arXiv subjects
Publications and source records attributed to Karmadeva Maharana.
Using the group theoretic method of spectrum generating algebras a class of differential equations is obtained whose eigenvalues are calculated without explicitly solving the equations. Solutions can be easily obtained by group theoretic methods for a certain type of potentials.
In an attempt to facilitate the construction of a quantum theory of gravity, 't Hooft has considered a chiral alternative to the vierbein field in general theory of relativity. These objects, $ f^a{}_{μν}$, behave like the "cube root" of the metric tensor. We try to construct specific representations of these tensors in terms of Dirac $γ$ matrices in Euclidean and Minkowski space and promote these to curved space through Penrose-Newman formalism. We conjecture that these new objects, with physical significance, are the analog of Killing-Yano tensors.
We use group theoretic methods to obtain the extended Lie point symmetries of the equations of motion for a charged particle in the field of a monopole. Cases with certain model magnetic fields and potentials are also studied. Our analysis gives the generators and Lie algebras generating the inherent symmetries. The equations of motion of a scalar particle probing the near horizon structure of a black hole is also treated likewise. We have also found the generators of Krause's complete symmetry groups for some of the above examples.
Using the method of $su(1,1)$ spectrum generating algebra, we analyze one dimensional Schroedinger equation with potential in the form ${C\over{x^2} + {D\over{x}}$ to obtain a class of potentials giving similar eigenvalues. By a group analysis of the differential equation it is found that the symmetry gets enhanced for particular values of $C$ and $D$. The generators of the Lie algebra do not close. The extension of the vector field gives rise to an interesting algebra.
We analyze the classical equations of motion for a particle moving in the presence of a static magnetic field applied in the $ z $ direction, which varies as $ {1\over{x^2}} $. We find the symmetries through Lie's method of group analysis. In the corresponding quantum mechanical case, the method of spectrum generating $su(1,1)$ algebra is used to find the energy levels for the Schroedinger equation without explicitly solving the equation. The Lie point symmetries are enumerated. We also find that for specific eigenvalues the vector field contains $ {1\over{x}} {{\p}\over{\p x}}$ and $ {1\over {x^2}} {{\p}\over{\p {x}}}$ type of terms and a finite Lie product of the generators do not close.
We perform the group analysis of Witten's equations of motion for a particle moving in the presence of a magnetic monopole, and also when constrained to move on the surface of a sphere, which is the classical Wess-Zumino-Witten model. We also consider variations of this model. Our analysis gives the generators of the corresponding Lie point symmetries. The Lie symmetry corresponding to Kepler's third law is obtained in two related examples.
We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain nonlinear form of these equations. The group classification through one parameter group of transformation for two of these equations is also carried out.
We investigate the tachyon-dilaton-metric system to study the "graceful exit" problem in string theoretic inflation, where tachyon plays the role of the scalar field. From the phase space analysis, we find that the inflationary phase does not smoothly connect to a Friedmann-Robertson-Walker (FRW) expanding universe, thereby providing a simple tachyonic extension of the recently proved stringy no-go theorem.