Plan of the talk: The space-cuto ff P ( ) 2 model at positive tem- - - PowerPoint PPT Presentation

Plan of the talk:

• The space-cutoff P(ϕ)2 model at positive tem-

perature.

• Euclidean approach.
• Thermodynamical limit and Nelson symme-

try.

• The relativistic KMS condition.

Relativistic KMS condition for the P(ϕ)2 model at positive temperature Christian G´ erard (Orsay), Christian J¨ akel (Talca) Brazilian Operator Algebras Conference Florian´

• polis, 24 - 28 July 2006

The free neutral field at temperature β−1:

• One particle space: h := H− 1

2 (I

R) with norm u2 =

• I

R |u(k)|2 dk 2(k2+m2)

1 2 ;

• Weyl algebra: W(h) := Weyl algebra over h;
• free dynamics:

τ 0

t (W(g)) := W(eitǫg), ǫ(k) = (k2 + m2)

1 2 ;

• KMS state: (τ, β) KMS state:

ω0

β(W(g)) := e− 1

4 (g,(2ǫ)−1(1+2ρ)g)L2,

for ρ = (eβǫ − 1).

Araki-Woods (GNS) representation:

• GNS Hilbert space: H := Γ(h) ⊗ Γ(h), Γ(h) = bosonic Fock

space over h;

• GNS representation of Weyl operators:

WAW(g) := WF((1 + ρ)

1 2 g) ⊗ WF(ρ 1 2 g);

• GNS vector: ΩAW = Ω ⊗ Ω;
• generator of dynamics (Liouvillean):

L0 = dΓ(ǫ) ⊗ 1 l − 1 l ⊗ dΓ(ǫ).

Space-cutoff P(ϕ)2 model at temperature β−1:

• P(λ) = 2n

j=0 ajλj, real, bounded below polynomial,

• space-cutoff interaction:

Vl := l

−l

: P(φAW(x)) : dx, l > 0, defined as Vl = lim

k→+∞

l

−l

: P(φAW(δκ(. − x))) : dx : : is the Wick ordering with respect to the 0−temperature covariance Cvac(g, g) = (g, (2ǫ)−1g)L2, δk(.) is an approximation of δ(.), φAW(g) is the field operator associated to AW representation.

One can show the following results:

• Hl = L0 + Vl is essentially selfadjoint on D(L0) ∩ D(Vl),
• the free KMS vector Ω belongs to D(e−βHl/2),
• if Ωl = e−βHl/2)Ω−1e−βHl/2)Ω and ωl(A) = (Ωl, AΩl),

τ l

t(A) = eitHlAe−itHl, then (B, τ l, ωl) is a β−KMS system.

B(I) := {WAW(g), supp g ⊂ I}”, I ⊂ I R bounded open interval. B :=

I⊂I R B(I) (∗).

Then [GJ]:

• (existence of limit dynamics): the limit

τt(A) := liml→∞ τ l

t(A) exists for A ∈ B and defines a group of

∗−automorphisms,

• (existence of limit state): the limit ω(A) := liml→∞ ωl(A)

exists for A ∈ B,

• (B, τ, ω) is a β−KMS system,
• ω is translation invariant and locally normal with respect to

the free KMS state ω0. The translation invariant P(ϕ)2 model at temperature

− :

β−1:

Stochastic positivity aka Euclidean approach to KMS states: Multi-time analyticity: Let (B, τt, ω) a KMS system at temperature β−1. Green’s functions: G(t1, · · · tn; A1, · · · , An) := ω(Πn

i=1τti(Ai)).

(Araki): the Green’s functions are holomorphic in In+

β

:= {(z1, · · · , zn)|Imzi < Imzi+1, Imzn − Imz1 < β} Euclidean Green’s functions:

EG(s1, · · · , sn; A1, · · · , An) := G(is1, · · · , isn; A1, · · · An),

defined for (s1, · · · , sn) with si ≤ si+1, sn − s1 ≤ β, (equivalent to n points on the circle Sβ ordered trigonometrically)

Stochastic positivity (Klein-Landau): there exists U ⊂ B abelian subalgebra, such that: i) EG(s1, · · · , sn; A1, · · · , An) ≥ 0, for Ai ∈ U, Ai ≥ 0, ii)B is generated by τt(U), t ∈ I R. (Klein-Landau): if (B, U, τ, ω) is stochastically positive, there exists:

• a probability space (Q, Σ, µ),
• a periodic stochastic process Xt, t ∈ Sβ, with values in σ(U)

(spectrum of U) such that EG(s1, · · · , sn; A1, · · · , An) =

• Q Πn

i=1Ai(Xti)dµ,

(U ∼ algebra of functions on σ(U).)

The space-cutoff P(ϕ)2 model at positive temperature is stochastically positive. Concrete choice of stochastic process:

• Q = S′

I R(Sβ × I

R), distributions on the cylinder Sβ × I R

• Σ = Borel σ−algebra on Q
• measure dµl defined by:

dµl := 1 Zl e

− β

l

−l:P (ϕ(t,x)):dtdxdφC,

for dφC Gaussian measure on Q with covariance C(u, u) = (u, (∂2

t + ∂2 x + m2)u),

Zl partition function so that

• Q dµl = 1,

Existence of thermodynamical limit: Key result to prove: lim

l→∞

• Q

F(φ)dµl =:

• Q

F(φ)dµ∞ exists and is a Borel probability measure on S′

I R(Sβ × I

R). Nelson symmetry (Hoegh-Krohn):

• exchanging the role of t and x one sees that

µ∞ = Euclidean measure of the P(ϕ)2 model on the circle Sβ at temperature 0!

• existence of limit ⇔ uniqueness of the ground state for the

P(ϕ)2 model on the circle.

Relativistic KMS condition (Bros-Buchholz): two point function: ω(ϕ(g)τtαxϕ(g)) =: g ⋆ g ⋆ Cβ(t, x), for Cβ(t, x)” = ”ω(ϕ(0)τtϕ(x)). relativistic KMS condition: C(t, x) should be holomorphic in I R2 + iVβ, for Vβ = {(t, x)||x| < inf(t, β − t)}. Nelson symmetry yields the formal identity: Cβ(it, x) = C0(t, ix), 0 ≤ t ≤ β, 0 ≤ x < ∞, where C0(t, y) is the two point function on the circle: C0(t, y) = (ϕ(0)ΩC, eiyHCeitPCϕ(0)ΩC), y ∈ I R, t ∈ Sβ.

• HC P(ϕ)2 Hamiltonian on the circle Sβ,
• PC momentum operator on Sβ,
• ΩC unique ground state of HC.

Properties of C0(t, y):

• Spectral condition on the circle (Haifets-Ossipov):
• ne has

P 2

C ≤ H2 C

hence C0(t, y) is the boundary value of a function holomorphic in Sβ × I R + iV+ for V+ = {|t| < y}

× V+ t x

• locality on the circle: if (t, y) ∈ Vβ then

[ϕ(0), eiyHCeitPCϕ(0)e−itPCe−iyHC] = 0 hence C0(t, y) valued on Vβ t1 t2 β By Schwarz reflection principle: C0 is holomorphic in Vβ + i(V+ ∪ V−).

is real

∈ ∈ t x

• edge of the wedge theorem:

C0 is holomorphic in Vβ + iB(0, δ) and satisfies C0(t, y) = C0(t, y), We note also that C0(t, iy) is real for t ∈ Sβ, y ∈ I R (Euclidean measure is real!) This implies C0(t, y) = C0(t, −y) for (t, y) ∈ Vβ + iB(0, δ).

end of the proof: Going back to Cβ, we get that Cβ is holomorphic in B(0, δ) + iVβ. We want to replace B(0, δ) by I R2. Set:

• P, L generators of space-time translations in the GNS rep.

for ω,

• Ωβ GNS vector for ωβ,

so that: Cβ(t, x) = (ϕ(0)Ωβ, ei(tL+xP )ϕ(0)Ωβ) Then ϕ(0)Ωβ is in D(e−(sL+yP )/2), (s, y) ∈ Vβ. Hence Cβ(t, x) is holomorphic in I R2 + iVβ.

On the relativistic KMS condition for the P(φ)2 model in Rigorous Quantum Field Theory: A Festschrift for Jacques Bros, G´ erard, C., and J¨ akel, C.D. , Birkhauser (2006), reprint available from christian.jaekel@mac.com