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1
Silk, James Brian. "Evaluation of correlation functions in integrable quantum field theories." Thesis, Durham University, 2012. http://etheses.dur.ac.uk/4447/.
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The aim of this thesis is to explore correlation functions in two dimensional quantum field theories in two distinct ways. In part I a new method for calculating the differential equations parametrising the correlation functions of twist fields associated with the U (1) symmetry of the Dirac model is presented. While developing this method a new family of descendent twist fields are identified and their form factors calculated. This provides a novel way of calculating the vacuum expectation values of the primary twist fields and is shown to be entirely consistent with known results. The method of calculating the correlation functions of twist fields provides a parametrisation of several other correlation functions for various quantum states. Since this method relies on the Ward identities found in a double copy model it is hoped to have wider applications in other free fermion models. Part II concerns the truncated conformal space approach which has been developed to approximate perturbed conformal field theories. In this part the theory underpinning the approach is discussed and a working algorithm is developed for both bulk and boundary perturbed minimal models. The energy levels, mass gaps and one point functions of various models are computed using the truncated conformal space approach and are shown to be in good agreement with previous calculations. A possible method for using this approach to approximate two point functions in perturbed conformal field theories is discussed.
2
Mattsson, Peter Aake. "Integrable quantum field theories, in the bulk and with a boundary." Thesis, Durham University, 2000. http://etheses.dur.ac.uk/4225/.
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In this thesis, we consider the massive field theories in 1+1 dimensions known as affine Toda quantum field theories. These have the special property that they possess an infinite number of conserved quantities, a feature which greatly simplifies their study, and makes extracting exact information about them a tractable problem. We consider these theories both in the full space (the bulk) and in the half space bounded by an impenetrable boundary at x = 0. In particular, we consider their fundamental objects: the scattering matrices in the bulk, and the reflection factors at the boundary, both of which can be found in a closed form. In Chapter 1, we provide a general introduction to the topic before going on, in Chapter 2, to consider the simplest ATFT—the sine-Gordon model—with a boundary. We begin by studying the classical limit, finding quite a clear picture of the boundary structure we can expect in the quantum case, which is introduced in Chapter 3. We obtain the bound-state structure for all integrable boundary conditions, as well as the corresponding reflection factors. This structure turns out to be much richer than had hitherto been imagined. We then consider more general ATFTs in the bulk. The sine-Gordon model is based on a(^(1))(_1), but there is an ATFT for any semi-simple Lie algebra. This underlying structure is known to show up in their S-matrices, but the path back to the parameters in the Lagrangian is still unclear. We investigate this, our main result being the discovery of a "generalised bootstrap" equation which explicitly encodes the Lie algebra into the S-matrix. This leads to a number of new S-matrix identities, as well as a generalisation of the idea that the conserved charges of the theory form an eigenvector of the Cartan matrix. Finally our results are summarised in Chapter 5, and possible directions for further study are highlighted.
3
Karabin, Svyatoslav. "Generalized hydrodynamics of a class of integrable quantum field theories with non-diagonal scattering." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18009/.
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In questo lavoro di tesi abbiamo analizzato alcuni modelli conformi con perturbazioni integrabili, in particolare il modello di Ising tri-critico e i successivi modelli minimali. Abbiamo costruito un protocollo che realizza questi modelli in un regime fuori dall'equilibrio termodinamico. Questo sistema è stato ottenuto connettendo due sistemi semi-infiniti termalizzati a due diverse temperature. In tempi e spazi grandi ci si aspetta che questo sistema evolva verso uno stato stazionario indipendente dal tempo. Le quantità fisiche di nostro interesse sono le correnti stazionarie generate in tale situazione. Per studiare questo sistema abbiamo utilizzato strumenti di integrabilità come il Bethe ansatz termodinamico, concetti di idrodinamica generalizzata e l'insieme di Gibbs generalizzato. Finora questo schema è stato formulato per le teorie di campo con un'interazione tra le particelle data da una matrice S diagonale, ovvero per i modelli con lo spettro di quasi-particelle prive di gradi di libertà interni. In questa tesi abbiamo proposto un'estensione di questo metodo a un modello dotato di uno spettro contenente quasi-particelle organizzate in multipletti di simmetrie e quindi dotate di gradi di libertà interni detti magnoni con processi d'urto descritti da matrici S non diagonali. Abbiamo quindi risolto numericamente le equazioni differenziali che descrivono il sistema di non equilibrio e abbiamo discusso questi risultati.
4
Belliard, Raphaël. "Geometry of integrable systems : from topological Lax systems to conformal field theories." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066175/document.
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Cette thèse de doctorat traite d’un cadre en géométrie complexe et de méthodes pouvant y être développées pour résoudre des ensembles d’équations différentielles compatibles venant de systèmes intégrables, classiques ou quantiques, dans le contexte de la géométrie d’espaces de modules de connexions au-dessus de courbes complexes, ou surfaces de Riemann. Elle vient de l’idée en physique mathématique que les symétries des systèmes intégrables imposent aux objets d’intérêt (fonctions de partitions ou de corrélations) des contraintes algébro-différentielles nommées équations de boucles. Le but est par la suite de résoudre ces contraintes par récurrence dans des régimes dits topologiques en utilisant une procédure nommée récurrence topologique. Leurs solutions sont en général les fonctions génératrices de quantités issues de problèmes de géométrie énumérative. Étant principalement déterminées par les conditions initiales de la récurrence, on produit au passage une classification algébro-géométrique de la famille de systèmes intégrables considéréeThis PhD thesis is about a framework in complex geometry and methods thereof for solving sets of compatible differential equations arising from integrable systems, classical or quantum, in the context of the geometry of moduli spaces of connections over complex curves, or Riemann surfaces. It is based on the idea in mathematical Physics that integrable systems posess symmetries that impose algebro-differential constraints, so-called loop equations, on the objects of interest (e.g. partition or correlation functions). In turn, we intend to solve these constraints recursively in certain topological regimes using a particular procedure called the topological recursion. Their solutions are in general generating functions of enumerative-geometric quantities. Since they are for the most part determined by the initial data of the recursive process, it realizes in the making an algebro-geometric classification of the family of integrable models under consideration
5
Longino, Brando. "Exact S-matrices for a class of 1+1-dimensional integrable factorized scattering theories with Uq(sl2) symmetry and arbitrary spins." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/20542/.
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In this thesis we will study the S-matrices associated to a new class of (1+1)-dimensional integrable models with Uq(sl2) symmetry, whose asymptotic particle states organize into a k/2 isospin multiplet, with k= 0,1,2,... Such S-matrices generalize the case study previously analyzed by S. R. Aladim and M. J. Martins, where it was only investigated the non-deformed limit q→1 of pure SU(2) symmetry. We check that the proposed S-matrix satisfies the constraints due to the the Yang-Baxter equation, crossing-symmetry requirement and unitarity and therefore defines a self-consistent integrable factorized scattering theory.
6
Parini, Robert Charles. "Classical integrable field theories with defects and near-integrable boundaries." Thesis, University of York, 2018. http://etheses.whiterose.ac.uk/20428/.
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In the first part of this thesis algebro-geometric solutions for the sine-Gordon and KdV equations in the presence of a type I integrable defect are found, generalising the previously known soliton solutions. Elliptic (genus one) solutions where the defect induces only a phase shift are obtained via ansätze for the fields on each side of the defect. Algebro-geometric solutions for arbitrary genus and involving soliton emission by the defect are constructed using a Darboux transformation, exploiting the fact that the defect equations have the form of a Bäcklund transformation at a point. All the soliton and phase-shifted elliptic solutions to the defect equations are recovered as limits of the algebro-geometric solutions constructed in this way. Certain energy and momentum conserving defects for the Kadomtsev-Petviashvili equation are then presented as a first step towards the construction of integrable defects in higher dimensions. Algebro-geometric solutions to the sine-Gordon equation on the half-line with an integrable two parameter boundary condition are obtained by imposing a corresponding restriction on the Lax pair eigenfunction or, alternatively, as a Darboux transformation of the known algebro-geometric solution for the Dirichlet boundary. Finally, the collision of sine-Gordon solitons with a Robin type boundary is examined. This boundary is typically non-integrable but becomes an integrable Neumann or Dirichlet boundary for certain values of a boundary parameter. Depending on the boundary parameter and initial velocity an antikink may be reflected into various combinations of kinks, antikinks and breathers. The soliton content of the field after the collision is numerically determined by computing the discrete scattering data associated with the inverse scattering method. A highlight of this investigation is the discovery of an intricate structure of resonance windows caused by the production of a breather which can collide multiple times with the boundary before escaping as a lighter breather or antikink.
7
Roa, Aguirre Alexis [UNESP]. "Type-II defects in integrable classical field theories." Universidade Estadual Paulista (UNESP), 2012. http://hdl.handle.net/11449/102532.
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roaaguirre_a_dr_ift.pdf: 584228 bytes, checksum: a9374484aaeff9b04bf55be78fb96d03 (MD5)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Nesta tese discutimos as propriedades de integrabilidade das teorias de campo clássicas em duas dimensões na presença de descontinuidades ou defeitos tipo-II, principalmente usando a linguagem do formalismo do espalhamento inverso. Um método geral para calcular a função geradora de um conjunto infinito de grandezas conservadas modificadas para qualquer equação de campo integrável é apresentado, uma vez que seus respetivos problemas lineares associados são dados e suas correspondentes matrices do defeito são calculadas. O método é aplicado no cálculo das contribuições dos defeitos para a energia e o momento para vários modelos e mostramos a relação entre as condições de defeito integráveis e suas respevtivas transformações de Bäcklund para cada modelo
In this thesis we discuss the integrability properties of two-dimensional classical field theories in the presence of discontinuities or type-II defects, mainly using the language of the inverses cattering approach. We present a general method to compute the generating function of an infinite set of modified conserved quantities for any integrable field equation givent heir associated linear problems and computing their corresponding defect matrices. We apply this method to derive in particular defect contributions to the energy and momentum for several models and show the relationship between the integrable defect conditions and the Bäcklund transformations for each model
8
Roa, Aguirre Alexis. "Type-II defects in integrable classical field theories /." São Paulo, 2012. http://hdl.handle.net/11449/102532.
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Orientador: Abraham Hirsz ZimermanBanca: Clisthenis Ponce Constantinidis
Banca: Harold Socrates Blas Achic
Banca: Andrei Mikhailov
Banca: Marcio José Martins
Resumo: Nesta tese discutimos as propriedades de integrabilidade das teorias de campo clássicas em duas dimensões na presença de descontinuidades ou defeitos tipo-II, principalmente usando a linguagem do formalismo do espalhamento inverso. Um método geral para calcular a função geradora de um conjunto infinito de grandezas conservadas modificadas para qualquer equação de campo integrável é apresentado, uma vez que seus respetivos problemas lineares associados são dados e suas correspondentes matrices do defeito são calculadas. O método é aplicado no cálculo das contribuições dos defeitos para a energia e o momento para vários modelos e mostramos a relação entre as condições de defeito integráveis e suas respevtivas transformações de Bäcklund para cada modelo
Abstract: In this thesis we discuss the integrability properties of two-dimensional classical field theories in the presence of discontinuities or type-II defects, mainly using the language of the inverses cattering approach. We present a general method to compute the generating function of an infinite set of modified conserved quantities for any integrable field equation givent heir associated linear problems and computing their corresponding defect matrices. We apply this method to derive in particular defect contributions to the energy and momentum for several models and show the relationship between the integrable defect conditions and the Bäcklund transformations for each model
Doutor
9
Hills, Daniel. "Generating boundary conditions for integrable field theories using defects." Thesis, University of York, 2016. http://etheses.whiterose.ac.uk/16379/.
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In this thesis, we examine the construction and characteristics of generalised reflection matrices, within the a_1^(1), a_2^(1) and a_2^(2) integrable affine Toda field theories. In doing so, we generalise the existing finite-dimensional reflection matrices because our construction involves the dressing of an integrable boundary with a defect. Within this framework, an integrable defect's ability to store an unlimited amount of topological charge is exploited, therefore all generalised solutions are intrinsically infinite-dimensional and exhibit interesting features. Overall, further evidence of the rich interplay between integrable defects and boundaries is provided. It is hoped that the generalised solutions presented in this thesis are potential quantum analogues of more general classical integrable boundary conditions.
10
Evangelisti, Stefano <1983>. "Quantum Correlations in Field Theory and Integrable Systems." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2013. http://amsdottorato.unibo.it/5161/1/Evangelisti_Stefano_Tesi.pdf.
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In this thesis we will investigate some properties of one-dimensional quantum
systems. From a theoretical point of view quantum models in one dimension are
particularly interesting because they are strongly interacting, since particles
cannot avoid each other in their motion, and you we can never ignore collisions.
Yet, integrable models often generate new and non-trivial solutions, which
could not be found perturbatively.
In this dissertation we shall focus on two important aspects of integrable one-
dimensional models: Their entanglement properties at equilibrium and their
dynamical correlators after a quantum quench. The first part of the thesis will
be therefore devoted to the study of the entanglement entropy in one-
dimensional integrable systems, with a special focus on the XYZ spin-1/2
chain, which, in addition to being integrable, is also an interacting model. We
will derive its Renyi entropies in the thermodynamic limit and its behaviour in
different phases and for different values of the mass-gap will be analysed.
In the second part of the thesis we will instead study the dynamics of correlators
after a quantum quench , which represent a powerful tool to measure how
perturbations and signals propagate through a quantum chain. The emphasis
will be on the Transverse Field Ising Chain and the O(3) non-linear sigma
model, which will be both studied by means of a semi-classical approach.
Moreover in the last chapter we will demonstrate a general result about the
dynamics of correlation functions of local observables after a quantum quench
in integrable systems. In particular we will show that if there are not long-range
interactions in the final Hamiltonian, then the dynamics of the model (non
equal- time correlations) is described by the same statistical ensemble that
describes its statical properties (equal-time correlations).
11
Evangelisti, Stefano <1983>. "Quantum Correlations in Field Theory and Integrable Systems." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2013. http://amsdottorato.unibo.it/5161/.
Full textAbstract:
In this thesis we will investigate some properties of one-dimensional quantum
systems. From a theoretical point of view quantum models in one dimension are
particularly interesting because they are strongly interacting, since particles
cannot avoid each other in their motion, and you we can never ignore collisions.
Yet, integrable models often generate new and non-trivial solutions, which
could not be found perturbatively.
In this dissertation we shall focus on two important aspects of integrable one-
dimensional models: Their entanglement properties at equilibrium and their
dynamical correlators after a quantum quench. The first part of the thesis will
be therefore devoted to the study of the entanglement entropy in one-
dimensional integrable systems, with a special focus on the XYZ spin-1/2
chain, which, in addition to being integrable, is also an interacting model. We
will derive its Renyi entropies in the thermodynamic limit and its behaviour in
different phases and for different values of the mass-gap will be analysed.
In the second part of the thesis we will instead study the dynamics of correlators
after a quantum quench , which represent a powerful tool to measure how
perturbations and signals propagate through a quantum chain. The emphasis
will be on the Transverse Field Ising Chain and the O(3) non-linear sigma
model, which will be both studied by means of a semi-classical approach.
Moreover in the last chapter we will demonstrate a general result about the
dynamics of correlation functions of local observables after a quantum quench
in integrable systems. In particular we will show that if there are not long-range
interactions in the final Hamiltonian, then the dynamics of the model (non
equal- time correlations) is described by the same statistical ensemble that
describes its statical properties (equal-time correlations).
12
Hausser, Frank. "Lattice quantum field theories with quantum symmetry." [S.l. : s.n.], 1998. http://darwin.inf.fu-berlin.de/1998/10/index.html.
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13
Schnakenburg, Igor. "Symmetries of supergravity theories and quantum field theories." Thesis, King's College London (University of London), 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397449.
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Spill, Fabian. "Yangians in integrable field theories, spin chains and gauge-string dualities." Thesis, Imperial College London, 2010. http://hdl.handle.net/10044/1/6128.
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In the following dissertation, we explore the applicability of Yangian symmetry to various integrable models, in particular, in relation with S-matrices. One of the main themes in this dissertation is that, after a careful study of the mathematics of the symmetry algebras one finds that in an integrable model, one can directly reconstruct S-matrices just from the algebra. It has been known for a long time that S-matrices in integrable models are fixed by symmetry. However, Lie algebra symmetry, the Yang-Baxter equation, crossing and unitarity, which are what constrains the S-matrix in integrable models, are often taken to be separate, independent properties of the S-matrix. Here, we construct scattering matrices purely from the Yangian, showing that the Yangian is the right algebraic object to unify all required symmetries of many integrable models. In particular, we reconstruct the S-matrix of the principal chiral field, and, up to a CDD factor, of other integrable field theories with su(n) symmetry. Furthermore, we study the AdS/CFT correspondence, which is also believed to be integrable in the planar limit. We reconstruct the S-matrices at weak and at strong coupling from the Yangian or its classical limit. This version of the thesis includes minor corrections following the viva on 17 September 2010.
15
Große, Johannes. "Quantum field theories coupled to supergravity." Diss., lmu, 2006. http://nbn-resolving.de/urn:nbn:de:bvb:19-67614.
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Wiegandt, Konstantin. "Superconformal quantum field theories in string." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2012. http://dx.doi.org/10.18452/16605.
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In dieser Dissertation werden Aspekte von superkonformen Quantenfeldtheorien untersucht, die für die sogenannte AdS/CFT Korrespondenz relevant sind. Die AdS/CFT Korrespondenz beschreibt eine Dualität zwischen Stringtheorien im Anti-de Sitter Raum und superkonformen Quantenfeldtheorien im Minkowskiraum. In diesem Kontext wurde die sog. Wilsonschleifen / Amplituden Dualität entdeckt, die die Übereinstimmung von n-Gluon MHV Amplituden und n-seitigen polygonalen Wilsonschleifen in der N=4 supersymmetrischen Yang-Mills (SYM) Theorie beschreibt. Im ersten Teil dieser Dissertation wird die Wilsonschleifenseite einer solchen möglichen Dualität in der N=6 superkonformen Chern-Simons (ABJM) Theorie untersucht. Das Hauptergebnis dieser Untersuchungen ist, dass der Erwartungswert der n-seitigen polygonalen Wilsonschleifen auf Einschleifenebene verschwindet, während er auf Zweischleifenebene in seiner funktionalen Form identisch zu der analogen Wilsonschleife in N=4 SYM auf Einschleifenniveau ist. Außerdem wird eine anomale konforme Wardidentität für Wilsonschleifen in Chern-Simons Theorie berechnet. Zudem werden die damit im Zusammenhang stehenden Entwicklungen für Amplituden und Korrelatoren in der ABJM Theorie diskutiert. Im zweiten Teil dieser Dissertation werden Dreipunktfunktionen von zwei geschützten Operatoren und einem Twist-Zwei Operator mit beleibigem Spin j in der N=4 SYM Theorie berechnet. Dafür werden die Indizes des Spin j Operators auf den Lichtkegel projiziert und der Korrelator wird in einem Grenzfall untersucht in dem der Impuls der bei dem Spin j Operator einfließt verschwindet. Dieser Grenzfall vereinfacht die perturbative Berechnung erheblich, da alle Dreipunktdiagramme effektiv auf Zweipunktdiagramme reduziert werden und die Abhängigkeit der Mischungsmatrix auf Einschleifenebene herausfällt. Das Ergebnis stimmt mit der Analyse der Operatorproduktentwicklung von Vierpunktfunktionen geschützter Operatoren von Dolan und Osborn aus dem Jahre 2004 überein.In this thesis aspects of superconformal field theories that are of interest in the so-called AdS/CFT correspondence are investivated. The AdS/CFT correspondence states a duality between string theories living on Anti-de Sitter space and superconformal quantum field theories in Minkowski space. In the context of the AdS/CFT correspondence the so-called Wilson loop / amplitude duality was discovered, stating the equality of the finite parts of n-gluon MHV amplitudes and n-sided lightlike polygonal Wilson loops in N=4 supersymmetric Yang-Mills (SYM) theory. It is the subject of the first part of this thesis to investigate the Wilson loop side of a possible similar duality in N=6 superconformal Chern-Simons matter (ABJM) theory. The main result is, that the expectation value of n-sided lightlike polygonal Wilson loops vanishes at one-loop order and at two-loop order is identical in its functional form to the Wilson loop in N=4 SYM theory at one-loop order. Furthermore, an anomalous conformal Ward identity for Wilson loops in Chern-Simons theory is derived. Related developments and symmetries of amplitudes and correlators in ABJM theory are discussed as well. In the second part of this thesis we calculate three-point functions of two protected operators and one twist-two operator with arbitrary even spin j in N =4 SYM theory. In order to carry out the calculations, the indices of the spin j operator are projected to the light-cone and the correlator is evaluated in a soft-limit where the momentum coming in at the spin j operator becomes zero. This limit largely simplifies the perturbative calculation, since all three-point diagrams effectively reduce to two-point diagrams and the dependence on the one-loop mixing matrix drops out completely. The result is in agreement with the analysis of the operator product expansion of four-point functions of half-BPS operators by Dolan and Osborn in 2004.
17
Söderberg, Alexander. "Renormalization in Field Theories." Thesis, Uppsala universitet, Teoretisk fysik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-251561.
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Several different approaches to renormalization are studied. The Callan-Symanzik equation is derived and we study its beta functions. An effective potential for the Coleman-Weinberg model is studied to find that the beta function is positive and that spontaneous symmetry breaking will occur if we expand around the classical field. Lastly we renormalize a non-abelian gaugetheory to find that the beta function in QCD is negative.
18
McDonald, Reuben. "Hamiltonian treatments of lattice quantum field theories." Thesis, University of Manchester, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.570530.
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Schritt, Dimitri. "Symmetries in quantum and classical field theories." Thesis, University of Canterbury. Physics and Astronomy, 2013. http://hdl.handle.net/10092/8032.
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The initial chapter of the thesis provides a review of Weinberg’s formalism for the derivation of quantum fields. The formalism is extended to allow for the derivation of quantum fields with more than one spin degree of freedom. It is conjectured that it may be possible to construct massive bosonic quantum field theories of any desired spin j that are consistent and unitary at all energies without the need for regulator terms by including j + 1
spin degrees of freedom: j, j - 1, down to j - j. The concept is then demonstrated in two subsequent chapters by the derivation of a quantum field with spin one and spin zero degrees of freedom followed the derivation of a quantum field with spin two, spin one, and spin zero degrees of freedom. Both field theories are found to be consistent and unitary at all energies without the need for regulator terms. The final two chapters are on unrelated topics. The penultimate chapter provides an explicit derivation of quantum fields for massless particles of spin one-half. In the final chapter, a derivation of the free-space Proca and Maxwell equations is provided via a consistent identification of the linear combinations of the classical fields of the (1,0) and (0,1) representations of the orthochronous
Lorentz group.
20
Hartley, David. "Supersymmetric quantum field theories from induced representations." Thesis, University of St Andrews, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329888.
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Klaczynski, Lutz. "Haag's theorem in renormalisable quantum field theories." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17448.
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Wir betrachten eine Reihe von Trivialitäts- resultaten und No-Go-Theoremen aus der Axiomatischen Quantenfeldtheorie. Von besonderem Interesse ist Haags Theorem. Im Wesentlichen sagt es aus, dass der unitäre Intertwiner des Wechselwirkungsbildes nicht existiert oder trivial ist. Als wichtigste Voraussetzung von Haags Theorem arbeiten wir die unitäre Äquivalenz heraus und unterziehen die kanonische Störungstheorie skalarer Felder einer Kritik um zu argumentieren, dass die kanonisch renormierte Quantenfeldtheorie Haags Theorem umgeht, da sie genau diese Bedingung nicht erfüllt. Der Hopfalgebraische Zugang zur perturbativen Quantenfeldtheorie bietet die Möglichkeit, Dyson-Schwinger-und Renormierungsgruppengleichungen mathematisch sauber herzuleiten, wenn auch mit rein kombinatorischem Ausgangspunkt. Wir präsentieren eine Beschreibung dieser Methode und diskutieren eine gewöhnliche Differentialgleichung für die anomale Dimension des Photons. Eine Spielzeugmodellversion dieser Gleichung lässt sich exakt lösen; ihre Lösung weist eine interessante nichtstörunsgtheoretische Eigenschaft auf, deren Auswirkungen auf die laufende Kopplung und die Selbstenergie des Photons wir untersuchen. Solche nichtperturbativen Beiträge mögen die Existenz eines Landau-Pols ausschliessen, ein Sachverhalt, den wir ebenfalls diskutieren. Unter der Arbeitshypothese, dass die anomale Dimension eines Quantenfeldes in die Klasse der resurgenten Funktionen fällt, studieren wir, welche Bedingungen die Dyson-Schwinger-und Renormierungsgruppengleichungen an ihre Transreihe stellen. Wir stellen fest, dass diese unter bestimmten Bedingungen kodieren, wie der perturbative Sektor den nichtperturbativen vollständig determiniert.We review a package of triviality results and no-go theorems in axiomatic quantum field theory. Of particular interest is Haag''s theorem. It essentially says that the unitary intertwiner of the interaction picture does not exist unless it is trivial. We single out unitary equivalence as the most salient provision of Haag''s theorem and critique canonical perturbation theory for scalar fields to argue that canonically renormalised quantum field theory bypasses Haag''s theorem by violating this very assumption. The Hopf-algebraic approach to perturbative quantum field theory allows us to derive Dyson-Schwinger equations and the Callan-Symanzik equation in a mathematically sound way, albeit starting with a purely combinatorial setting. We present a pedagogical account of this method and discuss an ordinary differential equation for the anomalous dimension of the photon. A toy model version of this equation can be solved exactly; its solution exhibits an interesting nonperturbative feature whose effect on the running coupling and the self-energy of the photon we investigate. Such nonperturbative contributions may exclude the existence of a Landau pole, an issue that we also discuss. On the working hypothesis that the anomalous dimension of a quantum field falls into the class of resurgent functions, we study what conditions Dyson-Schwinger and renormalisation group equations impose on its resurgent transseries. We find that under certain conditions, they encode how the perturbative sector determines the nonperturbative one completely.
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De, Renzi Marco. "Construction of extended topological quantum field theories." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCC114/document.
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La position centrale occupée par les Théories Quantiques des Champs Topologiques (TQFTs) dans l’étude de la topologie en basse dimension est due à leur structure extraordinairement riche, qui permet différentes interactions et applications à des questions de nature géométrique. Depuis leur première apparition, un grand effort a été mis dans l’extension des invariants quantiques de 3-variétés en TQFTs et en TQFT Étendues (ETQFTs). Cette thèse s’attaque à ce problème dans deux cadres généraux différents. Le premier est l’étude des invariants quantiques semi-simples de Witten, Reshetikhin et Turaev issus de catégories modulaires. Bien que les ETQFTs correspondantes étaient connues depuis un certain temps, une réalisation explicite basée sur la construction universelle de Blanchet, Habegger, Masbaum et Vogel apparaît ici pour la première fois. L’objectif est de tracer la route à suivre dans la deuxième partie de la thèse, où la même procédure est appliquée à une nouvelle famille d’invariants quantiques non semi-simples due à Costantino, Geer et Patureau. Ces invariants avaient déjà été étendus en TQFTs graduées par Blanchet, Costantino, Geer and Patureau, mais seulement pour une famille explicite d’exemples. Nous posons la première pierre en introduisant la définition de catégorie modulaire relative, un analogue non semi-simple aux catégories modulaires. Ensuite, nous affinons la construction universelle pour obtenir des ETQFTs graduées étendant à la fois les invariants quantiques de Costantino, Geer et Patureau et les TQFTs graduées de Blanchet, Costantino, Geer et Patureau dans ce cadre généralThe central position held by Topological Quantum Field Theories (TQFTs) in the study of low dimensional topology is due to their extraordinarily rich structure, which allows for various interactions with and applications to questions of geometric nature. Ever since their first appearance, a great effort has been put into extending quantum invariants of 3-dimensional manifolds to TQFTs and Extended TQFTs (ETQFTs). This thesis tackles this problem in two different general frameworks. The first one is the study of the semisimple quantum invariants of Witten, Reshetikhin and Turaev issued from modular categories. Although the corresponding ETQFTs were known to exist for a while, an explicit realization based on the universal construction of Blanchet, Habegger, Masbaum and Vogel appears here for the first time. The aim is to set a golden standard for the second part of the thesis, where the same procedure is applied to a new family of non-semisimple quantum invariants due to Costantino, Geer and Patureau. These invariants had been previously extended to graded TQFTs by Blanchet, Costantino, Geer an Patureau, but only for an explicit family of examples. We lay the first stone by introducing the definition of relative modular category, a non-semisimple analogue to modular categories. Then, we refine the universal construction to obtain graded ETQFTs extending both the quantum invariants of Costantino, Geer and Patureau and the graded TQFTs of Blanchet, Costantino, Geer and Patureau in this general setting
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Stenberg, Jacob. "Diagrammatic Representations in Quantum Theories." Thesis, Uppsala universitet, Teoretisk fysik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-201668.
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Starting from a mathematical basis where one analyses and developing different techniques in how to solve and represent different kinds of integrals with diagrams. Representing the integrals as n-valent vertices and introducing propagators is a great tool that helps with the book-keeping of the solutions and will sometimes do the calculations redundant. The symmetries of diagrams are analysed and how one extracts the symmetry factors from looking at a diagram by using some fairly simple combinatorics and cleverness. Introducing the probability amplitude and do some analysis of the path integral formulation is the step into physics. Discussing experiments as the double-slit experiment and deriving the Schrödinger equation from the generating functional. Looking at diagrams in Quantum Mechanics and Quantum Field Theory will explore the use of the crucial understanding of our generators for the diagrams. This thesis makes use of the so called generating functional almost throughout and to connect the first discussed mathematics to real physical theories is the aim.
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Pereira, Raul. "Correlation Functions in Integrable Theories : From weak to strong coupling." Doctoral thesis, Uppsala universitet, Teoretisk fysik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320811.
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The discovery of integrability in planar N=4 super Yang-Mills and ABJM has enabled a precise study of AdS/CFT. In the past decade integrability has been successfully applied to the spectrum of anomalous dimensions, which can now be obtained at any value of the coupling. However, in order to solve conformal field theories one also needs to understand their structure constants. Recently, there has been great progress in this direction with the all-loop proposal of Basso, Komatsu and Vieira. But there is still much to understand, as it is not yet possible to use that formalism to find structure constants of short operators at strong coupling. It is important to study wrapping corrections and resum them as the TBA did for the spectrum. It is also crucial to obtain perturbative data that can be used to check if the all-loop proposal is correct or if there are new structures that need to be unveiled. In this thesis we compute several structure constants of short operators at strong coupling, including the structure constant of Konishi with half-BPS operators. Still at strong coupling, we find a relation between the building blocks of superstring amplitudes and the tensor structures allowed by conformal symmetry. We also consider the case of extremal correlation functions and the relation of their poles to mixing with double-trace operators. We also study three-point functions at weak coupling. We take the OPE limit in a four-point function of half-BPS operators in order to shed some light on the structure of five-loop wrapping corrections of the Hexagon form factors. Finally, we take the first steps in the generalization of the Hexagon programme to other theories. We find the non-extremal setup in ABJM and the residual symmetry that it preserves, which we use to fix the two-particle form factor and constrain the four-particle hexagon. Finally, we find that the Watson equations hint at a dressing phase that needs to be further investigated.
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Lolli, Linda. "Prolegomena for a comparative study of correlation functions in 2D integrable field theories." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18144/.
Full textAbstract:
Questo lavoro di tesi si concentra sulla teoria dei Form Factors di operatori locali in due dimensioni. Si espone innanzitutto la teoria dello scattering fattorizzato, propria dei sistemi integrabili, per poi trattare sia classicamente che quantisticamente due modelli integrabili massivi, Sine-Gordon e Sinh-Gordon. Viene esposta l'algebra di Faddeev-Zamolodchikov, per mezzo di cui è possibile trovare lo spazio di Hilbert per modelli integrabili come spazio delle sue rappresentazioni. L'attenzione è infine posta sul regime in cui l'ampiezza relativa alla riflessione solitone-antisolitone nella matrice di scattering è nulla; tale caso si ottiene per valori particolari della costante di accoppiamento del modello di Sine-Gordon, coincidenti con i valori di soglia degli stati legati. Due proposte per l'espressione dei form factors per solitoni sono state prese in considerazione, avanzate una da Lukyanov, l'altra da Babelon, Bernard e Smirnov, tentando un confronto.
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Martini, Riccardo. "Tensorial group field theories." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/8941/.
Full textAbstract:
In questa tesi il Gruppo di Rinormalizzazione non-perturbativo (FRG) viene
applicato ad una particolare classe di modelli rilevanti in Gravit`a quantistica,
conosciuti come Tensorial Group Field Theories (TGFT). Le
TGFT sono teorie di campo quantistiche definite sulla variet`a di un gruppo G.
In ogni dimensione esse possono essere espanse in grafici di Feynman duali a com-
plessi simpliciali casuali e sono caratterizzate da interazioni che implementano una
non-localit`a combinatoriale. Le TGFT aspirano a generare uno spaziotempo in un
contesto background independent e precisamente ad ottenere una descrizione con-
tinua della sua geometria attraverso meccanismi fisici come le transizioni di fase.
Tra i metodi che meglio affrontano il problema di estrarre le transizioni di fase e un
associato limite del continuo, uno dei pi`
u efficaci `e il Gruppo di Rinormalizzazione
non-perturbativo.
In questo elaborato ci concentriamo su TGFT definite sulla variet`a di un
gruppo non-compatto (G = R) e studiamo il loro flusso di Rinormalizzazione. Identifichiamo con successo
punti fissi del flusso di tipo IR, e una superficie critica che suggerisce
la presenza di transizioni di fase in regime Infrarosso. Ci`o spinge ad uno stu-
dio per approfondire la comprensione di queste transizioni di fase e della fisica
continua che vi `e associata.
Affrontiamo inoltre il problema delle divergenze Infrarosse, tramite un processo di
regolarizzazione che definisce il limite termodinamico appropriato per le TGFT.
Infine, applichiamo i metodi precedentementi sviluppati ad un modello dotato
di proiezione sull’insieme dei campi gauge invarianti. L’analisi, simile a quella
applicata al modello precedente, conduce nuovamente all’identificazione di punti
fissi (sia IR che UV) e di una superficie critica. La presenza di transizioni di fasi
`e, dunque, evidente ancora una volta ed `e possibile confrontare il risultato col
modello senza proiezione sulla dinamica gauge invariante.
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Ashworth, Richard Michael. "Quantum field theories having conformal and chiral symmetry." Thesis, University of Cambridge, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.292952.
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Poole, C. W. "Gradient-flow equations for general Quantum Field Theories." Thesis, University of Liverpool, 2018. http://livrepository.liverpool.ac.uk/3015643/.
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Zhang, Ou. "Effective field theories for quantum chromo- and electrodynamics." Thesis, The University of Arizona, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10247445.
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Effective field theories (EFTs) provide frameworks to systematically improve perturbation expansions in quantum field theory. This improvement is essential in quantum chromodynamics (QCD) predictions, both at low energy in the description of low momentum hadron-hadron scattering and at high energy in the description of electron-positron, proton-proton, proton-electron collisions. It is also important in quantum electrodynamics (QED), when electrons interact with a high-intensity, long-wavelength classical field. I introduce the principles and methods of effective field theory and describe my work in three EFTs: First, in the perturbative QCD region, I use soft collinear effective theory (SCET) to prove that strong interaction soft radiation is universal and to increase the QCD accuracy to next-to-next-to-next-to leading logarithm order for new particle searches in hadron colliders. I also compute a new class of non-perturbative, large logarithmic enhancement arising near the elastic limits of deep inelastic scattering and Drell-Yan processes. Second, in the QCD confinement region, I use heavy hadron chiral perturbation theory to study near-threshold enhancements in the scattering of D and π mesons near the threshold for the excited D-meson state, D*, as well as in the scattering of D and D* mesons near the threshold for the exotic hadron X(3872). This work provides a clear picture of the hadronic molecule X(3872) and more profound understanding of the nuclear force between hadrons. Finally, inspired by SCET, I construct a new electron-laser effective field theory to describe highly-relativistic electrons traveling in strong laser fields, extract the universal distribution of electrons in strong electromagnetic backgrounds and its evolution in energy from the separated momentum scales of emitted photons and classical radiation, and predict the rate of wide angle photon emission. I conclude with limitations of EFT methods and some perspectives on what new work may be achieved with these EFTs.
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Zhang, Ou, and Ou Zhang. "Effective Field Theories for Quantum Chromo- and Electrodynamics." Diss., The University of Arizona, 2016. http://hdl.handle.net/10150/621825.
Full textAbstract:
Effective field theories (EFTs) provide frameworks to systematically improve perturbation expansions in quantum field theory. This improvement is essential in quantum chromodynamics (QCD) predictions, both at low energy in the description of low momentum hadron-hadron scattering and at high energy in the description of electron-positron, proton-proton, proton-electron collisions. It is also important in quantum electrodynamics (QED), when electrons interact with a high-intensity, long-wavelength classical field. I introduce the principles and methods of effective field theory and describe my work in three EFTs: First, in the perturbative QCD region, I use soft collinear effective theory (SCET) to prove that strong interaction soft radiation is universal and to increase the QCD accuracy to next-to-next-to-next-to leading logarithm order for new particle searches in hadron colliders. I also compute a new class of non-perturbative, large logarithmic enhancement arising near the elastic limits of deep inelastic scattering and Drell-Yan processes. Second, in the QCD confinement region, I use heavy hadron chiral perturbation theory to study near-threshold enhancements in the scattering of 𝐷 and 𝜋 mesons near the threshold for the excited 𝐷-meson state, 𝐷*, as well as in the scattering of 𝐷 and 𝐷* mesons near the threshold for the exotic hadron X(3872). This work provides a clear picture of the hadronic molecule X(3872) and more profound understanding of the nuclear force between hadrons. Finally, inspired by SCET, I construct a new electron-laser effective field theory to describe highly-relativistic electrons traveling in strong laser fields, extract the universal distribution of electrons in strong electromagnetic backgrounds and its evolution in energy from the separated momentum scales of emitted photons and classical radiation, and predict the rate of wide angle photon emission. I conclude with limitations of EFT methods and some perspectives on what new work may be achieved with these EFTs.
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Roy, Alan A. "Aspects of renormalisation in some quantum field theories." Thesis, Rhodes University, 1998. http://hdl.handle.net/10962/d1005214.
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Renormalisation is an important aspect of Quantum Field Theory. It is used to create physically meaningful theories and some major developments took place in the 1970's and onwards. We consider Renormalisation in its application to the theories of ψ⁴ , Quantum Electrodynamics, Quantum Chromodynamics and the Background Field Method. Feynman diagrams are used to illustrate many of the concepts.
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Schiappa, Ricardo. "Aspects of Riemannian geometry in quantum field theories." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85337.
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GORINI, NICOLA. "Aspects of Quantum Field Theories in Three Dimensions." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/364292.
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In questa tesi costruiamo il settore topologico unidimensionale della teoria ABJ(M) e studiamo la sua relazione con la funzione di partizione, deformata da parametri di massa, definita sulla sfera tridimensionale. La tecnica di localizzazione supersimmetrica consente di rappresentare tale funzione di partizione come un integrale finito-dimensionale (o integrale matriciale). E\' stata proposta una congettura secondo cui le funzioni di correlazione di operatori topologici possono essere ottenute prendendo certe derivate della funzione di partizione rispetto ai parametri di massa. In questa tesi presentiamo un'evidenza non banale della correttezza della congettura, calcolando le funzioni a tre e quattro punti ad un loop e la funzione a due punti a due loop, ritrovando correttamente i risultati ottenuti prendendo le derivate dell'integrale matriciale deformato una volta espanso in regime di accoppiamento debole. In aggiunta, otteniamo l'espressione della carica centrale della teoria a due loop, generalizzando i risultati noti in letteratura.
In seguito ci focalizziamo sullo studio delle fasi infrarosse di teorie tridimensionali di Chern-Simons quiver, ovvero il cui gruppo di gauge è composto da due fattori non Abeliani. Discutiamo in dettaglio i casi in cui i livelli di Chern-Simons sono uguali e il caso in cui sono opposti, proponendo alcune dualità riguardanti sia coppie di teorie quiver che teorie quiver con la versione supersimmetrica della QCD tridimensionale.In this thesis we construct the one-dimensional topological sector of $\mathcal N = 6$ ABJ(M) theory and study its relation with the mass-deformed partition function on $\mathbb S^3$. Supersymmetric localization provides an exact representation of this partition function as a matrix integral and it has been proposed that correlation functions of certain topological operators are computed through derivatives with respect to the masses. We present non-trivial evidence for this relation by computing the three- and four- point function up to one loop and the two-point function at two loops, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product, we obtain the two-loop explicit expression for the central charge $c_T$ of ABJ(M) theory. When then shift our attention to the study of the infrared phases of two-node quiver Chern-Simons theories with minimal supersymmetry in three dimensions. We discuss both the cases of Chern-Simons levels with the same and with opposite signs, where the latter case turn out to be more non-trivial. The determination of their phase diagrams allows us to conjecture certain infrared dualities involving either two quiver theories, or a quiver and adjoint QCD$_3$.
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Petrov, Dmitri. "Novel approaches to numerical solutions of quantum field theories /." View online version; access limited to Brown University users, 2005. http://wwwlib.umi.com/dissertations/fullcit/3174659.
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Omid, Hamid. "2+1d quantum field theories in large N limit." Thesis, University of British Columbia, 2016. http://hdl.handle.net/2429/60336.
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In Chapter 1, we present a brief introduction to the tight-binding model of graphene and show that in the low-energy continuum limit, it can be modeled by reduced QED₂₊₁ . We then review renormalization group technique which is used in the next chapters. In Chapter 2, we consider a quantum field theory in 3+1d with the defect of a large number of fermion flavors, N. We study the next-to-leading order contributions to the fermions current-current correlation function by performing a large N expansion. We find that the next-to-leading order contributions 1/N to the current-current correlation function is significantly suppressed. The suppression is a consequence of a surprising cancellation between the two contributing Feynman diagrams. We calculate the model's conductivity via the Kubo formula and compare our results with the observed conductivity for graphene. In Chapter 3, we study graphene's beta function in large N. We use the large N expansion to explore the renormalization of the Fermi velocity in the screening dominated regime of charge neutral graphene with a Coulomb interaction. We show that inclusion of the fluctuations of the magnetic field lead to a cancellation of the beta function to the leading order in 1/N. The first non-zero contribution to the beta function turns out to be of order 1/N². We perform a careful analysis of possible infrared divergences and show that the superficial infrared divergences do not contribute to the beta function. In Chapter 4, we study the phase structure of a Φ⁶ theory in large N. The leading order of the large N limit of the O(N) symmetric phi-six theory in three dimensions has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton. In this chapter, we show that this “light dilaton” is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate scale invariance. We rule out the existence of Bardeen-Moshe-Bander phase. In this thesis, we show that Large N expansion is a powerful tool which in regimes that the system is interacting strongly could be used as an alternative to coupling expansion scheme.Science, Faculty of
Physics and Astronomy, Department of
Graduate
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McAvity, David Marks. "The heat kernel for quantum field theories with boundary." Thesis, University of Cambridge, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260537.
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Perantonis, S. J. "Model calculations in quantum chromodynamics and other field theories." Thesis, University of Oxford, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.379929.
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Simms, R. "Exploring higher dimensional quantum field theories through fixed points." Thesis, University of Liverpool, 2018. http://livrepository.liverpool.ac.uk/3028491/.
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Renormalization was popularised in the 1940s following the appearance of non- sensical infinities in the calculation of the self-energy of the electron. Notably this led to Quantum Electrodynamics becoming a fully renormalizable quantum field theory. One useful tool that emerges from the technical aspects of renormal- ization is the Renormalization Group. In particular, the β-function defines the variation of the coupling constants with energy. The vanishing of the β-function at a particular value of the coupling is known as a fixed point, the location of which can be found using perturbation theory. Properties of quantum field the- ories such as ultraviolet behaviour can be studied using these fixed points. The calculation of two different types of fixed points forms the spine of this thesis. In Part I the d-dimensional Wilson-Fisher fixed point is used to connect scalar quantum field theories in different space-time dimensions. Specifically we look at dimensions greater than four and explore the property of universality through the Vasil'ev large N expansion. Different universality classes are examined, the first contains φ4 theory with O(N) symmetry while another incorporates O(N)×O(m) Landau-Ginzburg-Wilson theory. In the latter we perform a full fixed point sta- bility analysis and conformal window search which determines where conformal symmetry is present. Part I develops techniques that may later be applicable to calculations involving beyond the Standard Model physics including asymptotic safety, quantum gravity and emergent symmetries. Part II focuses on the non-trivial Banks-Zaks fixed point of four dimensional Quantum Chromodynamics. Using a variety of colour groups and representations we calculate the location of the fixed point and corresponding critical exponents to pinpoint exactly where the true value of the conformal window lies. Additionally a number of different renormalization schemes are used, including the momentum subtraction (MOM) and interpolating momentum subtraction (iMOM) schemes. This allows us to study where in the conformal window scheme dependence is most apparent. Both the Landau gauge and maximal abelian gauge are utilized to extend the analysis. Throughout this thesis we compare and contrast perturbative results with non-perturbative calculations such as those performed in lattice.
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Wang, Yifan Ph D. Massachusetts Institute of Technology. "Lessons on interacting quantum field theories from string theory." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/106452.
Full textAbstract:
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2016.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 185-194).
In this thesis, we use string theory constructions and dualities to explore various features of interacting quantum field theories. We begin with an overview in Chapter 1, of past and recent developments in quantum field theories, explaining the advantages of string theoretic techniques over traditional approaches in answering a range of questions about interacting dynamics. In Chapter 2 we study the holographic duality between the 6d (1, 1) Ak-1 little string theory (LST) and type II string theory in the double scaled limit. By identifying the low energy states, which are Cartan gluons in the 6d maximal super-Yang-Mills (SYM) that describes the massless sector of the LST, we compute the four-point amplitudes from both sides of the duality and demonstrate matching results. Since the two computations concern different regimes in the parameter space, their amazing agreement implies the presence of certain nonrenormalization theorems in the 6d SYM. In Chapter 3, motivated by the AdS/CFT duality, we develop a systematic procedure to derive an off-shell action for hydrodynamics from classical Einstein gravity. We first identity the boundary fluid degrees of freedom in the hydrodynamic regime, in terms of gapless modes of the metric in the bulk gravity. This allows us to derive an off-shell action, for relativistic fluids that have gravity duals, at leading order in derivative expansion, by explicitly integrating out gapped degrees of freedom in the bulk. We also explain the strategy to incorporate dissipation and higher order effects. In Chapter 4, we discuss 4d N = 2 superconformal field theories (SCFT) of the Argyres-Douglas (AD) type, which can be constructed in string/M theory by either wrapping M5 branes on punctured Riemann surface or probing 3-fold singularity by IIB string. We classify the punctures (irregular defects in Hitchin system) on the Riemann surface in the former construction, that will give rise to N = 2 SCFTs and demonstrate how to extract exact information about the Coulomb branch spectrum and central charges. We further identify these AD theories constructed from M5 branes with a special class of theories from IIB probing compound Du Val (cDV) singularities, thereby establishing a mathematical connection between singular Hitchin systems and cDV singularities through N = 2 SCFTs. We end with a short summary and outlook for future directions in Chapter 5.
by Yifan Wang.
Ph. D.
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Nolland, David John. "Quantum field theories with fermions in the Schrödinger representation." Thesis, Durham University, 2000. http://etheses.dur.ac.uk/4410/.
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This thesis is concerned with the Schrödinger representation of quantum field theory. We describe techniques for solving the Schrödinger equation which supplement the standard techniques of field theory. Our aim is to develop these to the point where they can readily be used to address problems of current interest. To this end, we study realistic models such as gauge theories coupled to dynamical fermions. For maximal generality we consider particles of all physical spins, in various dimensions, and eventually, curved spacetimes. We begin by considering Gaussian fields, and proceed to a detailed study of the Schwinger model, which is, amongst other things, a useful model for (3+1) dimensional gauge theory. One of the most important developments of recent years is a conjecture by Mal-dacena which relates supergravity and string/M-theory on anti-de-Sitter spacetimes to conformal field theories on their boundaries. This correspondence has a natural interpretation in the Schrödinger representation, so we solve the Schrödinger equation for fields of arbitrary spin in anti-de-Sitter spacetimes, and use this to investigate the conjectured correspondence. Our main result is to calculate the Weyl anomalies arising from supergravity fields, which, summed over the supermultiplets of type JIB supergravity compactified on AdS(_s) x S(^5) correctly matches the anomaly calculated in the conjecturally dual N = 4 SU{N) super-Yang-Mills theory. This is one of the few existing pieces of evidence for Maldacena's conjecture beyond leading order in TV.
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Ehrhardt, Mathieu. "Indices for supersymmetric quantum field theories in four dimensions." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/274322.
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In this thesis, we investigate four dimensional supersymmetric indices. The motivation for studying such objects lies in the physics of Seiberg's electric-magnetic duality in supersymmetric field theories. In the first chapter, we first define the index and underline its cohomological nature, before giving a first computation based on representation theory of free superconformal field theories. After listing all representations of the superconformal algebra based on shortening conditions, we compute the associated Verma module characters, from which we can extract the index in the appropriate limit. This approach only provides us with the free field theory limit for the index and does not account for the values of the $R$-charges away from free field theories. To circumvent this limitation, we then study a theory on $\mathbb{R}\times S^3$ which allows for a computation of the superconformal index for multiplets with non-canonical $R$-charges. We expand the fields in harmonics and canonically quantise the theory to analyse the set of quantum states, identifying the ones that contribute to the index. To go beyond free field theory on $\mathbb{R}\times S^3$, we then use the localisation principle to compute the index exactly in an interacting theory, regardless of the value of the coupling constant. We then show that the index is independent of a particular geometric deformation of the underlying manifold, by squashing the sphere. In the final chapter, we show how the matching of the index can be used in the large $N$ limit to identify the $R$-charges for all fields of the electric-magnetic theories of the canonical Seiberg duality. We then conclude by outlining potential further work.
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Kawai, Daisuke. "The study on quantum field theories from numerical approaches." Kyoto University, 2018. http://hdl.handle.net/2433/232236.
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Geyer, Yvonne Jasmin. "Ambitwistor strings : worldsheet approaches to perturbative quantum field theories." Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:ff543496-58bd-4818-ae0c-cf31eb349c90.
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Tree-level scattering amplitudes in massless theories not only exhibit a simplicity entirely unexpected from Feynman diagrams, but also an underlying structure remarkably reminiscent of worldsheet theory correlators, yet essentially algebraic. These features can be explained by ambitwistor strings - two-dimensional chiral conformal field theories in an auxiliary target space, the complexified phase space of null geodesics, known as ambitwistor space. The aim of this thesis is to explore the ambitwistor string approach to understand these structures in amplitudes, and thereby provide a new angle on quantum field theories. In the first part of this thesis, the wide-ranging impact of ambitwistor strings on the study of tree-level amplitudes is highlighted in three developments: an extension of ambitwistor string worldsheet models to an extensive family of massless theories, emphasising the universality of ambitwistor strings for massless theories; a beautiful proof of the duality between asymptotic symmetries and the low energy behaviour of a theory, relying on the contact geometry of the ambitwistor target space; and finally a twistorial representation of ambitwistor strings in four dimensions, leading to remarkably simple formulae for scattering amplitudes in Yang-Mills and gravity with any degree of supersymmetry. The second part of this thesis focusses on proving a conjectured ambitwistor string formula for loop amplitudes, and extending the formalism to more general theories. Remarkably, residue theorems reduce the computationally challenging ambitwistor higher-genus expressions to simple formulae on nodal Riemann spheres. This idea is developed into a widely applicable framework for loop integrands, that is shown to be applicable to both supersymmetric and non-supersymmetric theories. In the case of supergravity, this provides strong evidence for the validity of the ambitwistor string at loop level, and explicit proofs are given for non-supersymmetric theories. Notably, this leads to a proposal for an all-loop integrand for gravity and Yang-Mills.
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Gómez, Subils Javier. "Non-perturbative Aspects of Quantum Field Theories from Holography." Doctoral thesis, Universitat de Barcelona, 2021. http://hdl.handle.net/10803/672276.
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In this thesis we have employed the holographic duality to study the non-perturbative regime of a one-parameter family of theories with multi-scale dynamics. Normally, this (super)string theory motivated duality identifies gauge theories in flat space with string theories in a certain curved spacetime. Its relevance roots in its ability to relate the strongly-coupled regime of gauge theories with classical gravity governed by Einstein's equations. In the Introduction of the thesis, we have reviewed the main string theory ingredients that we used throughout the thesis and revisited some of the previous results which are the starting point of out study.
In Chapter 2, we gather the explicit form of the supergravity solutions whose dual are the gauge theories of interest. These are three-dimensional gauge theories. Generically, they share the same physics at high energies, given by Yang-Mills and Chern-Simons interactions. Remarkably, when the low energy regime of the theories is studied, a rich variety of non-perturbative phenomenology is discovered.
In particular, for a generic value of the parameter distinguishing the theories, they develop a mass gap in their spectrum. However, the two theories which are obtained at the limiting values of the parameter are special. On the one hand, the theory flows towards an infrarred fixed point, dual to a Conformal Field Theory. On the other hand, a confining theory is obtained, in the sense that the potential felt between quarks grows linearly with the distance between them for large separations.
All these phenomena, together with the computation of the spectrum of spin-0 and spin-2 states, are studied in Chapter 3. The fact that in this system the Renormalisation Group flow can pass close to a Conformal Field Theory motivated the search of a light dilaton in the spectrum. But such light state was not found, the reason for that being that the values of the source and the vacuum expectation value that prevented the flow from finishing at the fixed point where of the same order.
On top of that, in this Chapter some entanglement entropy measures were studied. This last investigation was motivated by the fact that in the literature some quantities extracted from such magnitudes where proposed as a probe for confinement. Our results show that, when these quantities are considered in our system, they are not able to discriminate between confining and non-confining gapped theories.
Not only did we consider the theories at zero temperature case, but we also studied thermal states by constructing numerical black brane solutions in the gravity side. Black branes are very much like black holes, with the peculiarity that their surface extends in non-compact directions. Such solutions are discussed in Chapter 4. As a result, we understood their phase diagram, exhibiting a rich structure endowed with first and second order phase transitions, as well as a triple point where three phases coexist and a critical point where the second order phase transition takes place.
Intrigued by the effect that the proximity of a Conformal Field Theory could have in the Renormalisation Group flow of a field theory, in Chapter 5 we carried out a study on complex Conformal Field Theories. We proposed their holographic dual, and analysed some of their properties in the strongly-coupled case.
Finally, in Chapter 6, we studied transport coefficients in holographic theories which model Quantum Chromodynamics. We concluded that the holographic results are quite different from the ones obtained using perturbative techniques. These studies could have phenomenological consequences and find their application in astrophysical observations concerning neutron stars.En esta tesis hemos utilizado la dualidad holográfica para entender el régimen no perturbativo de una familia uni-paramétrica de teorías con múltiples escalas. Primeramente, hemos repasado los ingredientes esenciales que necesitamos de teoría de cuerdas. A la vez, hemos introducimos algunos resultados previos que son el punto de partida de nuestras investigaciones. Tras dicha introducción, se recogen todas las soluciones de supergravedad duales a las teorías en tres dimensiones que estudiamos. Genéricamente, comparten la misma física a altas energías pero a bajas energías muestran una rica fenomenología. En particular, desarrollan un salto de masa en su espectro. Curiosamente, las teorías correspondientes a tomar los valores límites del parámetro son especiales. En un caso, la teoría fluye a una teoría de campos conforme. En el otro se obtiene una teoría confinante, con potencial lineal entre quarks. También se calcula el espectro de estados con espín 0 y espín 2. Además, se analizan diferentes medidas de entrelazamiento cuántico que en nuestro caso no son capaces de discriminar entre teorías con confinamiento y teorías con un salto de masa. Esto contrasta con algunas propuestas que se encuentran en la literatura. Adicionalmente hemos construido numéricamente soluciones de branas negras, que describen estados térmicos de las teorías. Hemos descubierto un diagrama de fases muy rico, con transiciones de fase de primer y segundo orden, junto a un punto crítico y un punto triple. Interesados por el efecto que una teoría conforme de campos pudiera tener si es cercana al flujo del grupo de renormalización de otra teoría, en el Capítulo 5 nos adentramos en el estudio de teorías conformes de campos complejas, dando su el dual holográfico. Finalmente, se calculan coeficientes de transporte en teorías holográficas que modelan Cromodinámica Quántica y que podrían tener consecuencias fenomenológicas en observaciones referentes a estrellas de neutrones.
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Levell, Jonathan. "Aspects of model building in noncommutative quantum field theories." Thesis, Durham University, 2004. http://etheses.dur.ac.uk/3029/.
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We review quantum field theories on noncommutative Minkowski spaces (NCQFTs), concentrating on the mixing between ultra-violet and infra-red degrees of freedom in such theories. We use background field perturbation theory at the one-loop level to calculate the three and four point functions in supersymmetric NCQFT. We use the results of this calculation to show that the infra-red logarithmic divergences that arise as a result of the UV/IR mixing can be reproduced by an explicitly gauge-invariant low-energy effective action expressed in terms of Wilson lines. We present a noncommutative gauge theory that has the ordinary Standard Model as its low-energy limit. The model is based on the gauge group U (4) x U (3) x U (2) and satisfies all the key constraints that are imposed by noncommutativity: the UV/IR mixing effects, restrictions on representations and charges of matter fields and the cancellation of noncommutative gauge anomalies. At energies well below the noncommutative mass scale our model flows to the commutative Standard model plus additional free U {1) degrees of freedom which decouple due to the UV/IR mixing. Our model also predicts the values of the hypercharges of the Standard Model fields. We find that in order to accomodate the matter content of the Standard Model it is necessary to introduce extra, as yet undetected, matter fields.
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Volkholz, Jan. "Nonperturbative studies of quantum field theories on noncommutative spaces." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2007. http://dx.doi.org/10.18452/15712.
Full textAbstract:
Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden.This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.
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Ludwig, Thomas. "Wick Rotation for Quantum Field Theories on Degenerate Moyal Space." Doctoral thesis, Universitätsbibliothek Leipzig, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-118795.
Full textAbstract:
In dieser Arbeit wird die analytische Fortsetzung von Quantenfeldtheorien auf dem nichtkommutativen Euklidischen Moyal-Raum mit kommutativer Zeit zu entsprechenden Moyal-Minkowski Raumzeit (Wick Rotation) erarbeitet. Dabei sind diese Moyal-Räume durch eine konstante Nichtkommutativiät
gegeben. Einerseits wird die Wick Rotation im Kontext der algebraischen Quantenfeldtheorie, ausgehend von einer Arbeit von Schlingemann, hergeleitet. Von einem Netz Euklidischer Observablen wird die Lorentz’sche Theorie durch alle Bilder der fortgesetzten Poincare Gruppenwirkung auf der Zeit-Null Schicht erhalten. Dabei wird gezeigt, dass die Vorgänge der nichtkommutativen Deformation und der Wick Rotation kommutieren. Andererseits ist so eine analytische Fortsetzung ebenfalls für Quantenfeldtheorien, die durch einen Satz von Schwingerfunktionen definiert ist, möglich. Durch die Gültigkeit einer Kombination aus Wachstumsbedinungen, die aus der Wick Rotation von Osterwalder und Schrader bekannt sind, kann der Übergang zu einer deformierten Wightman-Theorie gezeigt werden. Abschließend beinhaltet diese Arbeit ergänzende
Resultate zu den physikalischen Eigenschaften der Kovarianz und der Lokalität.
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Gasbarro, Andrew David. "Studies of Conformal Behavior in Strongly Interacting Quantum Field Theories." Thesis, Yale University, 2019. http://pqdtopen.proquest.com/#viewpdf?dispub=13851872.
Full textAbstract:
In this dissertation, we present work towards characterizing various conformal and nearly conformal quantum field theories nonperturbatively using a combination of numerical and analytical techniques. A key area of interest is the conformal window of four dimensional gauge theories with Dirac fermions and its potential applicability to beyond the standard model physics.
In the first chapter, we review some of the history of models of composite Higgs scenarios in order to motivate the study of gauge theories near the conformal window. In the second chapter we review lattice studies of a specific theory, SU(3) gauge theory with eight flavors of Dirac fermions in the fundamental representation of the gauge group. We place a particular emphasis on the light flavor-singlet scalar state appearing in the spectrum of this model and its possible role as a composite Higgs boson. We advocate an approach to characterizing nearly conformal gauge theories in which lattice calculations are used to identify the best low energy effective field theory (EFT) description of such nearly conformal gauge theories, and the lattice and EFT are then used as complementary tools to classify the generic features of the low energy physics in these theories. We present new results for maximal isospin ππ → ππ scattering on the lattice computed using Lüscher's finite volume method. This scattering study is intended to provide further data for constraining the possible EFT descriptions of nearly conformal gauge theory. In Chapter 3, we review the historical development of chiral effective theory from current algebra methods up through the chiral Lagrangian and modern effective field theory techniques. We present a new EFT framework based on the linear sigma model for describing the low lying states of nearly conformal gauge theories. We place a particular emphasis on the chiral breaking potential and the power counting of the spurion field.
In Chapter 4, we report on a new formulation of lattice quantum field theory suited for studying conformal field theories (CFTs) nonperturbatively in radial quantization. We demonstrate that this method is not only applicable to CFTs, but more generally to formulating a lattice regularization for quantum field theory on an arbitrary smooth Riemann manifold. The general procedure, which we refer to as quantum finite elements (QFE), is reviewed for scalar fields. Chapter 5 details explicit examples of numerical studies of lattice quantum field theories on curved Riemann manifolds using the QFE method. We discuss the spectral properties of the finite element Laplacian on the 2-sphere. Then we present a study of interacting scalar field theory on the 2-sphere and show that at criticality it is in close agreement with the exact c = 1/2 minimal Ising CFT to high precision. We also investigate interacting scalar field theory on [special characters omitted] x [special characters omitted]2, and we report significant progress towards studying the 3D Ising conformal fixed point in radial quantization with the QFE method. In the near future, we hope for the QFE method to be used to characterize the four dimensional conformal fixed points considered in the first half of this dissertation.
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Corrado, Richard Anthony. "Some aspects of the connection between field theories and gravity /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.
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Catterall, Simon Marcus. "Numerical studies of field theories on random lattices." Thesis, University of Oxford, 1988. http://ora.ox.ac.uk/objects/uuid:48c22ec6-0096-4a2c-be45-cbb20ba7a5b6.
Full textAbstract:
In this thesis we shall be concerned with the study of models which arise as a consequence of adopting discrete regularisations for various Euclidean space quantum field theories. Specifically, we employ a random triangulation of the continuum space, and define the fields only over nodes or links of the mesh. Lattice field theories, together with the Renormalisation Group, are introduced in the first chapter. Continuum physics is shown to depend on the positions and stabilities of zeroes of the β-function, which in turn requires a knowledge of the critical behaviour of the associated statistical model. In Chapter 2. we examine a theory of Dirac fermions in 2 + 1 dimensions on a random lattice. We investigate the behaviour of the 2-pt function and fermion condensate in the absence of any background gauge field. The results indicate certain doubling problems, generic to regular lattice formulations of fermion field theories, are evaded, at least at tree graph level. We then go on to examine the fermion vacuum currents in the presence of background fields with non-zero winding number. We are able to demonstrate the existence of a Chern-Simon's topological term in the gauge field effective action which yields parity violating vacuum currents. The magnitude of these are in agreement with certain continuum calculations. The final chapter concerns the properties of random surfaces. The particular class of models chosen originate as discretisations of Polyakov's string. The partition function is approximated by a sum over all possible random triangulations and an integral over vertex positions. The sum over random lattices is intended to mimick the functional integral over intrinsic metrics encountered in the continuum, and the model may also be pictured as 2D quantum gravity coupled to a scalar field. We consider the phase structure of the models when two forms of extrinsic curvature are added to the standard action. Monte-Carlo simulation indicates that with one type of curvature term a strong 2nd order phase transition exists at finite coupling, leading to a new continuum limit for the model possessing long-range correlation properties. With the other type a much weaker higher order transition is observed. In this case the surface will be crumpled at long distance. We discuss the implications of these results for continuum surfaces.