Compactification and Dimensional Reduction in Mathematics and The Effects On Space-Time Dimensions
We learn BEST when we learn by IMMERSION.
It’s NECESSARY to be cognitively overwhelmed, perhaps even so frustrated that we are ready to almost feel like crying!
Without that aggravation of being overwhelmed by some cognitive task, our cognitive faculties will atrophy. When we watch entertainment passively, it’s not just that our faculties are atrophying, but we are being programmed to seek comfort, avoid learning new things and to only IMAGINE that we think critically.
The syllabus for autodidactic study starts with by prompting an AI assistant for a year-long syllabus.
Develop a 200-module syllabus to study compactification [or de-compactification] and dimensional reduction [[or dimensional expansion] in mathematics and how this changes a theory with respect to one of its spacetime dimensions.
We take the results of from that prompt and then we tear it part critically as we read and research and then we re-factor and re-build a syllabus that is more comprehensive and more detailed … it will typically take at least one hundred additional prompts to the AI assistant to help us put leaves on the branches of the tree.
The following syllabus is an example of developing the result of the above prompt above.
Compactification and Dimensional Reduction in Mathematics and The Effects On Space-Time Dimensions
The following syllabus provides a comprehensive outline of an overview of compactification, de-compactification, dimensional reduction, and dimensional expansion in various areas of mathematics and theoretical physics. It starts with the foundational concepts of topology, compactness, and dimensional reduction in linear algebra and differential geometry.
The course then explores different compactification techniques and their applications, as well as the concepts of de-compactification and localization. It delves into the role of dimensional reduction in field theory, covering topics such as Kaluza-Klein theory, gauge theories, and supergravity.
The syllabus also includes a discussion of dimensional expansion and its applications in string theory and M-theory. It examines the cosmological implications of extra dimensions and the interplay between compactification, dimensional reduction, and cosmology.
The course covers dualities and their relation to dimensional reduction, as well as advanced topics such as flux compactifications, matrix models, and the role of compactification and dimensional reduction in quantum gravity.
Throughout the course, students will gain a deep understanding of how compactification, de-compactification, dimensional reduction, and dimensional expansion can alter the properties of spacetime dimensions and influence the behavior of physical theories. They will be exposed to cutting-edge research in mathematics and theoretical physics and will develop the skills needed to apply these concepts to a wide range of problems.
Introduction to Compactification and Dimensional Reduction (20 modules):
1-4: Topological Spaces and Continuous Functions
5-8: Compact Spaces and Their Properties
9-12: Dimensional Reduction in Linear Algebra and Tensor Analysis
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in D spacetime dimensions can be redefined in a lower number of dimensions d, by taking all the fields to be independent of the location in the extra D − d dimensions. Dimensional reduction also refers to a specific cancellation of divergences in Feynman diagrams.
The compact dimension imposes specific boundary conditions on all fields, for example periodic boundary conditions in the case of a periodic dimension, and typically Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient along this dimension are integer or half-integer multiples of 1/L (depending on the precise boundary conditions). In quantum mechanics this eigenvalue is the momentum of the field, and is therefore related to its energy. As L → 0 all eigenvalues except zero go to infinity, and so does the energy. Therefore, at this limit, with finite energy, zero is the only possible eigenvalue under gradient along the compact dimension, meaning that nothing depends on this dimension. This is the essence of dimensional reduction. The compact dimension is often called the compactified dimension, and the non-compact dimensions are called the uncompactified dimensions. The process of dimensional reduction is often called compactification, although this term is also used in a different sense in algebraic geometry. The process of dimensional reduction is often used in string theory and M-theory, where it is used to relate different theories in different dimensions. For example, the original 11-dimensional M-theory is related to the 10-dimensional type IIA string theory by compactifying one of the dimensions of M-theory on a circle. The resulting theory is then a 10-dimensional theory, and the extra dimension is the compactified dimension. The process of dimensional reduction is also used in the Kaluza-Klein theory, where it is used to unify gravity and electromagnetism in a higher-dimensional theory. In this case, the extra dimensions are compactified on a circle or a torus, and the resulting theory is a 4-dimensional theory with additional scalar fields that can be interpreted as the electromagnetic potential. The process of dimensional reduction is also used in the AdS/CFT correspondence, where it is used to relate a higher-dimensional theory of gravity in an anti-de Sitter space to a lower-dimensional theory of conformal field theory. In this case, the extra dimensions are compactified on an anti-de Sitter space, and the resulting theory is a lower-dimensional theory that describes the boundary of the anti-de Sitter space. The process of dimensional reduction is a powerful tool in theoretical physics, and has many applications in string theory, M-theory, and other areas of theoretical physics.
13-16: Dimensional Reduction in Differential Geometry
17-20: Applications of Dimensional Reduction in Physics
The discovery of the Higgs boson has heralded the era of precision in hadron collider physics. Disentangling potential new physics effects from the wealth of data requires a very high level of control over theoretical predictions for Standard Model cross sections which is very often limited by our ability to compute complicated Feynman diagrams. Feynman integrals are a rapidly developing field and there are many competing methods which each have their own merits and limitations and state-of-the art problems often require a combinations of various tools and basic concepts of dimensional regularization and Feynman parametrization, which provide the foundation for more advanced topics including sector [decomposition], Mellin Barnes representations, reduction to master integrals using integration-by-parts identities, solving master integrals by differential equations and the expansion by regions.