- It is possible to perform a complete experiment on the system. This means that it is possible to accurately measure a complete set of commuting quantum mechanical observables. A complete measurement involves determining the mass, momentum, spin, and one component of the spin vector for every particle in the initial and final states of a reaction.
- It is possible to calculate theoretical predictions of these experimental observables based on a few-body model with mathematically controlled errors to a precision that is better than experimental error.
- It should be a quantum theory (the relevant distance and energy scales are close to the limit defined by the uncertainty principle)
- Quantum mechanical probabilities should be relativistically invariant. (the momentum scales give kinetic energies comparable to the rest energy of strongly interacting particles)
- It should be possible to perform ab-initio computations of the theory to any pre-determined accuracy. (These need to be at least as accurate as experiment to constraint the theories)
- It should satisfy cluster properties (to provide a clear relation between the few and many-body problems).

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 predictions of the theory. These predictions are in the form of ``exact'' numerical calculations.

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 constrain 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).

The most compelling candidate for a theory of the strong interactions is quantum chromodynamics. It has a form that is similar to Quantum Electrodynamics, which is the very successful quantum theory describing the interactions of electrons with the electromagnetic field. In Quantum Chromodynamics the electrically charged electons are replaced by color charged quarks and the neutral photons are replacced by color charged gluons.

Quantum chromodynamics is difficult because (1) it is a theory of an infinite number of degrees of freedom (2) the interaction are strong (3) the elementary fields are not associated with particles that apper in the initial or final states of reactions.

While at present it is not known how to construct mathematically convergent algorithms to solve QCD, QCD is a relativistic quantum theory, and, at a given energy scale the theory should dominated by a finite number of degrees of freedom.

The challenge is to construct a mathematically convergent algorithm to solve the theory, where the leading approximation is is a few degree of freedom model. At present convergence has not been established for any algorithm. Lattice models replace spacetime by a lattice of finite volume and finite resolution. These models preserve local gauge symmetry of the exact theory, which is a very important consraint on the theory. On the other hand they break the Poincare invaraince of the theory.

Our interest is in models that are exactly relativistically invariant. Relativistic invariance is a symmetry of the exact theory. Sensitivity to physics on distance scales that are a fraction of the size of a proton requires energy and momentum transfers that are near or above the rest energy of a proton. This requires a relativistic theory. The number of particles that can be seen in initial or final states of of scattering reactions at a given energy scale is finite. This suggest that there should an approximation to QCD that has the sturcture of a theory of a finite number of interscting relativistic particles.

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.

While direct computation of model interactions from QCD are desirable, in general the theoretical models should satisfy the following minimal constraints:

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