Geometrical Aspects in Classical and Elementary Particle Physics
Theoretical elementary particle physics has witnessed remarkable advancements over the past few decades. Despite this progress, numerous fundamental questions remain open and continue to motivate our research. Key among these are:
- Is there a unifying geometrical principle from which all fundamental interactions can be derived?
- Does the quantum nature of these interactions necessitate an extension or revision of the classical geometrical framework?
- Are there deeper mathematical structures underlying the formal apparatus of quantum field theory?
- What are the theoretical and physical implications of noncommutative geometry?
- Can universal geometrical structures be identified that underpin the framework of quantum physics?
- How can the connection between classical and quantum physics be characterized, and what insights might this provide into unresolved issues in elementary particle physics?
Our investigations make extensive use of mathematical tools from differential geometry, algebra, and functional analysis. A distinctive feature of our approach is the application of the G-Theory Principle, a conceptual framework developed by members of our research group. This principle represents a comprehensive generalization of the relativity principle originally formulated by Poincaré and Einstein, enriched by ideas from Felix Klein’s Erlanger Programm, appropriately adapted to the context of modern physics.
The G-Theory Principle serves as a unifying foundation across various domains of theoretical physics, applicable in both classical and quantum regimes. It offers a promising avenue toward a deeper understanding of the quantization process and may provide a conceptual bridge between seemingly disparate theoretical structures.
Linear Algebra: A Foundational Framework in Mathematical and Physical Sciences
Linear algebra represents the most elementary mathematical theory in which geometric structures can be realized in a particularly intuitive and concrete manner. Moreover, it underpins virtually all areas of mathematics that are fundamental to theoretical physics.
Essential concepts of linear algebra, which form a critical foundation for the study of theoretical physics, are explored in a dedicated lecture series on linear algebra and differential geometry.
This newly established series follows the conceptual framework of the recent publication by Nikolaos Papadopoulos and Florian Scheck, Linear Algebra for Physics.