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Since the inventions of the theory of special relativity and quantum theory, people have tried to make use of both concepts at the same time, although in the beginning quantum theory was dealing with phenomena in the nonrelativistic regime. However, when the systems of physical interest became more sophisticated and the accelerators became more powerful, one had to find a way to describe a quantum mechanical system in accordance with special relativity. The first attempts in this direction were (in analogy to Schrödinger's equation) relativised wave equations, from which then quantum field theory originated. This is, however, not the only possibility. In 1949 Dirac published a paper in which he described three forms of Hamiltonian dynamics. These were first formulated in the context of classical mechanics, but they are also applicable to quantum mechanics as well as quantum field theory. While the most common form is the instant form, there are also two other main forms: the front form and the point form. Each form is characterized by the properties of its representation of the Poincare algebra. The Poincare group is the group of symmetry transformations fundamentally connected to special relativity. In each of the forms, different Poincare generators will contain interactions as soon as they are introduced in a physical system. These generators then determine the dynamics of the system and are called dynamical. The number of dynamical generators is 4 for the instant and point forms and 3 for the front form.
The front form has already been used in various fields including both quantum mechanics and quantum field theory. It has been successful especially in deep inelastic scattering, where the kinematic variables of phenomenological approaches coincide with the specific variables of the front form. Although also the point form was at first considered for an application in quantum field theory, it was put aside after people realized that its quantization surface (a space-time hyperpoloid) entails difficulties. However, for a quantum mechanical treatment the point form yields advantages. The Bakamjian-Thomas construction provides - for any of the different forms - a simple way, how to include interactions in the Poincare generators by imposing only linear conditions on the potential. In the point form, all interactions are put into the four-momentum operator, whereas the Lorentz generators remain free of interactions or kinematical. As a result, the point form is manifestly Lorentz covariant. The dynamical equations replacing the Schrödinger equation are then in general the eigenvalue equations for the four components of the four-momentum operator. Within the Bakamjian-Thomas framework the problem is reduced to one eigenvalue equation for the invariant-mass operator. Constructing the interaction part of the four-momentum operator from a quantum field theoretical interaction Lagrangian density does in general not yield a Bakamjian-Thomas form. One has to make the assumption that the interacting part, which we refer to as the potential, is diagonal in the four-velocity. This leads to the desired Bakamjian-Thomas structure and ensures validity of the Poincare algebra.
This thesis aims at an application of point-form relativistic quantum mechanics to the field of quark physics. As an instructive and comparatively simple system we have chosen a two-particle system of one (constituent) quark and one (constituent) antiquark which form a vector meson. The interaction in this simple model is the one of a chiral constituent quark model: pseudoscalar meson exchange. This model has been used for baryons in a semirelativistic form with great success and also for vector mesons, leading to a reasonable description of the spectra. In this thesis a fully relativistic calculation with dynamical mesons is done and compared to the usual semirelativistic approach in which the meson-exchange is treated in an instantaneous approximation.
The point form formalism is explained in detail after a general introduction to relativistic quantum mechanics. This introduction includes presentations of the three main forms of Hamiltonian dynamics and the Bakamjian-Thomas construction. We introduce velocity states, a suitable basis for the point-form treatment of few-particle systems, and use them to construct the elementary meson-(anti)quark vertex, which enters the invariant-mass operator, from a pseudoscalar interaction Lagrangian density. We let the mass operator act on a Hilbert space which is the direct sum of a two-particle and a two-plus-one particle Hilbert space. This leads to a coupled two-channel problem. In contrast to instant- and front-form dynamics such a truncation of the Fock space does not necessarily violate Poincare invariance in point-form dynamics. This is due to the kinematical natrue of the generators of Lorentz transformations and the locality in the overall velocity assumed in the construction of the meson-(anti)quark vertex. The fact that quarks and antiquarks are always confined is taken into account by introducing harmonic oscillator confinement terms in the diagonal parts of the square of the coupled-channel mass operator. The harmonic-oscillator eigenfunctions are then used as a basis for expanding the quark-antiquark wave functions. As a result, one can discretize the dynamical equation and obtains a set of coupled algebraic equations to solve; however, the coefficient matrix consists of high-dimensional integrals. It also turns out that, since there were no further approximations made concerning the potential, these integrals actually also depend on the eigenvalues themselves. To solve this nonlinear eigenvalue problem spectra are first generated by presetting the eigenvalue in the integral and solving the resulting linear eigenvalue problem. By doing this for different choices of the preset eigenvalue the actual eigenvalue can then be determined by a resonance condition, which just states that the preset eigenvalue (in the integral) and the eigenvalue of the resulting linear eigenvalue problem should be the same.
The actual calculation of the integrals and the various pieces of the integrands is presented in much detail in the appendices; these can also be viewed as a collection of point-form formulae. Results of the full relativistic calculations are presented for both masses and nonperturbative decay widths of the lowest lying excitations of J=1,P=- vector mesons. The calculations are performed using model parameters in the range suggested by the chiral constituent quark model for baryons as well as other calculations making use of pseudoscalar meson exchange between quarks. The value for the quark-meson coupling constant is, by physical arguments, restricted to a certain interval. We perform the calculation at the upper and lower borders of this interval and in this way also get a comparison of the effects of weaker an stronger couplings.
The results for the spectra are satisfactory in both the fully relativistic case and the instantaneous approximation - with some differences, of course. One interesting feature is the rate of convergence of the various states with increasing maximal n compared in these two different cases. For the omega, where the largest effect of the hyperfine interaction can be observed, the ground state shows a faster convergence in the relativistic result. The situation is also true for the excited states. A more general observation in this respect is that the convergence in the nonrelativistically reduced case seems close to a linear behaviour for small n, whereas in the fully relativistic case, the states with lowest n contribute significantly more than higher ones. It is also interesting to see that for the rho ground-state the binding energies are approximately the same in both the full calculation and the instantaneous approximation, whereas things are different for other ground states and level splittings. In the case of the Kstar, effects are hardly seen due to the weakness of the hyperfine ineraction.
The decay widths calculated in the full coupled-channel calculation are not so satisfactory. For the quark-antiquark systems we are investigating there are only a few experimentally observed decay widths that can be expected to be described within the present calculation, namely four. In three of the cases, the numbers calculated have the correct order of magnitude. However, in the most prominent case, the omega(1420) into pi+rho, in which the experimentally dominant decay channel has a rather large width, our theoretical predictions are more than one orders of magnitude below the data. This deficiency could perhaps be cured by the inclusion of loop graphs, which should also provide a contribution to the decay width. This point remains to be clarified.
There are also other open questions. Apart from the inclusion of loop graphs which yield above meson-production threshold also an absorptive contribution to the interacting mass operator, one can modify the interaction as long as one can write down its interaction Lagrangian density as a polynomial in free fields. Other systems of interest are, e.g., the positronium or baryons within a quark-diquark picture if one wants to stick to effective two-body problems and, of course, more complicated systems like baryons as three-quark systems coupled to a three-quark plus one-meson channel. Another important issue is, how one can establish and exploit the equivalence of the three main forms of Hamiltonian relativistic dynamics in practical calculations.