Study For Relativistic Positronium Equation And Ward-Takahashi Identities In Gauge Fields

Based on man observation on hadron physics, the dynamical chiral symmetry breaking (DCSB) and confinement are two crucial features of quantum chromodynamics (QCD). They are expected to occur on QCD. The precise origin of this nonperturbaive phenomenon as well as its relation to quark confinement is still little understood. Further studies of these issues gave to build on reliable nonperturvbative methods.A natural method for studying both DCSB and confinement in QCD is the complex of Dyson-Schwinger equations (DSE's) the manifestly relativistically covariant approach has provided the foundation for useful and successful understanding of the phenomena of low–energy QCD by facilitating the construction of realistic field–theoretic models.The knowledge of the three-point vertices iscrucial in the studies of gauge theories through the use of the Dyson-Schwinger equations (DSE'S), where the transverse part of the vertex plays the crucial role in ensuring multiplicative renormalizahility and in determining the propagator. Therefore, in the past years much effort has been devoted to constructing the transverse part of the vertex based on perturbative constraints on the transverse vertex through the one-loop evaluation of the vertex. Such perturbative constraints are useful because the physically meaningful solutions of the DSE'S must agree with perturbative results in the weak-coupling regime. However, such a constructed vertex is not unique since it is not fixed by the symmetry of the system. The latter provides the key point in determining the transverse part of the vertex; like the longitudinal part of the vertex , the transverse part of the vertex should be determined also by the WT-type relations.How to solve exactly the transverse part of the vertex and the full vertex function (Green functions) the becomes a curial problem. This is of the nature of our study.The work in this thesis is composed of four major parts:1. A rigorous three-dimensional relativistic equation satisfied by positronium bound states is derived from QED generating functional without making use of the four-dimensional Bethe-Salpeter equation and its reduction. The effective interaction kernel in the equation is given by a closed expression that includes all the interactions taking place in the positronium bound states, with the aid of the equations of motion satisfied by some kinds of Green's functions. Furthermore, it is shown that the equation of bound states can be equivalently reduced to a Pauli-Schrodinger equation. In order to show the applicability of the closed expressions derived, the effective potentials of the corresponding t-channel and s-channel one photon kernels are evaluated , which not only is relativistic but also takes account of the retardation effect properly, no ambiguity being involved. As an illustration, explicit expression for the effective potential is given for the lowest order approximation, which gives the annihilation interaction potential in the process of virtual pair annihilation and the common Breit interaction.2. In the second part of this thesis,The symmetries of gauge theories lead to a variety of exact relations among the Green's functions including the transverse WT relation. This provides an approach to determine the transverse part of the vertex. We are interested in deriving various full Green functions through general Ward-Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the variousWTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields. 3. The third part is mainly concerned with the quantum theory of the quantized Glashow-Weinverg-Salam model.. With the help of a postulate of gauge group parameter, the infinitesimal gauge transformation laws preserving the gauge–invariance of the generating functional itself of the quantized Glashow-Weinberg–Salam model have been established precisely. As viewed from the origin of the symmetry, the new transformations are the same as the standard Becchi-Rouet-Stora ones which also stem from the invariance of the full quantum Lagrangian itself. These new transformations are indispensable for both calculation of the anomaly in GWS model and derivation of the generalized WT identity. A straightforward application of the infinitesimal transformation generally leads the fermionic integral measure relating with the transformation to the path–integral derivation of the anomaly associated with WT identity.In the simplest case, by making use of a general path-integral prescription, a series of the expressions of the anomaly factor including the contribution for left–handed and right–handed fields have been carried out in coordinate space.Herein we explore the difference between the two integral- approaches. For purpose of comparison, by means of an alternative path–integral regulation scheme proposed in Ref.[Fu04], the possible effect of the Higgs coupling on the anomaly in the WS model, is precisely estimated. Thus we first present that, in general, the Higgs field gives rise to the influence on the anomaly with respect to tensor currents. As a consequence, it is shown that the two path–integral prescriptions are not completely equivalent to each other in the case of GWS model. 4. In addition, a new exposition of proper asymptotic behavior of Green Functions for large and small external momenta in QCD is presented in appendix B. Herein we try to demonstrate the confinement property of QCD.In appendix C, A topologcal way to distinguish the divergence of operator product of fermion current in quantum field theory is presented. By virtue of the topologically non-trivial Jacobian factor of the integration measure in the path-integral formulation of the theory, the singularity of operator product of fermion current linked with topological property of gauge field can be examined in a gauge background correctly.5.In appendix D, We make use of the symmetry of the generating functional to generate the identities for various full Green functions in quantized non- Abelian gauge field theory-QCD model. With the help of a postulate of gauge group parameter, such identities with the anomaly terms are exactly derived under a general local gauge transformation law maintaining the symmetry. Hence the usual BRS transformation is regarded as a special case in the new symmetry. A key point is that ghost field possesses a duel purpose. Thus the path-integral derivation of the anomaly factors associated with Ward-Takahashi identities for variant fermionic currents gives a viable possibility to express the complete vertex function. As a consequence, the complete expressions of the full tensor, the full axial-tensor vertex function, and so on are well expressed respectively in terms of the fermion propagators by solving the complete set of the WTIs in the momentum space in terms of these WTIs. The resulting expression contains only a few types of Green's functions which not only are easily calculated by the perturbation method, but also suitable for investigation by a certain nonperturbative approach. However, there are several problematic aspects to need further study in this paper;1) The application of the three-dimensional relativistic two-body wave equations presented in this thesis to calculating the meson spectra will be studied later.2) The derived complete expression of the transverse Ward-Takahashi(WT) relations for the fermion–boson vertex (vector vertex) in momentum space in four-dimensional Abelian gauge theory should lead to the same results one calculated in perturbation theory.