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A Summary on Solitons in Quantum Field Theory

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A Summary on Solitons in Quantum Field Theory 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 φ4-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 φ4-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 φ(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 = φ(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 ⇢φ˙(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 Lφ˙ @φ˙ −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.
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