Browsing by Subject "Quantum field theory"
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Item Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories(2008-08) Stirling, Spencer; Freed, Daniel S.Classical and quantum Chern-Simons with gauge group U(1)N were classified by Belov and Moore in [BM05]. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary (2 + 1)-dimensional TQFTs and modular tensor categories. We study group categories and extend them slightly to produce modular tensor categories that correspond to toral Chern-Simons. Group categories have been widely studied in other contexts in the literature [FK93],[Qui99],[JS93],[ENO05],[DGNO07]. The main result is a proof that the associated projective representation of the mapping class group is isomorphic to the one from toral Chern-Simons. We also remark on an algebraic theorem of Nikulin that is used in this paper.Item Non-supersymmetric holographic engineering and U-duality(2012-08) Young, Stephen Christopher; Fischler, Willy; Caceres, Elena; Paban, Sonia; Dicus, Duane; Freed, DanIn this Ph.D. thesis, we construct and study a number of new type IIB supergravity backgrounds that realize various flavored, finite temperature, and non-supersymmetric deformations of the resolved and deformed conifold geometries. We make heavy use of a U-duality solution generating procedure that allows us to begin with a modification of a family of solutions describing the backreaction of D5 branes wrapped on the S^2 of the resolved conifold, and generate new backgrounds related to the Klebanov-Strassler background. We first construct finite temperature backgrounds which describe a configuration of N_c D5 branes wrapped on the S^2 of the resolved conifold, in the presence of N_f flavor brane sources and their backreaction i.e. N_f/N_c ~ 1. In these solutions the dilaton does not blow up at infinity but stabilizes to a finite value. The U-duality procedure is then applied to these solutions to generate new ones with D5 and D3 charge. The resulting backgrounds are a non-extremal deformation of the resolved deformed conifold with D3 and D5 sources. It is tempting to interpret these solutions as gravity duals of finite temperature field theories exhibiting phenomena such as Seiberg dualities, Higgsing and confinement. However, a first necessary step in this direction is to investigate their stability. We study the specific heat of these new flavored backgrounds and find that they are thermodynamically unstable. Our results on the stability also apply to other non-extremal backgrounds with Klebanov-Strassler asymptotics found in the literature. In the second half of this thesis, we apply the U-duality procedure to generate another class of solutions which are zero temperature, non-supersymmetric deformations of the baryonic branch of Klebanov-Strassler. We interpret these in the dual field theory by the addition of a small gaugino mass. Using a combination of numerical and analytical methods, we construct the backgrounds explicitly, and calculate various observables of the field theory.Item A numerical study of relativistic fluid collapse(2003) Noble, Scott Charles; Morrison, Philip J.; Choptuik, Matthew WilliamItem The philosophical significance of unitarily inequivalent representations in quantum field theory(2008-05) Lupher, Tracy Alexander; Kronz, Frederick M.This dissertation gives a general account of the properties of unitarily inequivalent representations (UIRs) in both canonical quantum field theory and algebraic quantum field theory. A simple model is constructed and then used to show how to build a broad spectrum of UIRs including a version of Haag’s theorem. Haag and Kastler,P, two of the founding fathers of algebraic quantum field theory, argue that the problems posed by UIRs are solved by adopting a notion of equivalence that is weaker than unitary equivalence, which they refer to as physical equivalence. In the dissertation, it is shown that their notion does not provide a suitable classificatory schema. Some of the most important physical representations fail to satisfy the mathematical conditions of their notion. However, Haag and Kastler's notion has an unexpected connection with classical observables. A theorem is proven in which two representations make the same predictions with respect to all classical observables if and only if they satisfy their notion of physical equivalence. Following Haag and Kastler's lead, it was claimed by most proponents of algebraic quantum field theory that all physical content resides in a specific class of observables. It is shown in the dissertation that such claims are exaggerated and misleading. UIRs are used to elucidate the nature of quantum field theory by showing that UIRs have different expectation values for some classical observables of the system, such as temperature and chemical potential, which are not in Haag and Kastler’s specific class. It is shown how UIRs may be used to construct classical observables. To capture the physical content of quantum field theory it is shown that a much larger algebra than that of Haag and Kastler is necessary. Finally, the arguments that UIRs are incommensurable theories are shown to be flawed. The lesson of UIRs is that the mathematical structures in both canonical quantum field theory and Haag and Kastler’s version of algebraic quantum field theory are not sufficient to capture all of the physical content that UIRs represent. A suitable algebraic structure for quantum field theory is provided in the dissertation.Item Renormalization and effective field theory using the energy scale(2013-08) Russell, Tyler A.; Hamilton, Alastair; Gelca, Razvan; Drager, Lance D.In this thesis, we develop the notion of an effective field theory using the energy scale and present an algorithm for constructing counterterms based on the choice of a renormalization scheme.Item Renormalization group applications in area-preserving nontwist maps and relativistic quantum field theory(2002-05) Wurm, Alexander; DeWitt-Morette, CécileIn this thesis, we apply the ideas of the renormalization group to two different areas of physics. Extending the work of del-Castillo-Negrete, Greene and Morrison on the standard nontwist map, we study the break-up of an invariant torus with winding number different from the inverse golden mean, and interpret the result within the renormalization group framework. We construct a renormalization operator on the space of commuting map pairs, and study its fixed point. In addition, we present preliminary results about a new map that we call the piecewise-linear standard nontwist map. In the second part of this thesis, we extend to fields defined on Minkowski spacetime a functional integral formalism, developed in Euclidean Field Theory to study the long-distance behavior of scalar field theories. A new aspect of this approach is the use of an independent scaling variable. To compute some of the resulting integrals, we work out explicit formulae for the Fourier transform of Lorentz invariant functions in pseudo-spherical coordinates.Item Spin TQFTs and Chern-Simons gauge theory(2004-05) Jenquin, Jerome Anthony, 1975-; Freed, Daniel S.In the first half we construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-PatodiSinger η -invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed spin 3-manifolds and compact spin 3-manifolds with boundary where the action is properly thought of as a unitary element of a Pfaffian line of twisted Dirac operators. We then investigate the properties of the Hamiltonian field theory over 3-manifolds of the form R × Y , where Y is a closed spin 2-manifold. From the action we derive a unitary line bundle with connection over the moduli stack of flat gauge fields on Y . In the second half we conjecture the quantization of our classical gauge theory is a topological quantum field theory, or TQFT. To investigate its properties we apply the procedure of geometric quantization to our field theory when the gauge group is SO3 . We compare our results to the knot-theoretic spin TQFT constructed by Blanchet and Masbaum and find evidence that the two theories are isomorphic.Item State sums in two dimensional fully extended topological field theories(2011-05) Davidovich, Orit; Freed, Daniel S.; Ben-Zvi, David; Distler, Jacques; Reid, Alan; Rodriguez-Villegas, FernandoA state sum is an expression approximating the partition function of a d-dimensional field theory on a closed d-manifold from a triangulation of that manifold. To consider state sums in completely local 2-dimensional topological field theories (TFT's), we introduce a mechanism for incorporating triangulations of surfaces into the cobordism ([infinity],2)-category. This serves to produce a state sum formula for any fully extended 2-dimensional TFT possibly with extra structure. We then follow the Cobordism Hypothesis in classifying fully extended 2-dimensional G-equivariant TFT's for a finite group G. These are oriented theories in which bordisms are equipped with principal G-bundles. Combining the mechanism mentioned above with our classification results, we derive Turaev's state sum formula for such theories.Item Suppression of radiation damping in electromagnetic waveguide, signature of quantum decoherence in the field bath(2003) Ting, Chu Ong; Prigogine, I. (Ilya); Petrosky, Tomio Y.Recent development of spectral analysis of the Liouville-von Neumann equation has revealed the fact that irreversibility is a rigorous dynamical property of Poincaré non-integrable systems with an infinite degrees of freedom interacting among each other through resonance coupling. In the present work we discuss this role of resonance in some examples of matter-field coupling systems for both classical and quantum mechanics: the one is a classical motion of a charged particle in electromagnetic waveguide, and the other is the decoherence problem of quantum matter-field interacting systems. In the first part of this dissertation, we study an accelerated motion of a charged classical dipole molecule with frequency ω1 inside the rectangular waveguide. If the particle is in free space, it is well known that its accelerated motion will eventually stop by radiating the field through the resonance interaction. This result is the so-called radiation damping. For the case in the waveguide, there are two possible situations, due to the existence of the cut-off frequency ωc of the waveguide. Under the cut-off frequency electromagnetic wave cannot propagate inside the waveguide. The stability of the dipole depends on the relation between ω1 and ωc. For ω1 < ωc, the dipole cannot resonate with the field. This corresponds to the Poincaré integrable system. For this case the dipole keeps its accelerated motion without emitting the radiating field. Therefore the radiation damping of the dipole molecules is suppressed inside the waveguide under the absence of resonance interaction. The motion of this steady state somewhat resembles a quantum ground state. We show that this steady state is dressed by electromagnetic field. The overlap of the dressing field leads to a force analogous to van der Waals force in quantum mechanics. The critical frequency determined by ω1 = ωc gives a critical size of the waveguide. For heavy molecules, such as HCl, this is of order 10−5m. We show that the size of the dressing field is the same order of the size of the waveguide. Hence we have a macroscopic size of the dressing in the waveguide. For ω1 > ωc, the dipole can resonate with the field, and the system becomes non-integrable in the sense of Poincaré. As a result, the accelerated motion eventually stop by emitting the resonance field. This corresponds to the problem of classical radiation damping. We show that there is non-negligible deviation of exponential decay in a short time scale of the order t ∼ 1/ω1. This corresponds to the quantum Zeno effect, well known in quantum unstable systems. After this period, the dipole decays exponentially in time by emitting the resonance field. We found by choosing ωc very close to ω1, we can increase the decay rate 105 times faster than the case where the dipole is in the free space, at the same time the emitted field travel 10−4 times slower than the speed of light. This is again a consequence of the existence of the cut-off frequency in the waveguide. Indeed, the cut-off frequency leads to a non-linear dispersion relation for the electromagnetic field. To some extent, the electromagnetic field is sticky inside the waveguide. Due to the large decay rate and slower speed of light, the size of the wavepacket emitted by the dipole is significantly small (about 10cm for HCl). This is even smaller than the quantum case in free space, where the wavepacket of the field emitted by the decay of electron in hydrogen atom is about 1m. In the second part of this dissertation, we study a quantum matter-field coupled system. We focus our attention on the problem of quantum decoherence in a system of a particle coupled with a field, the Hamiltonian of which has a similar structure to the problem of classical radiation damping mentioned above. We apply the complex spectral representation of the Liouville-von Neumann operator that gives a rigorous approach to the irreversible processes. We focus our attention on the time evolution of the field, which is commonly neglected in phenomenological approaches to the decoherence problem. We found a signature of decoherence in the field which has a characteristic time dependence proportional to t that comes from the secular effect between the particle and the field through the resonance interaction that breaks time-symmetry.Item Tinkertoys for Gaiotto duality(2011-08) Chacaltana Alarcon, Oscar Chacaltana; Distler, Jacques; Kaplunovsky, Vadim; Paban, Sonia; Fischler, Willy; Freed, DanielWe describe a procedure for classifying 4D N=2 superconformal theories of the type introduced by Davide Gaiotto. Any punctured curve, C, on which the 6D (2,0) SCFT is compactified, may be decomposed into 3-punctured spheres, connected by cylinders. The 4D theories, which arise, can be characterized by listing the ``matter" theories corresponding to 3-punctured spheres, the simple gauge group factors, corresponding to cylinders, and the rules for connecting these ingredients together. Different pants decompositions of C correspond to different S-duality frames for the same underlying family of 4D \mathcal{N}=2 SCFTs. We developed such a classification for the A_{N-1} and the D_N series of 6D (2,0) theories. We outline the procedure for general A_{N-1} and D_N, and construct, in detail, the classification through A_4 and D_4, respectively.