AN EULER-MC LAURIN FORMULA FOR INFINITE DIMENSIONAL SPACES Jose Javer Garca Moreta Graduate Studet of Physcs ( Sold State ) at UPV/EHU Address: P.O 6 890 Portugalete, Vzcaya (Spa) Phoe: (00) 3 685 77 16 53 E-mal: josegarc00@yahoo.es MSC : 6-0, 6S99, 81Q30 ABSTRACT:=I ths paper we study a fte dmesoal geeralzato of Euler-Mc Laur sum formula to deal wth Fuctoal tegrato, our ma task s to produce a Remader formula for a aalogue of Trapezodal rule o fte dmesoal spaces usg the sum classcal to be able to calculate tegrals volvg fuctoals, m volvg F[ ( x y)] wth may applcatos to Mathematcal Physcs. o Keywords: = Euler-Mc Laur sum formula, Fuctoal tegral ad dervatve 1.INTRODUCTION Oe of the ope problems Mathematcal Physcs, s to defe Ifte dmesoal tegrals over fuctoal spaces, gve the form: D[ ]exp( as[ ]) Wth S[ ] d xl(, ) (1.1) Where a s ether a postve parameter (Wger Mathematcal formulato of Browa Moto), or a Pure magary Number (Feyma formulato of Quatum Mechacs). Oe of the forms to vew these tegrals s to dscretze the fte dmesoal space, ad calculate the Itegral: dx1 dx... dxn F( x1, x, x3,..., x ) D[ ] F[ ] (1.) Ufortuately to defe the Volume for a fte dmesoal space, we fd may problems maly, the Volume of a -dmesoal Hypercube of sde a : 1 ff u=1 V ( u) 0 ff u<1 ff u>1 for (1.3) 1
However f the parameter a s bg we could use a Saddle-pot expaso ear the extremal of S[ ], havg the expaso: 1 F S[ ] S[ ] classcal d x1 d x ( classcal )... ( x ) ( x ) classcal 1 classcal (1.) Wth F[ classcal ] 0, ths s the WKB Semclasscal approach, Its physcal meag s that we are just keepg terms upto, the our Fuctoal tegral (1.1) becomes just a Ifte dmesoal Gaussa aalogue of the expresso: dx e B j x x j (1.5) For the case of a Scalar Feld, we fd (C s a costat): 1 1 S[ ] d x m [ ] S / D e CDet A (1.5) A s the Fuctoal determat A m As a bref Summary before presetg our formula: o The fucto ( x) s a Vector of a Fuctoal space wth o Compoets ( ) R The fuctoal dervatve s the aalogue to the Gradet o The scalar product x u dtx( t) u( t). FUNCTIONAL EULER-MC LAURIN FORMULA I ths paper, we wll provde a approxmate Numercal method to defe the Fuctoal tegral, usg the seres F[ mt ( x y)] ad a Euler- m Mc-Laur formula to defe the Remader. classcal May dettes vald for fte-dmesoal spaces ca be geeralzed to ft them to a fte dmesoal space: x d x x 0 (scalar product) (.1) ( ) as I (.1) we have troduced the Fuctoal dervatve,whch ca be defed
the form: [ ( )] [ ] ( ( )) [ ] lm F x y F lm df x y F (.) 0 0 d ( y) (.) s the aalogue of the drectoal dervatve (fte dmesoal case) the drecto of the vector (,,,...), also we ca gve the aalogue to the Itegrato by parts o R : dxu v dxu v (vashg Boudary term dxu v 0 ) (.3) F[ ] G[ ] D[ ] G[ ] D[ ] F[ ] (.) Ad the Posso Sum-formula for Fuctoal tegrals: m k d x ( x ) T F[ classcal mt ( x y)] D[ ] F[ classcal ] e (.5) k k d x ( x ) T [ ] e [ T ( x y)] (.6) k To prove (.5) we ca use smply Fourer Aalyss ad the chage of varables: m ( x y) D[ ] D[ ] D[ ] (.7) So our measure must be traslatoal varat, for ay measure whch s ot traslatoal varat we could wrte the form: D[ ] M[ ] Ad the classcal acto becomes S[ ] S[ ] l M[ ] (.8) Takg T 0 sde (.5), oly the term wth k=0 s relevat so we have: lm F[ classcal mt ( x y)] D[ ] F[ classcal ] T 0 m (.9) The t seems that our asatz to approxmate the Fuctoal tegral by a sum F[ classcal m ( x y)] wth step seems accurate f we make ths step m small (as a aalogy to the Trapezodal rule o R ), for F[ ] exp( S[ ]/ ) classcal lm exp( S[ classcal mt ( x y)]/ e 0 m S [ ]/ (.10) 3
Whch s just the classcal result for the Feyma tegral, to mprove ths we would eed to fd some kd of Remader expresso, usg a geeralzato of the Euler-Mc Laur sum formula, volvg fuctoal dervatves. The codto (.) appled to QFT takg the Fuctoal tegral, settg S [ ] D[ ] e 1 (reormalzato codto) mples: S[ ] F[ ] S[ ] S[ ] D[ ] e D[ ] e F[ ] (.11) F[ ] S[ ] F[ ] (Where s a gve state of a QFT theory wth Classcal Acto S ) O codto F s a Polyomally Bouded Fuctoal. Usg the Geeratg Fuctoal for QFT Z[J] we ca wrte (.5) as Z[ J ] D[ ] e (.1) S[ ] d xj ( x) ( x) m k F[ classcal mt ( x y)] Z[ J ] T k To defe the Euler-Mc Laur aalogue for Ifte-dmesoal spaces, frst we Itroduce the Traslatoal operator : D e exp d x D e F[ ] F[ ( x y)] (.13) For 0, we must recover the Fuctoal dervatve: D F[ ] F lme 1 F[ ] d x ( x y) 0 (.1) ( y) Also usg the expaso (volvg Beroull Numbers B ) of the fuctoal: e d x d x 1 B d x d x! ( ) ( )... ( ) 1... 0 x1 x x (.15) Hece, we ca defe the geeralzato of Euler-Mc Laur formula for fte dmesoal spaces the form:
F[ ] D F F m x y R (.16) 0 0 [ ] [ ] [ 0 ( )] m1 B F[ ] d x d x R ( )! ( ) ( )... ( ) r 1 r r 1... r1 r1 r x1 x xr 1 (.17) r1 Where all the fuctoals dervatves F[ ] are evaluated at 0, ths formula wth the Remader gves a (approxmate) umercal value to the Fuctoal tegral usg the seres F[ 0 m ( x y)], as 0 the m1 Remader becomes less sgfcat, we thk ths formula ca be useful to defe Fuctoal tegrato wthout recallg to dscretzato of the space of Paths or Motecarlo tegrato, however for may cases the factor: r1 F[ ] ( x ) ( x )... ( x ) 1 r1 (.18) exsts oly the sese of dstrbuto theory ad eeds to be regularzed. Note, that for R volvg a smooth fucto f, the Remaeder R : 1 1. f ( x ) f ( x ) f 0 0. ( x ) R 1 0 e. f Drectoal dervatve o (1,1,1,...) R x0 R (.19) Expressos (.16) ad Remader R (.17) are the geeralzato of the formula: f ( x) B r r1 r f ( t) dt f ( ) x f ( x) (.0) x r 1 ( r)! x f ( x) 0 as x Ad we have the equvalece: D[ ] F[ ] dx dx... dx F( x, x,..., x ) (.1) 1 1 0 a1 a a Where the fucto (vector) 0 has compoets ( ) 0 a 5
3. REFERENCES [1] Abramowtz ad Stegu Hadbook of Mathematcal fuctos New York: Dover (197) [] Carter, Perre & DeWtt-Morette, Cécle ; A ew perspectve o Fuctoal Itegrato, Joural of Mathematcal Physcs 36 (1995) pp. 137-30. Full text avalable at : fuct-a/960005 [3] DeWtt-Morette, Cécle ; Feyma's path tegral - Defto wthout lmtg procedure, Commucato Mathematcal Physcs 8(1) (197) [] Feyma, R. P., ad Hbbs, A. R., Quatum Physcs ad Path Itegrals, New York: McGraw-Hll, 1965 [ISBN 0-07-00650-3]. [5] Kleert, Hage, Path Itegrals Quatum Mechacs, Statstcs, Polymer Physcs, ad Facal Markets, th edto, World Scetfc (Sgapore, 00) [6] Pesk M.E ad Schröeder D. V Itroducto to Quatum Feld Theory Addso-Wesley (1995) [7] Z Just, Jea ; Path Itegrals Quatum Mechacs, Oxford Uversty Press (00), [ISBN 0-19-85667-3]. 6