Reijiro Kubo

Geometry, Heat Equation and Path Integrals on the Poincare Upper Half-Plane

One of awkward but interesting features of string theories consists in the fact that quantization of strings is subject to the influence of topological and geometrical nature of the two-dimensional world sheet.1),2) Recent extensive studies on Polyakovs path integral of the bosonic string have revealed the geometrical nature of string interactions.2) String amplitudes defined by the path integral are invariant under diffeomorphism and Weyl scaling of the metric on a compact orientable twodimensional manifold. As is well known, a connected two-dimensional manifold with complex structure can be regarded as a one-dimensional complex manifold, a Riemann surface. The string amplitudes are subject to the complex structure of Riemann sufaces. Accordingly, the mathematics of Riemann surfaces has played an important role in the development of string theories. The partition function of the closed bosonic string is expanded in a perturbation series,2)

Infrared Phase Factors in Quantum Electrodynamics

Infrared singularities in Green functions in quantum electrodynamics are studied from the point of view of path-ordered phase factors in an infrared approximation. The singular exponential factor in the vertex part is reproduced by a simple calculation of the phase factor. The result is compared with the one obtained by the Kulish-Faddeev model.

Infrared singularity in the electron propagator in quantum electrodynamics

Abstract Infrared singularity in the electron propagator is studied by means of the perturbation calculation exploited by Grammer and Yennie. The result agrees with those derived by non-perturbative methods. The relation between cutoff-dependent normalization factors and conventional renormalization constants is discussed together with the Ward identity.