Description of bound states and scattering processes using Hamiltonian renormalization group procedure for effective particles in quantum field theory
Advisor: Professor Stanisław D. Głazek
Department of Physics
Institute of Theoretical Physics
AbstractThis thesis presents examples of a perturbative construction of Hamiltonians Hlambda for effective particles in quantum field theory (QFT) on the light front. These Hamiltonians (1) have a well-defined (ultraviolet-finite) 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 ultraviolet-finite 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 Hlambda 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 HDelta (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 HDelta and calculate the effective Hamiltonians Hlambda 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 Hlambda 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 Hlambda contains a form factor flambda generated by RGPEP, which eliminates overlapping divergences in the bound-state 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 flambda, 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 flambda also cause the bound state to be dominated by the lowest sectors in the Fock-space 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 phi3 type in 5+1 dimensions starts from constructing explicitly counterterms in HDelta by calculating Hlambda. I then use the Hamiltonian HDelta to calculate a scattering amplitude for a process analogous to e+e- -> hadrons in perturbation theory up to the order e2g2, 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 HDelta 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 HDelta is the same as the scattering amplitude obtained using Hlambda. Note that in the calculation using HDelta physical states of colliding particles are expressed in terms of bare particles and one uses the renormalized interaction Hamiltonian HIDelta. In the calculation using Hlambda physical states of colliding particles are expressed in terms of non-pointlike effective particles and one uses the effective interaction Hamiltonian HI lambda with form factors in all vertices. In the case of the considered amplitude of the type e+e- -> hadrons in a one-loop approximation, this theorem implies that the effective Hamiltonian Hlambda 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.
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