REMARKS ON THE GROUND STATE ENERGY OF THE SPIN-BOSON MODEL: AN APPLICATION OF THE WIGNER–WEISSKOPF MODEL

For the ground state energy of the spin-boson (SB) model, we give a new upper bound in the case with infrared singularity condition (i.e. without infrared cutoff), and a new lower bound in the case of massless bosons with infrared regularity condition. We first investigate spectral properties of the Wigner–Weisskopf (WW) model, and apply them to SB model to achieve our purpose. Then, as an extra result of the spectral analysis for WW model, we show that a non-perturbative ground state appears, and its ground state energy is so low that we cannot conjecture it by using the regular perturbation theory.

PERTURBATIVE QUANTUM GAUGE FIELDS ON THE NONCOMMUTATIVE TORUS

Using standard field theoretical techniques, we survey pure Yang–Mills theory on the noncommutative torus, including Feynman rules and BRS symmetry. Although in general free of any infrared singularity, the theory is ultraviolet divergent. Because of an invariant regularization scheme, this theory turns out to be renormalizable and the detailed computation of the one-loop counterterms is given, leading to an asymptotically free theory. Besides, it turns out that nonplanar diagrams are overall convergent when θ is irrational.

GAUSSIAN ANALYSIS OF SYMMETRY BEHAVIOR

We use the functional variational method in the Schrödinger picture to examine the existence of two phases of the scalar ϕ4 theory in Minkowski and Einstein spaces. In the static Einstein space one of the phases shows the possibility of spontaneous symmetry breaking even when the renormalized mass is positive, while in the other phase symmetry breaking is prohibited owing to the infrared singularity caused by the finiteness of the spatial volume.

NONPERTURBATIVE APPROACHES TO DETERMINING THE BEHAVIOR OF THE GLUON PROPAGATOR AND QUARK PROPAGATOR IN QUANTUM CHROMODYNAMICS BY SCHWINGER-DYSON EQUATIONS

The attempts to describe the behavior of the gluon propagator and quark propagator by using truncated Schwinger-Dyson equations and Slavnov-Taylor identities are reviewed. Special attention is paid to the problem of infrared behavior of Green’s functions. The most important attempts to calculate the gluon propagator using the axial as well as the covariant gauge are critically discussed. Furthermore, an approach concerning the gluon propagator is presented, with the background-field method as its basis. All the calculations confirm more or less the existence of an infrared singularity in the gluon propagator of the form q−4 in momentum space. The calculations to determine the behavior of the dynamical mass function of quarks, where the results concerning the gluon propagator are taken into account, show that chiral symmetry is dynamically broken. Furthermore, it turns out that there is no polelike singularity in the quark propagator. These results agree with the expectations from the confinement philosophy.