I am a BS-MS Dual Degree graduate (July 2020) from IISER-Kolkata with a major in Physical Sciences. I am currently taking a year away from Universities in order to embark on some self study. The main areas that I'm interested in learning are the mathematical foundations of theoretical physics (e.g. Differential Geometry, Operator Theory/Functional Analysis, Algebra, etc.) as well as some topics in physics (e.g. Quantum Field Theory, General Relativity, String Theory, etc.)Address:
In this work, we defined a new action principle for the atom of space-time-matter (STM atom) that not only led us to the equations for a quantum theory of gravity but also gave us the Yang-Mills action along with the Einstein-Hilbert action after spontaneous localisation. For this we made use of Adler's Trace Dynamics and Connes Non-Commutative Geometry along with the Connes time parameter to incorporate gravity and Yang-Mills theory into trace dynamics.
Trace dynamics is the classical dynamics of matrices on a space-time background whose elements are complex Grassmann numbers. These elements belong to even or odd sectors of the Grassmann algebra. One then takes the trace of the polynomial P that is constructed from these matrices as P = Tr P and defines a so-called trace derivative of P with respect to a given matrix so that one can then construct a trace Lagrangian from the matrices qr and their time derivatives q̇r. This paves way for the development of classical Lagrangian and Hamiltonian dynamics from the system of matrices in the conventional way. Trace Dynamics also turns out to be an underlying theory from which quantum theory and collapse models are emergent.
The main idea behind non-commutative geometry on the other hand is that we can trade spaces for algebras. It is a geometric approach to the study of non-commutative algebras and the construction of spaces that are locally presented by non-commutative algebras of functions. For this, the notion of a spectral triple was used from which the Lagrangian(s) can be derived by the so-called heat kernel expansion of the spectral action.
We also showed the invariance of the functional form of the trace Lagrangian under a transformation that leaves the gyromagnetic ration unchanged. This transformation should be thought of as a duality between Dirac fermions and Black Holes which then explains the same gyromagnetic ratio for a charged rotating Black Hole and an electron.
The work was summarised in the paper Why does the Kerr-Newman black hole have the same gyromagnetic ratio as the electron?
In the paper Canonical tensor model through data analysis : Dimensions, topologies, and geometries , the procedure to calculate the three-way tensor Pabc to construct a fuzzy space corresponding to a compact manifold M, was summarized and also the methodology was applied to various fuzzy spaces by using some low dimensional manifolds with various topologies. The main point there was to show the generality of the real symmetric three-way tensors by using those demonstrations and consequently the generality of the Canonical Tensor Model (CTM) in which unlike the tensors in other Euclidean tensor models where the number of ways (the amount of indices) of the tensors are supposed to be equivalent to the dimensions of the building simplicial blocks, the three-way tensors which are real and symmetric can in principle represent any space with free choice of dimensions and topologies.
In this project however, which can be thought of as an extension to the work done in the paper , we constructed the fuzzy spaces corresponding to a few more compact manifolds and saw if the idea can be generalized and extended to other topological spaces. In the first half, we constructed the fuzzy space corresponding to the disc after which we moved on to the cone which can be thought of as a slight generalization of the disc after which we constructed the fuzzy space corresponding to the cone with negative curvature which can be thought of as a slight generalization of the cone.
In the second half, we tried to extend the methodology to an orbifold after giving a very brief introduction to an orbifold followed by a few examples. We wanted to study the generality of the real symmetric three-way tensors by extending the methodology to construct the fuzzy spaces corresponding to orbifolds. So we considered the torus and studied the fuzzy space construction corresponding to the manifold first after which we constructed the fuzzy space corresponding to the orbifold obtained from the torus.
We also tried to understand the topological properties of these spaces by using the tools from data analysis. We first used the tensor-rank (or CP) decomposition to decompose a tensor into a number of vectors and then used the other data analysis technique called the persistent homology to understand the various topological properties by regarding the vectors as points forming a space as in .