Constructing dualities from quantum state manifolds

Van Zyl, Hendrik Jacobus Rust (2015-12)

Thesis (PhD)--Stellenbosch University, 2015.

Thesis

ENGLISH ABSTRACT: A constructive procedure to build gravitational duals from quantum mechanical models is developed with the aim of studying aspects of the gauge/gravity duality. The construction is simplified as far as possible - the most notable simplification being that quantum mechanical models are considered as opposed to quantum field theories. The simplifications allow a systematic development of the construction which provides direct access to the quantum mechanics / gravity dictionary. The procedure is divided into two parts. First a geometry is constructed from a family of quantum states such that the symmetries of the quantum mechanical states are encoded as isometries of the metric. Secondly, this metric is interpreted as the metric that yields a stationary value for the dual gravitational action. If the quantum states are non-normalisable then these states need to be regularised in order to define a sensible metric. These regularisation parameters are treated as coordinates on the manifold of quantum states. This gives rise to the idea of a manifold “bulk” where the states are normalisable and of a “boundary” where they are not. Asymptotically anti-de Sitter geometries arise naturally from non-normalisable states but the geometries can also be much more general. Time-evolved states are the initial interest. A sensible regularisation scheme for these states is a simple complexification of time so that the bulk coordinate has the interpretation of an energy scale. These two-dimensional manifolds of states are dual to models of dilaton gravity where the dilaton has the interpretation of the expectation value of a quantum mechanical operator. As an example, states time-evolving under an su(1, 1) Hamiltonian is dual to dilaton gravity on AdS2, in agreement with existing work on the AdS2/CFT1 correspondence. These existing results are revisited with the aid of the systematic quantum mechanics / dilaton gravity dictionary and extended. As another example, states time-evolving under an su(2) Hamiltonian are shown to be dual to dilaton gravity on dS2. The higher dimensional analysis is restricted, for computational reasons, to the example of states that possess full Schr¨odinger symmetry with and without dynamical mass. The time and spatial coordinates are complexified in order to both regularise the states and maintain the state symmetries as bulk isometries. Dictionaries are developed for both examples. It is shown that submanifolds of these state manifolds are studied in the existing AdS/CFT and AdS/NRCFT literature.

AFRIKAANSE OPSOMMING: ’n Konstruktiewe metode word ontwikkel wat swaartekragduale van kwantummeganiese modelle bou met die oog op die ondersoek van die yk / swaartekrag dualiteit. Die konstruksie word sover moontlik vereenvoudig en spesifiek word kwantummeganiese modelle beskou in plaas van kwantumveldeteorie¨e. Die vereenvoudigings laat ’n sistematiese ontwikkeling van die metode toe wat dus direkte toegang tot die kwantummeganika / swaartekrag woordeboek verleen. Die metode bestaan uit twee dele. Eers word ’n geometrie saamgestel vanaf ’n familie van kwantumtoestande wat die simmetrie¨e van die toestande as isometrie¨e behou. Daarna word ’n aksie gedefinieer wat deur hierdie metriek stasionˆer gelaat word. Indien die kwantumtoestande nie normaliseerbaar is nie moet hul op ’n gepaste wyse geregulariseer word. Die regularisasieparameters word dan as koordinate beskou. Dit gee dan aanleiding tot die idee van ’n “bulk”waar die toestande normaliseerbaar is en ’n “rand”waar hulle nie is nie. Asimptotiese anti-de Sitter geometrie¨e volg op natuurlike wyse vanaf nie-normaliseerbare toestande, maar die geometrie kan egter baie meer algemeen wees as dit. Tyd-ontwikkelde toestande is die eerste onderwerp. ’n Sinvolle regulariseringsmetode is bloot om tyd kompleks te maak wat dan die radiale koordinaat as ’n energieskaal giet. Die duale beeld van hierdie twee-dimensionele toestande is ’n model van dilaton-swaartekrag waar die dilaton die interpretasie van ’n kwantumoperatorverwagtingswaarde dra. As ’n voorbeeld hiervan - die duale beeld van toestande wat ontwikkel onder ’n su(1, 1) Hamiltoniaan is dilaton-swaartekrag op AdS2. Hierdie beeld strook met bestaande restultate uit the AdS2/CFT1 literatuur. Hierdie bestaande resultate word herondersoek en toevoegings word gemaak daaartoe met behulp van die sistematiese kwantummeganika / dilatonswaartekrag woordeboek. As nog ’n voorbeeld word dit aangetoon dat die duale beeld van toestande wat tydontwikkel onder ’n su(2) Hamiltoniaan ’n model van diltatonswaartekrag op dS2 is. Die ho¨er-dimensionele ondersoek word, ter wille van eenvoudigheid, beperk tot toestande wat oor volle Schr¨odinger simmetrie beskik met en sonder dinamiese massa. Die tyd- en ruimtelike koordinate word kompleks gemaak om die toestande te regulariseer en simmetrie¨e te behou. Woordeboeke word saamgestel vir beide gevalle. Dit word aangetoon dat submetrieke van hierdie metrieke in die AdS/CFT en AdS/NRCFT literatuur bestudeer word.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/97763
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