
 
Description of bound states and scattering processes using Hamiltonian renormalization group procedure for effective particles in quantum field theory
Marek WięckowskiJune 2005Doctoral thesisAdvisor: Professor Stanisław D. Głazek
Department of Physics Institute of Theoretical Physics Warsaw 2005 [full text of the thesis, pdf, 1.5MB] [abstract for old browsers]
AbstractThis thesis presents examples of a perturbative construction of Hamiltonians H_{lambda} for effective particles in quantum field theory (QFT) on the light front. These Hamiltonians (1) have a welldefined (ultravioletfinite) eigenvalue problem for bound states of relativistic constituent fermions, and (2) lead (in a scalar theory with asymptotic freedom in perturbation theory in third and partly fourth order) to an ultravioletfinite and covariant scattering matrix, as the Feynman diagrams do. lambda is a parameter of the renormalization group for Hamiltonians and qualitatively means the inverse of the size of the effective particles. The same procedure of calculating the operator H_{lambda} applies in description of bound states and scattering. The question of whether this method extends to all orders in QFT is not resolved here.The relativistic Hamiltonian formulation of QFT is based on a global regularization of all terms in a relativistic operator H^{Delta} (a canonical Hamiltonian with an ultraviolet cutoff Delta, plus counterterms). The renormalization group procedure for effective particles (RGPEP) makes it possible to find the structure of the counterterms in H^{Delta} and calculate the effective Hamiltonians H_{lambda} for lambda ranging from infinity down to lambda on the order of masses of bound states. I investigate bound states of two relativistic fermions using Yukawa theory as an example. I give an explicit form of the effective Hamiltonian H_{lambda} in the second order, and discuss the reduction of its eigenvalue equation to a Schroedinger equation for the wave function of the constituents. Every interaction term in the Hamiltonian H_{lambda} contains a form factor f_{lambda} generated by RGPEP, which eliminates overlapping divergences in the boundstate eigenvalue problem expressed in terms of effective particles. The overlapping divergences appear in the eigenvalue problem expressed in terms of pointlike particles and without the form factors f_{lambda}, and result from relativistic relative motion of fermions. Such divergences appear in all Hamiltonian theories of pointlike particles with spin, and in particular in quantum chromodynamics (QCD). The advantage of the Yukawa theory is that it allows one to investigate the ultraviolet behavior in bound states of fermions without additional complications of QCD. The form factors f_{lambda} also cause the bound state to be dominated by the lowest sectors in the Fockspace basis built with effective particles. The ultraviolet complications of local QFT are contained in a complex structure that emerges in the effective particles as a result of dressing of the bare particles of the initial canonical theory. My description of scattering in an asymptotically free scalar field theory of phi^{3} type in 5+1 dimensions starts from constructing explicitly counterterms in H^{Delta} by calculating H_{lambda}. I then use the Hamiltonian H^{Delta} to calculate a scattering amplitude for a process analogous to e^{+}e^{} > hadrons in perturbation theory up to the order e^{2}g^{2}, i.e., in one loop (e is an analogue of the electric charge of electrons in QED, and g of the color charge of quarks in QCD). I show that counterterms found using RGPEP without referring to the S matrix, remove the divergent regularization dependence from the calculated amplitude. I also give the explicit form of the finite parts of the counterterms in H^{Delta} that lead to a covariant result for the scattering amplitude. I show that the dependence of the amplitude calculated this way on the momenta of the colliding particles, is the same as the dependence derived from Feynman diagrams (the diagrams are regularized covariantly and without defining a regularized Hamiltonian ab initio). I prove a theorem that states that the scattering amplitude obtained using H^{Delta} is the same as the scattering amplitude obtained using H_{lambda}. Note that in the calculation using H^{Delta} physical states of colliding particles are expressed in terms of bare particles and one uses the renormalized interaction Hamiltonian H_{I}^{Delta}. In the calculation using H_{lambda} physical states of colliding particles are expressed in terms of nonpointlike effective particles and one uses the effective interaction Hamiltonian H_{I lambda} with form factors in all vertices. In the case of the considered amplitude of the type e^{+}e^{} > hadrons in a oneloop approximation, this theorem implies that the effective Hamiltonian H_{lambda} leads to the same predictions for the scattering matrix as the Feynman diagrams. I also present an alternative, simplified procedure for deriving the Hamiltonian counterterms needed for the description of the scalar analogue of e^{+}e^{} > hadrons. I illustrate the simplified procedure by giving mass and some vertex counterterms in QCD coupled to QED, in the Appendices.
[full text of the thesis, pdf, 1.5MB]
[abstract for old browsers]
 
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