Chintalpati Umashankar Shastry

Here I am undertaking the project, in the continuation of a project formulated during my internship, under the supervision of Dr Andrea Rocco. This research explores the Renormalization Group (RG) approach applied to open quantum systems, with a focus on incorporating both dissipation and decoherence phenomena. By extending the traditional Non-Perturbative Renormalization Group (NPRG) analysis, a generalized Wegner-Houghton equation is derived to describe the scale dependence of the effective action. The study builds on the Caldeira-Leggett model, which couples quantum systems to an environment approximated as a collection of harmonic oscillators. A novel ζ-function framework is introduced, allowing for the inclusion of off-diagonal density matrix elements, thereby providing a more comprehensive understanding of quantum-classical transitions. This approach addresses the limitations of earlier methods by integrating the effects of environmental interactions on both the diagonal and off-diagonal elements. The findings have implications for studying quantum tunneling, decoherence, and dissipation, offering insights into the interplay of quantum and thermal fluctuations across scales. Future work will validate the proposed framework through model-based analysis and explore its applications under the Local Potential Approximation.

In this project I am working under the supervision of Dr Andrea Rocco. This research investigates the origins of irreversibility in statistical mechanics, emphasizing the roles of particle correlations and the molecular chaos assumption within the Boltzmann Transport Equation (BTE). The work critically examines how the molecular chaos hypothesis introduces irreversible behavior in gas dynamics while neglecting particle correlations.

To address these limitations, the study explores alternative methods, including the incorporation of correlations through generalized entropy formulations, such as Tsallis entropy. Furthermore, it establishes a connection between irreversibility in statistical mechanics and the theory of open quantum systems. By integrating classical and quantum perspectives, this interdisciplinary approach aims to provide a deeper understanding of time irreversibility in molecular systems and proposes a robust theoretical framework to describe these phenomena.

This project, undertaken in collaboration with Dr Andrea Rocco, seeks to investigate the mechanisms by which systems attain equilibrium when interacting with a Non-Markovian bath. Currently in its conceptualization phase, this research has not yet commenced but aims to provide foundational insights into the role of memory effects and non-Markovian dynamics in equilibrium processes.

For my MSc dissertation, I investigated the Dirac equation for the hydrogen atom using finite basis expansion methods under the supervision of  Dr Paul Stevenson. The project focused on obtaining a matrix representation of the Dirac Hamiltonian within a finite set of basis spinors. While any infinite basis set can theoretically represent the Hamiltonian, improper kinetic balancing of the basis functions leads to the emergence of spurious energy states in the spectrum.

Two distinct methodologies were developed and analyzed: the Basis Expansion Method and the Variational Method. In the Basis Expansion Method, the matrix elements were calculated by taking the inner product of kinetically balanced Dirac basis spinors with the Dirac Hamiltonian. In contrast, the Variational Method involved expanding the large component in terms of orthonormal basis functions and the small component using kinetically balanced basis functions.

Both methods were found to produce spurious energy states; however, the Variational Method resulted in fewer such states for a given number of basis functions (or spinors, in the case of the Basis Expansion Method). Despite this advantage, the Variational Method exhibited slower computational performance and a less rapid convergence of the largest positive energy eigenvalue to the theoretical ground-state energy compared to the Basis Expansion Method.