Path Integrals and the Scale Anomaly

The theme of this thesis is scaling, units, and dimensional analysis, along with symmetry, and the failure of these concepts at the quantum level, along with implications of such a failure. Such a failure can be quantified by a quantity called the quantum anomaly. The term anomaly indicates deviation from expected behavior, where expected behavior is of course classical behavior, and deviant behavior is quantum behavior. In the introduction, we will give a short, simple, self-contained example explaining what is renormalization. This example will get to the heart of what we mean by the anomaly - the complete destruction of a system's symmetries due to quantum effects. The applications of the anomaly are enormous, spanning several branches of physics, from atomic to condensed matter to particle to gravitational physics. For example, just within particle physics, the chiral anomaly is responsible for the decay rate of the $\Pi^0$ meson to two photons. The scale anomaly is responsible for the Yang-Mills mass gap in pure QCD and the formation of glueballs, and its calculation is an intermediate step in lattice QCD to calculate the QCD phase diagram - indeed, the anomaly is responsible for $\Lambda_{\text{QCD}}$ itself, through dimensional transmutation. However, one of the most exciting applications of anomalies has been realized only in this decade in the study of ultracold gases, where the measurement of various manifestations of the anomaly has only now become experimentally accessible to atomic physicists. In 2008, a set of universal thermodynamic relations known as the Tan relations was published in a series of 3 back-to-back-to-back papers. In (2+1) dimensions, the Tan contact is merely the anomaly. In this thesis, we develop a novel framework for calculating anomalies using the path-integral and Fujikawa's determinant. In particular, we derive 4 results: the anomaly for a (3+1) relativistic Bose gas, the Tan-pressure relation for a (2+1) nonrelativistic Bose gas, a new derivation of the virial theorem, and the relationship between the Fujikawa determinant and the quantum effective potential using the background field method. Some unpublished results will also be discussed, and as how this all began, wildly speculative ideas end the thesis.Physics, Department o

A Primer for Black Hole Quantum Physics

The mechanisms which give rise to Hawking radiation are revealed by analyzing in detail pair production in the presence of horizons. In preparation for the black hole problem, three preparatory problems are dwelt with at length: pair production in an external electric field, thermalization of a uniformly accelerated detector and accelerated mirrors. In the light of these examples, the black hole evaporation problem is then presented. The leitmotif is the singular behavior of modes on the horizon which gives rise to a steady rate of production. Special emphasis is put on how each produced particle contributes to the mean albeit arising from a particular vacuum fluctuation. It is the mean which drives the semiclassical back reaction. This aspect is analyzed in more detail than heretofore and in particular its drawbacks are emphasized. It is the semiclassical theory which gives rise to Hawking's famous equation for the loss of mass of the black hole due to evaporation $dM/dt \simeq -1/M^2$. Black hole thermodynamics is derived from the evaporation process whereupon the reservoir character of the black hole is manifest. The relation to the thermodynamics of the eternal black hole through the Hartle--Hawking vacuum and the Killing identity are displayed. It is through the analysis of the fluctuations of the field configurations which give rise to a particular Hawking photon that the dubious character of the semiclassical theory is manifest. The present frontier of research revolves around this problem and is principally concerned with the fact that one calls upon energy scales that are greater than Planckian and the possibility of a non unitary evolution as well. These last subjects are presented in qualitative fashion only, so that this review stops at the threshold of quantum gravity.Comment: An old review article on black hole evaporation and black hole thermodynamics, put on the archive following popular demand, 178 pages, 21 figures (This text differs in slightly from the published version

Interpreting Theories without a Spacetime

In this paper we have two aims: first, to draw attention to the close connexion between interpretation and scientific understanding; second, to give a detailed account of how theories without a spacetime can be interpreted, and so of how they can be understood. In order to do so, we of course need an account of what is meant by a theory `without a spacetime': which we also provide in this paper. We describe three tools, used by physicists, aimed at constructing interpretations which are adequate for the goal of understanding. We analyse examples from high-energy physics illustrating how physicists use these tools to construct interpretations and thereby attain understanding. The examples are: the 't Hooft approximation of gauge theories, random matrix models, causal sets, loop quantum gravity, and group field theory

Interpreting Theories without a Spacetime

In this paper we have two aims: first, to draw attention to the close connexion between interpretation and scientific understanding; second, to give a detailed account of how theories without a spacetime can be interpreted, and so of how they can be understood. In order to do so, we of course need an account of what is meant by a theory `without a spacetime': which we also provide in this paper. We describe three tools, used by physicists, aimed at constructing interpretations which are adequate for the goal of understanding. We analyse examples from high-energy physics illustrating how physicists use these tools to construct interpretations and thereby attain understanding. The examples are: the 't Hooft approximation of gauge theories, random matrix models, causal sets, loop quantum gravity, and group field theory

Foundations of Multi-Paradigm Modelling for Cyber-Physical Systems

This open access book coherently gathers well-founded information on the fundamentals of and formalisms for modelling cyber-physical systems (CPS). Highlighting the cross-disciplinary nature of CPS modelling, it also serves as a bridge for anyone entering CPS from related areas of computer science or engineering. Truly complex, engineered systems—known as cyber-physical systems—that integrate physical, software, and network aspects are now on the rise. However, there is no unifying theory nor systematic design methods, techniques or tools for these systems. Individual (mechanical, electrical, network or software) engineering disciplines only offer partial solutions. A technique known as Multi-Paradigm Modelling has recently emerged suggesting to model every part and aspect of a system explicitly, at the most appropriate level(s) of abstraction, using the most appropriate modelling formalism(s), and then weaving the results together to form a representation of the system. If properly applied, it enables, among other global aspects, performance analysis, exhaustive simulation, and verification. This book is the first systematic attempt to bring together these formalisms for anyone starting in the field of CPS who seeks solid modelling foundations and a comprehensive introduction to the distinct existing techniques that are multi-paradigmatic. Though chiefly intended for master and post-graduate level students in computer science and engineering, it can also be used as a reference text for practitioners

Contextualizing generative design

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Architecture, 2004.Includes bibliographical references (p. 107-[110]).Generative systems have been widely used to produce two- and three-dimensional constructs, in an attempt to escape from our preconceptions and pre-existing spatial language. The challenge is to use this mechanism in real-world architectural contexts in which complexity and constraints imposed by the design problem make it difficult to negotiate between the emergent output, the context, and the controllability desired by the human designer. This thesis investigates how generative systems address contextual parameters, including the designer, client, user, meaning, aesthetics, environment, and function. This is demonstrated through my case studies, in which my aim was to avoid computerized unprocessed formalism that does not implicitly allow for any contextual and cultural content. I sought to extend simple algorithmic form-generation processes to allow for the subtleties of a given context to be effectively addressed. Some challenges and questions arose from these case studies. By interrogating different generative machines, common threads and challenges, similar to mine encountered in the case studies, were found. All of the processes that strove towards the creation of a generative system struggled with similar issues: How can we use rule-based systems without sacrificing meaning or function or the humanistic touch? How can we address contextual parameters without a loss?by Saeed Arida.S.M