{'fraction_non_alphanumeric': 0.07632695326853058, 'fraction_numerical': 0.014229882753782945, 'mean_word_length': 3.395463221259727, 'pattern_counts': {'":': 0, '<': 34, '<?xml version=': 0, '>': 31, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Thermal Quantum Fields without Cut-offs in 1+1 Space-time Dimensions

We construct interacting quantum fields in 1+1 dimensional Minkowski space, representing neutral scalar bosons at positive temperature. Our work is based on prior work by Klein and Landau and Hoegh-Krohn


Introduction
Constructive thermal field theory allows one to circumvent (at least in 1+1 space-time dimensions) the severe infrared problems (see e.g. [St]) of thermal perturbation theory. A class of models representing scalar neutral bosons with polynomial interactions was constructed by Høegh-Krohn [H-K] more than twenty years ago. Shortly afterwards, several related results on the construction of self-interacting thermal fields were announced by Fröhlich [Fr2].
Our first paper was devoted to the construction of neutral and charged thermal fields with spatially cutoff interactions in 1+1 space-time dimensions, using the notion of stochastically positive KMS systems due to Klein and Landau [KL1].
The construction of interacting thermal quantum fields without cutoffs presented here includes several of the original ideas of Høegh-Krohn [H-K], but instead of starting from the interacting system in a box we start from the Araki-Woods representation for the free thermal system in infinite volume. This 'algebraic' approach eliminates some cumbersome limiting procedures present in Høegh-Krohn's work due to the introduction of boxes. We provide complete proofs for a number of statements which where only touched upon in Høegh-Krohn's work. The list of 'new' contributions contains the Wick (re-)ordering with respect to different covariance functions, the existence of interacting sharp-time fields, the identification of local algebras, the existence and uniqueness of the solution of Høegh-Krohn's time dependent heat-equation, local normality of the interacting KMS state, uniqueness of the weak * accumulation point of the sequence of approximating KMS states, and a number of inequalities that enter into a rigorous construction at several points. Although some of our results were probably already known by the experts (most of our work is based on results by Glimm and Jaffe, Høegh-Krohn, Fröhlich, Klein and Landau, and Simon) more than twenty years ago, we feel that it is worth while to present the arguments in full detail.
We will provide a detailed description of the content of this paper in the next subsection. But before we do so, we give a rough outline of the main ideas.
A convenient realization of the GNS representation associated to the pair W(h), ω • β is the Araki-Woods representation defined by: , Here h is the conjugate Hilbert space to h, W F (.) denotes the Fock Weyl operator on Γ(h⊕ h) and Ω ∈ Γ(h⊕h) is the Fock vacuum. The von Neumann algebra generated by {π AW (W (h)) | h ∈ h} is denoted by R AW . The local von Neumann algebra generated by {π AW (W (h)) | h ∈ h I } is denoted by R AW (I). Here I ⊂ IR is an open and bounded interval and h I will be defined in (6.4).
The generator L AW of the free time evolution is called the (free) Liouvillean. Euclidean techniques were used in our first paper to define the operator sum : P (φ(0, x)) : C0 dx and to show that H l is essentially selfadjoint. Using Trotter's product formula as in [GJ2], a finite propagation speed argument shows that τ l t (A) = e iH l t Ae −iH l t is independent of l for t ∈ IR and A ∈ R AW (I) fixed, if I is bounded and l is sufficiently large. Thus there exists a limiting dynamics τ such that lim l→∞ τ l t (A) − τ t (A) = 0 (1.1) for all A ∈ R AW (I), I bounded. This norm convergence extends to the norm closure A := I⊂IR R AW (I) ( * ) of the local von Neumann algebras. The C * -algebra A is called the algebra of local observables.
It follows from general results of [KL1] that the vector Ω l ∈ H AW , induces a (τ l , β)-KMS state ω l for the W * -dynamical system (A, τ l ). Equation (1.2) should be compared with similar expressions which are well-known (see e.g. [BR,Theorem 5.4.4]) for bounded perturbations and which have recently been derived for a class of unbounded perturbations in [DJP,Theorem 5.6].
The existence of weak limit points (which are states) of the net {ω l } l>0 is a consequence of the Banach-Alaoglu theorem (see [BR,Theorem 2.3.15]).
That fact that all limit states satisfy the KMS condition w.r.t. the pair (A, τ ) follows from (1.1), which itself is a consequence of finite propagation speed.
Since A is the norm closure of the weakly closed local algebras, all limit points are locally normal KMS states w.r.t. the Araki-Woods representation [TW].
To prove that there is only one accumulation point is more delicate. Following Høegh-Krohn [H-K] we will use Nelson symmetry to relate the interacting vacuum theory on the circle to the interacting thermal model on the real line.

Content of this paper
In Section 2 we recall the notions of stochastically positive KMS systems and associated generalized path spaces, due to Klein and Landau [KL1]. The property corresponding to stochastic positivity in the 0-temperature case is called Nelson-Symanzik positivity.
In Subsection 2.1 we recall the characterization of the thermal equilibrium states of a dynamical system (B, τ ) by the KMS condition and the definition of Euclidean Green's functions. The notion of a stochastically positive KMS systems (B, U, τ, ω) rests on the introduction of a distinguished abelian sub-algebra U of the observable algebra B. In our case this algebra will be the algebra generated by the time-zero fields.
Klein and Landau (see [KL1]) have shown that for β > 0 there is a one to one correspondence between stochastically positive β-KMS systems and β-periodic OS-positive path spaces (for β = ∞ the object associated to an OS-positive path space is called a positive semigroup structure, see [K]). The role of OS-positivity is to ensure the positivity of the inner product in the Hilbert space H on which the real time quantum fields act.
The reconstruction theorem provides a concrete realization of the GNS triple (H ω , π ω , Ω ω ) associated to the pair (B, ω). The Liouvillean L implements the time evolution in the GNS representation π ω .
In Subsection 2.3 we recall some results from [KL1] (with some improvements in [GeJ]) concerning perturbations of generalized path spaces obtained from Feynman-Kac-Nelson kernels. The main examples of FKN kernels are those obtained from a selfadjoint operator V on the physical Hilbert space H ω , which is affiliated to U ∼ = L ∞ (K, ν ω ). If e −βV ∈ L 1 (K, ν ω ) and V ∈ L p (K, ν ω ), e − β 2 V ∈ L q (K, ν ω ) for p −1 + q −1 = 1 2 , 2 ≤ p, q ≤ ∞, then the operator sum L + V is essentially selfadjoint on D(L) ∩ D(V ) and the perturbed time-evolution τ V on B is given by The KMS state ω V for the pair (B, τ V ) is the vector state induced by The Liouvillean L V for the perturbed β-KMS system (B V , τ V , ω V ) equals L + V − JV J.
In Section 3 we recall some standard facts about Gaussian measures on distribution spaces and fix some notation. Gaussian measures are reviewed in Subsection 3.2. Sharp-time free fields are introduced in Subsection 3.3. If the space dimension d is 1, then it is possible to define similarly sharp-space free fields. This is done in Subsection 3.4.
In Section 4 we recall two well known path spaces supported by (S ′ IR (S β × IR), dφ C ), where S β is the circle of length β. In Subsection 4.1 we identify the generalized path space on (S ′ IR (S β × IR), dφ C ) corresponding to the free massive scalar field on the circle S β at temperature 0.
In Subsection 4.2 we identify the generalized path space on (S ′ IR (S β × IR), dφ C ) corresponding to the free massive scalar field on the real line IR at temperature β −1 . The physical Hilbert space associated to this path space can be unitarily identified with the Fock space Γ(h ⊕ h). The KMS vector Ω AW is identified with the Fock vacuum vector Ω in Γ(h ⊕ h). The dynamics τ • can be unitarily implemented in π AW : The (free) Liouvillean L AW is identified with dΓ(ǫ ⊕ −ǫ).
In Section 5 we describe perturbations of the two path spaces defined in Subsects. 4.1 and 4.2. The perturbed path spaces are obtained from FKN kernels corresponding to P (φ) 2 interactions.
In Subsection 5.1 we recall some well known facts concerning the Wick ordering of Gaussian random variables. In 1+1 space-time dimensions Wick ordering is sufficient to eliminate the UV divergences of polynomial interactions. As it turns out, the leading order in the UV divergences is independent of the temperature. Thus it is a matter of convenience whether one uses the thermal covariance function C 0 or the vacuum covariance function C vac to define the Wick ordering.
In Subsection 5.2 the P (φ) 2 model on the circle S β at temperature 0 is discussed. It is specified by the formal interaction Here P (λ) is a real valued polynomial, which is bounded from below. The time-evolution x → e ixH ren C is generated by H ren The operator H C is bounded from below and has a unique vacuum state ω C ( . ) = (Ω C , . Ω C ) such that (Ω C , Ω) > 0 and H ren C Ω C = 0. The renormalized energy operator H ren C is called the P (φ) 2 Hamiltonian on the circle S β .
Some bounds are provided in Proposition 5.4, which are used in the sequel to prove the existence of interacting sharp-time fields.
The spatially cutoff P (φ) 2 model on the real line IR at temperature β −1 is discussed in Subsection 5.3. It is specified by the formal interaction Here P (λ) is once again a real valued polynomial, which is bounded from below, and l ∈ IR + is a spatial cutoff parameter. The perturbed KMS state ω l turns out to be normal w.r.t. the Araki-Woods representation π AW . In fact, it is the vector state induced by where H l is the selfadjoint operator H l := L AW + V l . The perturbed time-evolution on B is given by τ l t (B) := e itH l Be −itH l , B ∈ B.
The following consequence of Lemma 5.3 will be important in Section 7: The analog of (1.3) in the zero temperature case is called Nelson symmetry (see e.g. [Si]).
The thermodynamic limit is discussed in Section 6. We prove that the limits lim l→+∞ τ l t (A) =: τ t (A) and lim l→+∞ ω l (A) =: ω β (A) exist for A in the C * -algebra of local observables A and that (A, τ, ω β ) is a β-KMS system, describing the translation invariant P (φ) 2 model at temperature β −1 . In Subsection 6.1 we recall localization properties of the classical solutions of the Klein-Gordon equation.
In Subsection 6.2 we introduce the net of local algebras I → R AW (I) for the free thermal field: for a bounded open interval I ⊂ IR, the symbol R AW (I) denotes the von Neumann algebra generated by {W AW (h) | h ∈ h I }. By a result of Araki [Ar1], the local von Neumann algebras for the free thermal scalar field are regular from the inside and from the outside: Moreover, if I is bounded, then the local algebra R AW (I) is * -isomorphic to the unique hyper-finite factor of type III 1 .
In Subsection 6.3 the existence of the limiting dynamics is discussed. For t ∈ IR fixed, the norm limit lim , where the I's are open and bounded. Finite propagation speed is used to show that τ l t (B), for B ∈ R AW (I) and |t| ≤ T , is independent of l for l > |I| + T . The proof uses Trotter's product formula, which requires that L AW + V l is essentially self-adjoint on D(L AW ) ∩ D(V l ).
In order to apply the results of Section 7 to the C * -algebra A, it is necessary to identify the local von Neumann algebra R AW (I) with the von Neumann algebra obtained by applying the interacting dynamics τ to the local abelian algebra of time-zero fields. This is done in Subsection 6.4: for I ⊂ IR a bounded open interval, we denote by U AW (I) the abelian von Neumann algebra generated by {W AW (h) | h ∈ h I , h real valued}. We denote by B α (I) the von Neumann algebra generated by We set B(I) := α>0 B α (I) and show that B(I) = R AW (I).
Taking the existence of the interacting path space (which we will construct in Section 7) for granted, we show that the net {ω l } l>0 has a unique accumulation point. This is done in Subsection 6.5, using the identification of algebras established in the previous subsection. Thus w − lim l→+∞ ω l =: ω β exists on A.
The state ω β is a (τ, β)-KMS state on A. It follows from a result of Takesaki and Winnink [TW] that ω β is locally normal, i.e., if I is an open and bounded interval, then ω β|RAW (I) is normal w.r.t. the Araki-Woods representation; thus ω β|RAW (I) is also normal with respect to the Fock representation. Moreover, ω β is invariant under spatial translations and satisfies the space-clustering property: Finally, the main results of this paper, namely the explicit construction of the translation invariant P (φ) 2 model at positive temperature is given in Section 7. Following ideas of Høegh-Krohn [H-K], Nelson symmetry is used to establish the existence of the model in the thermodynamic limit.
The first step is to construct the interacting path space supported by S ′ IR (S β × IR) describing the translation invariant P (φ) 2 model at temperature β −1 .
Following Høegh-Krohn [H-K] we consider the operator W [−∞,∞] (f ) solving the timedependent heat equation exists and that the map is the generating functional of a Borel probability measure µ on (Q, Σ). The measure µ is invariant under space translations, time translations and time-reflection. In Subsection 7.2 we prove the existence of interacting sharp-time fields. (Note that the necessary bounds (5.9) depend on the dimension of space-time.) This result allows us to equip the probability space (Q, Σ, µ) with an OS-positive β-periodic path space structure: -U (t) is the group of transformations generated by the time translations T s induced on Q by the map (t, x) → (t + s, x); -R is the transformation generated by the (euclidean) time reflection at t = 0; -Σ 0 is the sub−σ-algebra of Σ generated by the functions {φ(0, h) | h ∈ S IR (IR)}.
In Subsection 7.3 some properties of the associated interacting β-KMS system (B, U,τ ,ω) are discussed. We prove the convergence of sharp-time Schwinger functions and show that Here H ≥ 0 is a selfadjoint operator on a Hilbert space H and R(t), t ∈ IR, is a family of closed operators with D(H γ ) ⊂ D(R(t)) for some 0 ≤ γ < 1. We show that there exists a unique solution U (t, s) such that U (s, s) = 1l and U (t, r)U (r, s) = U (t, s) for s ≤ r ≤ t.
In Subsection A.2 we consider the dissipative case when R(t) is selfadjoint for t ∈ IR. We establish an approximation of U (t, s) by time-ordered products and prove some bounds on U (t, s), which are used in the main text to show the existence of interacting sharp-time fields and the convergence of sharp-time Schwinger functions.
Finally we establish a lemma which is used in the main text to prove spatial clustering for the translation invariant P (φ) 2 model at temperature β −1 . C. Jäkel wants to thank Hanno Gottschalk for discussing Høegh-Krohn's original paper. The second author was supported under the FP5 TMR program of the European Union by the Marie Curie fellowship HPMF-CT-2000-00881 and is currently supported by the IQN network of the DAAD. Both authors benefited from the IHP network HPRN-CT-2002-00277 of the European Union.

Stochastically positive KMS systems and generalized path spaces
In this section we briefly recall the notions of stochastically positive KMS systems and associated generalized path spaces, due to Klein and Landau [KL1]. We will also need the corresponding notions at 0-temperature, which can be found in [K].

Stochastically positive KMS systems
Let B be a C * -algebra and let {τ t } t∈IR be a one parameter group of * -automorphisms of B. We recall that a state ω on B is a (τ, β)-KMS state or (B, τ, ω) is a β−KMS system, if for each pair A, B ∈ B there exists a function F A,B (z) holomorphic in the strip I + β = {z ∈ C | 0 < Imz < β} and continuous on I + β such that For A i ∈ B and t i ∈ IR, 1 ≤ i ≤ n, the Green's functions are defined as follows: It is well known (see [Ar2,Ar3]) that the Green's functions are holomorphic in continuous on I n+ β and bounded there by n 1 A i . Therefore one can define the Euclidean Green's functions: E G(s 1 , . . . , s n ; A 1 , . . . , A n ) := G(is 1 , . . . , is n ; A 1 , . . . , A n ) for s 1 ≤ · · · ≤ s n , s n − s 1 ≤ β.
The following class of β-KMS systems has been introduced by Klein and Landau [KL1].
In applications it is more convenient to use a version of stochastic positivity, which is adapted to von Neumann algebras.
Definition 2.2 Let B ⊂ B(H) be a von Neumann algebra and let U ⊂ B(H) be a weakly closed abelian sub-algebra of B. Assume that the dynamics τ : B → B is given by where L is a selfadjoint operator on H. Moreover, assume that ω is a β-KMS state for the W * -dynamical system (B, τ ). Then the KMS system (B, U, τ, ω) is stochastically positive if (i) the von Neumann algebra generated by t∈IR τ t (U) is equal to B; (ii) the Euclidean Green's functions E G(s 1 , . . . , s n ; A 1 , · · · , A n ) are positive for all A 1 , . . . , A n in U + = {A ∈ U | A ≥ 0}.

Generalized path spaces
Stochastically positive β-KMS systems can be associated to generalized path spaces (see [KL1], [K]). Let us first recall some terminology. If Ξ i , for i in an index set I, is a family of subsets of a set Q, then we denote by i∈I Ξ i the σ-algebra generated by the sets i∈J Ξ i where J runs over all countable subsets of I.
A path space (Q, Σ, Σ 0 , U (t), R, µ) is said to be supported by the probability space (Q, Σ, µ). It follows from (iii) and (iv) that U (t) extends to a strongly continuous group of isometries of L p (Q, Σ, µ) and R extends to an isometry of L p (Q, Σ, µ) for 1 ≤ p < ∞.
For I ⊂ IR, we denote by E I the conditional expectation with respect to the σ-algebra Σ I := t∈I Σ t .
For simplicity of notation we will consider β as a parameter in ]0, +∞], the case β = +∞ corresponding to the 0-temperature case.
It is shown in [KL1] that for β > 0 there is a one to one correspondence between stochastically positive β-KMS systems and β-periodic OS-positive path spaces. For β = ∞ the object associated to an OS-positive path space is called a positive semigroup structure (see [K]).
Let us describe in more details one part of this correspondence, which is an example of a reconstruction theorem. Let (Q, Σ, Σ 0 , U (t), R, µ) be an OS-positive path space, β-periodic if β < ∞. We set H OS := L 2 (Q, Σ [0,β/2] , µ). where 1 ∈ H OS is the constant function equal to 1 on Q.
Finally, setting M t = L 2 (Q, The triple (P (t), D t , β/2) forms a local symmetric semigroup (see [Fr1,KL3]), and there exists a unique selfadjoint operator L on H such that P (s)u = e −sL u for u ∈ D t and 0 ≤ s ≤ t. The selfadjoint operator constructed in this way is said to be associated to the local symmetric semigroup (P (t), D t , β/2).
Next one defines: -B ⊂ B(H) as the von Neumann algebra generated by {e itL Ae −itL | t ∈ IR, A ∈ U}; -τ : t → τ t as the weakly continuous group of * -automorphisms of B, which is given by τ t (B) = e itL Be −itL for t ∈ IR and B ∈ B; -ω as the vector state on B given by ω(B) = (Ω, BΩ) for B ∈ B.

Perturbations of generalized path spaces
We now describe perturbations of generalized path spaces obtained from a Feynman-Kac-Nelson kernel. Unless stated otherwise, we will consider the case β < ∞.
Let (Q, Σ, Σ 0 , U (t), R, µ) be an OS-positive β-periodic path space. Let V be a selfadjoint operator on H, which is affiliated to U. Using the isomorphism between U and L ∞ (Q, Σ 0 , µ) we can view V as a real Σ 0 -measurable function on Q, which we still denote by V .
Assume that V ∈ L 1 (Q, Σ 0 , µ) and exp(−βV ) ∈ L 1 (Q, Σ 0 , µ). Then (see [KL1] or [GeJ,Proposition 6.2]) the function F := exp − β/2 −β/2 U (t)V dt belongs to L 1 (Q, Σ, µ). One can hence define the perturbed measure dµ V := ( Q F dµ) −1 F dµ. The perturbed path space (Q, Σ, Σ 0 , U (t), R, µ V ) is OS-positive and β-periodic (see [KL1]). Hence we can associate to this perturbed path space a stochastically positive β The following concrete realization of the perturbed β-KMS system (B V , U V , τ V , ω V ) has been obtained in [KL1] (with some improvements in [GeJ]): -the physical Hilbert space H V obtained from the reconstruction theorem outlined in the previous subsection is equal to the physical Hilbert space H of the unperturbed β-KMS system (B, U, τ, ω); -the von Neumann algebra B V and the abelian algebra U V are equal to B and U, respectively; -the operator sum L + V is essentially selfadjoint on D(L) ∩ D(V ) and if H V := L + V , then the perturbed time-evolution τ V on B is given by -the distinguished vector Ω of the unperturbed KMS system belongs to D e − β 2 HV and the perturbed KMS state ω V is given by ω The following result is shown in [GeJ,Theorem 6.12]: If e −βV ∈ L 1 (Q, Σ 0 , µ) and then the operator sum H V − JV J is essentially selfadjoint and the Liouvillean L V (for a general definition of Liouvilleans see, e.g., [DJP]) for the perturbed β Here J denotes the modular conjugation associated to the pair (B, Ω).

Perturbed dynamics associated to FKN kernels
Let us describe in more details the construction of H V = L + V given in [KL1] which is based on the Feynman-Kac-Nelson formula. Note that the results of this subsection are also valid in the 0-temperature case β = +∞. Let V be a real Σ 0 -measurable function such that and Setting finally is a well defined linear operator, and that P V (t), D t , inf(T, β)/2 is a local symmetric semigroup on H. Now let H V be the unique selfadjoint operator associated to the local symmetric semigroup (D t , P V (t), inf(T, β)/2). It follows (see [KL1]) that In the sequel we will need the following result.
Note that the selfadjoint operators H n are associated to local symmetric semigroups In order to prove (2.4), we recall the following bound from [KL4, Theorem 6.2 (i)]: Now let W be a real measurable function on Q. Using 1 − e −a = a 1 0 e −θa dθ we find where W − = sup(0, −W ) denotes the negative part of W . In the first line we have used the Cauchy-Schwarz inequality and the fact that U (t) is unitary on L 2 (Q, Σ, µ), in the second line the estimate (2.5).
Before we finish the proof, we extract a Lemma.
Lemma 2.6 Let P n (t), D (n) t , T for n ∈ IN∪{∞} be a family of local symmetric semigroups on a Hilbert space H. Let H n , n ∈ IN ∪ {∞}, denote the associated selfadjoint operators.
Assume that there exists a family Proof. Let us fix 0 < t ≤ T ′ and Ψ ∈ L t . From [KL3, Lemma 1], we know that there exist positive measures {ν n } on IR such that The family {f n } is uniformly bounded on ]0, t[+iIR by (2.8) and converges pointwise to f ∞ on ]0, T [ by (2.7). Applying Lemma B.3 we conclude that f n (z) converges to f ∞ (z) for all z ∈ iIR. This implies that on L t w − lim n→∞ e −iyHn = e −iyH∞ ∀y ∈ IR.
Since by hypothesis 0<t≤T L t is dense in H and for unitary operators weak convergence implies strong convergence, this completes the proof of the lemma ✷.
Proof of Proposition 2.5 (second part). Let us now fix a convenient family of sub- and sup n sup 0≤s≤t (Ψ, P n (s)Ψ) < ∞. Hence hypotheses (2.7) and (2.8) of Lemma 2.6 are satisfied. Thus we can apply Lemma 2.6 and this completes the proof of Proposition 2.5 ✷.

Gaussian measures
In this Section we recall some standard facts about Gaussian measures on distribution spaces.

Distribution spaces
The Fréchet space of Schwartz functions on IR d will be denoted by S(IR d ). For coherence of notation, the Fréchet space D(S β ) of smooth periodic functions on S β will also be denoted by S(S β ).
In addition, we denote by S(S β × IR d ) the Fréchet space of Schwartz functions on S β × IR, i.e., the space of smooth functions on S β × IR d , which are β-periodic in t and such that for The spaces of real elements in these spaces will be denoted by We now fix the notation concerning partial Fourier transforms. We first define the (unitary) partial Fourier transform with respect to t: (The coefficients ν n = 2πn/β, n ∈ IN, are called in physics Matsubara frequencies). Its inverse is The (unitary) partial Fourier transform with respect to x is For later use we fix two approximations of the Dirac δ functions in t and x. We set, for k ∈ IN,

Gaussian measures
We set Let F be a Borel function on IR. Then F (φ(f )) denotes the function We denote by dφ C the Gaussian measure on (Q, Σ) with covariance C defined by , F a Borel function on IR n and n ∈ IN, are dense in L p (Q, Σ, dφ C ) for 1 ≤ p < ∞.

Sharp-time fields
We now recall some standard results about the existence of sharp-time fields. We will make use of the following well known identity (see [KL2]): Using (3.4) we see that We set .

Sharp-space fields
If d = 1, then it is possible to define similarly sharp-space fields. We first recall another well-known identity, which is analogous to (3.4): (3.9) We note that φ(g, x) belongs to 1≤k<∞ L p (Q, Σ, dφ C ). For later use we define the 0temperature covariance on S β : , g 1 , g 2 ∈ S(S β ). (3.12)
We note that f t ∈ S IR (IR) and f x ∈ S IR (S β ).
Lemma 3.1 If f ∈ S IR (S β ×IR), then the following identity holds on 1≤p<∞ L p (Q, Σ, dφ C ): Proof. Let f ∈ S IR (S β × IR) and k ∈ IN. The map where the convolution product * acts only in the space variable It follows from (3.9) and (3.10) that This proves the first identity of the lemma. The second one can be shown by similar arguments ✷.

Path spaces supported by
In this section we recall two well known path spaces supported by (Q, Σ, dφ C ). The first is associated to the free neutral scalar field of mass m on S β at temperature 0; the second is associated to the free neutral scalar field of mass m on IR at temperature β −1 . We recall that (t, x) denotes a point in S β × IR, and refer to t as the (euclidean) time and to x as space variable. The time translation induced on Q by the map (t, x) → (t+s, x) will be denoted by T s : Q → Q and the spatial translations induced on Q by the map (t, x) → (t, x+y) will be denoted by a y : Q → Q.

The free massive euclidean field on the circle at 0-temperature
In this subsection we identify the generalized path space on (Q, Σ, dφ C ) corresponding to the free massive scalar field on the circle S β at temperature 0.
Let Σ C 0 be the sub σ-algebra of Σ generated by the functions {φ(g, 0) | g ∈ S IR (S β )}. We denote by {U C (x)} x∈IR the 1-parameter group generated by the spatial transla- is a strongly continuous unitary group on L 2 (Q, Σ, dφ C ), and hence extends to a group of measure-preserving automorphisms of L ∞ (Q, Σ, dφ C ) which is continuous in measure.
Let r C : Q → Q be the space reflection around x = 0. We denote by R C the measure preserving transformation of (Q, Σ, dφ C ) generated by r C . For It describes the free neutral scalar euclidean field of mass m on the circle S β at temperature 0.
Let us now briefly describe a well-known concrete form of the physical objects associated to this path space by the reconstruction theorem. Let H − 1 2 (S β ) be the Sobolev space of order − 1 2 equipped with its canonical complex structure i and scalar product . The abelian von Neumann algebra U C obtained from the reconstruction theorem can be identified with the von Neumann algebra generated by

4.2
The free massive euclidean field on IR at temperature β −1 We now identify the generalized path space on (S ′ IR (S β × IR), Σ, dφ C ) corresponding to the free massive scalar euclidean field on IR at temperature β −1 .
Let Σ 0 be the sub σ-algebra of Σ generated by the functions Hence it extends to a group of measure-preserving automorphisms of L ∞ (Q, Σ, dφ C ) which is continuous in measure.
Let r be the (euclidean) time reflection around t = 0. We denote by R the measure preserving transformation of (Q, Σ, dφ C ) generated by r.
For h ∈ S IR (IR) we have Again by Lemma 3.1, we see that Σ = t∈S β U (t)Σ 0 . Hence (Q, Σ, Σ 0 , U (t), R, dφ C ) is a generalized path space. Moreover, it is β-periodic and OS-positive (see e.g. [KL2]). It describes the free neutral scalar field of mass m on IR at temperature β −1 .
We now describe a well known concrete form of the β-KMS system associated to the generalized path space (Q, Σ, Σ 0 , U (t), R, dφ C ). Let h := H − 1 2 (IR) be the Sobolev space of order − 1 2 , equipped with its canonical complex structure i and scalar product ( On h we consider the unitary dynamics e −itǫ . On the Weyl algebra W(h) we define a state ω • β and a one-parameter group of automor- where ρ := (e βǫ − 1) −1 , β > 0. It can be easily seen that ω • β is a quasi-free (τ • , β)-KMS state on W(h).
Let us now recall some terminology. If h is a complex vector space, then the conjugate vector space h is the real vector space h equipped with the complex structure −i. We will denote by h ∋ h → h ∈ h the (anti-linear) identity operator. If a ∈ L(h), then we denote by a ∈ L(h) the operator ah := ah. If h is a Hilbert space, then h is equipped with the Hilbert space structure (h 1 , h 2 ) := (h 2 , h 1 ).
We recall a convenient realization of the GNS representation associated to W(h), ω • β , which is called the right Araki-Woods representation. It is specified by setting Here W F (.) denotes the Fock Weyl operator on Γ(h⊕h) and Ω ∈ Γ(h⊕h) is the Fock vacuum.
The abelian von Neumann algebra U AW obtained by the reconstruction theorem can be identified with the abelian von Neumann algebra generated by {W AW (h) | h ∈ H The von Neumann algebra B AW generated by t∈IR τ • t (U AW ) can be identified with the von Neumann algebra R AW generated by {W AW (h) | h ∈ H − 1 2 (IR)}.

Perturbations of path spaces
In this section we describe perturbations of the two path spaces defined in Subsects. 4.1 and 4.2, obtained from FKN kernels corresponding to P (φ) 2 interactions.

Interaction terms
We recall some well known facts concerning the Wick ordering of Gaussian random variables. Let (K, ν) be a probability space and X a real vector space equipped with a positive quadratic form f → c(f, f ) called a covariance. Let f → φ(f ) be a IR-linear map from X into the space of real measurable functions on K.
The Wick ordering : φ(f ) n : c with respect to the covariance c is defined by the following generating series: : e αφ(f ) : c := ∞ n=0 α n n! : where [.] denotes the integer part.

It will be denoted by
It will be denoted by S β g(t) : φ(t, 0) n : C β dt.
We recall that the covariances C, C 0 and C β have been defined in (3.1), (3.7) and (3.12), respectively. In Lemma 5.1 the probability space is (Q, Σ, dφ C ) and the real vector spaces are equal to S IR (S β × IR), S IR (IR) and S IR (S β ), respectively.
Remark 5.2 If P = P (λ) is a polynomial, then the functions : P (φ(t, 0)) : C β dt are well defined, by linearity. It can be easily shown (see [GeJ,Proposition 8.4]) using the socalled Wick reordering identities that there exists a linear invertible map between polynomials Here : : vac denotes Wick ordering with respect to the 0-temperature covariance (h, 1 2ǫ h) L 2 (IR) . Lemma 5.3 Let P be a polynomial, h ∈ L 1 (IR) ∩ L 2 (IR) and g ∈ L 1 (S β ) ∩ L 2 (S β ). Set as functions on Q.
Proof. Let W be a function in L p (Q, Σ, dφ C ) for some 1 ≤ p < ∞. The one parameter groups {U (t)} t∈S β and {U C (x)} x∈IR are strongly continuous groups of isometries of 1≤p<∞ L p (Q, Σ, dφ C ). Therefore the functions IR h(x)U C (x)W dx and S β g(t)U (t)W dt belong to L p (Q, Σ, dφ C ).
Together with Lemma 5.1 this implies that all three functions given in (5.4) belong to L p (Q, Σ, dφ C ). Let us now prove that they are identical. By linearity, we may assume that P (λ) = λ n . Using Lemma 5.1 and the Wick identity (5.2), it follows that the definition given in (3.5) of sharp-time fields implies that Applying Lemma B.1 with E = L p (Q, Σ, dφ C ) we obtain the first identity in (5.4). The second identity follows by the same argument, taking first the limit k ′ → ∞ and using then that lim

5.2
The P (φ) 2 model on the circle S β at temperature 0 Let P (λ) be a real valued polynomial, which is bounded from below. The P (φ) 2 model on the circle S β is specified by the formal interaction term This expression can be given two equivalent meanings: first of all, as recalled in Lemma 5.1, it can be viewed as a Σ C 0 measurable function V C ∈ 1≤p<∞ L p (Q, Σ C 0 , dφ C ). Secondly, V C can be considered as a selfadjoint operator on Γ H − 1 2 (S β ) affiliated to the abelian algebra U C . More precisely, for t ∈ S β and Λ ≫ 1 an UV cutoff parameter, we define an approximation It is then easy to verify, by adapting well-known results for the spatially cutoff P (φ) 2 model on the real line IR at 0-temperature (see [S-H.K] and hence G [a,b] ∈ 1≤p<∞ L p (Q, Σ, dφ C ). From the results recalled in Subsection 2.3, we obtain a selfadjoint operator The Hamiltonian H C is called the P (φ) 2 Hamiltonian on the circle S β .
Proof. The existence and uniqueness of the vacuum state can be shown by following the proofs of the corresponding results for spatially cutoff P (φ) 2 models. For example, one easily obtains (see e.g. [Si,Theorem V.20] or [DG,Theorem 6.4 (ii)]) that (dΓ(b) + 1) ≤ C(H C + c) for c ≫ 1. (5.10) Since dΓ(b) has compact resolvent on Γ H − 1 2 (S β ) , it follows that H C is bounded from below with a compact resolvent and hence has a ground state. The uniqueness of the vacuum (i.e., the ground state of H C ) follows from a Perron-Frobenius argument (see e.g. [Si, Theorem V.17]). Since b ≥ m > 0, we see that it suffices to check (5.7) and (5.8), with H C replaced by the number operator N , which is immediate. To prove (5.9) we use (5.10) and the well known bound (see e.g. [Ge, Appendix]) Without proof we quote the following result (see [HO]).
Theorem 5.5 Let H ren C := H C −E C , where E C := inf(σ(H C )) and let P C denote the generator of the translations along the circle S β . The joint spectrum of H ren C and P C is purely discrete and is contained in the forward light cone.

Consequently the correlation function
allows an analytic continuation to the tube IR 2 + iV + , where V + := {(t, x) | |t| < x; x > 0} denotes the forward light cone (with t and x reversed, due to our conventions).

5.3
The spatially cutoff P (φ) 2 model on IR at temperature β −1 Let P (λ) be a real valued polynomial, which is bounded from below (as in Subsection 5.2), and let l ∈ IR + be a spatial cutoff parameter. The spatially cutoff P (φ) 2 model on IR is specified by the formal interaction term (see (5.3)) Again this formal expression can be given two equivalent meanings: first of all, as recalled in Lemma 5.1, it can be viewed as a Σ 0 -measurable function V l ∈ 1≤p<∞ L p (Q, Σ 0 , dφ C ). Secondly, V l can be considered as a selfadjoint operator on Γ(h ⊕ h) affiliated to the abelian von Neumann algebra U AW . As in Subsection 5.2 we define an approximation h Λ,x ∈ H − 1 2 (IR) of the Dirac delta-function δ(. − x) ∈ H − 1 2 (IR). For x ∈ IR and Λ ≫ 1 we set and introduce cutoff fields φ Λ (0, x) := φ AW (h Λ,x ), where φ AW (h) is the selfadjoint field operator associated to W AW (h), h ∈ h. As before, the limit exists on a dense set of vectors in Γ(h ⊕ h). Since h Λ,x ∈ H − 1 2 IR (IR), one obtains that V l is a selfadjoint operator affiliated to U AW .
Adapting well known arguments (see [GeJ,Section 8.2]) it can be shown that e −T V l ∈ L 1 (Q, Σ 0 , µ) for all T > 0. Consequently, we can associate to V l the FKN kernel and the measure The generalized path space (Q, Σ, Σ 0 , U (t), R, µ l ) is β-periodic and OS-positive. The associated β-KMS system is called the spatially cutoff P (φ) 2 model on IR at temperature β −1 .
Applying the abstract results recalled in Subsection 2.3, we obtain the following facts: -the W * -algebra B V l and the abelian algebra U V l are equal to R AW and U AW , respectively; -the operator sum L AW + V l is essentially selfadjoint on D(L AW ) ∩ D(V l ) and if H l := L AW + V l , then the perturbed time-evolution on B is given by τ l t (B) := e itH l Be −itH l , B ∈ B; -the GNS vector Ω AW ∈ Γ(h ⊕ h) belongs to D e − β 2 H l and the perturbed KMS state ω l is given by ω l (B) = (Ω l , BΩ l ), where Ω l := e − β 2 H l Ω AW −1 e − β 2 H l Ω AW . The following consequence of Lemma 5.3 will be important in Section 7: where G [a,b] was defined in (5.5). The analog identity in the temperature zero case is called Nelson symmetry.

The thermodynamic limit
In this section we prove that the limits lim l→+∞ τ l t (A) =: τ t (A) and lim l→+∞ ω l (A) =: ω β (A) exist for A in the C * -algebra of local observables A and that (A, τ, ω β ) is a β-KMS system, describing the translation invariant P (φ) 2 model at temperature β −1 .

The net of local algebras
We start by recalling a result of Araki [Ar1, Thm. 1] which will be useful later on. Let us recall a standard notation: If H 1 , H 2 are two vector subspaces of a Hilbert space H, then H 1 ∨ H 2 denotes H 1 + H 2 . If R 1 , R 2 are two * -sub-algebras of B(H), then R 1 ∨ R 2 denotes the von Neumann algebra generated by R 1 ∪ R 2 .
Proposition 6.1 Let X be a Hilbert space and let Z be a real vector subspace of X. Let W(Z) ⊂ W(X) denote the C * -algebra generated by {W (x) | x ∈ Z} and let π F : W(X) → B(Γ(X)) be the Fock representation. Then where Z α is a family of real vector subspaces of X and Z ⊥ is the vector space orthogonal to Z for the symplectic form σ(x 1 , x 2 ) = Im(x 1 , x 2 ).
We now define the net of local von Neumann algebras I → R AW (I) describing free thermal scalar bosons. Let I ⊂ IR be a bounded open interval. We denote by R AW (I) the von Neumann algebra generated by Lemma 6.2 (i) The local von Neumann algebras for the free thermal field are regular from the inside and regular from the outside: (ii) The net of local von Neumann algebras for the free thermal field is additive: For each open and bounded interval I, the local observable algebra R AW (I) is * -isomorphic to the unique hyper-finite factor of type III 1 .
Proof. Recalling the definition of the Araki-Woods representation we see that, with the notation introduced above, Clearly J⊃I Z J = J⊂I Z J = Z I , which using (6.5) implies (i). Part (ii) is a direct consequence of (6.5). To prove (iii) we use (6.6) and (6.5) which implies that where Z ⊥ is the orthogonal space to Z in h ⊕ h for the symplectic form σ(f, g) = Im(f, g) on h ⊕ h. We claim that Z I ∩ Z ⊥ I = {0}, (6.7) which will imply that R AW (I) is a factor. To prove our claim we pick h ∈ h I such that (1 + ρ) I . This implies that Im(h, g) = 0 for all g ∈ h I . Hence to prove (6.7) it suffices to check that Letting α → 0 this yields Re(h, ǫh) = (h, ǫh) = 0, since ǫ is selfadjoint. Using that ǫ ≥ m > 0 this implies that h = 0, which proves (6.8) and hence (6.7). Thus R AW (I) is a factor, if I is bounded. Note that (6.8) shows that π F (W(h I )) ′′ is a factor, and it is well known (see e.g. [BD'AF] [L] and lit. cit.) that π F (W(h I )) ′′ is * -isomorphic to the unique hyper-finite factor of type III 1 . Thus Lemma 6.3 below completes the proof of the lemma ✷.
We now recall an easy fact about the restriction of the free KMS state ω • β to the local algebras W(h I ).
Lemma 6.3 Let I ⊂ IR be a bounded open interval. Then the representations π AW and π F of W(h I ) are quasi-equivalent.
This fact implies that if h 1 ⊂ h is a complex vector subspace, then the restriction of ω • β to W(h 1 ) is π F -normal iff Tr(Ee −βǫ E) < ∞, where E is the orthogonal projection onto h 1 . We will apply this remark to h = H − 1 2 (IR), ρ = (e βǫ − 1) −1 and h 1 = Ch I . Let E I denote the orthogonal projection on Ch I . Let χ ∈ C ∞ 0 IR such that χ ≡ 1 near I and x = i∂ k . If h ∈ h I , then Reh = χ(x)Reh and Imh = ǫχ(x)ǫ −1 Imh. Using pseudodifferential calculus, we see that the operators (1 + |x|) N χ(x) and (1 + |x|) N ǫχ(x)ǫ −1 are bounded on H − 1 2 (IR) for all N ∈ IN. This implies that , h ∈ h I . (6.9) Clearly (6.9) extends to Ch I , which implies that (1 + |x|) N E I is bounded for all N ∈ IN. Since e −βǫ (1 + |x|) −N is trace class for N large enough we see that E I e −βǫ E I is trace class. Using the arguments given above we obtain that ω • β restricted to W(Ch I ) (and hence also to W(h I )) is π F -normal.
Finally we have seen in the proof of Lemma 6.2 that π F (W(h I )) ′′ is a factor, hence π F is a factor representation of W(h I ). It is shown in [KR,Prop. 10.3.14] that if R is a C * -algebra and π is a factor representation of R, then π is quasi-equivalent to the GNS representation of any π-normal state ω. Since the restriction of π AW to W(h I ) is the GNS representation for the quasi-free state ω • β , this completes the proof of the lemma ✷.

Existence of the limiting dynamics
The C * -algebra of local observables A is defined as follows: where the union is over all open bounded intervals I ⊂ IR and the symbol I⊂IR R AW (I) denotes the C * -inductive limit (see e.g. [KR,Proposition 11.4.1.]). We denote by {α x } x∈IR the group of space translations on A, defined by where k is the momentum operator acting on h = H − 1 2 (IR).
Theorem 6.4 (Existence of limiting dynamics). Let I ⊂ IR be a bounded open interval. For t ∈ IR fixed, the norm limit lim l→∞ τ l t (B) =: τ t (B) exists for all B ∈ R AW (I). The map τ : t → τ t defines a group of * -automorphisms of A such that τ t • α x = α x • τ t for all t, x ∈ IR. Moreover, Proof. The proof follows the well-known proof in the 0-temperature case, which is based on finite propagation speed (see [GJ2,Theorem 4.1.2]). To prove the existence of the limit and the group property, it suffices to show that τ l t (B), for B ∈ R AW (I) and |t| ≤ T , is independent of l for l > |I| + T .
It follows from (6.2) and Huygens principle that The dynamics τ l t is unitarily implemented by e itH l , where H l = L AW + V l for Trotter's formula yields e itH l = s-lim n→∞ (e itLAW /n e itV l /n ) n and hence l, l] , we see that γ l t = γ l ′ t on R AW (I) for l, l ′ > |I|. Using (6.11) and (6.12), this implies that τ l t = τ l ′ t on R AW (I) for |t| ≤ T and l, l ′ > |I| + T . This proves our claim. The same argument using again (6.11) proves (6.10).
It remains to check that τ and α commute. Let T > 0 and I a bounded interval. For |t| ≤ T the time evolution is locally (i.e., applied to elements in R AW (I)) generated by H l if l > |I| + t. Now α x is implemented by e ixP with P = dΓ(k ⊕ k). It follows that It is easy to see that By the same argument as above, τ t is implemented by e itH l,x for |t| ≤ T if l > |I| + |T | + |x|, which implies that α x • τ t • α −1 x = τ t ✷.

An identification of local algebras
In order to apply the results of Section 7 to the algebra of local observables A, it is necessary to identify the local Weyl algebra R AW (I) with the von Neumann algebra B(I) obtained by applying the interacting dynamics τ to the local abelian algebra of time-zero fields U AW (I). This is done in Proposition 6.5 below. Note that by similar arguments the corresponding result holds also in the 0-temperature case.
For I ⊂ IR a bounded open interval, we denote by U AW (I) the von Neumann algebra generated by {W AW (h) | h ∈ h I , h real valued}. Note that U AW (I) ⊂ R AW (I) is abelian. We denote by B α (I) the von Neumann algebra generated by Proof. Let us first prove that B(I) ⊂ R AW (I). Using (6.10) and U AW (I) ⊂ R AW (I), we see that B α (I) ⊂ R AW I+] − α, α[ for all α > 0. According to Lemma 6.2 (i) this implies B(I) ⊂ R AW (I).
Let us now prove that R AW (I) ⊂ B(I). Using Lemma 6.2 (i) it suffices to show that for all J ⊂ I and α ≪ 1 one has R AW (J) ⊂ B α (I). (6.14) To this end we fix I and J with J ⊂ I and set δ = 1 2 dist(J, I c ). We will first prove that Using again Theorem 6.4 we obtain, for |t| < α, e itHI /p e −itV (n) where γ (n) is the dynamics implemented by the unitary group t → e −itV (n) I . Since V I is affiliated to U AW (I), e −itV (n) I ∈ U AW (I) and hence τ t/p • γ (n) t/p p (A) ∈ B α (I) for |t| < α.
Let us now prove (6.14). Clearly the operators W AW (h) for h ∈ h J and h real valued belong to U AW (J) and hence to B α (I). Let us now pick h ∈ h J ∩D(ǫ) and h real valued. (This is possible; see the discussion presented at the end of Subsection 6.1). Applying (6.15) to A = W AW (h), we obtain that W AW (e itǫ h) ∈ B α (I) for |t| < α. Hence W AW t −1 (e itǫ h − h) ∈ B α (I) for |t| < α. Letting t → 0 and using the fact that the map h ∋ h → W AW (h) is continuous for the strong operator topology, we obtain that W AW (iǫh) ∈ B α (I). But any vector h ∈ h J can be approximated in norm by vectors of the form h 1 + iǫh 2 , with h i ∈ h J real and h 2 ∈ D(ǫ). This implies that for all h ∈ h J the operators W AW (h) belong to B α (I) and hence R AW (J) ⊂ B α (I). This completes the proof of the proposition ✷.
6.5 Existence of the limiting state Theorem 6.6 (Existence of limiting state). Let {ω l } l>0 be the family of (τ l , β)-KMS states for the spatially cutoff P (φ) 2 models constructed in Subsection 5.3.
The state ω β on A has the following properties: (iv) ω β has the spatial clustering property, i.e., Remark 6.7 Let R be a C * -algebra, π i : R → B(H i ), i = 1, 2, two quasi-equivalent representations of R. Then there exists a * -isomorphism τ between π 1 (R) ′′ and π 2 (R) ′′ intertwining the two representations. This isomorphism is automatically weakly continuous. Therefore the representation π 2 extends uniquely from R to π 1 (R) ′′ and is quasi-equivalent to the concrete representation of π 1 (R) ′′ in B(H 1 ).
Applying this easy observation to the representations π AW and π F of W(h I ), which are quasi-equivalent by Lemma 6.3, we see that the Fock representation π F extends by weak continuity from π AW (W(h I )) to R AW (I) and is quasi-equivalent to the Araki-Woods representation. Since two quasi-equivalent representations have the same set of normal states, we obtain that ω β|RAW (I) is also normal with respect to the Fock representation.
Proof. The family {ω l } l>0 of states on A is weak * compact by the Banach-Alaoglu theorem. Let ω 1 be one of the limit points of {ω l } l>0 . Then we can find a subnet 1 {ω r } r∈R such that ω 1 = w − lim r∈R ω r .
We now apply a result of Takesaki and Winnink [TW]: clearly I → {R AW (I)} is a net of von Neumann algebras (see [TW,Section 2]). The algebras R AW (I) are σ-finite, since the Hilbert space Γ(h ⊕ h) on which they act is separable. Moreover, as factors on a separable infinite dimensional Hilbert space they are properly infinite. Applying [TW,Theorem 1], we obtain that the KMS state ω 1 is normal on R AW (I).
Let us now show that all limit states are identical. Let us denote by U 0 (I) the abelian C * -algebra generated by IR) and by T α (I) the * -algebra generated by τ t (A) | A ∈ U 0 (I), |t| < α .
From Theorem 6.4 and Proposition 7.6 we deduce that whereω andτ are defined in Subsection 7.2. Therefore all weak accumulation points of {ω l } l>0 coincide on the algebras T α (I) ⊂ R AW I+] − α, α[ . We note that T α (I) is weakly dense in the von Neumann algebra B α (I) defined in (6.13). Moreover, we have seen that all limit states are normal on the local algebras R AW (I), I open and bounded. Therefore they coincide on the von Neumann algebras B α (I), and hence by Proposition 6.5 on R AW (I).
Consequently, they also coincide on the norm closure A. Thus the weak * compact family {ω l } l>0 has a unique accumulation point, which implies that We have already seen that ω β is a locally normal (τ, β)-KMS state on A, which completes the proof of (i) and (ii). Property (iii) follows from the invariance of the stateω under space translations shown in Lemma 7.7 and the same density argument as above. It remains to prove (iv). Let (H β , π β , Ω β ) denote the GNS objects associated to (A, ω β ). The group {α x } x∈IR is implemented in H β by a strongly continuous group of unitary operators {e ixP β } x∈IR with P β Ω β = 0. Lemma 7.7 (ii) implies that, for A, B ∈ T α (I), Since R AW (I) is a factor, the representation π β provides a weakly continuous * -isomorphism between R AW (I) and R β (I) = π β (R AW (I)) = π β (R AW (I)) ′′ . Hence, by the same weak density argument as above, (6.18) extends to all A, B ∈ R AW (I). Thus the space clustering property holds on R AW (I) for all I, I open and bounded, and extends to A by norm density.

Construction of the interacting path space
In this section we construct the interacting path space supported by S ′ IR (S β × IR) describing the translation invariant P (φ) 2 model at temperature β −1 and study some of its properties.

Construction of the interacting measure
Let H ren C = H ren C −E C be the renormalized P (φ) 2 Hamiltonian on S β defined in Subsection 5.2. Let f ∈ S IR (S β ×IR). For x ∈ IR the function f x defined in Subsection 3.5 belongs to S IR (S β ). We will apply the results of Appendix A to the selfadjoint operator H = H ren C , R(x) = φ F (f x ) (replacing the variable t in Appendix A by the variable x).
It follows from the bound (5.7) in Proposition 5.4 and the fact that the map is infinitely differentiable that the hypothesis (A.3) in Subsection A.1 is satisfied. Similarly, using the bound (5.8) and the fact that the map We will set for −∞ ≤ a ≤ b ≤ +∞: Then where Ω • C is the free vacuum on Γ H − 1 2 (S β ) .
Proof. Let us first introduce a notation which we will use throughout the proof. If A is a Σ-measurable function on Q, the image of A under U C (x) for x ∈ IR will be denoted by U C (x)(A). On the other hand, the expression U C (x)A will denote the operator product of the operator U C (x) and the operator of multiplication by A, acting on L 2 (Q, Σ, dφ C ). Using Lemma 3.1 we find We will approximate the above integral using Riemann sums. Let n, p ∈ IN and 0 ≤ j ≤ 2np.
We set x j = −a + j a np and z j = −a + [j/p] a n , where [.] denotes the integer part. It follows from (3.15) that the map U C (x j ) e i(xj+1−xj)φ(fz j ,0) (7.1) in 1≤p≤∞ L p (Q, Σ, dφ C ), where in the last line we use the fact that U C (x j ) is an automorphism of L ∞ (Q, Σ, dφ C ). Since G [a,b] is a FKN kernel, Next let A j , 0 ≤ j < 2np−1, be the multiplication operators by Σ-measurable functions. Using the identity U C (x j )(A j ) = U C (x j )A j U C (−x j ) and the fact that U C (x) is an automorphism of L ∞ (Q, Σ, dφ C ), we obtain as an operator identity on L 2 (Q, Σ, dφ C ): In the above identity the product on the l.h.s. is the operator of multiplication by the product of the functions U C (x j )(A j ) and the product on the r.h.s. is an operator product. Using that Let us now set for 0 ≤ k ≤ 2n, y k = −a + k a n . We note that z j = y [j/p] and that (x j+1 − x j ) = a np = (y k+1 − y k )/p. We obtain that Taking into account the construction of H C recalled in Subsection 5.2 we find Letting now n and p tend to ∞ and using Proposition A.5, we obtain the proposition ✷.
where Ω C is the unique vacuum state of H C .
is the generating functional of a Borel probability measure µ on (Q, Σ).
Proof. Note first that applying Proposition 7.1 for f = 0, we obtain W [−a,a] (0) = e −2aH ren a] for some a ∈ IR. Using Proposition 7.1 we find . Now lim l→+∞ e −(l−a)H ren where Ω C is the eigenvector for the simple eigenvalue {0} of H ren C . Thus a], we see that Ω C , W [s,t] (f )Ω C is constant for s ≤ −a, t ≥ a, which proves (i).
To prove (ii) we apply Minlos theorem (see e.g. [GV]). As a limit of functionals of Borel probability measures on (Q, Σ) the functional f → Ω C , W [−∞,∞] (f )Ω C is of positive type. It remains to show that the map is continuous. Using the bound (5.8) we obtain where C > 0 is some constant and .
Clearly r L 2 (IR) ≤ C f 1 −f 2 p , where . p is a Schwartz semi-norm on S(S β ×IR). Applying Lemma A.7 for δ = 1 2 , we obtain which proves the desired continuity result.
Let us now verify (iii). The measure µ is invariant under time translations and time reflection as the weak limit of the time translation and time reflection invariant measures µ l . The fact that µ is invariant under space translations follows directly from (i) and Remark A.9.
It remains to prove a]. We consider the family of functions u l (λ) = e iλφ(f ) dµ l for λ ∈ C.

Existence and properties of sharp-time fields
Proposition 7.3 Let h ∈ S IR (IR) and t ∈ S β . Then the sequence φ δ k (. − t) ⊗ h is Cauchy in 1≤p<∞ L p (Q, Σ, µ) and hence is continuous for each h ∈ S IR (IR).
Proof. For p ≥ 1 we have . It follows then from the estimate (5.9) in Proposition 5.4 that Applying now Lemma A.8 we obtain that . A similar argument shows that t → φ(t, h) ∈ L 2p (Q, Σ, µ) is continuous, using the fact that t → δ(. − t) ∈ H −1 (S β ) is continuous ✷.
Using the existence of sharp-time fields, we can equip the probability space (Q, Σ, µ) with an OS-positive β-periodic path space structure: We recall that U (t) is the group of transformations generated by the (euclidean) time translations T t and R is the transformation generated by time reflection, and Σ 0 is the sub−σ-algebra of Σ generated by the functions {φ(0, h) | h ∈ S IR (IR)}.
Proof. Since the measure µ is invariant under time translations and time reflection, we see that U (t) and R are measure preserving automorphisms of L ∞ (Q, Σ, µ). Proposition 7.3 implies that the map S β ∋ t → e iφ(t,h) ∈ L 2 (Q, Σ, µ) is continuous. Hence U (t) is a strongly continuous group on L 2 (Q, Σ, µ). This implies that U (t) is strongly continuous in measure on L ∞ (Q, Σ, µ). Clearly it is β-periodic.
The generalized path space (Q, Σ, Σ 0 , U (t), R, µ) is OS-positive, since µ is the weak limit of the measures µ l , which are associated to OS-positive path spaces. Finally we have already seen that Σ = t∈S β Σ t . This completes the proof of the theorem ✷.

Properties of the interacting β-KMS system
We first prove the convergence of sharp-time Schwinger functions.
Proposition 7.5 Let h i ∈ C ∞ 0 IR (IR) and t i ∈ S β for 1 ≤ i ≤ n. Then Proof. Let a > 0 such that supp h j ⊂ [−a, a]. By Proposition 7.3, we know that After extracting a subsequence, this implies that (7.5) by Theorem 7.2 (i). Note that for all l > 0 because this convergence holds in L 2 (Q, Σ, dφ C ) and where G [−l,l] ∈ L 2 (Q, Σ, dφ C ) as a consequence of (5.6). By the same arguments as above, we obtain Applying now Lemma B.1 (ii) and using (7.5) we obtain the proposition ✷.
This completes the proof of the proposition ✷.
Let us denote by {α x } x∈IR the group of space translations on U 0 defined by α Lemma 7.7 Let A j ∈ U 0 and t j ∈ IR, 1 ≤ j ≤ n. Set A = k j=1 τ tj (A j ) and B = n j=k+1 τ tj (A j ). It follows that Proof. Property (i) follows directly from the invariance of the measure µ under the space translations {α x } x∈IR shown in Theorem 7.2 (iii). It remains to prove (ii). We set Due to the KMS condition, the functions G x and G ∞ are holomorphic in I n+ β and bounded by n j=1 A j . We claim that, for s 1 ≤ · · · ≤ s n and s n − s 1 ≤ β, lim x→∞ G x (is 1 , . . . , is n ) = G ∞ (is 1 , . . . , is n ). (7.7) Let us prove (7.7). Let us first assume that A j = e iφ(0,hj ) for h j ∈ C ∞ 0 IR (IR). Then G x (is 1 , . . . , is n ) = Q l j=1 e iφ(δ(.−sj)⊗hj ) n j=l+1 e iφ(δ(.−sj )⊗hj ( . −x)) dµ, G ∞ (is 1 , . . . , is n ) = Q l j=1 e iφ(δ(.−sj)⊗hj ) dµ × Q n j=l+1 e iφ(δ(.−sj)⊗hj ) dµ. By Proposition 7.3 we have As before (see Section 4.1), the group of spatial translations induced on Q by the map (t, y) → (t, y + x) has been denoted by {a x } x∈IR . Applying Lemma A.10 we find where a > 0 is the spectral gap of H ren Using functional calculus, we conclude that (7.7) holds for all A j ∈ U 0 . To complete the proof of the lemma, we can now argue as in the proof of Proposition 7.6, using Lemma B.3 ✷.

A A time-dependent heat equation
Let H ≥ 0 be a selfadjoint operator on a Hilbert space H and let R(t), t ∈ IR, be a family of closed operators with D(H γ ) ⊂ D(R(t)) for some 0 ≤ γ < 1. We consider the following time-dependent heat equation: This equation is (formally) equivalent to the following integral equation: In the main text we only use the results of this section in the dissipative case, i.e., when R(t) is selfadjoint for all t ∈ IR. However, part of the results are valid and will be proved in the general case.
The solution of (A.1) will be denoted by U (t, s) or U λ (t, s). If we want to display its dependence on the family R(t), then the solution of (A.1) will be denoted by U (t, s; R).

A.1 Existence of solutions
We assume that the maps are Hölder continuous of some order ǫ > 0.
In the sequel we will use the following result.
The following result is shown in [H,Theorem 7.1.3].

A.2 The dissipative case
We now consider the dissipative case when R(t) is selfadjoint for t ∈ IR. We first prove a result about approximation by time-ordered products. We will make use of an extension of Gronwall's inequality to integral equations, shown in [H,Lemma 7.1.1].
Lemma A.4 Let b ≥ 0 and γ > 0. Let a(t) and u(t) be non negative locally integrable functions on s ≤ t ≤ T < ∞ such that Then Proposition A.5 Assume that R(t) is selfadjoint and that (A.3) holds. Then for s ≤ t there exists a sequence {p n } n∈IN with lim n→∞ p n = +∞ such that Proof. Let R i , i = 1, 2, be two families of closed operators satisfying (A.3) and let U (i) (t, s) be the associated propagators. Then This implies, using Lemma A.1, that We now apply Gronwall's inequality, as given in Lemma A.4, with We obtain Let us now prove the proposition. For s < t fixed, n ∈ IN, we set Note that H + iR(t j ) with domain D(H) is the generator of a C 0 -semigroup of contractions, since it is closed and maximal accretive, using (A.3). If U (n) (t, s) is the solution of (A.1) for the piecewise constant family of operators {R n (t)}, then one can easily verify that: Applying next [Ch] we obtain e −(tj+1−tj )(H+iR(tj )) = s-lim p→∞ e −(tj+1−tj )H/p e −i(tj+1−tj )R(tj )/p p .
Using the fact that e −τ (H+iR(tj )) , e −τ H and e −iτ R(tj) are all contractions, we conclude that there exists a sequence p n → ∞ such that This completes the proof of the proposition ✷.
From Lebesgue dominated convergence we deduce that lim s→−∞ d n dλ n U λ (t, s) |λ=0 = n!(−i) n −∞<t1≤···≤tn≤t Applying Lemma B.1 we obtain (iii) ✷. We will use the following lemma to show that the limiting functional obtained in Theorem 7.2 defines a Borel measure on S ′ (S β × IR).
Integrating in λ from 0 to 1 we obtain the lemma ✷.

B Miscellaneous results
Lemma B.1 Let F : IR 2 → E be a map with value in a metric space E.
The proof is easy and left to the reader.
Lemma B.2 Let (Q, Σ, µ) be a probability space. Let f be a real measurable function on Q and set C(t) := Q e itf dµ.
Then f ∈ 1≤p<∞ L p (Q, Σ, µ) if and only if sup t∈IR |∂ n t C(t)| < ∞ for all n ∈ IN. If this is the case, then Proof. The ⇒ part and the formula for ∂ n t C(t) is obvious by differentiating under the integral sign. It remains to prove the ⇐ part. Let χ(τ ) = e −τ 2 /2 and let p ≥ 1. By monotone convergence it suffices to prove that sup n∈IN Q f 2p χ f n dµ < ∞ in order to show that f ∈ L 2p (Q, Σ, µ). We have τ 2p χ τ n = n 2p+1 2π e itτ ∂ 2p tχ (nt) dt.
Then lim α∈I u α = u exists in the compact-open topology on Ω and u is a holomorphic function in Ω. (ii) Assume moreover that Ω is bounded with a smooth boundary and that sup α∈I sup z∈Ω |u α (z)| < ∞.
Proof. Let us first prove (i). By Vitali's theorem the family {u α } is compact for the compact-open topology. Let {u β } β∈J be a subnet converging to a continuous function u. Assume that the net {u α } α∈I does not converge to u. Then there exists a bounded open set Ω 1 ⊂ Ω and a subnet {u γ } γ∈J1 such that sup z∈Ω1 |u γ (z) − u(z)| ≥ ǫ 0 > 0 for γ ∈ J 1 . Applying again Vitali's theorem to the net {u γ } γ∈J1 , we obtain another subnet {u δ } δ∈J2 such that lim δ∈J2 u δ = v, with v = u. But u and v are holomorphic in Ω, as limits of holomorphic functions for the compact-open topology and coincide on Γ by hypothesis. Since Γ has an accumulation point in Ω, we have u = v which gives a contradiction.