Teorija umeritvenih polj, Yang-Mills-Higgsove enačbe in spinske strukture na Lorentzovih mnogoterostih

magistrsko delo

Abstract

Teorija umeritvenih polj ponuja geometrijsko bogat teoretičnofizikalni skelet, v katerem simetrije narekujejo interakcije. V pričujočem delu začnemo pri osnovah glavnih svežnjev in natančno predstavimo splošen pojem povezave na njih. Pokažemo, da povezava naravno porodi pojma ukrivljenosti in kovariantnega odvoda na pridruženem vektorskem svežnju glede na dano upodobitev strukturne Liejeve grupe, raziščemo njune lastnosti ter natančno definiramo pojem umeritvenega polja ter njegove jakosti. Pri tem izrazimo vpliv umeritvenih transformacij nanju in pokažemo, kako lahko ukrivljenost na glavnem svežnju identificiramo z diferencialno formo, ki ima vrednosti v adjungiranem svežnju. Ukrivljenost povezave je -- skupaj s Hodge-* operatorjem, vnanjim kovariantnim odvodom in variacijskim principom -- osnova naše razprave o Yang--Millsovi teoriji, ki jo interpretiramo kot teorijo interakcije polja umeritvenih bozonov (tj. nosilcev fundamentalnih fizikalnih sil) s samim sabo. V tem kontekstu pokažemo umeritveno invariantnost Yang--Millsovega Lagrangiana in izpeljemo Yang--Millsovo enačbo, ki jo interpretiramo (skupaj z Bianchijevo identiteto) kot posplošitev Maxwellovih enačb. Z umestitvijo posplošenega Klein--Gordonovega Lagrangiana v ta kontekst izpeljemo Yang--Mills--Higgsove enačbe, ki opisujejo interakcijo polja umeritvenih bozonov s skalarnim poljem. Posledica teh enačb je proslavljeni Brout--Englert--Higgsov mehanizem, ki ga na kratko orišemo. Skozi fizikalno motivirano študijo Lorentzove grupe in njenega univerzalnega krova nazadnje naravno vpeljemo koncept spinske strukture na Lorentzovo mnogoterost -- to nam omogoča, da umestimo Levi--Civitajevo afino povezavo na pridružen spinorski sveženj, kar nam dalje omogoča, da natančno definiramo Diracov operator na Lorentzovi mnogoterosti. S pridobitvijo osnovnih lastnosti Cliffordovega množenja dokažemo sebiadjungiranost Diracovega operatorja glede na skalarni produkt, ki ga porodi Diracova hermitska forma. Z uporabo variacijskega računa rigorozno izpeljemo Diracovo enačbo na spinski Lorentzovi mnogoterosti. Za konec orišemo konstrukcijo spoja glavnega spin svežnja z glavnim umeritvenim svežnjem in povemo, kako nas to privede do nehomogene Yang--Millsove enačbe, ki v fiziki opisuje interakcije fermionov z umeritvenimi polji.

Keywords

simetrija;glavni sveženj;umeritev;povezava;ukrivljenost;pridružen vektorski sveženj;upodobitev Liejeve grupe;vzporedni prenos;kovariantni vnanji odvod;bozoni;Hodge-* operator;kodiferencial;Yang-Millsova teorija;Lagrangian;variacijski princip;Klein-Gordonova enačba;fermioni;Diracova enačba;Lorentzova grupa;spin grupa;spinorsko polje;Cliffordovo množenje;Diracov operator;interakcija;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[Ž. Grad]
UDC:	514.7
COBISS:	28180995
Views:	1779
Downloads:	463
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary title:	Gauge field theory, Yang-Mills-Higgs equations and spin structures on Lorentzian manifolds
Secondary abstract:	Gauge field theory provides a geometrically rich theoretico-physical framework in which symmetries dictate interactions. We begin the present work by examining the fundamental properties of principal bundles and introduce on them the general notion of a connection. We show that the latter naturally gives rise to the concept of curvature; furthermore, given a representation of the structure Lie group, we conceive the notion of a covariant derivative on the associated vector bundle. We precisely define gauge fields, express the influence of gauge transformations on them and show how curvature on a principal bundle may be identified with a differential form with values in the adjoint bundle. Together with the notions of the Hodge-* operator, the exterior covariant derivative and the variational principle, the curvature of a connection is the basis for the Yang--Mills theory, which we physically interpret as the theory of self-interaction of the gauge boson fields (i.e. the carriers of fundamental forces). In this context, we prove gauge invariance of the Yang--Mills Lagrangian and derive the Yang--Mills equation, which we interpret (together with the Bianchi identity) as a generalization of the Maxwell's equations. By including the Klein--Gordon Lagrangian into this theory, we derive the Yang--Mills--Higgs equations, which describe the interaction of gauge boson fields and scalar fields. A consequence of these equations is the celebrated Brout--Englert--Higgs mechanism; we provide a short sketch thereof. Through a physically motivated study of the Lorentz group and its universal cover, we naturally introduce the notion of a spin structure on a Lorentz manifold. The latter enables us to induce from the affine Levi--Civita connection the covariant derivative on the associated spinor bundle, which further enables us to precisely define the Dirac operator on a Lorentz manifold. By acquiring the basic properties of the Clifford multiplication, we prove self-adjointness of the Dirac operator with respect to the hermitian Dirac form. By means of the variational principle, we arrive to the Dirac equation on a spin Lorentz manifold. We conclude our investigation by splicing the principal spin bundle with the principal gauge bundle and sketch how this tool is used to describe interactions of fermions with gauge fields.
Secondary keywords:	symmetry;principal bundle;gauge;connection;curvature;associated vector bundle;representation of a Lie group;parallel transport;covariant exteriror derivative;bosons;Hodge-* operator;codifferential;Yang-Mills theory;Lagrangian;variational principle;Klein-Gordon equation;fermions;Dirac aquation;Lorentz group;spin group;spinors field;Clifford multiplication;Dirac operator;interaction;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages:	XI, 126 str.
ID:	12033212