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Physical Properties of Quantum Field Theory Measures 1 0 J.M. Mourãoa), T. Thiemannb) and J.M. Velhinhoc) 0 2 n Abstract a J Wellknownmethodsofmeasuretheoryoninfinitedimensional spaces 9 areusedtostudyphysicalpropertiesofmeasuresrelevanttoquantumfield theory. Thedifferenceoftypicalconfigurationsoffreemassivescalarfield 2 theories with different masses is studied. We apply the same methods to v studytheAshtekar-Lewandowski(AL)measureonspacesofconnections. 9 In particular we prove that the diffeomorphism group acts ergodically, 3 with respect to the AL measure, on the Ashtekar-Isham space of quan- 1 tum connections modulo gauge transformations. We also prove that a 1 typical, with respect to the AL measure, quantum connection restricted 1 toa(piecewise analytic)curveleadstoaparalleltransport discontinuous 7 9 at every point of the curve. / h t - p Published in: J. Math. Phys. 40 (1999) 2337 e h : v i X r a 1 Quantum Field Theory Measures 2 I Introduction Path integrals play an important role in modern quantum field theory. The application in this context of methods of the mathematical theory of measures oninfinite dimensionalspacesis due to constructive quantumfield theorists1-5. With the help of these methods important physicalresults have been obtained, especially concerning two and three dimensional theories. Recently analogous methods have been applied within the frameworkof Ashtekar non-perturbative quantumgravitytogivearigorousmeaningtotheconnectionrepresentation6-10, solvethediffeomorphismconstraint11anddefinetheHamiltonianconstraint12-18. These works all crucially depend on the use of (generalized) Wilson loop variables which have been considered for the first time, within a Hamiltonian for- mulationofgaugetheories,byGambini,Triasandcollaborators19,20,21andwere rediscovered for canonical quantum gravity by Rovelli and Smolin22. In fact, instead of working in the connection representation for which Wilson loops are justconvenientfunctions,onecanuseaso-calledlooprepresentation(seeRef.23 and referencestherein) by means ofwhich a richarsenalof (formal)results was obtained which complement those obtained in the connection representation. For previous works on measures on spaces of connections see e.g. Refs. 24 and 25. Thepresentpaperhastwomaingoals. Thefirstconsistsinstudyingphysical properties of the support of the path integral measure of free massive scalar fields. We use a system with a countable number of simple random variables which probe the typical scalar fields over cubes, with volume Ld+1, placed far away in (Euclidean) space-time. With these probes we are able to study the difference between the supports of two free scalar field theories with different masses. The results that we obtain provide a characterization of the supports which is physically more transparent than those obtained previously26-30. Our second goal consists in providing the proof of analogous results for the Ashtekar-Lewandowski(AL)measureonthespace / ofquantumconnections A G modulo gauge transformations6,7. First we prove that the group of diffeomor- phisms acts ergodically, with respect to the AL measure, on / . Second we A G show that the AL measure is supported on connections which, restricted to a curve, lead to parallel transports discontinuous at every point of the curve. Quantum scalar field theories have been intensively studied both from the mathematical and the physical point of view. Divergences in Schwinger functions (which, in measure theoretical terminology, are the moments of the measure)aredirectlyrelatedwiththefactthattherelevantmeasuresaresupported notonthespaceofnicesmoothscalarfieldconfigurations,thatenterthe classi- calaction,but ratheronspacesofdistributions. This givesa strongmotivation formore detailedstudies ofthe supportofrelevantmeasures. In physicalterms this corresponds to finding “typical (quantum) scalar field configurations”, the set of which has measure one. Thepresentpaperisorganizedasfollows. InsectionIIwerecallsomeresults Quantum Field Theory Measures 3 from the theory of measures on infinite dimensional spaces. Namely Bochner- Minlostheoremsarestatedandtheconceptofergodicityof(semi-)groupactions is introduced. In sectionIII we study propertiesofsimple measures: the count- ableproductofidenticalone-dimensionalGaussianmeasuresandthewhitenoise measure. In the first example we illustrate the disjointness of the support of differentmeasures,whichareinvariantunder a fixedergodicactionofthe same group. For the white noise measure we choose random variables which probe the support and which will be also used in section IV to study the support of massivescalarfieldtheories. InsectionVweobtainpropertiesofthesupportof the AL measure that complement previously obtained results8 and prove that the diffeomorphism group acts ergodically on the space of connections modulo gauge transformations. In section VI we present our conclusions. II Review of Results From Measure Theory II.1 Bochner-Minlos Theorems Inthecharacterizationoftypicalconfigurationsofmeasuresonfunctionalspaces the so called Bochner-Minlos theorems play a very important role. These theorems areinfinite dimensionalgeneralizationsofthe Bochnertheoremfor proba- bilitymeasuresonRN. Letus,fortheconvenienceofthereader,recallthelatter result. Considerany(Borel)probabilitymeasureµonRN,i.e. afinitemeasure, normalized so that µ(RN) = 1. The generating functional χ of this measure µ is its Fourier transform, given by the following function on RN (= (RN), the ∼ ′ prime denotes the topological dual, see below) χ (λ)= dµ(x)ei(λ,x) , (1) µ RN Z where (λ,x) = N λjx . Generating functionals of measures satisfy the fol- j=1 j lowing three basic conditions, P (i) Normalization: χ(0)=1; (ii) Continuity: χ is continuous on RN; (iii) Positivity: m c c χ(λ λ ) 0, for all m N, c ,...,c C and k,l=1 k l k − l ≥ ∈ 1 m ∈ λ ,...,λ RN. 1 m ∈P Thelastconditioncomesfromthefactthat||f||µ ≥0,forf(x)= mk ckei(λk,x), where denotes the L2(RN,dµ) norm. The finite dimensional Bochner ||· ||µ P theoremstatesthattheconverseisalsotrue. Namely,foranyfunctionχonRN satisfying (i), (ii) and (iii) there exists a unique probability measure on RN such that χ is its generating functional. Quantum Field Theory Measures 4 Both in statistical mechanics and in quantum field theory one is interested in the so-called correlators,or in probabilistic terminology, the moments of the measure µ, <(xi1)p1... (xik)pk >:= RN dµ(x)(xi1)p1... (xik)pk . (2) Z For the correlatorsofany orderto existthe measureµ musthavea rapiddecay at x infinity, in order to compensate the polynomial growth in (2) (examples − areGaussianmeasuresandmeasureswithcompactsupport). Intheλ-spacethe latter condition turns out to be equivalent to χ being infinitely differentiable (C ). The correlators are then just equal to partial derivatives of χ at the ∞ origin, multiplied by an appropriate power of i. − Let us now turn to the infinite dimensional case. The role of the space of λ’s will be played by (Rd+1), the Schwarz space of C -functions on (Eu- ∞ S clideanized) space-time with fast decay at infinity. So we have the indices λ(i) := λi replaced by f(x). The space (Rd+1) has a standard (nuclear) S topology. Its elements are functions with regularity properties both for small andforlargedistances. Thephysicallyinterestingmeasureswill“live”onspaces dualto (Rd+1). Considerthespace ′(Rd+1)ofallcontinuouslinearfunction- alson S(Rd+1)(i.e. thetopologicalduSalof (Rd+1)). Thisisthesocalledspace S S of tempered distributions, which includes delta functions and their derivatives, as well as functions which grow polynomially at infinity. We will consider also the even bigger space a(Rd+1) of all linear (not necessarily continuous) functionals on (Rd+1). TShen the simplest generalization of the Bochner theorem states thatSa function χ(f) on (Rd+1) satisfies the following conditions, S (i’) Normalization: χ(0)=1; (ii’) Continuity: χiscontinuousonanyfinitedimensionalsubspaceof (Rd+1); S (iii’) Positivity: m c c χ(f f ) 0, for all m N, c ,...,c C and k,l=1 k l k − l ≥ ∈ 1 m ∈ f ,...,f (Rd+1), 1 m ∈PS ifandonlyifitistheFouriertransformofaprobabilitymeasureµon a(Rd+1), S i.e. χ(f)= dµ(φ)eiφ(f) . (3) ZSa(Rd+1) The topology of convergence on finite dimensional subspaces of (Rd+1) is S unnaturallystrong. Demandingin(ii’)continuityofχwithrespecttothemuch weaker standard nuclear topology on (Rd+1) yields a measure supported on the topological dual ′(Rd+1) of (RSd+1)31. This is the first version of the S S Bochner-Minlostheorem. Further refinementcanbe achievedif χ iscontinuous withrespecttoaevenweakertopologyinducedbyaninnerproduct. Wepresent a special version of this result, suitable for the purposes of the present work; Quantum Field Theory Measures 5 for different, more general formulations see Refs. 3 and 32. Let P be a linear continuousoperatorfrom (Rd+1)onto (Rd+1),withcontinuousinverse. Sup- pose further that P is posSitive when viewSed as an operator on L2(Rd+1,dd+1x) and that the bilinear form <f ,f > :=(P1/2f ,P1/2f ) , f ,f (Rd+1) (4) 1 2 P1/2 1 2 1 2 ∈S defines an inner product on (Rd+1), where ( , ) denotes the L2(Rd+1,dd+1x) S inner product. Let χ satisfy (i’), (iii’) and be continuous with respect to the norm associated with the inner product < , > . Natural examples are pro- P1/2 vided by Gaussian measures µ with covariance C and Fourier transform C χC(f)=e−12(f,Cf) , (5) inwhichcaseonecantakethepositiveoperatorP tobe thecovarianceC itself. Aparticularcaseis the pathintegralmeasurefor freemassivescalarfields with mass m, which is the Gaussian measure with covariance Cm =( ∆+m2)−1 , (6) − where ∆ denotes the Laplacian on Rd+1. In the general (not necessarily Gaus- sian) case let ( ) denote the completion of (Rd+1) with respect to the inner prHoPdu1/c2t <HP,−>1/2 (< , > ). Then theSmeasure on ′(Rd+1) corresponding to χ is actualPly1/2supportePd−o1n/2a proper subset of ′(Rd+S1) given S byanextensionof definedbyaHilbert-Schmidtoperatoron . We HP−1/2 HP1/2 seethatinthescalarfieldcase isthespaceoffiniteactionconfigurations HCm−1/2 and thereforetypical quantum configurationslive in a bigger space. In order to define the extension mentioned above, recall that an operator H on a Hilbert space is said to be Hilbert-Schmidt if given an (arbitrary) orthonormal basis e one has k { } ∞ <He ,He > < . k k ∞ k=1 X Given such a Hilbert-Schmidt operator H on , which we require to be invertible, self-adjoint and such that H( (Rd+H1)P)1/2 (Rd+1), define the new inner product < , > on (Rd+1) bSy ⊂ S P−1/2H S <f ,f > :=(P 1/2Hf ,P 1/2Hf ) , f ,f (Rd+1). (7) 1 2 P−1/2H − 1 − 2 1 2 ∈S Consider , the completion of (Rd+1) with respect to the inner product < , >HP−1/2H, and identify its elemSents with linear functionals on (Rd+1) through thPe−1L/22H(Rd+1,dd+1x) inner product. Under the above conditiSons, the (second version of the) Bochner-Minlos Theorem states that Quantum Field Theory Measures 6 (Bochner-Minlos) A generating functional χ, continuous with respect to the inner product < , > , is the Fourier transform of a unique P1/2 measuresupportedon ,foreveryHilbert-Schimdt operatorH on such that (Rd+H1P)−1/2RHanH, H 1( (Rd+1)) is dense and H 1 : H(PR1d/2+1) S is co⊂ntinuous. − S − S →H< , >P1/2 InsectionIII.1wewillalsouseanobviousadaptationofthisresulttothespace of infinite sequences RN. A common feature of the two versions of the Bochner-Minlos theorem is that they give the support as a linear subspace of the original measure space. Nonlinear properties of the support have to be obtained in a different way. In particular one can show (see section III.2) that the white noise measures with χσ1(f) := e−σ21(f,f) (8) and χσ2(f) := e−σ22(f,f) (9) havedisjointsupportsforσ ,σ >0andσ =σ ,whileBochner-Minlostheorem 1 2 1 2 6 would give the same results in both cases. II.2 Ergodic Actions We review here some concepts and results from ergodic theory32,33. Let ϕ denote an action of the group G on the space M, endowed with a probability measure µ, by measure preserving transformations ϕ : M M , g G, i.e. g → ∈ ϕ µ= µ , g G or, equivalently, for every measurable set A M and for g eve∗ry g G, t∀he∈measure of A equals the measure of the pre-image⊂of A by ϕ . g ∈ The action ϕ is said to be ergodic if all G-invariant sets have either measure zeroorone. Thefactthatϕismeasurepreservingimpliesthatthe(right)linear representation U of G on L2(M,dµ) induced by ϕ (U ψ)(x):=ψ(ϕ x) (10) g g is unitary. The action ϕ on M is ergodic if and only if the only U -invariant G vectors on L2(M,dµ) are the (almost everywhere) constant functions. This followseasilyfromthefactthatthelinearspacespannedbycharacteristicfunc- tions of measurable sets (equal to one on the set and zero outside) is dense in L2(M,dµ). The abovealsoappliesforadiscretesemi-groupgeneratedbyasin- gle (not necessarily invertible) measure preserving transformation T, in which case the role of the group G is played by the additive semi-group N, T being identified with ϕ . Notice that in the latter case the linear representation on 1 L2(M,dµ) may fail to be unitary, although isometry still holds. For actions of R and N, respectively, the following properties are equivalent to ergodicity33 1 τ lim dtψ(ϕ x )= dµ(x)ψ(x) (11a) t 0 τ 2τ →∞ Z−τ ZM Quantum Field Theory Measures 7 N 1 lim ψ(ϕ x )= dµ(x)ψ(x) , (11b) n 0 N→∞N +1n=0 ZM X where ϕ := ϕn and the equalities hold for µ-almost every x and for all ψ n 1 0 ∈ L1(M,dµ). One important consequence of (11) is that if µ and µ are two 1 2 different measures and a given action ϕ is ergodic with respect to both µ and 1 µ then these measures must have disjoint supports (points x for which (11) 2 0 holds). Recall also that the action of R and N, respectively, is called mixing if for every ψ ,ψ L2(M,dµ) we have 1 2 ∈ lim <ψ ,U ψ > = <ψ ,1><1,ψ >, (12a) 1 t 2 1 2 t →∞ lim <ψ ,U ψ > = <ψ ,1><1,ψ > . (12b) 1 n 2 1 2 n →∞ It follows from (12) that every U-invariant L2(M,dµ)-function is constant almost everywhere and therefore every mixing action is ergodic (see Ref. 33 for details). If M is a linear space then (11) gives a non-linear characterization of the support. Indeed if x and x are typical configurations, in the sense that 1 2 (11) holds for them, then x +x and λx for λ=1, are in general not typical 1 2 1 6 configurations. The nonlinearity of supports is best illustrated by the action of N on the space of infinite sequences endowed with a Gaussian measure that we recall in the next subsection. III Support Properties of Simple Measures III.1 Countable Product of Gaussian Measures We will consider here the simplest case of a Gaussian measure in an infinite dimensionalspace. WewillseehoweverthatmanyaspectsofGaussianmeasures on functional spaces can be rephrased in this simple context. Let M =RN, the set of all real sequences (maps from N to R) x= x ,x ,... 1 2 { } andconsideronthisspacethemeasuregivenbytheinfiniteproductofidentical Gaussian measures on R, of mean zero and variance ρ dµρ(x)= ∞ e−x22nρ dxn . (13) √2πρ n=1 Y As we saw above, an equivalent way of defining µ is by giving its Fourier ρ Transform. Let <y,z > :=ρ(y,z) , y,z , (14) √ρ ∈S Quantum Field Theory Measures 8 where (y,z)= ∞n=1ynzn and S is the space of rapidly decreasing sequences P := y RN : ∞ nky2 < , k >0 . S ∈ n ∞ ∀ n nX=1 o Then χρ(y):=e−12<y,y>√ρ = ei(y,x)dµρ(x) , (15) Z where y , x RN. ∈S ∈ Considernowtheergodic(infactmixing)actionϕofNgeneratedbyT ϕ , 1 ≡ (ϕ (x)) :=x , x RN (16) 1 n n+1 ∈ which,aswewillsee,isadiscreteanalogueoftheactionofRbytranslationson thequantumconfigurationspaceofafreescalarfieldtheory. Thetransformation ϕ is clearly measurable and measure preserving, since all the measures in the 1 product (13) are equal. It is also not difficult to see that ϕ is mixing. In fact, invoking linearity and continuity, one only needs to show that (12b) is verifiedforasetofL2-functions whosespanis dense. The functions ofthe form exp(i(y,x)), y form such a set, and one has for them ∈S lim <ei(y,x),Unei(z,x) > = e−21(<z,z>√ρ+<y,y>√ρ) n →∞ = <ei(y,x),1><1,ei(z,x) > , y,z , ∀ ∈S where, in the firstandlast terms, < , > denotes the L2(RN,dµ ) inner product ρ and U denotes the isometric representation associated with ϕ. Thus, ϕ is an ergodic action with respect to µ , for any ρ. Of course, ρ=ρ implies µ =µ ρ ′ ρ ρ 6 6 ′ andone concludesthatµ andµ mustbe mutually singular(i.e. havedisjoint ρ ρ ′ supports). Infact,takingin(11b) ψ(x)=exp(i(y,x))forbothµ andµ leads ρ ρ ′ to a contradiction, unless a x satisfying (11b) for both µ and µ cannot be 0 ρ ρ ′ found. The aim now is to find properties of typical configurations which allow to distinguish the supports of µ and µ . Unfortunatelly the Bochner-Minlos ρ ρ′ theoremcannothelp,duetothefactthattheinnerproducts(14),thatdefinethe measures µ and µ via (15), are proportional to each other and therefore the ρ ρ ′ correspondingextensions in the Bochner-Minlostheorem are equal: = Hρ−1/2H (= ) for any ρ,ρ. Hρ′−1/2H HH ′ Letusnowfindabettercharacterizationofthesupport,forwhichthemutual singularity of µ and µ becomes explicit. In order to achieve this we use a ρ ρ ′ slightmodificationofanargumentgiveninRef.1,thatprovidesconvenientsets, both of measure zero and one. Proposition 1 Given a sequence ∆ , ∆ >1 the µ -measure of the set j j ρ { } Z ( ∆ ):= x : N N s.t. x < 2ρln∆ , for n N (17) ρ j x n n x { } { ∃ ∈ | | ≥ } is one (zero) if 1/ ∆ ln∆ converges (divperges). j j P (cid:0) p (cid:1) Quantum Field Theory Measures 9 This can be proven as follows. For fixed integer N and positive sequence Λ j { } define sets Z ( Λ ) by N j { } Z ( Λ ):= x : x <Λ , for n N (18) N j n n { } { | | ≥ } The µ -measure of each of these sets is ρ ∞ Λn µ (Z ( Λ )) = Erf , (19) ρ N j { } √2ρ n=N (cid:18) (cid:19) Y where Erf(x) = 1/√π x e ξ2dξ is the error function. The sequence of sets x − Z ( Λ )isanincreasin−gsequenceforfixed Λ ,andthesetZ ( ∆ )defined N j j ρ j { } R { } { } in (17) is just their infinite union, for ∆ =exp Λ2/2ρ : j j (cid:0) (cid:1) Z ( ∆ )= Z ( Λ ) . (20) ρ j N j { } { } N N [∈ From σ-additivity one gets µ (Z ( ∆ ))= lim µ (Z ( Λ )) (21) ρ ρ j ρ N j { } N { } →∞ and therefore ∞ µ (Z ( ∆ ))=exp lim ln(Erf( ln∆ )) . (22) ρ ρ j n { } N ! →∞n=N X p Notice that the exponent is the limit of the remainder of order N of a series. Since only divergent sequences ∆ may lead to a non-zero measure, one can j { } use the asymptotic expression for Erf(x), which gives ∞ 1 µ (Z ( ∆ ))=exp lim . (23) ρ ρ j { } −N→∞n=N ∆n√ln∆n! X Depending on the sequence ∆ , only two cases are possible: either the series j { } exists,orit divergesto plus infinity, since ∆ >1, j. Inthe firstcasethe limit j ∀ of the remainder is zero, and so the measure of Z ( ∆ ) will be one. If the ρ j { } sum diverges so does the remainder of any order and therefore the measure of Z ( ∆ ) is zero. 2 ρ j { } Let us now discuss the meaning of this result. To begin with, it is easy to present disjoint sets A, A s.t µ (A)=1, µ (A)=0 and µ (A)=0, µ (A)= ′ ρ ρ ρ ′ ρ ′ ′ ′ 1, for ρ = ρ. Without loss of generality, take ρ = aρ, a > 1. The set Z ( n ) ′ ′ ρ 6 { } has µ -measure zero, since 1/(n√lnn) diverges. But Z ( n ) = Z ( na ) ρ ρ ρ { } ′ { } (see (17)) andZ ( na ) hasµ -measureone,since 1/(na√lnna) converges, ρ′ { } Pρ′ for a>1. On the other hand the sets Z ( n1+ǫ ) are such that ρ { } P µ (Z ( n1+ǫ ))=µ (Z ( n1+ǫ ))=1 , ǫ>0 ρ ρ ρ ρ { } ′ { } ∀ Quantum Field Theory Measures 10 and since Z ( n ) Z ( n1+ǫ ), ǫ>0, the difference sets ρ ρ { } ⊂ { } ∀ Aǫ :=Z ( n1+ǫ ) Z ( n ) (24) ρ ρ { } \ ρ { } are such that µ (Aǫ)=1 and µ (Aǫ)=0, ǫ>0. ρ ρ ρ′ ρ ∀ Notice that the “square-root-of-logarithm” nature of the support Aǫ of µ ρ ρ does not mean that the typical sequence x approaches √2ρlnn, as n . To →∞ clarifythispointletusappealtotheBochner-MinlosTheorem(seesectionII.1). Take tobeℓ2,thecompletionof withrespecttotheinnerproduct(, ): HP1/2 S ℓ2 = y RN : ∞ y2 < . ∈ n ∞ n nX=1 o Consider a vector a = a ,a ,... ℓ2 and a Hilbert-Schmidt operator H 1 2 a { } ∈ defined by (H (x)) :=a x , x ℓ2. (25) a n n n ∈ Then the Bochner-Minlos Theorem leads to the conclusion that a typical sequence x in the support of the measure must satisfy ∞ a2x2 < , (26) n n ∞ n=1 X whichiscertainlynottrueforasequencebehavingasymptoticallylike√2ρlnn, if we choose appropriately a ℓ2. However, the Bochner-Minlos theorem does ∈ notforbidtheappearanceofasubsequencebehavingasymptoticallyevenworse than √2ρlnn. Therefore proposition 1 means that in a typical sequence x n { } no subsequence x can be found such that, n , x > 2(1+ǫ)ρlnn , { nk} ∀ k | nk| k for any arbitrary but fixed ǫ greater than zero, and that one is certainly found p if ǫ is taken to be zero. But this subsequence is rather sparse, as demanded by Bochner-Minlos Theorem; from a stochastic point of view the occurrence of values x greater than √2ρlnn is a rare event. The typical sequence in n | | the support is one that is generated with a probability distribution given by the measure. The measure in this case is just a product of identical Gaussian measuresinR,sothetypicalsequenceisoneobtainedbythrowinga“Gaussian dice” an infinite number of times. Noticethatatypicalµ sequencecanbeobtainedfromaµ typicalsequence ρ ρ ′ simply by multiplying by ρ/ρ. This follows from the fact that the map ′ x ρ/ρ x is a isomorphism of measure spaces (RN,µ ) (RN,µ ). 7→ ′ p ρ → ρ′ p III.2 The White Noise Measure Weconsidernowthesocalled“whitenoise”measure,whichinsomesenseisthe continuous analogue of the previous case4. Again, we will look for convenient