My research interests are in is the area of theoretical relativistic few-body quantum mechanics.
A few-body system is an isolated system that is sufficiently simple that:
The above describes ideal properties of a few-body system. What constitutes a few-body system is a moving target. It depends on the current state of the art in experimental physics, theoretical physics, and computational physics.
The goal of few-body physics is to constrain theories by comparing the results of complete experimental measurements to ``exact'' numerical calculations. Once the relevant quantum mechanical degrees of freedom are identified, the application of few-body methods determines the Hamiltonian of the few-body system.
Cluster properties, which relate the physics of many-body systems to the physics of isolated few-body subsystems, provide a means to use few-body methods to determine the Hamiltonian of complex systems with great confidence.
Most of my research is involved with applications of few-body methods to understand systems of strongly interacting particles. The most familiar strongly interacting particles are neutrons and protons. There are many more unstable strongly interacting particles and they are all believed to be bound states of systems of non-observable particles called quarks and gluons. The interaction between the observable particles are complex and short ranged. In addition, most of the strongly interacting particles have short lifetimes. Few-body methods provide a useful tool to develop an understanding of the structure of simple hadronic or sub-hadronic systems and the reactions between the constituent particles.
Understanding the transition from a description of strongly interacting particles in terms of nucleon and meson degrees of freedom to one in terms of quark and gluon degrees of freedom is a problem of interest in both nuclear and particle physics. The quark substructure of a nucleon becomes relevant at distance scales that are a fraction of the size of a proton (about a femptometer). According to the uncertainty principle, a De Broglie wavelength on this scale requires a minimum momentum transfer comparable to the rest energy of the proton (in units where the speed of light is 1).
These considerations mean that a theoretical description of strong interaction physics on these scales must be consistent with quantum mechanics (because the sizes and momentum transfers are at the limit dictated by the uncertainty principle) and special relativity (because the kinetic and rest energy of the strongly interacting particles are comparable).
The types of problems of general interest are understanding the quark-gluon sub-structure of mesons and nucleons (strongly interacting particles), understanding the decays of unstable strongly interacting particles, understanding the relation between the forces between nucleons and the interactions between the their constituent quarks, understanding the distribution of charges and currents in nucleons, nuclei and mesons, understanding reactions with particle production, and understanding the structure of few-nucleon systems with high precision.
For most of the applications of interest, in order to carry out the few-body program, it is necessary to have a theoretical description satisfying the following minimal constraints:
One of the the exciting aspects of this line of research it that there are no theories that are completely consistent with all of these constraints for all systems of interest.
Local quantum field theory is the most popular theoretical framework for studying these problems. It has many nice features, but the standard expression for the Hamiltonian has an empty domain on the Hilbert space of the non-interacting system. This means that when the Hamiltonian is applied to any vector in the Hilbert space, the result is not in the Hilbert space. What this means is that the precise definition of the theory is not yet known, making it impossible to compute ab-initio error bounds. For interesting comments on local field theory see arxiv:hep-th/0405105
My research uses relativistic quantum mechanics of particles. It is a more conservative approach than quantum field theory, but it has a complementary set of difficulties. The philosophy is to treat strongly interacting systems with the most general quantum mechanical theory consistent with the symmetry of special relativity. Relativistic quantum mechanics of particles is also not well developed and an important part of my research program is to understand the structure of relativistic quantum theories and their relation to quantum field theories.
Most of the computational work in few-body physics involves the numerical solution of integral and partial differential equations for bound states and scattering observables. In general the systems of equations are large systems of linear equations. In scattering problems the equations either involve complicated boundary conditions or singular integral kernels. Because the problems are both large and complicated, they cannot be solved by commercial software packages. Some our our research involves both the development and applications of computational methods to these problems. Most of the programs are in Fortran or C
Recently we have developed numerical method based on wavelets to solve scattering integral equations.Click here to return to main page
If you would like more information about my research program, or about the Physics and Astronomy or the Applied Mathematical and Computational Science Programs at Iowa, please contact me at polyzou@uiowa.edu