Σχεδόν τέσσερις δεκαετίες πριν, η μαθηματική αναλογία μεταξύ των νόμων της μηχανικής των μελανών οπών και των νόμων της θερμοδυναμικής έδωσε ισχυρά επιχειρήματα στους Bekenstein και Hawking να υποθέσουν ότι, αν τα κβαντικά φαινόμενα λαμβάνονται υπόψη, οι μελανές οπές θα πρέπει να θεωρηθούν ως θερμοδυναμικά συστήματα, που χαρακτηρίζονται από θερμοκρασία και την εντροπία τους. Πιο συγκεκριμένα, ότι η εντροπία των μελανών οπών είναι S = A / 4, όπου Α είναι η περιοχή του ορίζοντα γεγονότων. Ακόμη και σήμερα, υπάρχουν δύο σημαντικά ερωτήματα σχετικά με τους βαθμούς ελευθερίας: α) ποιοι είναι οι βαθμοί ελευθερίας υπεύθυνοι για την εντροπία των μελανών οπών; β) που βρίσκονται αυτοί οι βαθμοί ελευθερίας; Πριν από τριάντα χρόνια, η απάντηση και στα δύο παραπάνω ερωτήματα ήταν: "Δεν γνωρίζουμε" Σήμερα, αντίθετα, το πρόβλημα είναι ότι, υπάρχουν πάρα πολλές απαντήσεις για τα παραπάνω ερωτήματα. Σε εργασία αυτή, μελετώ την παραγωγή των Bekenstein-Hawking φόρμουλα για την εντροπίας των μελανών οπών στην κβαντική θεωρία της βαρύτητας. Ειδικότερα, επικεντρώνομαι στις υποθέσεις, τη μεθοδολογία και τα αποτελέσματα κάθε προσέγγισης. Πρώτα, μελετώ την παραγωγή της Bekenstein-Hawking φόρμουλα σε μεθόδους που δεν βασίζονται σε μια θεμελιώδη θεωρία της κβαντικής βαρύτητας, όπως η στατιστική μηχανική προέλευση της εντροπίας μιας περιστρεφόμενης, φορτισμένη μελανής οπής, η μέθοδος του ολοκληρώματος διαδρομών στο πλαίσιο της Ευκλείδεια κβαντική βαρύτητα και τη μέθοδος του κανονικού συνόλου για τις μελανές οπές σε μια σφαιρική κοιλότητα στο πλαίσιο της Ευκλείδεια κβαντικής βαρύτητας. Τα τελευταία είκοσι χρόνια περίπου, oι αποδείξεις γίνονται στα πλαίσια θεμελιωδών θεωριών της κβαντικής βαρύτητας. Εδώ, θα δούμε την απόδειξη στα πλαίσια της σύμμορφης θεωρίας πεδίου του ορίζοντα, της επαγόμενης βαρύτητας, της κβαντικής βαρύτητας των βρόγχων και της θεωρίας των χορδών. / Almost four decades ago, the mathematical analogy between the laws of black hole mechanics and the laws of thermodynamics gave strong arguments to Bekenstein and Hawking to assume that, if quantum effects are taken into account, black holes should be viewed as thermodynamic systems, characterized by temperature and entropy. In particular, that the black hole entropy is S=A/4, where A is the area of the event horizon. Even today, there are two important questions about the degrees of freedom: a) which are the degrees of freedom responsible for the black hole entropy? b) Where are the degrees of freedom localized? Thirty years ago, the answer to both questions above was: "we do not know". Today, by contrast, the problem is that, there are too many answers for the questions above. In this thesis, I study the derivation of the Bekenstein-Hawking formula for the black hole entropy in quantum gravity theories. In particular, I focus on the hypotheses, the methodology and the results of each approach. First, I consider the derivation of the Bekenstein-Hawking formula in methods which are not based on a fundamental theory of quantum gravity such as, the statistical mechanical origin of the entropy of a rotating, charged black hole, the path integral method in the Euclidean quantum gravity framework and the canonical ensemble method for black holes in a spherical cavity in the Euclidean quantum gravity framework. Then, I consider the derivations of the black hole entropy which are made in the framework of candidate quantum gravity theories such as, the horizon conformal field theory, the induced gravity theory, loop quantum gravity and string theory.

Despite a general recognition of the importance of complex systems, there is a dearth of general models capable of describing their dynamics. This is attributed to a complexity scale; the models are attempting to describe systems at different parts of the scale and are hence not compatible. We require new models capable of describing complex behaviour at different points of the complexity scale. This work identifies, and proceeds to examine systems at the high end of the complexity scale, those which have not to date been well understood by our current modelling methodology. It is shown that many such models exhibit what might be termed contextual dependency, and that it is precisely this feature which is not well understood by our current modelling methodology. A particular problem is discussed; our apparent inability to generate systems which display high end complexity, exhibited by for example the general failure of strong ALife. A new model, Process Physics, that has been developed at Flinders University is discussed, and arguments are presented that it exhibits high end complexity. The features of this model that lead to its displaying such behaviour are discussed, and the generalisation of this model to a broader range of complex systems is attempted.

The theory of decoherence aims to explain how macroscopic quantum objects become effectively classical. Understanding this process could help in the search for the quantum theory underlying gravity, and suggest new schemes for preserving the coherence of technological quantum devices. The process of decoherence is best understood in terms of information flow within a quantum system, and between the system and its environment. We develop a novel way of characterizing this information, and give a sufficient condition for its classicality. These results generalize previous models of decoherence, clarify the process by which a phase-space based on non-commutative quantum variables can emerge, and provide a possible explanation for the universality of the phenomenon of decoherence. In addition, the tools developed in this approach generalize the theory of quantum error correction to infinite-dimensional Hilbert spaces. We characterize the nature of the information preserved by a quantum channel by the observables which exist in its image (in the Heisenberg picture). The sharp observables preserved by a channel form an operator algebra which can be characterized in terms of the channel's elements. The effect of the channel on these observables can be reversed by another physical transformation. These results generalize the theory of quantum error correction to codes characterized by arbitrary von Neumann algebras, which can represent hybrid quantum-classical information, continuous variable systems, or certain quantum field theories. The preserved unsharp observables (positive operator-valued measures) allow for a finer characterization of the information preserved by a channel. We show that the only type of information which can be duplicated arbitrarily many times consists of coarse-grainings of a single POVM. Based on these results, we propose a model of decoherence which can account for the emergence of a realistic classical phase-space. This model supports the view that the quantum-classical correspondence is given by a quantum-to-classical channel, which is another way of representing a POVM.

The theory of decoherence aims to explain how macroscopic quantum objects become effectively classical. Understanding this process could help in the search for the quantum theory underlying gravity, and suggest new schemes for preserving the coherence of technological quantum devices. The process of decoherence is best understood in terms of information flow within a quantum system, and between the system and its environment. We develop a novel way of characterizing this information, and give a sufficient condition for its classicality. These results generalize previous models of decoherence, clarify the process by which a phase-space based on non-commutative quantum variables can emerge, and provide a possible explanation for the universality of the phenomenon of decoherence. In addition, the tools developed in this approach generalize the theory of quantum error correction to infinite-dimensional Hilbert spaces. We characterize the nature of the information preserved by a quantum channel by the observables which exist in its image (in the Heisenberg picture). The sharp observables preserved by a channel form an operator algebra which can be characterized in terms of the channel's elements. The effect of the channel on these observables can be reversed by another physical transformation. These results generalize the theory of quantum error correction to codes characterized by arbitrary von Neumann algebras, which can represent hybrid quantum-classical information, continuous variable systems, or certain quantum field theories. The preserved unsharp observables (positive operator-valued measures) allow for a finer characterization of the information preserved by a channel. We show that the only type of information which can be duplicated arbitrarily many times consists of coarse-grainings of a single POVM. Based on these results, we propose a model of decoherence which can account for the emergence of a realistic classical phase-space. This model supports the view that the quantum-classical correspondence is given by a quantum-to-classical channel, which is another way of representing a POVM.

There is a growing consensus among physicists that the classical notion of spacetime has to be drastically revised in order to nd a consistent formulation of quantum mechanics and gravity. One such nontrivial attempt comprises of replacing functions of continuous spacetime coordinates with functions over noncommutative algebra. Dynamics on such noncommutative spacetimes (noncommutative theories) are of great interest for a variety of reasons among the physicists. Additionally arguments combining quantum uncertain-ties with classical gravity provide an alternative motivation for their study, and it is hoped that these theories can provide a self-consistent deformation of ordinary quantum field theories at small distances, yielding non-locality, or create a framework for finite truncation of quantum field theories while preserving symmetries. In this thesis we study the gauge theories on noncommutative Moyal space. We nd new static solitons and instantons in terms of the so-called generalized Bose operators (GBO). GBOs are constructed to describe reducible representation of the oscillator algebra. They create/annihilate k-quanta, k being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of GBOs. The Nielsen-Olesen vortex solutions found in terms of these operators also reduce to the ones known in the literature. On the other hand, we nd a class of new instanton solutions which are unitarily inequivalent to the ones found from ADHM construction on noncommutative space. The charge of the instanton has a description in terms of the index representing the reducibility of the Fock space representation, i.e., k. After studying the static soliton solutions in noncommutative Minkowski space and the instanton solutions in noncommutative Euclidean space we go on to study the implications of the time-space noncommutativity in Minkowski space. To understand it properly we study the time-dependent transitions of a forced harmonic oscillator in noncommutative 1+1 dimensional spacetime. We also provide an interpretation of our results in the context of non-linear quantum optics. We then shift to the so-called DSR theories which are related to a different kind of noncommutative ( -Minkowski) space. DSR (Doubly/Deformed Special Relativity) aims to search for an alternate relativistic theory which keeps a length/energy scale (the Planck scale) and a velocity scale (the speed of light scale) invariant. We study thermodynamics of an ideal gas in such a scenario. In first chapter we introduce the subjects of the noncommutative quantum theories and the DSR. Chapter 2 starts with describing the GBOs. They correspond to reducible representations of the harmonic oscillator algebra. We demonstrate their relevance in the construction of topologically non-trivial solutions in noncommutative gauge theories, focusing our attention to flux tubes, vortices, and instantons. Our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the GBO. When used in conjunction with the noncommutative ADHM construction, we nd that these new instantons are in general not unitarily equivalent to the ones currently known in literature. Chapter 3 studies the time dependent transitions of quantum forced harmonic oscillator (QFHO) in noncommutative R1;1 perturbatively to linear order in the noncommutativity . We show that the Poisson distribution gets modified, and that the vacuum state evolves into a \squeezed" state rather than a coherent state. The time evolutions of un-certainties in position and momentum in vacuum are also studied and imply interesting consequences for modelling nonlinear phenomena in quantum optics. In chapter 4 we study thermodynamics of an ideal gas in Doubly Special Relativity. We obtain a series solution for the partition function and derive thermodynamic quantities. We observe that DSR thermodynamics is non-perturbative in the SR and massless limits. A stiffer equation of state is found. We conclude our results in the last chapter.