No abstract

No abstract

Using Penrose's binor calculus for SU (2) (SL(2, C)) tensor expressions, a graphical method for the connection representation of Euclidean quantum gravity (real connection) is constructed. It is explicitly shown that: (i) the recently proposed scalar product in the looprepresentation coincides with the Ashtekar-Lewandowski cylindrical measure in the space of connections; (ii) it is possible to establish a correspondence between the operators in the connection representation and those in the loop representation. The construction is based on an embedded spin network, the Penrose's graphical method of SU (2) calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations. ‡ It is worthwhile recalling that Loll proposed [5] a very interesting lattice regularization for canonical quantum gravity. It amounts, basically, to considering graphs γ that constitute a finite cubic lattice. § The term cylindrical function comes from the language of Wiener integration on an infinite-dimensional space.

In the paper a canonical isomorphism between the space of cusp forms S W+2 (T) of weight w + 2 with respect to the modular group Γ and the space of holomorphic differential forms of higher degree on Kuga's modular variety Βγ is constructed. Bibliography: 6 titles.

No abstract