1 CBPF-NF-03/08 Some commets o rigorous quatum field path itegrals i the aalytical regularizatio scheme Luiz C.L. Botelho Departameto de Matemática Aplicada, Istituto de Matemática, Uiversidade Federal Flumiese, Rua Mario Satos Braga , Niterói, Rio de Jaeiro, Brazil Abstract Through the systematic use of the Milos theorem o the support of cylidrical measures o R, we produce several mathematically rigorous path itegrals i iteractig euclidea quatum fields with Gaussia free measures defied by geeralized powers of the Laplacea operator. 1 Itroductio Sice the result of R.P. Feyma o represetig the iitial value solutio of Schrodiger Equatio by meas of a aalytically time cotiued itegratio o a ifiite - dimesioal space of fuctios, the subject of Euclidea Fuctioal Itegrals represetatios for Quatum Systems has became the mathematical - operatioal framework to aalyze Quatum Pheomea ad stochastic systems as showed i the previous decades of research o Theoretical Physics ( ). Oe of the most importat ope problem i the mathematical theory of Euclidea Fuctioal Itegrals is that related to implemetatio of soud mathematical approximatios to these Ifiite-Dimesioal Itegrals by meas of Fiite-Dimesioal approximatios outside of the always used [computer orieted] Space-Time Lattice approximatios (see ,  - chap. 9). As a first step to tackle upo the above cited problem it will be eeded to characterize mathematically the Fuctioal Domai where these Fuctioal Itegrals are defied. The purpose of this ote is to preset the formulatio of Euclidea Quatum Field theories as Fuctioal Fourier Trasforms by meas of the Bocher-Marti-Kolmogorov theorem for Topological Vector Spaces (,  - theorem 4.35) ad suitable to defie ad aalyze rigorously Fuctioal Itegrals by meas of the well-kow Milos theorem ( - theorem 4.31 ad  - part ) ad preseted i full i Appedix 1.
2 CBPF-NF-03/08 We thus preset ews results o the difficult problem of defiig rigorously ifiitedimesioal quatum field path itegrals i geeral space times Ω R ν (ν =, 4,... ) by meas of the aalytical regularizatio scheme. Some rigorous quatum field path itegral i the Aalytical regularizatio scheme Let us thus start our aalysis by cosiderig the Gaussia measure associated to the (ifrared regularized) α-power (α > 1) of the Laplacea actig o L (R ) as a operatioal quadratic form (the Stoe spectral theorem or the aalytical regularizig schem i QFT) ( ) α ε = (λ) α de(λ) (1-a) ε IR λ α,ε IR [j] = exp 1 Z (0) = j, ( ) α ε α,εµ [ϕ] exp ( i j, ϕ j L (R ) L (R ) ) (1-b) Here ε IR > 0 deotes the ifrared cut off. It is worth call the reader attetio that due to the ifrared regularizatio itroduced o eq (1-a), the domai of the Gaussia measure ( ) is give by the space of square itegrable fuctios o R by the Milos theorem of Appedix 1, sice for α > 1, the operator ( ) α ε IR defies a classe trace operator o L (R ), amely T r H (( ) α 1 ε IR ) = d 1 k ( K α + ε IR ) < (1-c) This is the oly poit of our aalysis where it is eeded to cosider the ifra-red cut off cosidered o the spectral resolutio eq (1-a). As a cosequece of the above remarks, oe ca aalize the ultra-violet reormalizatio program i the followig iteractig model proposed by us ad defied by a iteractio g bare V (ϕ(x)), with V (x) deotig a fuctio o R such, that it posseses a essetially bouded (compact support) Fourier trasform ad g bare deotig the positive bare couplig costat. Let us show that by defiig a reormalized couplig costat as (with g re < 1) g bare = g re (1 α) 1/ () oe ca show that the iteractio fuctio exp g bare (α) d x V (ϕ(x)) (3) is a itegrable fuctio o L 1 (L (R ), α,ε IR µ [ϕ]) ad leads to a well-defied ultra-violet path itegral i the limit of α 1.
3 CBPF-NF-03/08 3 The proof is based o the followig estimates. Sice almost everywhere we have the poitwise limit exp g bare (α) d x V (ϕ(x)) N ( 1) (g bare (α)) lim dk 1 dk Ṽ (k 1 ) N! Ṽ (k ) =0 R dx 1 dx e ik 1ϕ(x 1) ik ϕ(x) e R we have that the upper-boud estimate below holds true with Zε α IR [g bare ] ( 1) (g bare (α)) =0 Zε α IR [g bare ] =! R dx 1 dx α,ε IR µ[ϕ] exp g bare (α) R dk 1 dk Ṽ (k 1 ) Ṽ (k ) NP (4) α,ε IR µ[ϕ](e i k l ϕ(x l ) l=1 ) (5-a) d x V (ϕ(x)) (5-b) we have, thus, the more suitable form after realizig the d k i ad α,ε IR µ[ϕ] itegrals respectivelly Zε α IR =0[g bare ] (g bare (α)) =0! dx 1 dx det 1 ( ) Ṽ L (R) [ ] G (N) α (x i, x j ) 1 i N 1 j N (6) Here [G (N) α (x i, x j )] 1 i N deotes the N N symmetric matrix with the (i, j) etry 1 j N give by the Gree-fuctio of the α-laplacea (without the ifra-red cut off here! ad the eeded ormalizatio factors!). (1 α) Γ(1 α) G α (x i, x j ) = x i x j Γ(α) (7) At this poit, we call the reader attetio that we have the formulae o the asymptotic behavior for α 1. lim det 1 [G (N) α (x i, x j )] α 1 α>1 ( (1 α) N/ (N 1)( 1) N π N/ ) 1 After substitutig eq.(8) ito eq.(6) ad takig ito accout the hypothesis of the compact support of the oliearity Ṽ (k), oe obtais the fiite boud for ay value (8)
4 CBPF-NF-03/08 4 g rem > 0, ad producig a proof for the covergece of the perturbative expasio i terms of the reormalized couplig costat. lim Zε α ( Ṽ α 1 IR =0[g bare (α)] ( ) L (R)) g re (1 ) (1 α) /! (1 α) 1 =0 e πg re Ṽ L (R) < (9) Aother importat rigorously defied fuctioal itegral is to cosider the followig α-power Klei Gordo operator o Euclidea space-time L = ( ) α + m (10) with m a positive mass parameters. Let us ote that L 1 is a operator of class trace o L (R ν ) if ad oly if the result below holds true T r L (R ν )(L 1 ) = d ν 1 k k α + m = C(ν) m ( ν π α ) α cosecνπ < (11) α amely if α > ν (1) I this case, let us cosider the double fuctioal itegral with fuctioal domai L (R ν ) Z[j, k] = exp G β[v(x)] ( ) α +v+m µ[ϕ] i d ν x (j(x) ϕ(x) + k(x) v(x)) where the Gaussia fuctioal itegral o the fields V (x) has a Gaussia geeratig fuctioal defied by a -itegral operator with a positive defied kerel g( x y ), 1 amely Z (0) [k] = G i β[v(x)] exp d ν x k(x)v(x) = exp 1 d ν x d ν y (k(x) g( x y ) k(x)) (14) By a simple direct applicatio of the Fubbii-Toelli theorem o the exchage of the itegratio order o eq.(13), lead us to the effective λϕ 4 - like well-defied fuctioal itegral represetatio Z eff [j] = (( ) α +m ) µ [ϕ(x)] exp 1 d ν xd ν y ϕ(x) g( x y ) ϕ(y) exp i d ν x j(x)ϕ(x) (15) (13)
5 CBPF-NF-03/08 5 Note that if oe itroduces from the begiig a bare mass parameters m bare depedig o the parameters α, but such that it always satisfies eq.(11) oe should obtais agai eq.(15) as a well-defied measure o L (R ν ). Of course that the usual pure Laplacea limit of α 1 o eq.(10), will eeded a reormalizatio of this mass parameters (lim m bare (α) = +!) as much as doe i the previous example. α 1 Let us cotiue our examples by showig agai the usefuless of the precise determiatio of the fuctioal - distributioal structure of the domai of the fuctioal itegrals i order to costruct rigorously these path itegrals without complicated limit procedures. Let us cosider a geeral R ν Gaussia measure defied by the Geeratig fuctioal o S(R ν ) defied by the α-power of the Laplacea operator actig o S(R ν ) with a of small ifrared regularizatio mass parameter µ Z (0) [j] = exp = 1 E alg (S(R ν )) j, (( ) α + µ 0) 1 j L (R ν ) α µ[ϕ] exp(i ϕ(j)) (16) A explicitly expressio i mometum space for the Gree fuctio of the α-power of ( ) α + µ 0 give by ( ) d (( ) +α + µ 0) 1 ν k 1 (x y) = (π) ν eik(x y) (17) k α + µ 0 Here C(ν) is a ν-depedet (fiite for ν-values!) ormalizatio factor. Let us suppose that there is a rage of α-power values that ca be choose i such way that oe satisfies the costrait below α µ[ϕ]( ϕ L j (R ν )) j < (18) E alg (S(R ν )) with j = 1,,, N ad for a give fixed iteger N, the highest power of our poliomial field iteractio. Or equivaletly, after realizig the ϕ-gaussia fuctioal itegratio, with a space-time cutt off volume Ω o the iteractio to be aalyzed o eq.(16) Ω ( d ν x[( ) α + µ 0] j d ν k (x, x) = vol(ω) k α + µ 0 ( = C ν (µ 0 ) ( ν π ) α ) α cosecνπ < (19) α For α > ν, oe ca see by the Milos theorem that the measure support of the Gaussia measure eq.(16) will be give by the itersectio Baach space of measurable Lebesgue fuctios o R ν istead of the previous oe E alg (S(R ν )) ( ). L N (R ν ) = ) j N (L j (R ν )) (0) j=1
6 CBPF-NF-03/08 6 I this case, oe obtais that the fiite - volume p(ϕ) iteractios N exp λ j (ϕ (x)) j dx 1 (1) j=1 Ω is mathematically well-defied as the usual poitwise product of measurable fuctios ad for positive couplig costat values λ j 0. As a cosequece, we have a measurable fuctioal o L 1 (L N (R ν ); α µ[ϕ]) (sice it is bouded by the fuctio 1). So, it makes sese to cosider mathematically the well-defied path - itegral o the full space R ν with those values of the power α satisfyig the cotrait eq.(17). N Z[j] = α µ[ϕ] exp λ j ϕ j (x)dx exp(i j(x)ϕ(x)) () L N (R ν ) Ω R ν j=1 Fially, let us cosider a iteractig field theory i a compact space-time Ω R ν defied by a iteger eve power of the Laplacea operator with Dirichlet Boudary coditios as the free Gaussia kietic actio, amely Z (0) [j] = exp 1 j, ( ) j L (Ω) = W (Ω) () µ[ϕ] exp(i j, ϕ L (Ω)) (3) here ϕ W (Ω) - the Sobolev space of order which is the fuctioal domai of the cylidrical Fourier Trasform measure of the Geeratig fuctioal Z (0) [j], a cotiuous biliear positive form o W (Ω) (the topological dual of W (Ω)) ( ). By a straightforward applicatio of the well-kow Sobolev immersio theorem, we have that for the case of k > ν (4) icludig k a real umber the fuctioal Sobolev space W (Ω) is cotaied i the cotiuously fractioal differetiable space of fuctios C k (Ω). As a cosequece, the domai of the Bosoic fuctioal itegral ca be further reduced to C k (Ω) i the situatio of eq.(4) Z (0) [j] = () µ[ϕ] exp(i j, ϕ L (Ω)) (5) C k (Ω) That is our ew result geeralizig the Wieer theorem o Browia paths i the case of = 1, k = 1 ad ν = 1 Sice the bosoic fuctioal domai o eq.(5) is formed by real fuctios ad ot distributios, we ca see straightforwardly that ay iteractio of the form exp g F (ϕ(x))d ν x (6) with the o-liearity F (x) deotig a lower bouded real fuctio (γ > 0) Ω F (x) γ (7)
7 CBPF-NF-03/08 7 is well-defied ad is itegrable fuctio o the fuctioal space (C k (Ω), () µ[ϕ]) by a direct applicatio of the Lebesque theorem exp g Ω F (ϕ(x)) d ν x exp+gγ (8) At this poit we make a subtle mathematical remark that the ifiite volume limit of eq.(5) - eq.(6) is very difficult, sice oe looses the Gardig - Poicaré iequalite at this limit for those elliptic operators ad, thus, the very importat Sobolev theorem. The probable correct procedure to cosider the thermodyamic limit i our Bosoic path itegrals is to cosider solely a volume cut off o the iteractio term Gaussia actio as i eq.() ad there search for vol(ω) ( ). As a last remark related to eq.(3) oe ca see that a kid of fishet expoetial geeratig fuctioal Z (0) [j] = exp 1 j, exp α j (9) L (Ω) has a Fourier trasformed fuctioal itegral represetatio defied o the space of the ifiitelly differetiable fuctios C (Ω), which physically meas that all field cofiguratios makig the domai of such path itegral has a strog behavior like purely ice smooth classical field cofiguratios. As a last importat poit of this ote, we preset a importat result o the geometrical characterizatio of massive free field o a Euclidea Space-Time (). Firstly we aoucig a slightly improved versio of the usual Milos Theorem (). Theorem 3. Let E be a uclear space of tests fuctios ad dµ a give σ-measure o its topologic dual with the strog topology. Let, 0 be a ier product i E, iducig a Hilbertia structure o H 0 = (E,, 0 ), after its topological completatio. We suppose the followig: a) There is a cotiuous positive defiite fuctioal i H 0, Z(j), with a associated cylidrical measure dµ. b) There is a Hilbert-Schmidt operator T : H 0 H 0 ; ivertible, such that E Rage (T ), T 1 (E) is dese i H 0 ad T 1 : H 0 H 0 is cotiuous. We have thus, that the support of the measure satisfies the relatioship support dµ (T 1 ) (H 0 ) E (30) At this poit we give a o-trivial applicatio of ours of the above cited Theorem 3. Let us cosider a differetial iversible operator L: S (R N ) S(R), together with a positive iversible self-adjoit elliptic operator P : D(P ) L (R N ) L (R N ). Let H α be the followig Hilbert space H α = S(R N ), P α ϕ, P α ϕ L (R N ) =, α, for α a real umber. (31) We ca see that for α > 0, the operators below P α : L (R N ) H +α ϕ (P α ϕ) (3)
8 CBPF-NF-03/08 8 P α : H +α L (R N ) are isometries amog the followig sub-spaces ϕ (P α ϕ) (33) sice ad D(P α ),, L ) ad H +α P α ϕ, P α ϕ H+α = P α P α ϕ, P α P α ϕ L (R N ) = ϕ, ϕ L (R N ) (34) P α f, P α f L (R N ) = f, f H +α (35) If oe cosiders T a give Hilbert-Schmidt operator o H α, the composite operator T 0 = P α T P α is a operator with domai beig D(P α ) ad its image beig the Rage (P α ). T 0 is clearly a ivertible operator ad S(R N ) Rage (T ) meas that the equatio (T P α )(ϕ) = f has always a o-zero solutio i D(P α ) for ay give f S(R N ). Note that the coditio that T 1 (f) be a dese subset o Rage (P α ) meas that T 1 f, P α ϕ L (R N ) = 0 (36) has as uique solutio the trivial solutio f 0. Let us suppose too that T 1 : S(R N ) H α be a cotiuous applicatio ad the biliear term (L 1 (j))(j) be a cotiuous applicatio i the Hilbert spaces H +α S(R N ), L amely: if j j, the L 1 : P α L j L 1 P α j, for j Z ad j S(R N ). By a direct applicatio of the Milos Theorem, we have the result Z(j) = exp 1 [L 1 (j)(j)] = dµ(t ) exp(it (j)) (37) (T 1 ) H α Here the topological space support is give by [ (P (T 1 ) H α = α T 0 P α) ( ) 1] (P α (S(R N ))) = [ (P α ) (T 1 0 ) (P α) ) ] P α (S(R N )) = P α T 1 0 (L (R N )) (38) I the importat case of L = ( + m ): S (R N S(R N ) ad T 0 T0 = ( + m ) β ( ) N 1 (L (R N )) sice T r(t 0 T0 1 m Γ( N ) = )Γ(β N ) < for β > (m ) β 1 Γ(β) N 4 with the choice P = ( + m ), we ca see that the support of the measure i the path-itegral represetatio of the Euclidea measure field i R N may be take as the measurable sub-set below supp d ( + m ) u(ϕ) = ( + m ) α ( + m ) +β (L (R N )) (39) sice L 1 P α = ( + m ) 1 α is always a bouded operator i L (R N ) for α > 1.
9 CBPF-NF-03/08 9 As a cosequece each field cofiguratio ca be cosidered as a kid of fractioal distributioal derivative of a square itegrable fuctio as writte below [ ( ϕ(x) = ) + m N 4 f] +ε 1 (x) (40) with a fuctio f(x) L (R N ) ad ay give ε > 0, eve if origially all fields cofiguratios eterig ito the path-itegral were elemets of the Schwartz Tempered Distributio Spaces S (R N ) certaily very rough mathematical objects to characterize from a rigorous geometrical poit of view. We have, thus, make a further reductio of the fuctioal domai of the free massive Euclidea scalar field of S (R N ) to the measurable sub-set as give by eq.(130) deoted by W (R N ) exp [ 1 ( + m ) 1 j ] (j) = d ( +m )µ(ϕ) e i ϕ(j) S (R N ) = d ( +m ) µ(f) e i f,( +m ) N 4 +ε 1 f L (R N ) W (R N ) S (R N ) (41)
10 CBPF-NF-03/08 10 Refereces  B. Simo, Fuctioal Itegratio ad Quatum Physics - Academic Press, (1979).  B. Simo, The P (φ) Euclidea (Quatum) Field Theory - Priceto Series i Physics, (1974).  J. Glimm ad A. Jaffe, Quatum Physics - A Fuctioal Itegral Poit of View Spriger Verlag, (1988).  Y. Yamasaki, Measure o Ifiite Dimesioal Spaces - World Scietific Vol. 5, (1985).  Xia Dao Xig, Measure ad Itegratio Theory o Ifiite Dimesioal Space Academic Press, (197).  L. Schwartz, Radom Measure o Arbitrary Topological Space ad Cylidrical Measures - Tata Istitute - Oxford Uiversity Press, (1973).  K. Symazik, J. Math., Phys., 7, 510 (1966).  B. Simo, Joural of Math. Physics, vol. 1, 140 (1971).  Luiz C.L. Botelho, Phys., Rev. 33D, 1195 (1986).  E Nelso, Regular probability measures o fuctio space; Aals of Mathematics (), 69, , (1959).
11 CBPF-NF-03/08 11 APPENDIX Some Commets o the Support of Fuctioal Measures i Hilbert Space Let us commet further o the applicatio of the Milos Theorem i Hilbert Spaces. I this case oe has a very simple proof which holds true i geeral Baach Spaces (E, ). Let us thus, give a cylidrical measures d µ(x) i the algebraic dual E alg of a give Baach Space E ( ). Let us suppose either that the fuctio x belogs to L 1 (E alg, d µ(x)). The the support of this cylidrical measures will be the Baach Space E. The proof is the followig: Let A be a sub-set of the vectorial space E alg (with the topology of potual covergece), such that A E c (so x = + ). Let be the sets A µ = x E alg x. The we have the set iclusio A =0 A, so its measure satisfies the estimates below: µ(a) lim if = lim if lim if = lim if µ(a ) µx E alg x 1 x d µ(x) E alg x L 1 (E alg,d µ ) = 0. (1) Leadig us to the Milos theorem that the support of the cylidrical measure i E alg is reduced to the ow Baach Space E. Note that by the Mikowisky iequality for geeral itegrals, we have that x L 1 (E alg, d µ(x)). Now it is elemetary evaluatio to see that if A 1 (M), whe 1 E = M, a give Hilbert Space, we have that d A µ(x) x = T r M (A 1 ) <. () M alg This result produces aother criterium for supp d A µ = M (the Milos Theorem), whe E = M is a Hilbert Space. It is easy too to see that if x d µ(x) < (3) the the Fourier-Trasformed fuctioal Z(j) = is cotiuous i the orm topology of M. M M e i(j,x) M d µ(x) (4)
12 CBPF-NF-03/08 1 Otherwise, if Z(j) is ot cotiuous i the origi 0 M (without loss of geerality), the there is a sequece j M ad δ > 0, such that j 0 with δ Z(j ) 1 e i(j,x) M 1 d µ(x) M (j, x) d µ(x) M ( ) j x d µ(x) 0, (5) a cotradictio with δ > 0. Fially, let us cosider a elliptic operator B (with iverse) from the Sobelev space M m (Ω) to M m (Ω). The by the criterium give by eq.() if M T r L (Ω)[(I + ) + m B 1 (I + ) + m ] <, (6) we will have that the path itegral below writte is well-defied for x M +m (Ω) ad j M m (Ω). Namely exp( 1 (j, B 1 j) L (Ω)) = d B µ(x) exp(i(j, x) L (Ω)). (7) M +m (Ω) By the Sobolev theorem which meas that the embeeded below is cotiuous (with Ω R ν deotig a smooth domai), oe ca further reduce the measure support to the Hölder α cotiuous fuctio i Ω if m ν > α. Namely, we have a easy proof of the famous Wieer Theorem o sample cotiuity of certai path itegrals i Sobolev Spaces M m (Ω) C α (Ω) (8-a) The above Wieer Theorem is fudametal i order to costruct o-trivial examples of mathematically rigorous euclideas path itegrals i spaces R ν of higher dimesioality, sice it is a trivial cosequece of the Lebesgue theorem that positive cotiuous fuctios V (x) geerate fuctioals itegrable i M m (Ω), d B µ(ϕ) of the form below exp V (ϕ(x))dx L 1 (M m (Ω), d B µ(ϕ)). (8-b) Ω As a last importat remark o Cylidrical Measures i Separable Hilbert Spaces, let us poit at to our reader that the support of such above measures is always a σ- compact set i the orm topology of M. I order to see such result let us cosider a give dese set of M, amely x k k I +. Let δ k k I + be a give sequece of positive real umbers with δ k 0. Let ε aother sequece of positive real umbers such that =1 ε < ε. Now it is straightforward to see that M h=1 B(x k, δ k ) M ad thus lim sup µ ( k=1 B(x k, δ k ) = µ(m) = 1. As a cosequece, for each, there is a k, such k ) that µ k=1 B(x k, δ k ) 1 ε. Now the sets K µ = [ k ] =1 k=1 B(x k, δ k ) are closed ad totally bouded, so they are compact sets i M with µ(m) 1 ε. Let is ow choose ε = 1 ad the associated
13 CBPF-NF-03/08 13 compact sets K,µ. Let us further cosider the compact sets ˆK,µ = l=1 K l,µ. We have that ˆK,µ ˆK +1,µ, for ay ad lim sup µ( ˆK,µ ) = 1. So, Supp dµ = ˆK =1,µ, a σ-compact set of M. We cosider ow a eumerable family of cylidrical measures dµ i M satisfyig the chai iclusio relatioship for ay I + Supp dµ Supp dµ +1. () Now it is straightforward to see that the compact sets ˆK, where Supp dµ m = =1 is such that Suppdµ m () =1 ˆK, for ay m I +. Let us cosider the family of fuctioals iduced by the restrictio of this sequece of measures i ay compact ˆK (). Namely µ L () (f) = ˆK () f(x) dµ p (x). ˆK (m), (8-c) () Here f C b ( ˆK ). Note that all the above fuctioals i =1 C () b( ˆK ) are bouded by 1. ( By the Alaoglu-Bourbaki theorem they form a compact set i the weak star topology of =1 C (), b( ˆK )) so there is a sub-sequece (or better the whole sequece) covergig to a uique cylidrical measure µ(x). Namely f(x)dµ (x) = f(x)d µ(x) (8-d) for ay f =1 C () b( ˆK ). lim M M