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REFLECTION POSITIVE AFFINE ACTIONS AND STOCHASTIC PROCESSES 6 1 P.E.T.JORGENSEN,K.-H.NEEB∗,ANDG.O´LAFSSON 0 2 Abstract. In this note we continue our investigations of the representation n theoreticaspectsofreflectionpositivity,alsocalledOsterwalder–Schraderpos- a itivity. We explain how this concept relates to affine isometric actions on J real Hilbert spaces and how this is connected with Gaussian processes with 5 stationaryincrements. 2 ] h p - 1. Introduction h t a In this note we continue our investigations of the mathematical foundations of m reflection positivity, also called Osterwalder–Schrader positivity, a basic concept [ in constructive quantum field theory [1, 2, 3, 4, 5]. It arises as a requirement on theeuclideansidetoestablishadualitybetweeneuclideanandrelativisticquantum 1 fieldtheories[6]. Itiscloselyrelatedto“Wickrotations”or“analyticcontinuation” v in the time variable from the real to the imaginary axis. 8 6 The underlying concept is that of a reflection positive Hilbert space, introduced 6 in [7]. This is a triple ( , ,θ), where is a Hilbert space, θ : is a unitary + 6 involution and is aEcloEsed subspaceEof which is θ-positiveEi→n tEhe sense that + 0 E E the hermitian form θu,v is positive semidefinite on .1 We write for the 1. h i E+ E corresponding Hilbert space and q: for the canonical map. 0 E+ →E b To see how this relates to group representations, let us call a triple (G,S,τ) 6 b 1 a symmetric Lie semigroup if G is a Lie group, τ is an involutive automorphism : of G and S G a unital subsemigroup (with dense interior) invariant under the v ⊆ involutions s♯ :=τ(s) 1. TheLiealgebragofGdecomposesintoτ-eigenspaces Xi g=h qand7→weobtainth−eCartandualLiealgebragc =h iq. Gcthenstandsfora ⊕ ⊕ r LiegroupwithLiealgebragc,oftentakensimplyconnectedtomakeitunique. The a prototypicalpair(G,Gc)consistsoftheeuclideanmotiongroupE(d)=Rd⋊O (R) d and the simply connected covering of the orthochronous Poincaré group P(d) = ↑ Rd⋊O (R) . 1,d 1 ↑ − If (G,H,τ) is a symmetric Lie group and ( , ,θ) a reflection positive Hilbert + E E space,then we say that a unitary representationπ: G U( ) is reflection positive → E with respect to (G,S,τ) if π(τ(g))=θπ(g)θ for g G and π(S) . + + ∈ E ⊆E Key words and phrases. reflection positivity, reflection positive function, reflection positive representation,reflectionpositiveaffineactionreflectionnegativefunction,Bernsteinfunction. 1Ascustomaryinphysics,wefollowtheconvenientthattheinnerproductofacomplexHilbert spaceislinearinthesecondargument. 1 2 P.E.T.JORGENSEN,K.-H.NEEB∗,ANDG.O´LAFSSON If(π, )isareflectionpositiverepresentationofGon( , ,θ),thenπ(s)q(v):= + E E E q(π(s)v) defines a representation (π, ) of the involutive semigroup (S,♯) by con- E b tractions (Lemma 1.4 in Ref. [7] or Ref. [4]). However, we would like to have a unitary representation πc of the simbpbly connected Lie group Gc with Lie algebra gc on whose derived representation is compatible with the representation of S. E If such a representation exists, then we call (π, ) a euclidean realization of the b E representation(πc, ) of Gc. Sufficient conditions for the existence of πc have been E developed in [8] (see also [3, 9]). b The main new aspects introduced in this short note is a notion of reflection positivity for affine actions of G on a real Hilbert space. Here is generated + E by the S-orbit of the origin. On the level of positive definite functions, this leads to the notion of a reflection negative function. For (G,S,τ) = (R,R+, idR), − reflection negative functions ψ are easily determined because reflection negativity is equivalent to ψ being a Bernstein function. We conclude this note with (0, ) a brief discussion|of∞connections with some stochastic processes on the real line. Proofs, more details, and more complete results will appear in [10]. 2. From kernels to reflection positive representations Since our discussion is based on positive definite kernels, the corresponding Gaussian processes and the associated Hilbert spaces, we first recall the pertinent definitions. Definition 2.1. (a) Let X be a set. A kernel Q: X X C is called hermitian × → if Q(x,y)=Q(y,x). A hermitian kernel Q is called positive, respectively negative, definite, if for x ,...,x X,c ,...,c C, we have n c c Q(x ,x ) 0, 1 n ∈ 1 n ∈ j,k=1 j k j k ≥ respectively if in addition c =0, we have n c cPQ(x ,x ) 0 [11]. j j,k=1 j k j k ≤ (b)IfGisagroup,thenPafunctionϕ: G PCiscalledpositive(negative)definite → ifthe kernel(ϕ(gh 1)) is positive(negative)definite. Moregenerally,if(S, ) − g,h G is an involutive semigrou∈p, then ϕ: S C is called positive (negative) definite∗if → the kernel (ϕ(st )) is positive (negative) definite. ∗ s,t S ∈ Remark 2.2. Let X be a set, K: X X C be a positive definite kernel and = CX the corresponding r×eprod→ucing kernel Hilbert space. This is K the uEniqueHHil⊆bert subspace of CX on which all point evaluations f f(x) are 7→ continuous and given by f(x) = K ,f for K (y) = K(x,y). Then the map x x h i γ: X ,γ(x) = K has total range and satisfies K(x,y) = γ(x),γ(y) . The K x → H h i latter property determines the pair (γ, ) up to unitary equivalence. K H Example 2.3. (a) Suppose that K: X X C is a positive definite kernel and × → τ: X X is an involution leaving K invariant and that X X is a subset with + → ⊆ the property that the kernel Kτ(x,y) := K(τx,y) is also positive definite on X . + Then the closed subspace := generated by (K ) is θ-positive for E+ ⊆ E HK x x∈X+ (θf)(x):=f(τx). We thus obtain a reflectionpositive Hilbert space ( , ,θ). We + E E call such kernels K reflection positive with respect to (X,X ,τ). + (b) If ( , ,θ) is a reflection positive Hilbert space, then the scalar product + E E defines a reflection positive kernel K(v,w) := v,w with respect to ( , ,θ). In + h i E E this sense all reflection positive Hilbert spaces can be obtained in the context of (a), which provides a “non-linear” setting for reflection positive Hilbert spaces. REFLECTION POSITIVE AFFINE ACTIONS AND STOCHASTIC PROCESSES 3 ForsymmetricLiesemigroups(G,S,τ), weobtainnaturalexamplesofreflection positive kernels: A function ϕ: G C is called reflection positive if the kernel → K(x,y) := ϕ(xy 1) is reflection positive with respect to (G,S,τ) in the sense of − Example 2.3. These are two simultaneous positivity conditions, namely that the kernel ϕ(gh 1)) is positive definite on G and that the kernel ϕ(st♯)) is − g,h G s,t S ∈ ∈ positive definite on S. Here X =G and X =S. + Prototypical examples are the functions ϕ(t) = e−λ|t|, λ 0, for (R,R+, idR) ≥ − with onedimensional. Forthistripleeverycontinuousreflectionpositivefunction E has anb integral representation ϕ(t) = 0∞e−λ|t|dµ(λ) for a positive measure µ on [0, ) (see Cor. 3.3 in Ref. [7]). In ReRf. [7] we also discuss generalizations of this ∞ conceptto distributionsandobtainintegralrepresentationsforthe casewhere Gis abelian. 3. Second quantization and Gaussian processes Definition3.1. [12]LetT beasetandK=RorC. AK-valuedstochasticprocess (X ) is said to be Gaussian if, for all finite subsets F T, the corresponding t t T distri∈bution of the random vector X =(X ) with valu⊆es in KF is Gaussian. F t t F ∈ Definition 3.2. (Secondquantization)(a)ForarealHilbertspace ,wewrite ∗ H H foritsalgebraicdual,i.e.,thesetofall(notnecessarilycontinuous)linearfunction- als R, continuous or not. Let Γ( ) := L2( ,γ,C), where ( ,Σ,γ) is the ∗ ∗ H → H H H canonical Gaussian measure space on (Ex. 4.3 in Ref. [13]; see also Ref. [14]). ∗ HereΣisthesmallestσ-algebraforwhHichallevaluationsφ(v): R,α α(v), ∗ H → 7→ v , are measurable and the probability measure γ is determined uniquely by ∈H E(eiφ(v))=e v 2/2 for v . (3.1) −k k ∈H Considering the φ(v) as random variables, we thus obtain the canonical Gaussian process (φ(v)) over . It satisfies v ∈H H E(φ(v))=0 and E(φ(v)φ(w)) = v,w for v,w . h i ∈H Remark 3.3. There are many realizations of Gaussian measure spaces over real Hilbert spaces. The one chosen above has the advantage that it directly leads to a natural unitary representationof the motion group Mot( )= ⋊O( ) of by H ∼H H H (ρ(b,g)F)(α)=eiφ(b)(α)F(g∗α) for which the map v eiφ(v) = ρ(v,1)1 is equivariant (cf. Remark 3.7). Its range 7→ is total, and the corresponding positive definite function on Mot( ) is given by e H ϕ(b,g)= 1,ρ(b,g)1 =E(eiφ(b))=e−kbk2/2. (3.2) h i Remark3.4. Supposethat isrealorcomplexHilbertspace,T aset,γ: T a H →H map,and(φ(v)) the canonicalGaussianprocessindexedby (Definition 3.2). v ∈H H Then (φ(γ(t))) is a centered Gaussian process indexed by T with covariance t T ∈ kernel C(s,t)=E φ(γ(s))φ(γ(t)) = γ(s),γ(t) . h i In view of Remark 2.2, a kerne(cid:0)l on T is the co(cid:1)variance kernel of a Gaussian process if and only if it is positive definite. If γ(T) is total in , then the corre- H sponding Gaussian process is full in the sense that, up to sets of measure 0, Σ is the smallest σ-algebra for which every X is measurable. Conversely, for ev- t ery full and centered Gaussian process (X ) with covariance kernel C on the t t T ∈ 4 P.E.T.JORGENSEN,K.-H.NEEB∗,ANDG.O´LAFSSON probability space (Q,Σ,µ), there exists a uniquely determined unitary operator U: Γ( ) L2(Q,Σ,µ) with U(φ(γ(t)))=X for t T (Thm. 1.10 in Ref. [12]). t H → ∈ Definition 3.5. (Normalization) If (X ) is a centered Gaussian process with t t T E(X 2)>0 for every t T and covarianc∈ekernel C, then X :=X / E(X 2) is t t t t | | ∈ | | called the associated normalized process. Its covariance kernel has thepform e C(t,s) C(t,s)= for t,s T. C(t,t)C(s,s) ∈ e p Definition 3.6. Let (X ) be a centered real-valued Gaussian process with co- t t T variance kernel C: T T ∈ R and σ: G T T,(g,t) g.t a group action. × → × → 7→ (a) The process (X ) is called stationary if t t T ∈ C(g.t,g.s)=C(t,s) for g G,t,s T. ∈ ∈ For any realizationγ: T with total range as in Remark 3.4, we thus obtain a →H uniquely determined orthogonal representation U: G O( ) satisfying Uγ(t) = → H γ(g.t) for g G,t T. ∈ ∈ (b) The process (X ) is said to have stationary increments if the kernel t t T ∈ D(t,s):=E (X X )2 =C(t,t)+C(s,s) 2C(t,s), (3.3) t s − − (cid:0) (cid:1) is G-invariant. For any realization γ: T with total range as in Remark 3.4, → H we thus obtain a unique affine action α: G Mot( ) satisfying α γ(t) = γ(g.t) g → H for g G,t T [15, 16]. ∈ ∈ Remark 3.7. (a) The canonicalGaussian process (φ(v)) over the real Hilbert v ∈H space isstationaryfortheorthogonalgroupO( )andhasstationaryincrements H H for the motion group Mot( ) because the kernel H D(v,w)=E((φ(v) φ(w))2)= v 2+ w 2 2 v,w = v w 2 − k k k k − h i k − k is invariant under all isometries. (b) We also observe that K(v,w):=E(eiφ(v)eiφ(w))=E(eiφ(w−v))=e−kv−2wk2. The functions (eiφ(v)) form a total subset of Γ( ) and the map η: v Γ( ),η(v) = eiφ(v) is M∈Hot( )-equivariant with respeHct to the representatiHon→ρ. H H This reflects the Mot( )-invariance of the kernel K. H (b) The randomfield(eiφ(v)) arisesby normalizationofthe processgivenby v ∈H Γ(v):=ekvk2/2eiφ(v) with E(Γ(v)Γ(w))=ekv2k2+kw2k2e−kv−2wk2 =ehv,wi. 4. Reflection positive affine actions Let (G,S,τ) be a symmetric Lie group and be a real Hilbert space, endowed E with an isometric involution θ. We consider an affine isometric action α v =U v+β for g G,v , g g g ∈ ∈E where U: G O( ) is an orthogonal representation and β: G a 1-cocycle, → E → E i.e., β = β +U β for g,h G. We further assume that θα θ = α , which is gh g g h g τ(g) ∈ equivalent to θU θ =U and θβ =β for g G. g τ(g) g τ(g) ∈ REFLECTION POSITIVE AFFINE ACTIONS AND STOCHASTIC PROCESSES 5 Then the positive definite kernel C(s,t) := β ,β and the negative definite func- s t h i tion ψ(g):= β 2 on G are related by g k k 1 C(s,t)= ψ(s)+ψ(t) ψ(s−1t) , ψ(s−1t)=C(s,s)+C(t,t) 2C(t,s)= βs βt 2 2 − − k − k (cid:0) (cid:1) (cf. (3.3)). In particular, the affine action α can be recovered completely from the function ψ (cf. Definition 3.6, [15, 16]). We also note that θβ =β implies that g τ(g) ψ τ = ψ. The action of G on preserves the positive definite θ-invariant kernel Q(◦x,y):=e x y 2/2 (Remark 3E.7). −k − k Definition 4.1. (Reflection positive affine actions) We now consider the closed subspace generated by (β ) and observe that it is invariant under the affine + s s S E ∈ action of S on because α β =β for s,t S. We call the affine action (α, ) is s t st E ∈ E reflection positive if is θ-positive. + E The θ-positivity of is equivalent to the positive definiteness of the kernel + E 1 Cτ(s,t)=C(s,τ(t))=C(τ(s),t)= ψ(s)+ψ(t) ψ(s♯t) 2 − (cid:0) (cid:1) on S. This in turn is equivalent to the negative definiteness of the function ψ S | (Definition2.1). Thene ψ(s)/2 ispositivedefinite onS andtheGelfand–Naimark– − Segalconstructionleadstothecorrespondingcontractionrepresentation(Lem.1.4, Prop. 1.11(ii) in Ref. [7]). This leads us to the following concept: Definition 4.2. We calla continuousfunction ψ: G R reflection negative (with → respect to (G,S,τ)) if ψ is a negative definite function on G and ψ is a negative S | definite function on the involutive semigroup (S,♯) (Definition 2.1). Remark 4.3. According to Schoenberg’s Theorem for kernels (Thm. 3.2.2 in Ref. [11]), a function ψ: G C is reflection negative if and only if, for every → λ>0, the function e λψ is reflection positive[7]. − From now on we focus on some specific aspects of the special case (G,S,τ) = (R,R , id). Recall that a smooth function ψ: (0, ) R is called a Bernstein + function−if ψ 0 and ( 1)k 1ψ(k) 0 for every k∞ N→. Combining Cor. 3.3 in − ≥ − ≥ ∈ Ref. [7] with Bernstein’s Theorem (Thm. 3.2 in Ref. [17]) we obtain: Theorem 4.4. A symmetric continuous function ψ: R R is reflection negative → if and only if ψ is a Bernstein function. In particular, this is equivalent to the (0, ) | ∞ existence of a,b 0 and a positive measure µ on (0, ) with ∞(1 λ)dµ(λ)< ≥ ∞ 0 ∧ ∞ such that R ψ(t)=a+bt + ∞(1 e λt)dµ(λ) − | | | | Z − 0 (Lévy–Khintchine representation). Here a,b and µ are uniquely determined by ψ. Example 4.5. (Cor. 3.2.10 in Ref. [11]) For α 0, the function ψ(t) := tα is reflection negative on R if and only if 0 α 1.≥For 0<α<1, this follows|f|rom ≤ ≤ the integral representation tα = Γ(1αα) 0∞(1−e−λt)λ−1−αdλ for t>0. − R 5. Relations to stochastic processes In this last section, we discuss the abstract concepts from above in the context of natural Gaussian processes on the real line, such as Brownian motion and its relatives. 6 P.E.T.JORGENSEN,K.-H.NEEB∗,ANDG.O´LAFSSON Example 5.1. Two sided Brownian motion is a real-valued centered Gaussian process (Bt)t R with covariance kernel ∈ 1 C(s,t)=E(B B )= (s + t s t) for s,t R s t 2 | | | |−| − | ∈ and D(s,t)=E((B B )2)=C(s,s)+C(t,t) 2C(s,t)= s t. s t − − | − | In particular, it has stationary increments. A natural realization in Γ(L2(R)) is given by B = φ(b ) for b := sgn(t)χ . Then the corresponding affine iso- t t t [t 0,t 0] metric action of the additive group R on∧L2∨(R) is given by α f =S f +b , where (S f)(x)=f(x t). t t t t − Thisactionisreflectionpositivebecausethefunctionψ(t):= b 2 = t isreflection t k k | | negative for (R,R+, idR). Since C(s, t) = 0 for t,s 0, we get = 0 if =L2(R) and =−L2(R ). − ≥ E { } E More generalEly+, for a re+flection positive affine action (α, ) of R onebcan show E that is trivial if and only if β ,β =0 for ts< 0 (see Ref. [10]). This property s t E h i characterizesBrownianmotion(uptopositivemultiples)amongGaussianprocesses b with stationary increments. Example 5.2. For 0<H <1, fractionalBrownianmotion is a centeredGaussian process (Xt)t R with stationary increments, covariance kernel ∈ 1 C(t,s)= (t2H+s2H t s2H), D(t,s)=C(t,t)+C(s,s) 2C(t,s)= t s2H. 2 | | | | −| − | − | − | Therefore it corresponds to the negative definite function ψ(t) := t2H, which is | | reflection negative if and only if 0 H 1 (Example 4.5). Note that H = 1/2 ≤ ≤ 2 corresponds to Brownian motion. Example 5.3. One sided Brownian motion (B ) has the covariance kernel t t>0 C(s,t) = s t and the corresponding normalized process on (0, ) has the ker- ∧ ∞ nel C(s,t)= √s∧stt =qss∧∨tt. This kernelis invariantunder the dilation groupR×+, so thatethe process (B ) is stationary with respect to dilations. t t>0 Accordingly, the natural realization by B = φ(b ) Γ(L2(R )) with b = χ e t t ∈ + t [0,t] and its normalization b = t 1/2b are compatible with the unitary representation t − t of the dilation group oen L2(R+) by (τtf)(x) = et/2f(etx) in the sense that be−t = τ χ =τ b . t [0,1] t 1 e The invareiance of the kernel C under the involutionτ(t)=t−1 of R× implies on L2(R ) the existence of a unique unitary involution θ with θ(b )=b . On (0,1), + e t 1/t the reflected kernel C(s,t 1)= s t−1 = s =√st is posietive deefinite, so that we obtain with =eL2(R−+) andq+s∧∨t=−1L2([p0,t1−])1 a reflection positive Hilbert space on which the dilEation representatEion of R is reflection positive. The corresponding reflection positive function on R is ϕ(t)=hχ[0,1],τtχ[0,1]i=hb1,be−ti=C(1,e−t)=e−|t|/2 e e e and is one dimensional. E b REFLECTION POSITIVE AFFINE ACTIONS AND STOCHASTIC PROCESSES 7 Acknowledgments The research of P. Jorgensen was partially supported by the Binational Science Foundation Grant number 2010117. K. H. Neeb was supported by DFG-grant NE413/7-2,Schwerpunktprogramm“Darstellungstheorie”andGesturO´lafssonby NSF grant DMS-1101337. References [1] J. Glimm and A. Jaffe, Quantum Physics–A Functional Integral Point of View (Springer- Verlag,NewYork,1981) [2] A.Klein,Gaussian OS-positiveprocesses, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40:2,115(1977) [3] A.KleinandL. Landau, Fromthe Euclidean groupto the Poincarégroup viaOsterwalder- Schraderpositivity,Comm. Math. Phys.87,469(1982/83) [4] P.E.T.Jorgensen, andG.O´lafsson,Unitaryrepresentations andOsterwalder-Schraderdu- ality, inThe Mathematical Legacy of Harish–Chandra, R. S. Doranand V. 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Math.deFrance,1989) [17] R.Schilling,R.SongandZ.Vondracek,BernsteinFunctions,StudiesinMath.(deGruyter, 2010) Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA, E- mail: [email protected] DepartmentMathematik,FAUErlangen-Nu¨rnberg,Cauerstrasse11,91058-Erlangen, Germany,∗E-mail: [email protected] Department of mathematics, Louisiana State University, Baton Rouge, LA 70803, USA,E-mail: [email protected]