Read e-book online Functional Integration and Semiclassical Expansions PDF

By F. Langouche, D. Roekaerts, E. Tirapegui (auth.)

ISBN-10: 9048183774

ISBN-13: 9789048183777

ISBN-10: 940171634X

ISBN-13: 9789401716345

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30) Inserting N times Ictqlq>R= I N N+l IT dqi qk \1 IT lq. > = J J' r l rl In the limit N ~ oo I (21T) -d . llj y - dp. • >tn and therefore, product to o(q(t)-q)o(q(to)-qo) forti r r tj • i j. 34) q~{t') LR ,to< t' < we can write this as N t. _ 1> ). (qk-qk-1) - ie hy(pk,qk,qk-1'tk)] qk In the limit N + ~. q- Hy(p,q,T) Y ~ to } o(q(t)-q) o(q(to)-qo) so CHAPI'ER Ill = y (3. 36) there follows that (3. 37) for all times ti' i =l, ••• ,n different from one another.

104c) The symbol ,;, denotes equality as far as use in the functional integral is con- cerned. 104) is related to the commutation relations in the operator formalism and is proved in cha0 ter 5. There we will also study analogous replacements for powers of ~ in the configuration space functional integral. integral changes but because H(p,q,T) changes. The new discretization is the prepoint discretization, since the resulting h depends on p,q and It T only. Therefore we obtain in fact immediately H(p,q,T).

1scret1zat1ons • . h ji\) ji\) 1 . 1 t1on y and y- sue h tat g[ji\) 1-g[y]=eg 1[ y,y_ 1 are a so. equ1va ent. \) \) y . (We deliberately don'-t,·Wt-ite gL~~-gi'yJ=O(€), since in fact the result 1s not true for discretizations such that g~y]- g[~]= 6K6AaKA ). ,t. 1 ~t. 84) it follows that equivalent for a = 0 and y 1(a) and y 2 (a) are a = I (then they coincide) and for the set of phase space functions which are at most quadratic in ~ and have a coefficient of the quadratic term which is constant or linear in q.

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Functional Integration and Semiclassical Expansions by F. Langouche, D. Roekaerts, E. Tirapegui (auth.)

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