kink will in fact describe interaction with the kink. In section 3.1 we encountered the classical sine-Gordon Lagrangian density (3.5). If the following substitution is made 0 = ,@µ0 = @µ the Lagrangian density transforms to

1 1 µ 2 = (@ 0)(@ 0) m (1 cos 0) L 2 2 µ ✓ ◆ is the weak coupling and is an arbitrary constant. is therefore not important when considering solutions of the sine-Gordon equation since if we can find solutions without considering it is possible to find solutions when considering by scaling the solution. However the situation is different when considering the quantum version of the theory. As explained by [6]; in quantum physics the relevant quantity is not the Lagrangian density, , but the Lagrangian density divided by the reduced Planck’s constant L . This gives the L ~ following form of the Lagrangian density for the sine-Gordon model.

1 1 µ 2 L = (@ 0)(@ 0) m (1 cos 0) 2 2 µ ~ ~ ✓ ◆ We can now ask ourselves what happens to other interesting quantities such as the canonical momentum. The canonical momentum is defined as in (2.1).

@ @00 ⇡0 = L = 2 @(@00) To generalize the quantum Lagrangian density for all scalar field’s the sine-Gordon potential is simply interchanged with an abstract potential U(0).Atthesametimeas~ 1 and ! standard quantum field theory units is adopted.

1 1 µ = (@ 0)(@ 0) U(0) L 2 2 µ ✓ ◆ Let’s consider the quantum Hamiltonian for such a system.

2 2 @ 1 4 2 1 2 H = dx L @0 = dx ⇡0 + (@ 0) + U(0) (3.13) @(@ ) 0 L 2 2 x Z ✓ 0 ◆ Z ✓ ◆ 1 4 2 = dx ⇡0 + V (0) (3.14) 2 Z ✓ ◆

–22– This Hamiltonian is rather curious. The infinitesimal coupling constant is multiplied with the kinetic energy of the system. A rather common approach in physics is now to see if we can recognise such a system. In the section 2.5 regarding second quantization we did a similar approach when we realised the Klein-Gordon equation in momentum space was in fact an equation with standard harmonic oscillators as solutions. Then we choose to interpret the wave functional as quantum harmonic oscillators which turned out to be the correct interpretation. This time we realise that a similar model is the Hamiltonian for the diatomic molecule. pˆ2 H = + U(r) di 2µ Where µ is the reduced mass of the molecule m m µ = 1 2 m1 + m2 4 1 The exact solution is however a polyatomic with infinitely many atoms. = mi is identified with the individual mass of the atoms. 2 1 pˆi Hpoly = + U(ri) 2mi Xi=1 3.8.3 Recapitulation of the diatomic molecule One approach to find the energy perturbations of the scalar field theories is to use the resemblance between the diatomic molecule and soliton solutions single scalar fields. The energy levels of the diatomic model were found in the early twentieth century and have been studied rigorously ever since. After finding the energy levels for the diatomic molecule it is possible to proceed by generalizing the diatomic energy levels to the polyatomic system and finally apply the field language to the description of our system. Beginning with the diatomic molecule the potential as a function of distance between the atoms looks as follows. The asymptotic behaviour of the Morse potential towards zero as r is a consequence !1 of that the Coulomb potential dominates at large length scales. See figure 2. The Pauli exclusion principle is the reason for the sharp increase of the potential as the radius goes to zero. Between these two asymptotes there are a minima, this is the ground state energy. The zeroth order of the perturbation of the energy is exactly this ground state energy. If we consider harmonic oscillations around this minima we get the first level approximation. The energy eigenvalue for the first order perturbation is the quantum harmonic oscillator eigen- value. The second order approximation is including the rotational energy levels of the di- atomic molecule. The energy degeneracy is lifted yet one level with the corresponding eigen- value of the angular momentum operator energy eigenvalue, Lˆ2 n, l, m = l(l + 1) n, l, m . | i | i In the first order perturbation we can consider the solutions to follow

(x, t)=vac + ⌘(x, t)

4 Considering the example of the - model the vacuum, vac, would be defined as.

v vac ⌘±

–23– U(r) 2 U(r)=D (1 e (r r0) 1) e

r

r0 E0

Figure 2. Graph over the potential for the diatomic molecule i.e. the Morse potential. The dashed line is the harmonic first order perturbation of the potential. The first order approximation is accurate if the domain is close to the domain of the minima of the Morse potential.

Order of approximation Energy eigenstate Energy eigenvalue 0 r ,✓,' E | 0 i 0 1 n, ✓,' + n + 1 U 00(r0) | i 2 m l(lq+1) 2 n, l, m +2mr2 + ... | i 0

Table 1.Energyapproximationsuptosecondorderforthediatomicmolecule.

Looking at the partitions of the total Lagrangian of the system gives

L = LCoulomb + Lvibrational + Linteractions + ... The small oscillations around the ground state of U() would have to follow the equations of motion determined by the following harmonic Lagrangian found by Taylor expanding the potential U(r) to second order approximation around the minima r0.

1 µ 1 2 L = dx (@ ⌘)(@ ⌘) U 00(r )⌘ , (3.15) vib. 2 µ 2! 0 Z  Finding the Euler-Lagrange equation’s for this Lagrangian gives. µ @⌫@ ⌘ + U 00(r0)⌘ =0

–24– We want to diagonalize the Lagrangian. This is done by considering the time independent solutions, assume periodic solution’s in time and doing the change of variables that follows the time independent Schrödinger equation.

2 2 @ + U 00(r ) (x)=! (x) x 0 i i i i!it ⌘i(x, t)= i(x)e

Generalizing this further gives

1 i!it i!it ⌘(x, t)= ci(t) i(x)e + ci(t) i(x)†e Xi=1 ⇣ ⌘ Inserting the expanded solution into equation 3.15 gives the energy eigenvalues of the quan- tum harmonic oscillator. n 1 E = ! n + n i i 2 Xi=1 ✓ ◆ Now to the rotational energy levels. The diatomic molecule can be viewed as a rigid rotator l(l+1) with an angular momentum. The rotational energy levels are given by 2 , where the 2mr0 numerator is the moment of inertia for the diatomic molecule and l are the angular quantum numbers.

3.8.4 Transition to the Infinitely polyatomic molecule To make the transition to the polyatomic molecule is simple. We have to remember that for each energy level there are an infinite number of atom interactions to include. This means that the integers in the diatomic model becomes infinite string’s of integers in the infinitely diatomic case. Consider the first perturbation energy for the ground state. The following holds.

1 U 00(r0) 1 1 ki n + ni + (3.16) 2 r m ! 2 rm ✓ ◆ Xi=1 ✓ ◆

Where ki is the individual spring constants between the atoms.

3.8.5 Quantized kink energy levels The transition to the infinitely polyatomic molecule at the same time as incorporation our kink description produces the following table. Defining the finite energy solutions as

0(x)=f(x b) Where b is the centre of mass of the soliton. The ground state energies for the hamiltonian 3.13 is defined as.

E V (f) 0 ⌘

–25– Order of approximation Energy eigenstate Energy eigenvalue E0 0 b 2 | i 1 1 n1,n2,...; b + ni + 2 !i | i 2p2 2 n1,n2,...; b + + ... | i P 2E0

Table 2.Quantizedkinkenergylevels.

Considering the ground state the zeroth order approximation for the kink gives the first line in the table. The first order approximation is achieved by letting the integers n transform to the infinite string of integers n and taking the sum over the entire string as described { i}i1=1 by 3.16. The !i is the same as from the stability analysis of the kink 3.10. The occupation numbers is interpreted as interaction with the kink. The state where all the n’s vanish is an alone kink, the state where n =1is an kink interacting with one meson and so forth. The ground state with the alone kink can be written as.

0,b | i The standard normalization of two such ground state is.

0; b0 0; b = (b0 b) h | i Which implies that.

0; b0 0(x) 0; b = (b0 b)f(x b) h | | i As in second quantization 2.6 the solution can be written in momentum space as.

db ipb n1,...; p = e n1,...; b | i p2⇡ | i Z As was shown in section 3.5 the solitons obey the standard relativist energy relation (3.11). The relativistic energy has a minimum at p =0. Maclaurin expansion around that minimum gives the following expression. @E p @2E m2 = = 2 2 2 3 @p p + m @p (m2 + p2) 2 1 pE0(0) = 0 E00(0) = m

2 p 3 E(p)=m + + (m ) 2m O

E0 In the field formalism this becomes by replacing m with 2

2 2 E0 p 6 2 + + ( ) 2E0 O

–26– So the degeneracy has been lifted yet another level. The frequencies in table 1 are discrete but they form a continuous set which means that the sum is in fact an integral. Up to this point we have studied the system in a box a now we shall let the box go to infinity. From the stability analysis we remember that equation (3.10) holds for the kink. This means that the squares of the eigenfrequencies are the eigenvalues to the K-operator.

2 K = @ + U 00(f) x

For the ground state the K-operator is defined as

2 K = @ + U 00( ) 0 x 0

This means that we can write the energy difference between the ground state and the kink as

(! ! )=tr(pK K ) (3.17) i|kink i|vacuum 0 i X p

Since the trace operator is the sum of all the diagonal elements. Letting the box go the infinity the right hand side of equation (3.17) becomes the energy value in table 2. To reach a model of the quantized soliton we started offfrom the model of the diatomic molecule to the infinitely polyatomic molecule incorporating the field formalism and finally letting the box the system was placed in go to infinity. The leading term in the expansions of the kink is the classical term. Higher order terms correspond to quantum corrections such as the quantum harmonic oscillator.

3.9 Soliton interactions

The main property separating a soliton from a kink is that a soliton will be unperturbed in a collision with another soliton [7]. A kink on the other hand will not conserve it’s shape in collisions. In figure 3 and 4 showing soliton interaction it is clearly visible that the soliton survives the collision and maintains it’s shape.

–27– 3.9.1 Soliton-soliton interaction

1.5 t = -3 t = -2 t = -1 t = 0 t = 1 1 t = 2 t = 3

0.5

/4 0 φβ

-0.5

-1

-1.5 -6 -4 -2 0 2 4 6 X-axis

Figure 3.Soliton-solitoncollisionforu =0.1

4 1 u sinh mx (x, t)= tan ss cosh mut 

The soliton-soliton interaction seen in figure 3 begins at t = . The soliton with initial 1 position x = has a positive velocity and the soliton with initial position x = has 1 1 a equal but negative velocity. The two solitons moves towards each other and collide at t =0, the force interacting between the solitons is repulsive and conservative meaning that it will be a totally elastic collision and the kinks will return the way the came and at t = 1 find themselves at their initial position.

–28– 3.9.2 Soliton-anstisoliton interaction

1.5 t = -3 t = -2 t = -1 t = 0 t = 1 1 t = 2 t = 3

0.5

/4 0 φβ

-0.5

-1

-1.5

-6 -4 -2 0 2 4 6 X-axis

Figure 4.Soliton/Anti-solitoncollisionforu=0.1

4 1 sinh mut (x, t)= tan sa u cosh mx 

The soliton/anti-soliton interaction visible in figure 4 also begins at t = . The soliton’s 1 initial position is at x = with positive initial velocity and the anti-soliton begins at 1 x = with negative velocity. The interaction between the soliton and anti-soliton is 1 attractive meaning that they will speed up as they get close to one another. They collide at t =0and pass through each other unperturbed with the exception of a phase shift. At t = the soliton is placed at the anti-soliton’s initial position and vice versa. 1

–29– 3.9.3 Breather interaction

1 t = -4 t = -3 0.8 t = -2 t = -1 t = 0 0.6 t = 1 t = 2 t = 3 t = -4 0.4

0.2

/4 0 φβ

-0.2

-0.4

-0.6

-0.8

-1 -6 -4 -2 0 2 4 6 X-axis

Figure 5.Breathersoliton:Asolitonandaanti-solitonoscillationaroundacommoncenterof mass.

4 1 sin mvt0 (x, t)= tan v v cosh mx  0 2 0 = 1+v p The analytic solution of the breather soliton can be found by letting the real velocity u become a imaginary velocity in sa(x, t).

u = iv, v R 2 The period of the breather soliton is given by

2⇡ ⌧ = mv0

The breather soliton in figure 5 is composed of a soliton and a anti-soliton oscillating around a common centre of mass. The oscillation is stable. There also exist moving breather solitons where the common centre of mass has a velocity.

–30– 4 Geometrical methods in the theory of solitons

Here we will introduce the notion of the no curvature condition representation as a method of determining the equations of motion for a model. As in many cases there exist interesting ways to think about certain constants of motions. For the sine-Gordon model it is possible to use the no curvature condition construction. We will follow [8]. Consider the auxiliary linear problem.

@ = U @x @ = V @t Where U and V are two 2 2 differential matrix operators and functions of space, time and ⇥ the spectral parameter. For the sine-Gordon model U and V is defined as follows

U(x, t, )= i @ ik ()sin ik () cos 4 t 3 0 2 1 1 2 2 V (x, t, )= i @ ik ()sin ik () cos 4 x 3 1 2 1 0 2 2 m 1 k ()= ( + ) 0 4 m 1 k ()= ( ) 1 4

Where a is the Pauli spin matrices. The compatibility condition for the auxiliary linear problem xt = tx is equivalent to the no curvature condition.

U V +[U, V ]=0 (4.1) t x The first diagonal element of the no curvature condition gives the sine-Gordon equation of motion 2 U V +[U, V ] = U V + @ @ k k sin2 ik2 sin cos t,1,1 x.1,1 1,1 t x 16 t x 0 1 2 o 2 2 2 +ik2 cos sin k k cos2 @ @ + k k sin2 + ik2 sin cos 1 2 2 0 1 2 16 x t 0 1 2 1 2 2 ik2 cos sin + k k cos2 = U V ik2 sin + ik2 sin 0 2 2 0 1 2 t x 0 1 im2 = i @2 + i @2 + i sin (k2 k2)= i @2 + i @2 sin =0 4 t 4 x 1 0 4 t 4 x 4 m2 @2 @2 + sin =0 , t x

So a question that can be asked is how does the other elements of the no curvature condition look. As will be shown the only elements that survive are those in the main diagonal and they are generators of the equations of motion. In the following computations the canonical

–31– momenta and the derivative of the soliton with respect to x is identified below for simplicity.

⇡ = @t ⇧= @ x Let’s begin by consider the offdiagonal elements.

Element 1, 2 { } ⇡ ⇧ U V +[U, V ] = ik cos + k sin ik cos + k sin t,1,2 x,1,2 1,2 0 2 1 2 2 1 2 0 2 2 ✓ ◆ ✓ ◆ ⇡ ⇧ + k sin + ik cos + k sin + ik cos =0 2 1 2 0 2 2 0 2 1 2 ✓ ◆ ✓ ◆ Element 2, 1 { } ⇡ ⇧ U V +[U, V ] = ik cos k sin ik cos k sin t,2,1 x,2,1 2,1 0 2 1 2 2 1 2 0 2 2 ✓ ◆ ✓ ◆ ⇡ ⇧ + k sin + ik cos + k sin + ik cos =0 2 1 2 0 2 2 0 2 1 2 ✓ ◆ ✓ ◆ So the offdiagonal elements are identically equal to zero.

Element 2, 2 { } U V +[U, V ] = i @2 i @2 + i(k2 sin k2 sin ) t,2,2 x.2,2 2,2 4 t 4 x 0 1 im2 = i @2 i @2 + sin 4 t 4 x 4 This shows that the only elements that survive are the diagonal elements and the diagonal elements generate the sine-Gordon equation’s of motion.

4.1 Parallel transport and Gauge transformations If we transform the wave function as follows the corresponding change happens to U and V.

(x, t, ) G(x, t, ) (x, t, ) 7! @G 1 1 U G + GUG 7! @x @G 1 1 V G + GV G 7! @t These transformations acting on U and V can be though of as Gauge transformations. In vector calculus the only determining property of a vector is it’s end point, so if we would transport an arbitrary vector from an initial position to a final position we would only need to transport the end point. However in the notion of parallel transport it is

–32– different. A parallel transport can be visualised as a transformation of a vector tangent to a sphere preserving the vectors property as a tangent. We want to translate the tangent on the surface of the sphere while still letting it be a tangent to the sphere. This means that we need to do a translation and a rotation of the tangent. If this were in ordinary vector calculus we would only need a translation since we would have no condition that the vector should be tangent to the sphere. The translation is done by the Monodromy matrix TL(, t0) and the rotation via the rotational matrix Q(✓).

Paralell transport induced by the (U,V)-connection. Let be a curve in R2 with initial point (x0,t0) and final point (x, t).Theparalleltransportfromtheinitialpointto the final point is defined as

⌦ =exp (Udx+ Vdt) Z If the parallel transport is acted on by a gauge transformation it transforms as follows

1 ⌦ G(x, t)⌦ G (x ,t ) 7! 0 0 The superposition of two curves follow the classical exponential formula

⌦1+2 =⌦1 ⌦2 (4.2)

If the curvature is zero it means that ⌦ only depend on the end point of the transformation. If additionally is a closed curve the parallel transport is trivial.

⌦ = I (4.3)

Faddeev and Takhtajan shown in [9] that that models that accept a no curvature repre- sentation have infinitely many integrals of motion. I will go through the argument briefly. Consider a parallel transport along the space-axis symmetric with respect to the time axis in 2-dimensional space time. is covariantly constant if @ = U(x, t ,) @x 0 Using the analogy of the tangent of the sphere we know that we need to rotate the the tangent under a translation. This is represented in matrix form as

1 U(x +2L, t, )=Q (✓)U(x, t, )Q(✓) (4.4) 1 V (x +2L, t, )=Q (✓)V (x, t, )Q(✓) (4.5)

These expressions are called the quasi periodicity conditions. Where 2L is the entire dis- tance of the symmetric translation. Q(✓) is defined as.

i✓ e 2 0 Q(✓)= i✓ 0 e 2 !

–33– The monodromy matrix is the matrix of parallel transport symmetric with respect to the time axis at constant time and is defined as

L TL(, t0)=exp U(x, t0,)dx L Z Consider a closed rectangular curve in 2 dimensional space-time symmetric with respect to the time axis as in figure 6. From the superposition property (4.2) combined with the consequence of no curvature (4.3) gives. x

(t1,x2) (t2,x2)

t

(t1,x1) (t2,x1)

Figure 6. Closed rectangular curve in (1 + 1)-dimensional spacetime symmetric with respect to the time axis.

1 1 S TL (t2)S+TL(t1)=I (4.6) Where

t2 S (, t1,t2)=exp V ( L, t, )dt (4.7) ± ± Zt1 Using the quasi-periodic condition (4.4)onS shows that S+ is conjugate to S . ± 1 S+ = Q (✓)S Q(✓) We can therefore rewrite equation (4.6)as.

1 1 1 S TL (t2)S+TL(t1)=I TL (t2)S+TL(t1)=S , 1 S+TL(t1)=TL(t2)S = TL(t2)Q(✓)S+Q (✓) , 1 T (, t )Q(✓)=S (t ,t )T (, t )Q(✓)S (t ,t ) , L 2 + 1 2 L 1 + 1 2

Which in turn says that TL(, t2)Q(✓) is conjugate to TL(, t1)Q(✓). This implies that the traces of the transports are equal.

tr TL(, t2)Q(✓)=tr TL(, t1)Q(✓)

–34– This means that the traces are time independent and we have found our integrals of motion. The functional FL is hence a generation function for infinitely many integrals of motion.

FL()=tr TL()Q(✓)

4.2 An interesting duality for the sine-Gordon model A duality in physics is when a physical phenomena can be described by two different mod- els without loss of information when switching between the two models. In a paper by Coleman [10] a duality between the massive Thirring model and the sine-Gordon model was discovered. I will begin by briefly go through the basics of the massive Thirring model and then continue with describing the duality with the sine-Gordon model. The mass- less Thirring field is a one dimensional Dirac field defining the massless Thirring model Lagrangian density as.

1 = ¯i @µ gj jµ L µ 2 µ With the current defined as.

jµ = ¯µ where g is a coupling constant and µ are the Dirac matrices. The Dirac gamma matrices are defined as follows. 10 0 0 0001 000 i 0010 01 0 0 0010 00i 0 000 1 0 = 0 1 ,1 = 0 1 ,2 = 0 1 ,3 = 0 1 00 10 0 100 0 i 00 100 0 B C B C B C B C B00 0 1C B 1000C B i 00 0C B 0100C B C B C B C B C @ A @ A @ A @ A The Dirac matrices follows the anti-commutator relation

µ,⌫ = µ⌫ + ⌫µ =2gµ⌫ { } Where gµ⌫ is the metric. The fifth Dirac matrix is defined as

5 = i0123

Both the fifth and the zeroth Dirac matrix are hermitian matrices. The hamiltonian density for the massless Thirring model becomes. From the Noether theorem the Hamiltonian density can be found. @ ⇡ = L = ¯i0 @(@0 ) 1 = ¯i @i + gj jµ H i 2 µ Remember that the latin indices span 1,n where n counts the total spacetime dimension. { } The massive Thirring model is constructed by adding a mass term to the hamiltonian density for the massless Thirring model.

–35– + H!H M Where the mass term is given by. M 1 = m0(+ + ) M 2 And the eigendensities is given by. ±

= Z ¯(1 5) ± ± Where Z is a cutoffdependent constant. A sine-Gordon breather has the following mass [5].

n⇡ 8⇡m2 Mn =2Mkink sin " 2 1 2 # 8⇡m This expression tells us that.

n< 1 8⇡m2 This means that no breather states can exist if.

4⇡m2 (4.8) Since it would force n to be zero yielding a zero mass for the breather. If we assume that the breather is in fact in the weak coupling regime, 2 1 ,thegroundstatemassofthe m ⌧ breather can be expanded as.

1 2 3 M = m 1 + 1 6 16m2 O m6 " ✓ ◆ ✓ ◆# Rewriting equation (4.8) on the following form makes it possible to analyse the coupling for values slightly below the boundary. 1 4⇡m2 1 = 4⇡m2 1+ ⇡ Then the ground state breather mass transforms to.

43 M = M 2 2 + + (4) (4.9) 1 kink ⇡ O 

–36– Now consider the mass of a fermion-antifermion bound state in the massive Thirring model. It has the following mass [5].

4g3 M = M 2 g2 + + (g4) (4.10) bound ⇡ O  The mass of the ground state breather (4.9) has exactly the same mass as a fermion- antifermion bound state (4.10)if is equal to g. Suppose that is identified with g. Then it is possible to express the breather coupling as a function of the fermion-antifermion coupling and vice versa.

1 2 = g 4⇡m 1+ ⇡ 4⇡2m2 4⇡ g = ⇡, (g)= m2 m2 1+ g ✓ ◆ ⇡ For the weak coupling limit this becomes lim g 2 = 0 m 1 m2 ! ✓ ◆ lim (g)=4⇡ g 0 m2 ! This suggest the following correspondence between the massive Thirring model and the sine-Gordon model [5].

1. kink , fermion () 2. anti-kink ¯, anti-fermion () 3. boson ¯, fermion-antifermion pair () 4. topological charge fermion number ()

5Summary

Solitons are interesting objects with several applications in modern science. Solitons in quantum field theory can act as a method for the discovery of new physical phenomena. This text begins with a brief introduction to quantum field theory to give the tools needed for understanding solitons in quantum field theory and continues with a deeper analysis of finite energy solution to field equations. Explicit computation of the time-independent non-trivial finite energy solution for the 4- and the sine-Gordon field is made using the relation between the field potential and the position. After finding the time-independent solution the time- dependent solution is found by Lorentz transformation of the time-independent. A brief walkthrough of the interesting quantities of the soliton is made such as the energy density and the soliton mass. Solitons in a low dimensional quantum field theory is an interesting

–37– tool for theorists for finding new physical phenomena. The soliton in low dimension can be extended to higher dimensions however the is a limitation for this extension. Derrick’s theorem states that the spatial dimension can not transgress two if non-trivial finite energy solutions are to exist. The stability analysis of the kink shows that the kink is stable for all real eigenfrequencies. The same eigenfrequencies used in the energy eigenvalue for first order perturbation of the quantized kink. In the examination of soliton interaction it was visualised the repulsive force in soliton-soliton interaction and the attractive force in the interaction between a soliton and a anti-soliton and that the solitons are unperturbed in collisions with other solitons. It was shown breather soliton, a soliton and a anti-soliton oscillating around a common centre of mass, was identical to the soliton for the soliton anti- soliton collision except with an imaginary eigenfrequency. The period of the breather soliton was also found. The fact that the sine-Gordon model is a no curvature representation was shown. With a theorem regarding parallel transport it was shown that models that accept a no curvature representation have infinite integrals of motion. Finally the equivalence between the sine-Gordon model and the massive Thirring model was presented.

Acknowledgments

Thanks to my project mentor Luigi Tizzano for his patience, his large time investment in this project, since the autumn of 2015, and his ability to make me work harder. And to my father, Fredrik Laurell, who helped me understand spatial solitons in non-linear optics. Additionally thanks to Erik Orvehed Hiltunen for helping me understand math I had not yet mastered. And of course thanks to my girlfriend Gjertrud Louise Langaas.

References

[1] B. Kath, B. Kath, “Making Waves: Solitons and Their Optical Applications”. SIAM News, Volume 31, Number 2. [2] M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory,” Reading, USA: Addison-Wesley (1995) 842 p [3] H. Goldstein, “Classical Mechanics, third edition”, Columbia, South Carolina, July (2000) [4] O. Temkin, M.D., “Forerunners of Darwin, 1745-1859”, Johns Hopkins University Press, (1968) [5] E. J. Weinberg, “Classical Solutions in Quantum Field Theory", Cambridge University Press, (2012) [6] S. Coleman, “Aspects of Symmetry," Cambridge University Press, (1985) [7] R. Rajaraman, “Solitons and Instantons," Amsterdam, North-Holland, (1987) [8] V. Caudrelier, “Multisymplectic approach to integrable defects in the sine-Gordon model”, arXiv:1411.5171, J. Phys. A 48 (2015) 195203 [9] L. D. Faddeev and L. Takhtajan, “Hamiltonian Methods in the Theory of Solitons”, doi:10.1007/978-3-540-69969-9, (1987)

–38– [10] S. Coleman, “The quantum sine-Gordon equation as the massive Thirring model”, Phys. Rev. D11,2088(1975).

–39–