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Nontrivial models of quantum fields with indefinite metric 5 0 0 Sergio Albeverio and Hanno Gottschalk 2 Institut fu¨r angewandte Mathematik n Rheinische Friedrich-Wilhelms-Universität Bonn a D-53115 Bonn, Germany J [email protected]/[email protected] 2 1 February 7, 2008 1 v 2 3 Key Words: Indefinite metric quantum fields, gauge fields, non trivial scat- 0 tering, Wightman and Schwinger functions, random fields, stochastic PDE’s, 1 uncertainty relations. 0 5 0 Introduction / h p - Thenonperturbativeconstructionofquantumfieldmodelswithnontrivialscat- h teringinarbitrarydimensiondofthe underlyingMinkowskispace-timeismuch t a more simple in the framework of quantum field theory with indefinite metric m thaninthepositivemetriccase. Inparticular,thereexistanumberofsolutions : inthephysicaldimensiond=4,whereuptonownopositivemetricsolutionsare v i known. Here we review, why this is so, and we discuss some examples obtained X by analytic continuation from the solutions of Euclidean covariant stochastic r partial differential equations (SPDEs) driven by non-Gaussian white noise. a Main text IthasbeenprovenbyF.Strocchithataquantumgaugefieldinalocal,covariant gauge cannot act on a Hilbert space with a positive definite inner product. But it is possible to overcome this obstacle by passing from a Hilbert space representationofthealgebraofthequantumfieldtoKreinspacerepresentations in order to preserve locality and covariance under the Poincaré group. A Krein K space is an inner product space which also is a Hilbert space with respect to some auxilary scalar product. The relation between the inner producth.,.iandtheauxilaryscalarproduct(.,.)isgivenbyaself-adjointlinear operator J : K → K with J2 = 1 and h.,.i = (.,J.). J is called the metric K 2 S. Albeverio, H. Gottschalk operator. Aquantumfieldactingonsuchaspaceiscalledaquantumfieldwith indefinite metric. The formal definition is as follows: Let D ⊆K be a dense linear space and Ω∈D a distinguished vector hence- forthcalledthevacuum. LetS =S(Rd,CN)bethespaceofSchwartztestfunc- tions with values in CN. A quantum field φ by definition is a linear mapping from S to the linear operators on D. One usually assumes that D is generated asthelinearspanofvectorsgeneratedbyrepeatedapplicationoffieldoperators to the vacuum. The following properties should hold for the quantum field φ: 1. (Temperednes) f →f in S ⇒ hΨ,φ(f )Φi→hΨ,φ(f)Ψi ∀Ψ,Φ∈S; n n 2. (Covariance) ∃ a weakly continuous representation U of the covering of the orthochronous, proper Poincaré group P˜↑ by linear operators on D + which is J-unitary, i.e. U[∗] = U−1 with U[∗] = JU∗J| and leaves Ω D invariant. φ is said to be covariant with respect to U and a representation τ of the covering of the orthochonous, proper Lorentz group L˜↑ if + U(g)φ(f)U(g)−1 = φ(f ), where f (x) = τ(Λ)f(Λ−1(x−a)), g = {Λ,a}, g g Λ∈L˜↑, a∈Rd; + 3. (Spectrality) Let U(a), a ∈ Rd, be the representation of the translation group and let σ = ∪ suppF(hΨ,U(.)Φi) with F the Fourier trans- Ψ,Φ∈D form (in the sense of tempered distributions). Formally, σ is the joint spectrum of the generators of space-time translations U(a). The spectral condition then demands that σ ⊆V¯+ the closed forward lightcone in 0 energy-momentum space; 4. (Locality) There is a decompositionCN =⊕ V suchthat for eachf,h∈ κ κ S taking values in one ofthe V andhaving space-likeseparatedsupports κ we either have [φ(f),φ(h)] = 0 or {φ(f),φ(h)} = 0, where [.,.] is the commutator and {.,.} the anti-commutator; 5. (Hermiticity) There is an involution ∗ on S such that φ(f)[∗] =φ(f∗). The quantum mechanical interpretation of the inner product of two vectors in K as a probability amplitude however gets lost. It has to be restored by the construction of a physical subspace of K where the restriction of the inner product is non-negative. This is called the Gupter-Bleuler gauge procedure. Typically,one considersthe problemofconstructing quantumfields with indefinite metric first. This is being seen as the dynamic side of the problem. The constructionofthe physicalstates,whichcanalsobe seenasimplementationof quantum constraints, is often postponed. The vacuum expectations values (VEVs), also called Wightman functions, of the quantum field theory with indefinite metric (IMQFT) are defined as: W (f ⊗···⊗f )=hΩ,φ(f )···φ(f )Ωi, f ,...,f ∈S. (1) n 1 n 1 n 1 n An axiomatic framework for (unconstrained) IMQFT has been suggested by G. Morchio and F. Strocchi in terms of the Wightman functions W ∈ S′, n Nontrivial models with indefinite metric 3 n ∈ N . Previous work on the topic had been done by J. Yngvason. These 0 generalized Wightman axioms of Morchio and Strocchi replace the positivity condition on the Wightman functions by a so-called Hilbert space structure condition (HSSC): For n∈N ∃p a Hilbert seminorm on S⊗n such that 0 n |W (f ⊗h)|≤p (f)p (h) ∀n,m∈N , f ∈S⊗n, h∈S⊗m. (2) n+m n m 0 This condition makes sure that a field algebra on a Krein space with VEVs equal to the given set of Wightman functions can be constructed. The remain- ing axioms of the Wightman framework – temperedness, covariance, spectral condition,localityandHermiticity –remainthe same. ClusteringofWightman functions is assumed at least for massive theories: lim W (f ⊗h )=W (f)W (h) ∀n,m∈N , f ∈S⊗n, h∈S⊗m, (3) n+m ta n m 0 t→∞ forspace-likea∈Rd. Itfailstoholdincertainphysicalcontextswheremultiple vacua(alsocalledΘ-vacua)accompaniedwithmasslessGoldstoneBosonsoccur due to spontaneous symmetry breaking. IntheoriginalWightmanaxiomsthereareessentiallytwononlinearaxioms: Positivity and clustering. Non-linear here means that checking that condition involvesmorethanoneVEVwithagivennumberoffieldoperators. Thecluster condition can be linearized by an operation on the Wightman functions called ’truncation’. The equations W (f ⊗···⊗f )= WT(f ⊗···⊗f ) (4) n 1 n n j1 jl I∈XP(n) j{1j<1,j.Y2..<.j.l..}<∈jIl recursively define the truncated Wightman functions WT for n∈N. Here P(n) n stands for the set of all partitions of {1,...,n} into disjoint, nonempty sets. Unfortunately,the positivity condition(atleastwhen combinedwith nontrivial scattering) becomes highly non-linear for truncated Wightman functions. This can be seen as one explanation, why it is so difficult to find non trivial (i.e. corresponding to nontrivial interactions) solutions to the Wightman axioms. But it turns out that in contrast to positivity the Hilbert space structure condition is essentially linear for truncated Wightman functions. Theorem 1 If ∃ a Schwartz norm k.k on S such that WT is continuous with n respect to k.k⊗n for n∈N then the associated sequence of Wightman functions {W } fulfills the Hilbert space structure condition (2). n Note that k.k⊗n is well-defined as S is a nuclear space. This theorem makes it much more easy to construct quantum fields with indefinite metric. In particular, all known solutions of the linear program for truncated Wightman func- tionsleadto anabundanceofmathematicalsolutionsto the axiomsofIMQFT, as long as the singularities of truncated Wightman functions in position and energy-momentum space do not become stronger and stronger with growing n. E.g. the perturbative solutions to Wightman functions of A. Ostendorf and O. 4 S. Albeverio, H. Gottschalk Steinmanprovidesolutionswhenthe perturbationseriesistruncatedatagiven order. In the classical work on constructive quantum field theory relativistic fields in space-time dimensions d = 2 and 3 have been constructed by analytic continuation from Euclidean random fields. This in particular has led to firm connections between quantum field theory and equilibrium statistical mechanics. Let us discuss one specific class of solutions of the axioms of IMQFT for d arbitrarywhich alsostem fromrandomfields relatedto anensemble ofstatisti- cal mechanics of classical, continuous particles. On the mathematical side, this is conneted with using random fields with Poisson distribution. As usually in constructiveQFT,themoments,alsocalledSchwingerfunctions,oftherandom fieldcanbeanalyticallycontinuedfromEuclideanimaginarytimetorelativistic real time. That this is possible results from an explicit calculation. Axiomatic results cannot be used as they depend on positivity or reflection positivity in the Euclidean space-time, respectively. By definition, a mixing Euclidean covariant random field ϕ is an almost surely linear mapping from SR = S(Rd,RN) to the space of real valued mea- surable functions (random variables) on some probability space that fulfills the following properties: L 1. (Temperedness) fn →f in SR ⇒ ϕ(fn)→ϕ(f); L 2. (Covariance) ϕ(f) = ϕ(fg) ∀ f ∈ SR, g = {Λ,a}, Λ ∈ SO(d), a ∈ Rd, f (x) = τ(Λ)f(Λ−1(x−a)) for some continuous representation τ : g SO(d)→GL(N). 3. (Mixing) lim E[AB ] = E[A]E[B] for all square integrable random t→∞ ta variables A = A(ϕ),B = B(ϕ) and B = B(ϕ ), ϕ (f) = ϕ(f ) ∀f ∈ ta ta ta ta SR, a∈Rd\{0}. The mixing condition in the Euclidean space-time plays the same rôle as the cluster property in the generalized Wightman axioms. In particular, we consider such random fields ϕ obtained as solutions of the stochastic partial differential equation Dϕ = η. In this equation, η is a noise field, i.e. η is τ-covariant for some representation, η(f) has infinitely divisible probabilitylawandη(f),η(h)areindependent∀f,h∈SR withsuppf∩supph= ∅. D is a τ-covariant (i.e. τ(Λ)Dτ(Λ)−1 = D ∀Λ ∈ SO(d)) partial differential operator with constant coefficients (also pseudo differential operators D could be considered). From the classification of infinitely divisible probability laws it is known that η essentially consists of Gaussian white noise and Poisson fields andderivativesthereof. SuchaGauss-PoissonnoisefieldbytheBochner–Minlos theorem is characterized by its Fourier transform. Direct relations with QFT arise if one chooses E[eiη(f)]=exp ψ(f)−f ·σ¯2p(−∆)fdx , f ∈SR (5) (cid:26)ZRd (cid:27) Nontrivial models with indefinite metric 5 where ψ :RN →C is a Lévy function, t·σ2t ψ(t)=ia·t− +z (eit·s−1)dr(s), t∈RN. (6) 2 ZRN\{0} Here·isaτ-invariantscalarproductonRN,σapositivesemidefiniteτ-invariant N ×N matrix, z ≥0 a real number and r is a τ-invariant probability measure on Rn\{0} with all moments. σ¯2 =(∂2ψ(t)/∂t ∂t )| . p:[0,∞)→[0,∞) α,β α β t=0 is a polynomialdepending on D. If Dˆ−1, the Fourier transformed inverseof D, exists, it can be represented by Q (k) Dˆ−1(k)= E (7) P (|k|2+m2)νl l=1 l Here Q (k) is a complex N × NQmatrix with polynomial entries being τ- E covariant, τ(Λ)Q (Λ−1k)τ(Λ)−1 = Q (k) ∀Λ ∈ SO(d), k ∈ Rd. ν ∈ N and E E l m ∈C\(−∞,0) are parameters with the interpretation of the mass spectrum l (m ,...,m ) and (ν ,...,ν ) the dipole degrees of the related masses. We 1 P 1 P restrict ourselves to the case of positive mass spectrum where m > 0 and in l this case P (t+m2)νl p(t)=p(t,D)= l=1 l , t>0. (8) P m2νl Q l=1 l OnecanshowthatϕobtainedastheunQiquesolutionoftheSPDEDϕ=η isan Euclidean covariant, mixing random field. The Schwinger functions (moments) of ϕ are given by Sn(f1⊗···⊗fn)=E[ϕ(f1)···ϕ(fn)], f1,...fn ∈SR. (9) One then gets that the Schwinger functions can be calculated explicitly. They are determined by the truncated Schwinger functions, cf. (4), as follows: For n=2 QE (−i∇ ) N ST (x ,x )= 2,α1,α2 2 (−∆+m2)−νl (x −x ) (10) 2,α1,α2 1 2 N m2νl " l # 1 2 l=1 l l=1 Y Q and for n≥3 ST (x ,...,x ) = QE (−i∇ ) n,α1···αn 1 n n,α1···αn n n N × (−∆+m2)−νl (x −x)dx (11) l j ZRdj=1"l=1 # Y Y where n ∂ QE (−i∇ )=Cβ1···βn Q (−i ) (12) n,α1···αn n E,βl,αl ∂x l l=1 Y 6 S. Albeverio, H. Gottschalk with ∂nψ(t) C =(−i)n (13) β1···βn ∂t ···∂t β1 βn(cid:12)t=0 (cid:12) and the Einstein conventionof summation and uppe(cid:12)ring/loweringof indices on (cid:12) RN w.r.t. the invariant inner product · is applied. The Schwinger functions fulfill the requirements of τ-covariance, symmetry, clustering and Hermiticity from the Osterwalder-Schraderaxioms of Euclidean QFT. While there isnoknowngeneralreason,whyarelativisticQFTshouldexist for the given set of Schwinger functions, one can take advantage of the explicit formulae (10)–(13) in order to calculate the analytic continuation from Euclidean to relativistic times explicitly. Itsimplifiestheconsiderationstoexcludedipolefields,i.e. oneassumesthat ν = 1 for l = 1,...,n. In physical terms, the no-dipole condition guaran- l teesthattheasymptoticfieldsinMinkowskispace-timefulfilltheKlein-Gordon equation and thus generate particles in the usual sense if applied to the vacuum. If this condition is not imposed, asymptotic fields might only fulfill a dipole equation ((cid:3)+m2)2φin/out = 0 or a related hyperbolic equation of even higher order and the particle states generated by application of such fields to the vacuum require a gauge fixing (constraints) in order to obtain a physical interpretation. Given the no-dipole condition, one obtains by expansion into partial fractions N 1 b l = (14) P (|k|2+m2) (|k|2+m2) l=1 l l=1 l X with b ∈(0,∞) uniquQely determined and b 6=0. For the truncated Schwinger l l functions this implies (n≥3) P ST (x ,...,x ) = QE (−i∇ ) n,α1···αn 1 n n,α1···αn n l1,.X..,ln=1 n n × b (−∆+m2)−1(x−x ) dx. (15) r=1 lrZRdj=1 lj j Y Y At this point, a lengthy calculation yields a representation of the functions n (−∆+m2)−1(x−x ) dx as the Fourier–Laplacetransformof a distri- Rd j=1 j j bution WˆT that fulfills the spectralcondition. This is equivalent to the R Q n,m1,...,mn statement that the analytic continuation of such functions to relativistic times yieldsWT ,wherethelatterdistributionistheinverseFouriertransform n,m1,...,mn of WˆT . This distribution up to a constant that can be integrated into n,m1,...,mn QE is given by n j−1 n n (−1) δ− (k ) δ+ (k ) δ( k ) (16)  ml l k2−m2 ml l  l Xj=1Yl=1 j l=Yj+1  Xl=1   Nontrivial models with indefinite metric 7 Here δ±(k) = θ(±k0)δ(k2 −m2), where θ is the Heaviside step function and m k2 =k02−|~k|2. On the other hand the partial differential operator QE can be n analytically continued in momentum space QM((k0,~k ),...,(k0,~k ))=QE((ik0,~k ),...,(ik0,~k )), (17) n 1 1 n n n 1 1 n n k ,...,k ∈Rd. With the definition 1 n QM (k ,k ) N WˆT (k ,k )=(2π)(d+1) 2,α1α2 1 2 b δ− (k )δ(k +k ) (18) 2,α1α2 1 2 N m2 l ml 1 1 2 l=1 l l=1 X and Q WˆT (k ,...,k ) = QM (k ,...,k ) n,α1···αn 1 n n,α1···αn 1 n N n × b WˆT (k ,...,k ), (19) lj n,ml1,...,mln 1 n l1,.X..,ln=1jY=1 the analytic continuationof Schwinger functions can be summarized as follows: Theorem 2 ThetruncatedSchwingerfunctionsST haveaFourier-Laplacerep- n resentation with WˆT defined in Eqs. (18) and (19). Equivalently, ST is the n n analytic continuation of WT from purely real relativistic time to purely imagi- n nary Euclidean time. The truncated Wightman functions WT fulfill the require- n ments of temperedness, relativistic covariance w.r.t. the representation of the orthochronous, proper Lorentz group τ˜:L↑(d)→Gl(L), locality, spectral prop- + erty and cluster property. Here τ˜ is obtained by analytic continuation of τ to a representation of the proper complex Lorentz group over Cd (which contains SO(d) as a real submanifold) and restriction of this representation to the real orthochronous proper Lorentz group. MakingagainuseoftheexplicitformulainTheorem2,theconditionofTheorem 1canbe verified. This provesthe existenceofIMQFT modelsassociatedto the class of random fields under discussion. Theorem 3 The Wightman functions defined in Theorem 2 fulfill the HSSC. In particular, there exists a QFT with indefinite metric s.t. the Wightman functions are given as the vacuum expectation values of that IMQFT. TheoriesasdescribedinTheorem2obviouslyhavetrivialscatteringbehaviorif the noisefieldη is Gaussian,i.e. ifin(7)z =0. Inthe casewhenthereis alsoa Poissoncomponent in η, i.e. z >0, higher order tuncated Wightman functions do not vanish and such theories have non-trivial scattering. Beforethe scatteringofthe models canbe discussed,somewordsneedto be saidabout scattering in IMQFT in general. The scattering theory in axiomatic QFT, Haag-Ruelle theory, relies on positivity. In fact, one can show that in the classofmodels under discussionthe LSZ asymptoticconditionis violatedif 8 S. Albeverio, H. Gottschalk dipole degrees of freedom are admitted. In that case more complicated asymptotic conditions have to be used. In any case, Haag–Ruelle theory cannot be adapted to IMQFT. Nevertheless, asymptotic fields and states can be constructed in IMQFT if one imposes a no-dipole condition in a mathematically precise way. Then the LSZ asymptotic condition leads to the construction of mixed VEVs of asymptotic in- and out-fields with local fields. The collection of such VEVs is called the form factor functional. After constructing this collection of mixed VEVs, onecantrytocheckthe HSSCforthisfunctionalandtheobtainsaKreinspace representation for the algebra generated by in- local and out-fields. Followingthisline,asymptoticin-andout-particlestatescanbeconstructed for the given mass spectrum (m ,...,m ). If ain/out†(k), l = 1,...,P denotes 1 P α,l the creation operator for an incoming/outgoing particle with mass m , spin l component α and energy-momentum k, the following scattering can be derived for r incoming particles with masses m ,...,m and n−r outgoing particles l1 lr with masses m ,...,m lr+1 ln T ain† (k )···ain† (k )Ω,aout† (k )···aout† (k )Ω α1,l1 1 αr,lr r αr+1,lr+1 r+1 αn,ln n D n E = −(2π)iQM (−k ,...,−k ,k ,...,k ) δ+ (k )δ(Kin−Kout). α1,...αn 1 r r+1 n mlj j j=1 Y (20) Kin/outstandforthetotalenergy-momentumofin-andout-particles,i.e. Kin = r k and Kout = n k . j=1 j j=r+1 j Two immediate consequences can be drawn from (20). Firstly, choosing a P P model with non-vanishing Poissonpart suchthat C 6=0 anda differential β1β2β3 operator D containing in its mass spectrum the masses m and µ with m > 2µ one gets a non-vanishing scattering amplitude for the process µ (cid:0)(cid:18) (cid:0) - (cid:19)(cid:16) m (21) @ (cid:18)(cid:17) @Rµ even though in- and out-particle states consist of particles with well-defined sharpmasses. Thus, for the incoming particle the energy uncertainty which for a particle at rest is proportional to the mass uncertainty vanishes but still the particle undergoes a non-trivial decay and must have a finite decay time. This appears to be a contradiction to the energy-time uncertainty relation, which therefore seems to have a unclear status in IMQFT, (i.e. in QFT including gauge fields). The origin of this inequality, which of course is experimentally very well tested, apparently has to be located in the constraints of the theory and not in the unconstrained IMQFT. Secondly, one can replace somewhat artificially the polynomials QM in (17) n by any other symmetric and relativistically covariant polynomial. If the se- Nontrivial models with indefinite metric 9 quenceofthe”new”QM isofuniformlyboundeddegreeinanyofthearguments n k ,...k ,there-definedWightmanfunctionsin(17)stillfulfilltherequirements 1 n of Theorem 1 and thus define a new relativistic, local IMQFT. The scattering amplitudes of such a theory are again well-defined and given by (20). E.g. in the case of only one scalar particle with mass m one can show that arbitrary LorentzinvariantscatteringbehaviorofBosonicparticlescanbe reproducedby such theories for energies below an arbitrary maximal energy up to arbitrary precision. This kind of interpolation theorem shows that the outcome of an arbitraryscatteringexperimentcanbe reproducedwithintheformalismof(unconstrained)IMQFTaslongasitisinagreementwiththegeneralrequirements of Poincaré invariance and statistics. References [1] S. Albeverio,H. Gottschalk: Scattering theory for quantum fields with indefinite metric, Commun. Math. Phys. 216 (2001) 491-513. [2] S. Albeverio, H. Gottschalk, J.-L. Wu, Models of local relativistic quantum fields with indefinite metric (in all dimensions), Commun. Math. Phys. 184, 509–531,(1997). [3] H.Gottschalk,Onthestabilityofoneparticlestatesgeneratedbyquan- tum fields fulfilling Yang-Feldman equations, Rep. on Math. Phys. 47 No. 2 (2001), 241-246. [4] M. Grothaus, L. Streit, Construction of relativistic quantum fields in the framework of white noise analysis, J. Math. Phys. 40 No.11 5387 (1999). [5] G. Morchio, F. Strocchi, Infrared singularities, vacuum structure and pure phases inlocalquantum fieldtheory,Ann. Inst.H. Poincaré,Vol. 33, p. 251, (1980). [6] O. Steinmann, Perturbative quantum electrodynamics and axiomatic field theory, Springer Berlin/Heidelberg/N.Y., 2000. [7] F. Strocchi, Selected topics on the general properties of quantum field theory, Lecture Notes in Physics 51, Singapore–New York–London– Hong Kong: World Scientific, 1993.