Schwinger-Bargmann generating function of the $6j$-symbol.

In Aquilanti (2007) we applied multidimensional WKB theory for integrable systems to the asymptotics of the $3j$-symbol, and in this paper we apply similar techniques to the $6j$-symbol. These methods bear the closest relationship to the works of Roberts (1999) and of Charles (2008). The point of this paper is not another derivation of the Ponzano-Regge formula, although one is provided, but rather to clarify the relationship among some of the methods used in the past, to reveal useful calculational techniques, and to lay the basis for the development of new results. Among the latter we mention our own work on uniform approximations for the $6j$-symbol (Littlejohn and Yu, 2009), our recent derivation of the asymptotic form for the $9j$-symbol (Haggard and Littlejohn, 2010), both of which relied on techniques explained in this paper, and our work on the Bohr-Sommerfeld quantization of the volume operator in loop quantum gravity (Bianchi and Haggard, 2011). Previous and current work on the volume operator includes Chakrabarti (1964), Lévy-Leblond and Lévy-Nahas (1965), Lewandowski (1996), Major and Seifert (2001), Carbone (2002), Neville (2006), Brunnemann and Rideout (2008, 2010) and Ding and Rovelli (2010).

In addition, this paper is distinguished 