Two-dimensional models as testing ground for principles and concepts of local quantum physics

In the past two-dimensional models of QFT have served as theoretical laboratories for testing new concepts under mathematically controllable condition. In more recent times low-dimensional models (e.g. chiral models, factorizing models) often have been treated by special recipes in a way which sometimes led to a loss of unity of QFT. In the present work I try to counteract this apartheid tendency by reviewing past results within the setting of the general principles of QFT. To this I add two new ideas: (1) a modular interpretation of the chiral model Diff(S)-covariance with a close connection to the recently formulated local covariance principle for QFT in curved spacetime and (2) a derivation of the chiral model temperature duality from a suitable operator formulation of the angular Wick rotation (in analogy to the Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational chiral theories. The SL(2,Z) modular Verlinde relation is a special case of this thermal duality and (within the family of rational models) the matrix S appearing in the thermal duality relation becomes identified with the statistics character matrix S. The relevant angular Euclideanization'' is done in the setting of the Tomita-Takesaki modular formalism of operator algebras. I find it appropriate to dedicate this work to the memory of J. A. Swieca with whom I shared the interest in two-dimensional models as a testing ground for QFT for more than one decade. This is a significantly extended version of an ``Encyclopedia of Mathematical Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section

STUDIES TO IMPROVE EXHAUST SYSTEM ACOUSTIC PERFORMANCE BY DETERMINATION AND ASSESSMENT OF THE SOURCE CHARACTERISTICS AND IMPEDANCE OPTIMIZATION

It is shown that the relationship between an impedance change and the dynamic response of a linear system is in the form of the Moebius transformation. The Moebius transformation is a conformal complex transformation that maps straight lines and circles in one complex plane into straight lines and circles in another complex plane. The center and radius of the mapped circle can be predicted provided that all the complex coefficients are known. This feature enables rapid determination of the optimal impedance change to achieve desired performance. This dissertation is primarily focused on the application of the Moebius transformation to enhance vibro-acoustic performance of exhaust systems and expedite the assessment due to modifications. It is shown that an optimal acoustic impedance change can be made to improve both structural and acoustic performance, without increasing the overall dimension and mass of the exhaust system. Application examples include mufflers and enclosures. In addition, it is demonstrated that the approach can be used to assess vibration isolators. In many instances, the source properties (source strength and source impedance) will also greatly influence exhaust system performance through sound reflections and resonances. Thus it is of interest to acoustically characterize the sources and assess the sensitivity of performance towards source impedance. In this dissertation, the experimental characterization of source properties is demonstrated for a diesel engine. Moreover, the same approach can be utilized to characterize other sources like refrigeration systems. It is also shown that the range of variation of performance can be effectively determined given the range of source impedance using the Moebius transformation. This optimization approach is first applied on conventional single-inlet single-outlet exhaust systems and is later applied to multi-inlet multi-outlet (MIMO) systems as well, with proper adjustment. The analytic model for MIMO systems is explained in details and validated experimentally. The sensitivity of MIMO system performance due to source properties is also investigated using the Moebius transformation

Complex Formalism of the Linear Beam Dynamics

It has long been known that the ellipse normally used to model the phase space extension of a beam in linear dynamics may be represented by a complex number which can be interpreted similar to a complex impedance in electrical circuits, so that classical electrical methods might be used for the design of such beam transport lines. However, this method has never been fully developed, and only the transport transformation of single particular elements, like drift spaces or quadrupoles, has been presented in the past. In this article, we complete the complex formalism of linear beam dynamics by obtaining a general differential equation and solving it, to show that the general transformation of a linear beam line is a complex Moebius transformation. This result opens the possibility of studying the effect of the beam line on complete regions of the complex plane and not only on a single point. Taking advantage of this capability of the formalism, we also obtain an important result in the theory of the transport through a periodic line, proving that the invariant points of the transformation are only a special case of a more general structure of the solution, which are the invariant circles of the one-period transformation. Among other advantages, this provides a new description of the betatron functions beating in case of a mismatched injection in a circular acceleratorThis work was supported in part by the Ministerio de Asuntos Economicos y Transformacion Digital (MINECO) under Grant DPI2017-82373-R and in part by Universidad del Pais Vasco/Euskal Herriko Univertsitatea (UPV/EHU) under Grant GIU18/19

Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics

After recalling episodes from Pascual Jordan's biography including his pivotal role in the shaping of quantum field theory and his much criticized conduct during the NS regime, I draw attention to his presentation of the first phase of development of quantum field theory in a talk presented at the 1929 Kharkov conference. He starts by giving a comprehensive account of the beginnings of quantum theory, emphasising that particle-like properties arise as a consequence of treating wave-motions quantum-mechanically. He then goes on to his recent discovery of quantization of ``wave fields'' and problems of gauge invariance. The most surprising aspect of Jordan's presentation is however his strong belief that his field quantization is a transitory not yet optimal formulation of the principles underlying causal, local quantum physics. The expectation of a future more radical change coming from the main architect of field quantization already shortly after his discovery is certainly quite startling. I try to answer the question to what extent Jordan's 1929 expectations have been vindicated. The larger part of the present essay consists in arguing that Jordan's plea for a formulation without ``classical correspondence crutches'', i.e. for an intrinsic approach (which avoids classical fields altogether), is successfully addressed in past and recent publications on local quantum physics.Comment: More biographical detail, expansion of the part referring to Jordan's legacy in quantum field theory, 37 pages late