Appendix 7B Proof of Eqs 764 and 765
Here we prove, for a Gaussian stochastic processes z t and a general function of time x t the results 7.64 and 7.65 . Our starting point is cf. Eq. 7.63 e'T,jxjzA jT.jX zj gt - 1 2 EjEkxj Szi Szk xk 7.119 where the sums are over the n random variables. Noting that this relationship holds for any set of constants xj , we redefine these constants by setting and take the limit Atj 0 and n in the interval to lt t1 lt t2 lt lt tn. Then
V2
fn 17 -nfn-1 - -n 1fn 1 2.157 a n -n n - 1 at n Vn 1 n 1 2.158 where we have used n to denote f n. The operators ci and a are seen to have the property that when operating on an eigenfUnction of the Harmonic oscillator Hamiltonian they yield the eigenfUnction just above or below it, respectively. a and a will therefore be referred to as the harmonic oscillator raising or creation and lowering or annihilation operators, respectively.8 Equation 2.152 also leads to hence the name number operator...
Acknowledgements
In the course of writing the manuscript I have sought and received information, advice and suggestions from many colleagues. I wish to thank Yoel Calev, Graham Fleming, Michael Galperin, Eliezer Gileadi, Robin Hochstrasser, Joshua Jortner, Rafi Levine, Mark Maroncelli, Eli Pollak, Mark Ratner, Joerg Schr der, Zeev Schuss, Ohad Silbert, Jim Skinner and Alessandro Troisi for their advice and help. In particular I am obliged to Misha Galperin, Mark Ratner and Ohad Silbert for their enormous help...