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G. F ( X ) = ~ anX n consider ](x, ti) = ao + ~ ( x n / 1 - I ~ +1 ti). 2. VERTEX OPERATORS AND COHERENT STATES 37 and one can choose the Ti recursively so that 1/T1 = al, 1/TIT2 = a 2 , - ' - , leading to F(X) = F(X, Ti). In any event we have now another situation where a QM problem is related to a dispersionless soliton equation, namely dKdV. 2. A priori there is no reason why KdV should be connected to QM except for the accidental occurance of a second order differential operator 02 - v common to both theories.

20)). Note also (Q) fa/g IA, ~ > d# < A, 121 = I (here apparently < A, fll ~ < 12, A[). 22). 29) while /U f(z)g(z)dpH(Z, 5) < oc The closed subspace s generated by such f corresponds to T/A. For an operator A one has the coherent state diagonal element (Q representation or covariant symbol) (S) A(p, q) = A(z, 2) = < A, f~[AIA, ~t > which uniquely describes A. ; z', 5') = < A, ~[A[A, ~' >. 31) As for algebra one must determine whether the P bracket of the phase space representatives of two arbitrary operators satisfy the same algebraic structure.

VERTEX OPERATORS AND COHERENT STATES 41 in powers of e (cf. here [153, 169] and remarks below for more details on this). q at once and we will see how this goes below. Actually one can arrive at many different formulations of dispersionless Hirota equations for F ~ (and of enhanced equations for the F k) by working with any of the dKdV hierarchy equations, upon stipulating the canonical forms for S ~ P0 (and S j, Pj). 96) (although perhaps expressed differently). qk of course and here F2nl,2m -- 0 is entirely natural (with F~ "-- 0 mandatory since we will preserve the dKdV origins in the F ~ terms).

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