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On a c-number quantum $\tau$-function

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arxiv hep-th/9312213 v2 pith:NED6UJX5 submitted 1993-12-31 hep-th nlin.SIsolv-int

classification hep-thnlin.SIsolv-int
keywords fieldsfreefunctionalgebrasassociatedbetterc-numberconcept
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $\tau$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.

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