By Philippe Blanchard, Arkadiusz Jadczyk

This quantity provides specific discussions of a couple of unsolved conceptual and technical concerns coming up, particularly, within the foundations of quantum thought and the philosophy of technology. The 14 contributions seize a large choice of viewpoints and backgrounds. a few chapters deal basically with the most experimental matters; others specialise in theoretical and philosophical questions. additionally, makes an attempt are made to systematically research ways that quantum physics will be attached to the neurosciences and cognizance examine.

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727, 748]. We now look briefly at [987] as developed in [159]. 2. V E R T E X O P E R A T O R S A N D C O H E R E N T S T A T E S 31 group (Z) Gh = G = {Uh = exp[(i/h)(a+(1/2)Tr~)]U(Tc,~) where U(~r, ~) ~ exp[(i/h)(~rQ~/5)] ~ D(z) for z = ( 1 / x f ~ ) ( ~ + i~r). One generates coherent states ]u > h = Iu > = Uh[O >h where ]0 >h corresponds to a peaked vacuum for Q (cf. 17)). 38) The variables (Tr, ~) provide coordinates on F and there is a standard symplectic structure. g. 4iu > / < ulu > generate so called classical operators via (AB) f~ = f d)~ f()~)exp[h3,] and covariant symbols (AC) Ah(u) = < uif~iu >h leading to functions a(~) on F.

VERTEX OPERATORS AND COHERENT STATES 35 which can be further expanded. One is thus given various /)(A) = exp(/ka t - A a ) and the insertion of e could be thought of as a way of introducing peaked states and coadjoint orbit variables to provide a geometrical background for dispersionless KP theory. g. fluid dynamics analogous to the control of quantum fluctuations expressed via Q ~ v/-hq = eq ---. ~. 4. We want to indicate next a possible connection of the Maslov canonical operator (cf. [694]) with semiclassical soliton theory.

3. The restriction A E S 1 with p2 + q2 = 1 is not pleasing. One could in principle work with general A and exp[Aat(e) - A-la(e)] but the nice geometry of coherent state theory would seem to be lost or seriously compromised. 63) 4)@d4 This enters then the realm of coherent state transforms, oscillatory integrals, Weyl-WignerMoyal theory, etc. (cf. [84, 85, 350, 438, 694] and Chapter 4). Further development in [159] includes some elaboration of the multisoliton situation of ( A M ) . One inserts e following [902] to obtain (AP) ~-~r ,,- 1-IIN[1+ (aj/e)X((T/E), s ~j)].