E. C. G. Sudarshan’s research while affiliated with University of Texas at Austin and other places

Time is a part of our immediate awareness.

We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized tomograms. Some of the new tomograms may have interesting applications in quantum optical tomography.

We construct a relativistic Hamiltonian model for Higgs resonance. Resonance trajectories are studied as a function of the parameters of the model. The possibility of a narrow width Higgs resonance in 1 TeV region cannot be ruled out from general considerations. Our model may serve as a phenomenological guide in the search for Higgs resonance.

Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.

The notion of f-oscillators generalizing q-oscillators is discussed. For the classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of the oscillation depends on the energy. The f-coherent states generalizing the q-coherent states are constructed. Applied to quantum optics, the photon distribution function and photon number means and dispersions are calculated for the f-coherent states as well as the Wigner-Moyal function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck's distribution formula.

We construct a non-Markovian dynamical map that accounts for systems correlated to the environment. We refer to it as a canonical dynamical map, which forms an evolution family. The relationship between inverse maps and correlations with the environment is established. The mathematical properties of complete positivity is related to classical correlations, according to quantum discord, between the system and the environment. A generalized non-Markovian master equation is derived from the canonical dynamical map.

After a general description of the tomographic picture for classical systems, a tomographic description of free classical scalar fields is proposed both in a finite cavity and the continuum. The tomographic description is constructed in analogy with the classical tomographic picture of an ensemble of harmonic oscillators. The tomograms of a number of relevant states such as the canonical distribution, the classical counterpart of quantum coherent states and a new family of so called Gauss--Laguerre states, are discussed. Finally the Liouville equation for field states is described in the tomographic picture offering an alternative description of the dynamics of the system that can be extended naturally to other fields.

Metastable states are associated with discrete complex eigenvalues of the Hamiltonian in the analytically continued spaces of quantum mechanics formulated in dual spaces. The necessary modifications when two resonances approach each other and coincide are discussed and illustrated in some exactly soluble models. The Hamiltonian no longer can be diagonalized completely but can be reduced to a Jordan block form. The duals are identified and a resolution of unity established. Some remarks are made about secular terms in the survival amplitude.