Prof. Glazek is interested in developing a precise theory of hadronic observables. In particular, he contributed to the foundation of the similarity renormalization group procedure for Hamiltonians [1,2,3]. More recently, he developed the renormalization group procedure for effective particles (RGPEP) in quantum field theory and studied it with his students [4,5,6,7,8]. His method is designed to produce a successive approximation scheme for solving relativistic quantum field theories numerically. The key element of the new procedure is the introduction of a well-defined dynamical concept of a relativistic effective particle. The effective particles can play the role of virtual constituent quarks and gluons in hadrons. But there remain fascinating questions to answer concerning gauge symmetry, confinement, small-x singularities, how spontaneous chiral symmetry breaking occurs, and how condensates form [9], when one applies the new light-front Hamiltonian approach to QCD and when one develops the corresponding numerical procedures. One of the intriguing questions is if recent developments concerning symmetries in complex theories formulated using Lagrangians and path integral techniques are able to provide information about how the new computational strategy using Hamiltonians may be implemented most efficiently and as a result provide quantitative understanding of relativistic non-perturbative quantum phenomena that continue to puzzle theorists. The other question is whether the new precisely defined Hamiltonian approach can shed light on the nature of the basic symmetries that physicists use in building their theories, independently of the methods they apply.
Numerical accuracy of the new computational scheme that can be applied in solving QCD Hamiltonian eigenvalue problem has been studied in simple model examples [10]. In one-flavor QCD in the leading non-relativistic approximation for the relative motion of quarks in heavy quarkonia and for distances between the quarks not significantly larger than the strong Bohr radius, the RGPEP in light-front Hamiltonian scheme allows one to finesse a harmonic oscillator potential with definite strength as the leading candidate for the first correction term to the strong Coulomb potential in the effective Schroedinger equation for boost-invariant wave functions of the heavy quarkonia [11]. The finessed harmonic term introduces a positive contribution to the quarks' energy and the resulting bound-state mass of a quarkonium can be larger than the sum of the masses of the quarks, a phenomenon not occurring in Abelian gauge theories or atomic quantum mechanics, where the binding mechanism we know how to describe quantitatively works only below the mass threshold and is always associated with a mass defect, instead of the surplus required in the quark physics. Pilot quantitative studies show that the finessed theoretical picture can match experimental data for masses of heavy quarkonia intriguingly well [12].
Current work of prof. Glazek is focused on extension of his techniques that are already found workable for heavy quark dynamics to light quarks, which means quarks whose mass parameters are assigned values much smaller than the QCD scale parameter Lambda. This work aims at solving problems with combining Hamiltonian quantum mechanics, special relativity, gauge symmetry, and chiral symmetry in the context of renormalization group procedure for effective particles in quantum field theory, which is a major challenge. One of the recent steps in this direction is analysis of non-local interaction densities in renormalized Hamiltonians [13]. Extension of RGPEP beyond perturbation theory is offered in [14], the associated perturbative expansion is generally presented in [15], and a special example of exact solution which indicates that RGPEP may handle the vacuum problem in quantum field theory in a new way is described in [16] in a case with bosons and further illustrated in a case with fermions in [17]. Application of the RGPEP to QED concerning the proton radius in atomic physics is described in Ref. [18]. Derivation of asymptotic freedom using the same method in front-form Hamiltonian formulation of QCD is available in [19]. Recent applications of the RGPEP to bound states of quarks and gluons in QCD are described in Refs. [20] and [21]. Collisions of gluon strings in colliders are discussed in Ref. [22].