On the relativistic KMS condition for the P (φ)2

P ( φ )

model

Christian G´erard1and Christian D. J¨akel2 1 Universit´e Paris Sud XI, F-91405 Orsay, France

christian.gerard@math.u-psud.fr

2 ETH Z¨urich, H¨onggerberg, CH-8093 Z¨urich, Switzerland christian.jaekel@itp.phys.ethz.ch

Summary. The relativistic KMS condition introduced by Bros and Buchholz pro- vides a link between quantum statistical mechanics and quantum field theory. We show that for theP(φ)2 model at positive temperature, the two point function for fields satisfies the relativistic KMS condition.

1 Introduction

The operator algebraic framework allows to characterize the equilibrium states of a quantum system by first principles: when the dynamical law is changed by a local perturbation, which is slowly switched on and slowly switched off again, an equilib- rium state returns to its original form at the end of this procedure. In a pioniering work Haag, Kastler and Trych-Pohlmeyer [15] showed that this characterization of an equilibrium state leads to a sharp mathematical criterion, the so-calledKMS con- dition, first encountered by Haag, Hugenholtz and Winnink [14] and more implicitly by Kubo [23], Martin and Schwinger [24].

On the other hand, the vacuum states of a relativistic QFT are characterized by Poincar´e invariance and the spectrum condition [27]. Both the KMS and the spec- trum condition can be formulated in terms of analyticity properties of the correlation functions, but for almost 40 years the connection between these two conditions was not investigated in depth, and algebraic quantum statistical mechanics and algebraic quantum field theory were treated in this respect as disjoint subjects. But finally increased interest in cosmology and heavy-ion collisions led to a need to combine QFT and quantum statistical mechanics.

It was first recognized by Bros and Buchholz [6] that the thermal equilibrium states of a relativistic QFT should have stronger analyticity properties in configu- ration space than those imposed by the traditional KMS condition. The result of their careful analysis is a relativistic KMS condition which can be understood as a remnant of (and, in applications, as a substitute for) the relativistic spectrum condition in the vacuum sector.

1.1 Content of the Paper

The content of this short paper is as follows. In Section 2 we recall the construction of theP(φ)2model at positive temperature presented in [12]. In Section 3 we review the main ingredients of the Euclidean approach to thermal field theory. In Section 4 we show that the Wightman two-point function for fields satisfies the relativistic KMS condition. Section 5 is devoted to a brief discussion of the implications of the relativistic KMS condition.

2 The spatially-cutoff P ( φ )

model at positive temperature

The construction of interacting thermal quantum fields in [12] includes several of the original ideas of Høegh-Krohn [17], but instead of starting from the interacting system in a box the authors started from the Araki-Woods representation for the free thermal system in infinite volume. We briefly recall the main steps of this construction.

2.1 Preliminary Material The KMS condition

A stateωβ over aC^∗-algebraAequipped with aC^∗−dynamicsτ^tis called a (τ, β)- KMS state for someβ∈IR∪ {±∞}, if for allA, B∈Athere exists a functionFA,B

which is continuous in the strip{z∈C|0≤ =z≤β}and analytic and bounded in the open strip{z∈C|0<=z < β}, with boundary values given by

FA,B(t) =ωβ Aτt(B)

, FA,B(t+iβ) =ωβ τt(B)A

,∀t∈IR.

Thus KMS states are characterized by a real parameterβ, which has the meaning of inverse temperature.

The relativistic KMS condition

Lorentz invariance is always broken by a KMS state. A KMS state might also break spatial translation or rotation invariance, but the maximal propagation velocity of signals is not affected by such a lack of symmetry.

Definition 1.A stateωβ over a C^∗-algebra Asatisfies therelativistic KMS condi- tionat inverse temperatureβ >0if there exists some positive timelike vectore∈V+, e²= 1, such that for every pair of elementsA, B ofAthere exists a functionFA,B

which is analytic in the tube domain

T^β:={z∈C²| =z∈V+∩(βe+V₋)},

and continuous at the boundary sets IR² andIR²+ iβe with boundary values given by

FA,B(t, x) =ωβ Aαt,x(B)

and FA,B(t+ iβe, x) =ωβ αt,x(B) for all (t, x) ∈ IR². Here V± = {(t, x) ∈ IR² | |t| < |x|, ±t ≥ 0} denote the for- ward/backward light-cones.

Remark 1.Note that in the limitβ→ ∞the tubeT^βtends toward the forward tube IR²+ iV⁺.

As we will discuss in Section 5, the relativistic KMS condition has numerous applications. For instance, the famous Reeh-Schlieder property (in the thermal rep- resentation) is a direct consequence of the relativistic KMS condition (and weak additivity), no matter if the spatial translation or rotation invariance is broken by the KMS state or not [19].

2.2 The free neutral scalar field at temperatureβ⁻¹

Let h = H⁻¹²(IR) be the complex Sobolev space of order −¹2 equipped with the norm

khk²= (h,(2ν)⁻¹h)L2(IR,dx)

forν:= (D²x+m²)¹².

LetW⁽h) be the abstract WeylC^∗-algebra overh^{. On}W⁽h) we define the free time evolution{τ^t◦}^t∈IRby

τt^◦ W(h)

=W(e^itνh), h∈h, t∈IR.

Form >0 the unique (τ^◦, β)-KMS state on the Weyl algebraW(h) is given by ωβ^◦ W(h)

:= e⁻¹⁴(h,(1+2ρ)h)

, h∈h^{, (1)}

whereρ:= (e^βν−1)⁻¹,β >0.

The Araki-Woods representation

A realization of the GNS representation associated to the pair W(h), ωβ^◦

is pro- vided by the Araki-Woods representation. We set:

HAW :=Γ(h⊕h^), ΩAW :=ΩF,

πAW(W(h)) =WAW(h) :=WF (1 +ρ)¹²h⊕ρ¹²h

, h∈h^.

Herehis the conjugate Hilbert space toh^,WF(.) denotes the Fock Weyl operator on the bosonic Fock spaceΓ(h⊕h^{) andΩ}F ∈Γ(h⊕h) is the Fock vacuum.

The generator of the time evolution

Since ωβ^◦ is τ^◦-invariant, the time evolution can be unitarily implemented in the representationπAW:

e^itLAWπAW(A)ΩAW :=πAW τt^◦(A)

ΩAW, A∈W(h). (2) The generatorLAW of the free time evolution, uniquely fixed by (2) and the request LAWΩAW = 0, is called the (free) Liouvillean. It is equal to dΓ(ν⊕ −ν).

Thermal fields

The Araki-Woods representationπAW is a regular representation, i.e., the map IR3 λ7→Wπ(λh) is strongly continuous for anyh∈h. Thus one can definefield operators

φAW(h) :=−i d

dλWAW(λh)

λ=0, h∈h,

which satisfy in the sense of quadratic forms on D(φAW(h¹))∩ D(φAW(h²)) the commutation relations

[φAW(h1), φAW(h2)] = i=(h¹, h2), h1, h2∈h^. The net of local von Neumann algebras

The von Neumann algebra generated by{πAW(W(h))|h∈h}is denoted byRAW. To define the net of local von Neumann algebras, we introduce the IR−linear map

U: h^=H−12(IR) → S⁰(IR) h 7→ <h+ iν⁻¹=h.

For I ⊂ IR a bounded open interval, we define the following real vector subspace ofh

h^I:={h∈h|suppUh⊂I}. (3) We denote byRAW(I) the von Neumann algebra generated by{WAW(h)|h∈h^I}.

The algebra

A^:= [

I⊂IR

RAW(I)

(C∗)

is called thealgebra of local observables. The union is over all open bounded inter- valsI⊂IR and the symbolS

I⊂IRRAW(I)^(C∗) denotes theC^∗-inductive limit (see e.g. [20, Proposition 11.4.1.]).

Remark 2.We note that the Araki-Woods representation is locally normal w.r.t. the vacuum representation of the free field. Consequently, theC^∗-algebraAis identical (up to∗-isomorphisms) to the algebra

A^F:= [

I⊂IR

πF W⁽h^I)00(C^∗)

,

whereπFis the Fock representation.

2.3 The spatially-cutoff P(φ)₂ model at temperatureβ⁻¹ LetP(λ) =P2n

j=0ajλ^j be a real valued polynomial, which is bounded from below, and letl∈IR⁺ be a spatial cutoff parameter. The spatially cutoffP(φ)2 model on the real line IR at temperatureβ⁻¹ is then defined by the formal interaction term

Vl= Z l

−l

:P(φAW(x)) : dx, l >0.

Here : : denotes the Wick ordering (see e.g. [13]) with respect to thetemperatureβ⁻¹ covariance onIR:

C0(h1, h2) :=

h1, (1 + e^−βν) 2ν(1−e^−βν)h2

L2(IR), h1, h2 ∈ S(IR). (4) Using a sequence of functions{δ^k}approximating the delta-function, the limit

Vl:= lim

k→∞

Z l

−l

:P(φ^AW(δk(.−x)) : dx

exists on a dense set of vectors inΓ(h⊕h). An approximation of the Diracδfunction can be fixed by settingδk(x) := kχ(kx), whereχis a function in C0 IR^∞ (IR^d) with Rχ(x)dx= 1.

The perturbed KMS system

It can be shown (see [12]) that the operator sumLAW+V^lis essentially selfadjoint onD(LAW)∩D(V^l) and if we setHl:=LAW+Vl, then the perturbed time-evolution onAis given by

τt^l(A) := e^itHlAe^−itHl, A∈A.

The perturbed KMS state ωl onA is normal w.r.t. the Araki-Woods representa- tionπAW. In fact, the GNS vectorΩAW ∈Γ(h⊕h) belongs to D e^−β2Hl

and the perturbed KMS stateωlis the vector state induced by the state vector

Ωl:= e^−β2HlΩ_AW ke^−β2HlΩAWk.

These results are in complete analogy to the analytic perturbation theory for bounded perturbations due to Araki (see e.g. [5]). Identical formulas, valid for a certain class of unbounded perturbations, have recently been derived in [8].

2.4 The translation invariant P(φ)₂ model at temperatureβ⁻¹ Existence of the dynamics

LetI⊂IR a bounded open interval. Fort∈IR fixed, the norm limit

llim→∞τt^l(A) =:τt(A)

exists for allA∈RAW(I). In fact, forAandtfixed,τt^l(A) is independent oflforl sufficiently large.

By construction the elements of the local von Neumann algebrasRAW(I),Iopen and bounded, are norm dense inA. Thus the mapτ:t7→τt extends to a group of∗- automorphisms ofA. Moreover, if{α^x}^x∈IRdenotes the group of space translations overA, defined by

αx WAW(h)

:=WAW(e^ix.kh), x∈IR, wherekis the momentum operator acting onh^{, then}

τt◦αx=αx◦τt

for allt, x∈IR. Consequently the time evolution is translation invariant.

Existence of the thermodynamic limit

Let{ω^l}^l>0be the family of (τ^l, β)-KMS states for the spatially cutoffP(φ)2models constructed in Subsection 2.3. Then it has been shown in [12] that

w− lim

l→+∞ωl=:ωβ exists overA^{. (5)} Moreover, the stateωβ has the following properties:

(i) ωβ is a (τ, β)-KMS state overA^;

(ii) ωβ is locally normal, i.e., if I ⊂IR is an open and bounded interval, thenωβ|RAW(I) is normal w.r.t. the Araki-Woods representation;

(iii) ωβ is invariant under spatial translations, i.e., ωβ(αx(A)) =ωβ(A), x∈IR, A∈A^; (iv) ω^β has thespatial clustering property, i.e.,

xlim→∞ωβ(Aα^x(B)) =ωβ(A)ωβ(B) ∀A, B∈A.

The rate of the spatial clustering is related to the infrared properties of the Liouvillean [18].

3 Euclidean approach

As far as the formulation of the spatially cut-off thermalP(φ)2model is concerned, the Euclidean approach is only used to show that the sum of the operators L^AW and Vl (which are both unbounded from below) is essentially selfadjoint on the intersection of their domains.

However, for the existence of the thermodynamic limit (5) the Euclidean ap- proach is used in a more sophisticated manner. The key argument in the proof of (5),Nelson symmetry, will be crucial for the proof of the relativistic KMS con- dition too. In order to formulate it, we briefly recall the Euclidean approach, in a framework adapted to the thermalP(φ)² model (see [11], [21] for a more general abstract framework).

3.1 Euclidean reconstruction theorem Euclidean measures on the cylinder

LetS^β = [−β/2, β/2] be the circle of lengthβ. The points in the cylinderS^β×IR are denoted by (t, x). LetQ:=SIR⁰ (Sβ×IR) be the space of real valued,β−periodic Schwartz distributions onSβ×IR and letΣ be the Borelσ-algebra onQ. Letµbe a Borel probability measure on (Q, Σ).

Forf∈ S^IR(Sβ×IR), we denote byφ(f) the function φ(f): Q → IR

q 7→ hq, fi.

ForT ≥0, we denote byΣ[0,T], the subσ-algebra ofΣgenerated by the functions e^iφ(f) for suppf ⊂ [0, T[×IR. Let r:Q → Q be the time-reflection around t = 0 defined by rφ(t, x) = φ(−t, x) and let τt^E:Q→Q be the group of euclidean time translations defined byτt^Eφ(s, x) =φ(s−t, x). The maprlifts to a∗−automorphism RofL^∞(Q, Σ) defined byRF(q) :=F(rq), and the groupτt^Elifts to a groupU(t) of

∗-automorphisms ofL^∞(Q, Σ). This group is unitary onL²(Q, Σ, µ) ifµis invariant underτt^E.

Reconstruction theorem

Let H^OS := L²(Q, Σ[0,β/2],dµ). We assume that the measure µ satisfies the Osterwalder-Schrader positivity

(F, F) :=

Z

Q

R(F)Fdµ≥0 ∀F ∈ H^OS. LetN ⊂ H^OS be the kernel of (., .). We set

H^phys:=H^OS/N,

where the completion ofH^OS/N is done w.r.t. the norm (., .)¹². The canonical pro- jectionH^OS toH^phys is denoted byV.

InH^physwe have the distinguished vector Ωphys:=V(1), where 1 is the constant function equal to 1 onQ.

The unitary groupU(t) fort≥0 doesnotpreserveH^OS(contrary to 0-tempera- ture theories), because distributions supported in the ‘future’ [0, β/2[×IR come back in the ‘past’ ]−β/2,0]×IR by time translations. NeverthelessU(s) for 0≤s ≤t sendsL²(Q, Σ[0,β/2−t[,dµ) intoH^OS.

Using the theory oflocal symmetric semigroups(see [10], [22]), it is possible to define a selfadjoint operatorL^phys onH^physsuch that forF∈L²(Q, Σ[0,β/2−t[,dµ) and 0≤s≤tone has

V(U(s)F) = e^−sLphysV(F).

Finally ifA∈L^∞(Q, Σ_{0}), then multiplication byApreservesH^OSandN, and one obtains a representationπphysof thealgebra of time-zero fieldsL^∞(Q, Σ_{0}):

π^phys(A)V(F) :=V(AF), F ∈ H^OS.

From this reconstruction procedure one obtains aβ-KMS system defined as follows:

(i) the C^∗-algebra A^phys is the von Neumann algebra generated by the operators e^itLphysπphys(A)e^−itLphys,t∈IR,A∈L^∞(Q, Σ_{0});

(ii) the dynamics τt onA^phys is the dynamics generated by the unitary group e^−itLphys;

(iii) theβ-KMS state onA^physis the state generated by the vectorΩphys.

3.2 Euclidean measure for the translation invariant P(φ)₂ model The spatially-cutoffP(φ)2 model at positive temperature allows an Euclidean for- mulation: the measure dµlfor the spatially cutoffP(φ)2 model is given by

dµl:= 1 Zle⁻

R_β/2

−β/2

R_l

−l:P(φ(t,x)):_Cdtdx

dφC, (6)

where dφ^C denotes the Gaussian measure on (Q, Σ) with covariance C(u, u) = (u,(D²t +D²x+m²)⁻¹u)

(withβ-periodic b.c.) defined by Z

Q

e^iφ(f)dφ^C= e^−C(f,f)/2, f∈ S^IR(Sβ×IR). (7) The partition functionZlis chosen such thatR

Qdµl= 1.

Existence of limiting measure

In order to show that one can remove the spatial cutoff one has to show that

l→lim+∞

Z

Q

F(q)dµl=:

Z

Q

F(q)dµ_∞ (8)

exists and defines a Borel probability measure onSIR⁰ (Sβ×IR).

Nelson Symmetry

Formally exchanging the role oftandxin (6) one notices that dµ∞is the Euclidean measure of theP(φ)2model on the circle at temperature zero. This formal argument can be made rigorous (see [12], [17]). In particular one has:

e⁻ R_β/2

−β/2(R_l

−l:P(φ(t,x)):_C 0dx)dt

= e⁻ R_l

−l(R_β/2

−β/2:P(φ(t,x)):

Cβdt)dx

. (9)

Note that on the r.h.s. Wick ordering is done w.r.t. the covariance Cβ(g1, g2) :=

g1, 1 2νg2

L2(S_β,dt), g1, g2∈ S(Sβ).

The analog of (9) in the zero temperature case is called Nelson symmetry (see e.g. [26]).

It was first noticed by Høegh-Krohn [17] that the existence of the limit (8) is equivalent to the uniqueness of the vacuum state for theP(φ)2 model on the circle.

Using Nelson symmetry, the existence of the limit (8) is proved in [12]. Moreover it is shown in [12] thatµ∞is OS positive, invariant under space-time translations, and that sharp-time fieldsare well defined: this means that if δk ∈ C0^∞(Sβ) is a sequence of functions tending to the Dirac distributionδ⁰, then the limits

φ(t, h) := lim

k→∞φ(δk(.−t)⊗h) exist inT

1≤p<∞L^p(Q, Σ,dµ_∞) for anyh∈C0^∞(IR).

4 The relativistic KMS condition

In this section we show that two point functions for fields in the thermal P(φ)2

model satisfy the relativistic KMS condition.

Identification of physical objects

Let (H^β, πβ, Ωβ) be the GNS triple associated to the KMS state ωβ over theC^∗- algebraA. Let alsoPβ, Lβ be the unique generators of space-time translations such thatLβΩβ=PβΩβ = 0.

As we saw in Subsect. 3.1, the Euclidean reconstruction theorem provides a Hilbert spaceH^phys, a unit vectorΩphys, a representation πphys of the abelian von Neumann algebraU^AW generated by{W^AW(h)|h∈h^{, h}real valued}, and a self- adjoint operator Lphys. Since the Euclidean measure µ∞ is invariant under space translations, we obtain also a selfadjoint operator Pphys implementing the space translations.

Let us briefly check that these two families of objects are identical (up to unitary equivalence). In the sequel we will freely identify them.

Let U(I) be the abelian von Neumann algebra generated by {W^AW(h) | h ∈ h^{, h}real valued,supph⊂I}. It was shown in [12] that forAj∈U^{(I), 1}≤j≤n,

(Ωphys,Qn

j=1e^itjLphysπphys(Aj)e^−itjLphysΩphys)

=ωβ(Qn

j=1τt_j(Aj))

= (Ωβ,Qn

j=1e^itjLβπβ(Aj)e^−itjLβΩβ).

(10)

Thus we can define a mapU:H^phys→ H^β by

Ue^itLphysπphys(A)Ωphys:= e^itLβπβ(A)Ωβ, t∈IR, A∈U(I), I⊂IR. (11) From (10) we see that U preserves the scalar product, hence it can be uniquely extended by linearity to

E = Vect{e^itLphysπphys(A)Ωphys| t∈IR, A∈U^{(I), I}⊂IR}

as an isometry. It follows from the Euclidean reconstruction theorem, thatEis dense inH^phys. Moreover it is clear from (11) thatU intertwinesπphysandπβrestricted to Uand also intertwinesLphysandLβ. FinallyU also intertwinesPphysandPβ (note that the algebra of time-zero fields is clearly invariant under space translations).

To check thatU is unitary, we use a result in [12]: letB^α(I) be the von Neumann algebra generated by{τ^t(A)|A∈U(I),|t|< α}. Then it was shown in [12, Prop.

6.5] that

RAW(I) = \

α>0

B^α(I).

Since by construction Ωβ is cyclic for πβ(A), this implies that the range of U is dense inH^β. ThereforeU is unitary.

4.1 Wightman two point function for the thermal P(φ)₂ model LetI⊂IR be a bounded open interval andh^I the real subspace ofhdefined in (3).

By restriction to the local algebraR^AW(I),πβ defines a CCR representation of the real symplectic spaceh^I.

Lemma 1.i) The representationπβ restricted toR^AW(I)is quasi-equivalent to the concrete representation ofR^AW(I);

ii) the CCR representationh^I3h7→πβ(WAW(h))∈ B(H^β) is regular.

Proof. It is well known (see e.g. [12, Lemma 6.2]) thatR^AW(I) is a factor. Now it is shown in [20, Prop. 10.3.14] that ifRis aC^∗-algebra andπis a factor representation ofR, thenπis quasi-equivalent to the GNS representation of anyπ-normal stateω.

Applying this fact to R^AW(I) (with its concrete representation) and to ωβ, we obtain that πβ is quasi-equivalent to the concrete representation ofR^AW(I). This proves i). We know then that there exists a ∗-isomorphism γ from R^AW(I) into πβ(R^AW(I))⁰⁰extendingπβ. This isomorphism is automatically weakly continuous.

Since the Araki-Woods representation is regular, the same holds true for the GNS representationπ^β restricted toh^I. ut

Since πβ is a regular CCR representation, we can define for h∈ h^{I the Segal} field operators

φβ(h) :=−id

dsπβ(WAW(sh))

s=0. In the sequel we will consider only thetime-zero fieldsϕβ: ϕβ(h) :=φβ(h), forh∈h^{I, hreal.}

Remark 3.If we restrict ourselves to time-zero fields, it is possible to give a direct proof that the CCR representation is locally regular, avoiding the use of the local normality of the stateωβ. In fact, let us show that

the map IR3s7→πβ WAW(sh)

is strongly continuous forh∈h^{I, hreal, (12)} using the Euclidean approach.

It suffices to prove the weak continuity on a dense subspace ofH^β. From the reconstruction theorem, we see that we may take as a dense subspace of H^β the linear span of the vectorsV(Qk

1e^iφ(tj,hj)) for hj∈C0IR^∞(IR), 0≤tj< β/2.

We see that it suffices to prove the continuity of the map IR3s7→

Z

Q

Yn 1

e^iφ(tj,hj)

e^isφ(0,h)dµ_∞

forhj∈C0IR^∞(IR),tj∈Sβ. But this follows from the fact thatφ(0, h)∈L¹(Q,dµ_∞), shown in [12].

Lemma 2.Ωβ∈ D(ϕ^β(h)),∀h∈h^{I, h}real valued.

Proof. Clearly it suffices to prove that

2−(Ωβ,e^isϕβ(h)Ωβ)−(Ωβ,e^−isϕβ(h)Ωβ)≤C|s|², 0≤s≤1. (13) By the reconstruction theorem, the r.h.s. is equal to

Z

Q

2−e^isφ(0,h)−e^−isφ(0,h) dµ_∞. Sinceφ(0, h)∈L²(Q,dµ∞), we obtain (13). ut

We now define Wightman two point functions. For a functionh ∈C0^∞(IR) we denote byh⁻the functionh⁻(x) =h(−x).

Proposition 1.There exists a unique W^β(t, .) ∈ C⁰(IRt,D⁰(IR)) such that for h1, h2∈C^∞0IR(IR):

ϕβ(h1)Ωβ,e^itLβϕβ(αxh2)Ωβ

=h1? h⁻2 ?W^β(t, x), (t, x)∈IR². (14) Proof. For fixedtthe l.h.s. is a bilinear formQ^tw.r.t.h¹andh². Moreover it is shown in [12] thatkϕ^β(h)Ωβk ≤Ckhk^S, wherek.k^S is a Schwartz seminorm. ThereforeQt

is continuous for the topology ofC0^∞(IR), which by translation invariance, implies the existence ofW^β(t, .). The continuity w.r.t. the variabletofW(t, .) follows from the obvious continuity intof the l.h.s. of (14). ut

4.2 Relativistic KMS condition

The rest of this section is devoted to the proof of the following theorem:

Theorem 1.The distributionWβ(t, x)extends holomorphically toIR²+ iVβ, where Vβ := {(t, y) | |y|< inf(t, β−t)}. Therefore for Aⁱ =ϕβ(hⁱ), hⁱ ∈ C0IR^∞(IR)the two-point functionFA₁,A₂(t, x)is holomorphic inIR²+ iVβ.

4.3 Proof of Thm. 1

Let us briefly recall a few facts concerning the 0−temperatureP(φ)² model on the circleSβ. The Hilbert space is the Fock spaceΓ(H⁻¹²(Sβ)), whereH⁻¹²(Sβ) is the Sobolev space of order−¹2 onSβ with norm

kgk²= (g,(2b)⁻¹g)L2(S_β,dt)

with b = (D²t +m²)¹². For g ∈ H⁻¹²(Sβ), we denote by φ^C(g) the (Fock) field operator acting onΓ(H⁻¹²(Sβ)).

The operator sum

dΓ(b) + Z

S_β

:P(φC(t)) :C_β dt

is essentially selfadjoint and bounded below, and the Hamiltonian of the P(φ)2

model onS^β is

HC:= dΓ(b) + Z

S_β

:P(φ(t)) :C_β dt−EC,

where E^C is an additive constant such that inf σ(H^C) = 0. The HamiltonianH^C has a unique (up to a phase) ground state (i.e., vacuum state) induced by the state vectorΩ^C. Another fact we shall need is the following: ifg¹, g² arerealelements of H⁻¹²(Sβ), then (φ^C(g¹)Ω^C,e^−yHCφ^C(g²)Ω^C) is real fory∈IR⁺. This follows from the representation of e^−yHC using the Feynman-Kac-Nelson (FKN) formula.

Finally we note that

PC:= dΓ(Dt) is the momentum operator on the circleSβ.

The two-point function for the P(φ)₂ model on the circle

Now consider the two-point functionW^C for theP(φ)² model on the circle W^C(t, y) = Ω^C, φ^C(δ⁰)e^iyHCe^itPCφ^C(δ⁰)Ω^C

, t∈Sβ, y∈IR.

W^C is a tempered distribution onS^β×IR rigorously defined by hW^C, f⊗gi:= (2π)²¹ φ^C(δ⁰)Ω^C,˜g(−H^C)φ^C(f)Ω^C

, forf∈C^∞(Sβ),g∈ S(IR), and ˜gthe Fourier transform ofg.

To check thatW^C is well defined as a tempered distribution, we use the bound k(H^C+ 1)⁻¹²φC(h)(HC+ 1)⁻¹²k ≤CkhkH−1(S_β), (15) which using thatδ0∈H⁻¹(Sβ) yields

|hW^C, f⊗gi| ≤Ck(H^C+ 1)˜g(−H^C)kkfkH−1(S_β)k(H^C+ 1)¹²ΩCk². The r.h.s. can clearly be estimated in terms of Schwartz seminorms off andg.

Analytic continuation

We first recall thespectrum condition on the circle[14]:

|P^C| ≤HC. Set

V±:={(t, y)∈IR² | |t|<|y|, ±y≥0}.

Usingke^−HC(HC+ 1)k ≤C⁻¹ and the bound (15) we conclude that F(τ, z) := ΩC, φC(δ0)e^izHCe^{iτ PC}φC(δ0)ΩC

is holomorphic inSβ×IR + iV+, has a moderate growth when=(τ, z)→0 and RF(t+ i, y+ i)f(t)g(y)dtdy

= (2π)¹² φ^C(δ⁰)Ω^C,˜g(−H^C)e^−(HC+PC)φ^C(f)Ω^C

→ hW^C, f⊗gi, when→0, i.e.,

lim→0F(.+ i, .+ i) =W^C(., .) inS⁰(Sβ×IR).

Locality on the circle Clearly

W^C= lim

k→+∞Gk inS⁰(Sβ×IR), where

Gk(t, y) := ΩC, φC(δk)e^iyHCe^itPCφC(δk)ΩC ,

and δk is a sequence in C^∞(Sβ) with support in{t∈IR| |t| ≤k⁻¹}and tending toδ0 whenk→ ∞.

Now using locality (i.e., finite speed of light) on the circle, we see that if (t, y)∈ Vβ,k,

Vβ,k={(t, y)| |y|<inf(t, β−t)−2k⁻¹}, then

[φC(δk),e^iyHCe^itPCφC(δk)e^−itPCe^−iyHC] = 0,

because no signal can go from suppδ^k to supp (δ^k+t) in timey if (t, y) ∈V^β,k. This fact can be shown by exactly the same arguments as those used for theP(φ)2

model on IR. Thus for (t, y)∈Vβ,k, the functionGk is real valued.

Edge of the wedge

According to the Schwarz reflection principle,W^Ccan now be view as the boundary value of a function holomorphic inVβ−iV+:

W^C(t, x) =H(t, x) where

H(τ, z) := φC(δ0)ΩC,e^−izHCe^{−iτ PC}φC(δ0)ΩC . Thus

W^C(t, y) =W^C(¯t,y),¯ (t, y)∈Vβ+i(V⁺∪V−). (16) We can now apply the edge of the wedge theorem [27]. It implies that there exists an open ballB(0, d) :={z∈C²| |z|< d}such thatW^C(t, y) is holomorphic inVβ+iΓ, where

Γ :=V+∪V₋∪B(0, d).

Moreover,W^C(t, iy) is real for t ∈ Sβ, y >0 (by using the representation of e^−yHC as a Feynman-Kac-Nelson (FKN) kernel). ThusW^C(t, iy) is real fort∈Sβ, y∈IR, by (16). Applying the Schwarz reflection principle one more time, we conclude that

W^C(t, y) =W^C(t,−y) ∀(t, y)∈V^β+iΓ.

Schwinger two-point function for the thermalP(φ)₂ model

Leth∈C0IR^∞(IR) and set I(t, x) :=

Z

S⁰(S_β×IR)

φ(0, h)φ(t, αxh)dµ_∞, t∈Sβ, x∈IR, whereαxh(y) =h(y−x). By [13, Thm. 7.2], we get:

I = 2R

−∞<x₁≤x₂<∞h(x1) ΩC, φC(δ0)e^{−(x2−x1)HC}φC(δt)ΩC

h(x2−x)dx1dx2

= 2R

−∞<x₁≤x₂<∞h(x1−x) ΩC, φC(δt)e^{−(x2−x1)HC}φC(δ0)ΩC

h(x2)dx1dx2. But ΩC, φC(δ0)e^{−(x2−x1)HC}φC(δt)ΩC

is real by the FKN formula and hence ΩC, φC(δ0)e^{−(x2−x1)HC}φC(δt)ΩC

= ΩC, φC(δt)e^{−(x2−x1)HC}φC(δ0)ΩC , which yields:

I(t, x) = 2R

−∞<x₁≤x₂<∞ h(x1)r(t, x2−x1)h(x2−x) +h(x¹−x)r(t, x²−x¹)h(x²)

dx¹dx², (17) for

r(t, x) = ΩC, φC(δ0)e^−xHCφC(δt)ΩC

=W^C(t,ix), x≥0.

We have seen thatW^C(t, y) =W^C(t,−y) for (t, y)∈V^β+ iΓ, which when restricted tot∈Sβ,<y= 0, yieldsW^C(t,ix) =W^C(t,−ix).

Therefore exchanging the variablesx1andx2 in the r.h.s. of (17), we obtain:

2I(t, x) = Z

IR2

h(x1)W^C(t, x2−x1)h(x2−x)dx1dx2=h ? h⁻?W^C(t, x), (18) whereh⁻(x) :=h(−x).

Wightman two-point function for the thermal P(φ)₂ model

Let Ωβ be the GNS vector for the thermal state ωβ, and let Pβ and Lβ be the generators of space-time translations such thatPβΩβ =LβΩβ = 0. We recall that ϕβ(h) forh∈C0IR^∞(IR) is the field operator in the GNS representation.

LetW^β(t, x) be the two-point function for the thermalP(φ)2 model, defined by ϕβ(h1)Ωβ,e^itLβϕβ(αxh2)Ωβ

=h1? h⁻2 ?W^β(t, x), (19) forhi∈C0IR^∞(IR). Since dµ_∞is the Euclidean measure for the thermalP(φ)2model on the line, we have

W^β(it, x) =W^C(t,ix), 0< t < β/2, x∈IR. (20) As we have seen,W^C(t, y) is holomorphic inVβ+iΓ. Thus we deduce from (20) that W^β(t, x) is holomorphic inΓ + iV^β. Applying (19) withh¹ =h² =:h, we deduce from this fact that

ϕβ(h)Ωβ ∈ D(e^−(sL+yP)/2) ∀(s, y)∈Vβ.

But by the spectral theorem this clearly implies that W^β(t, x) is holomorphic in IR²+ iVβ. ut

5 Outlook

The condition of locality leads to strong constraints on the general form of the thermal two-point functions which allow one to apply the techniques of the Jost–

Lehmann–Dyson representation. As has been shown by Bros and Buchholz [6][7], the interacting two-point functionW^β can be represented in the form

W^β(t, x) = Z _∞

0

dmD^β(x, m)Wβ⁽⁰⁾(t, x, m).

HereD^β(x, m) is a distribution inx, mwhich is symmetric inx, and Wβ⁽⁰⁾(x, m) = (2π)⁻¹

Z

dνdp ε(ν)δ(ν²−p²−m²)(1−e^−βν)⁻¹e^i(νt−px) is the two-point correlation function of the free field of massmin a thermal equilib- rium state at inverse temperature β. In contrast to the vacuum case, the damping factors D^β(x, m) depend in general in a non-trivial way on the spatial variables x.

They describe the dissipative effects of the thermal background on the propagation of sharply localized excitations.

If the underlying equilibrium state satisfies the relativistic KMS condition, the function Dβ(x, m) is regular in x and admits an analytic continuation into the domain{z∈C| |=z|< β/2}.

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