By Mikio Namiki (auth.)
ISBN-10: 0387555633
ISBN-13: 9780387555638
ISBN-10: 3540472177
ISBN-13: 9783540472179
ISBN-10: 3540555633
ISBN-13: 9783540555636
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Extra info for Stochastic Quantization
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T) qJ(t') qk(t") .. 23b) 23 t< standing for the earliest of all times and T being the chronological-ordering symbol. 16). We can easily rewrite the eigenvalue problem of the Fokker-planck operator, already discussed - -t in Section 3, in the abstract form. ,. of Section 3. In particular, the abstract Fokker-Planck operator is p = ~]ai: + i,K,(q)] . 27) the right-hand sides being the eigenfunctions introduced in Section 3. 28b) hold because of (; = exp[V(q)/(2aJ)] (with af = kaT). Note that I J > is proportional to I Vo > -t and both belong to the zero eigenvalue of F .
8) and T can also prove a sort of adiabatic theorem, _ -, _(0) luo>=T(O,-oo)luo (t» 14>eq >= 14>eq >. 3). This theorem holds for the case in which the common highest eigenvalue of l' and 1'0 is non-degenerate discrete zero. 14) 14>' >t<= T'(t<, -00) 1 u~O) > jC; for an asymptotic system in thermal equilibrium. 10b) (for quantities with superscript or subscript (0)) and < fiFo = or < J I 1 1' (t,t') =< J I for any t and t'. ·. 1=< V~O) 1 T 1'1(00, -00) q~(t) q~(t'), .. 12) present our starting point to develop the stochastic "FeynmanDyson" perturbative-approach to expectation values and correlation functions.
This is also one of the important merits of SQM. However, many workers have so far been using the asymptotic behavior of the correlation function in ordinary space-time but not in the new (fictitious) time. One may expect that the latter business will be developed in near future. Although the operator formalism is convenient to our SQM procedure, we should remark that the stochastic quantization itself belongs to a sort of c-number quantization as well as the Feynman path-integral method. Anyway, SQM enables us to calculate field-theoretical propagators through a classical random process.
Stochastic Quantization by Mikio Namiki (auth.)
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