The Quantum Geometer's Universe: Particles, Interactions and Topology

Abstract

With the two most profound conceptual revolutions of XX th century physics, quantum mechanics and relativity, which have culminated into relativistic spacetime geometry and quantum gauge field theory as the principles for gravity and the three other known fundamental interactions, the physicist of the XXI st century has inherited an unfinished symphony: the unification of the quantum and the continuum. As an invitation to tomorrow's quantum geometers who must design the new rulers by which to size up the Universe at those scales where the smallest meets the largest, these lectures review the basic principles of today's conceptual framework, and highlight by way of simple examples the interplay that presently exists between the quantum world of particle interactions and the classical world of geometry and topology.

Nonlinear Spinor Field Equations in Gravitational Theory: Spherical Symmetric Soliton-Like Solutions

“…where r stands for the radial component of the spherical symmetric metric, and satisfying the coordinate condition 2 . …”

“…Such a picture lends itself most ideally to a perturbative understanding of the fundamental interactions, which has proved to be so powerful beginning with quantum electrodynamics, up to the modern Standard Model of the strong and electroweak interactions. For more explanation on these profound concepts, quantum theory and relativity, which have culminated into relativistic spacetime geometry and quantum gauge theory as the principles for gravity and the three other known fundamental interactions, see notes [2] on The quantum geometer's universe: Particles, interactions and topology delivered in 2001 by Govaerts at the Second International Workshop on Contemporary Problems in Mathematical Physics.…”

“…All these activities, diverse and complementary, made in this field [1][2][3][4][5][6][7][8][9][10][11][12][13][14], are also mainly motivated by the wide roles of Einstein and Dirac equations in modern physics, for example, for investigating the spin particle and for the necessity of analysis of synchrotronic radiation [11]. To this purpose, many systems have been subjects of considerable interest and studies.…”

On the Road Towards the Quantum Geometer’s Universe: An Introduction to Four-Dimensional Supersymmetric Quantum Field Theory

“…1 Quantised fermionic field theories are briefly dealt with in Sec. 3. Besides such physics motivations just hinted at, supersymmetry has also proved to be of great value in mathematical physics, in the understanding of nonperturbative phenomena in quantum field theories and M-theory, 3,4 and for uncovering deep connections between different fields of pure mathematics.…”

“…Our aim here will thus be very modest. Namely, starting from the contents of the previous lecture notes, 1 build a bridge reaching the entry roads and the shores towards supersymmetric field theories and the fundamental concepts entering their construction. Not that the lectures delivered at the Workshop did not discuss the general superfield approach over superspace as the most efficient and transparent techniques for such constructions in the case of N = 1 supersymmetry, but the latter material being so widely and in such detailed form available from the literature, it is felt that rather a detailed introduction to the topics missing from Ref.…”