By Gerald W Johnson
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Extra info for Generalized Dyson Series, Generalized Feynman's Diagrams, the Feynman Integral, and Feynman's Operational Calculus
Example text
1 , f o r every X > 0 , F(A x+£) i s d e f i n e d f o r m x L e b e s g u e - a . e . F). '3) V U= ^ where { T p . ^i MVUII - We write VU^T p=l ' *> c and its p;U C and S |« P=l | < + co. 1) as an absolutely convergent infinite series. series together. ,m}. ,cu } € [k;m], we shall always write 26 If GENERALIZED DYSON SERIES AND FEYNMANlS OPERATIONAL CALCULUS {a 27 k + l ' * • ' 'am} " U , . . ,m}\{alf . . ,ak>. Now, i n view of ( 2 . 2 ) and ( 2 . 4) F(x) = - n [ / 6 ( s , x ( s ) ) d u u ( s) + I a)p u 8u ( T p u u-1 (0,t) u p-1 ' ' - m z Z k=0 { a 1 , .
W. dsn-e ° I5lfl_e ° *| n (f) 1 1 / m / T-I m . -T(H n /X) (e n T m -i T , . n -T(H0/X) ° T+i _ n m < (l ? ) n / ~ 1 m [ 9 . d8n m -(t-x)(Hn/X) ^ P - « ° _, _ m n n (j) f ... d Sn | Z f &l> d s ^ . d s ^ 1 n . dsn i n _ _ m n 2 < (5)n / / ... 41) = (j) J J (=) - J J . Hence ( 4 . 3 8 ) i s e s t a b l i s h e d and ( 4 . 3 7 ) h o l d s f o r a l l A c C + . REMARK 4 . 6 . The p r o o f above r e v e a l s r a t h e r n i c e l y some of t h e i n t u i t i v e ideas underlying the present theory.
A n I , Z f|^| n n n=0 q 0 + q i + . +qh=n ^ - " V j ^ . . dy(s_ An . , , q0,J1,. ,s ) l qQ T-, < S . 2 < T, (0,t) e < . . < S. < s. 1 j . , . Jl+J2+1 < s. integers , Jl4"*--4"^ . < . . < s^ + 1