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

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´ opolis, 24 - 28 July 2006 Plan of the talk: • The space-cuto ff P ( ϕ ) 2 model at positive tem- perature. • Euclidean approach. • Thermodynamical limit and Nelson symme- try. • The relativistic KMS condition.

The free neutral field at temperature β − 1 : • One particle space: h := H − 1 2 (I R) with norm � u � 2 = R | u ( k ) | 2 d k � 2 ; 1 I 2( k 2 + m 2 ) • Weyl algebra: W ( h ) := Weyl algebra over h ; • free dynamics: t ( W ( g )) := W (e i t ǫ g ) , ǫ ( k ) = ( k 2 + m 2 ) 1 τ 0 2 ; • KMS state: ( τ , β ) KMS state: β ( W ( g )) := e − 1 4 ( g, (2 ǫ ) − 1 (1+2 ρ ) g ) L 2 , ω 0 for ρ = (e βǫ − 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: 1 1 2 g ) ⊗ W F ( ρ 2 g ); W AW ( g ) := W F ((1 + ρ ) • GNS vector: Ω AW = Ω ⊗ Ω ; • generator of dynamics (Liouvillean): L 0 = d Γ ( ǫ ) ⊗ 1 l − 1 l ⊗ d Γ ( ǫ ).

Space-cuto ff P ( ϕ ) 2 model at temperature β − 1 : • P ( λ ) = � 2 n j =0 a j λ j , real, bounded below polynomial, • space-cuto ff interaction: � l V l := : P ( φ AW ( x )) : d x, l > 0 , − l defined as � l V l = lim : P ( φ AW ( δ κ ( . − x ))) : d x k → + ∞ − l : : is the Wick ordering with respect to the 0 − temperature covariance C vac ( g, g ) = ( g, (2 ǫ ) − 1 g ) L 2 , δ k ( . ) is an approximation of δ ( . ), φ AW ( g ) is the field operator associated to AW representation.

One can show the following results: • H l = L 0 + V l is essentially selfadjoint on D ( L 0 ) ∩ D ( V l ), • the free KMS vector Ω belongs to D (e − β H l / 2 ), • if Ω l = � e − β H l / 2 ) Ω � − 1 e − β H l / 2 ) Ω and ω l ( A ) = ( Ω l , A Ω l ), τ l t ( A ) = e i tH l A e − i tH l , then ( B , τ l , ω l ) is a β − KMS system.

β − 1 : The translation invariant P ( ϕ ) 2 model at temperature − : B ( I ) := { W AW ( g ) , supp g ⊂ I } ”, I ⊂ I R bounded open interval. ( ∗ ) . B := � R B ( I ) I ⊂ I Then [GJ]: • (existence of limit dynamics): the limit τ t ( A ) := lim l →∞ τ l t ( A ) exists for A ∈ B and defines a group of ∗− automorphisms, • (existence of limit state): the limit ω ( A ) := lim l →∞ ω l ( 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 ω 0 .

Stochastic positivity aka Euclidean approach to KMS states: Multi-time analyticity: Let ( B , τ t , ω ) a KMS system at temperature β − 1 . Green’s functions: G ( t 1 , · · · t n ; A 1 , · · · , A n ) := ω ( Π n i =1 τ t i ( A i )) . (Araki): the Green’s functions are holomorphic in I n + := { ( z 1 , · · · , z n ) | Im z i < Im z i +1 , Im z n − Im z 1 < β } β Euclidean Green’s functions: E G ( s 1 , · · · , s n ; A 1 , · · · , A n ) := G (i s 1 , · · · , i s n ; A 1 , · · · A n ) , defined for ( s 1 , · · · , s n ) with s i ≤ s i +1 , s n − s 1 ≤ β , (equivalent to n points on the circle S β ordered trigonometrically)

Stochastic positivity (Klein-Landau): there exists U ⊂ B abelian subalgebra, such that: i ) E G ( s 1 , · · · , s n ; A 1 , · · · , A n ) ≥ 0 , for A i ∈ U , A i ≥ 0, ii ) B is generated by τ t ( 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 X t , t ∈ S β , with values in σ ( U ) (spectrum of U ) such that E G ( s 1 , · · · , s n ; A 1 , · · · , A n ) = Q Π n � i =1 A i ( X t i )d µ , ( U ∼ algebra of functions on σ ( U ).)

The space-cuto ff P ( ϕ ) 2 model at positive temperature is stochastically positive. Concrete choice of stochastic process : • Q = S ′ R ( S β × I R), distributions on the cylinder S β × I R I • Σ = Borel σ − algebra on Q • measure d µ l defined by: − � β � l d µ l := 1 − l : P ( ϕ ( t,x )):d t d x d φ C , e 0 Z l for d φ C Gaussian measure on Q with covariance C ( u, u ) = ( u, ( ∂ 2 t + ∂ 2 x + m 2 ) u ), � Z l partition function so that Q d µ l = 1,

Existence of thermodynamical limit: Key result to prove: � � lim F ( φ )d µ l =: F ( φ )d µ ∞ l →∞ Q Q exists and is a Borel probability measure on S ′ R ( S β × I R). I 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 S β 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 ) τ t α x ϕ ( g )) =: g ⋆ g ⋆ C β ( t, x ) , for C β ( t, x )” = ” ω ( ϕ (0) τ t ϕ ( x )) . relativistic KMS condition: C ( t, x ) should be holomorphic in R 2 + i V β , for I V β = { ( t, x ) || x | < inf( t, β − t ) } . Nelson symmetry yields the formal identity: C β (i t, x ) = C 0 ( t, i x ) , 0 ≤ t ≤ β , 0 ≤ x < ∞ , where C 0 ( t, y ) is the two point function on the circle: C 0 ( t, y ) = ( ϕ (0) Ω C , e i yH C e i tP C ϕ (0) Ω C ) , y ∈ I R , t ∈ S β . • H C P ( ϕ ) 2 Hamiltonian on the circle S β , • P C momentum operator on S β , • Ω C unique ground state of H C .

Properties of C 0 ( t, y ) : • Spectral condition on the circle (Haifets-Ossipov): one has P 2 C ≤ H 2 C hence C 0 ( t, y ) is the boundary value of a function holomorphic in S β × I R + i V + for V + = {| t | < y } × t V + x

• locality on the circle: if ( t, y ) ∈ V β then [ ϕ (0) , e i yH C e i tP C ϕ (0)e − i tP C e − i yH C ] = 0 hence C 0 ( t, y ) is real valued on V β 0 t 1 t 2 β By Schwarz reflection principle: C 0 is holomorphic in V β + i( V + ∪ V − ).

• edge of the wedge theorem: C 0 is holomorphic in V β + i B (0 , δ ) and satisfies C 0 ( t, y ) = C 0 ( t, y ), We note also that C 0 ( t, i y ) is real ∈ ∈ for t ∈ S β , y ∈ I R (Euclidean measure is real!) t x This implies C 0 ( t, y ) = C 0 ( t, − y ) for ( t, y ) ∈ V β + i B (0 , δ ).

end of the proof: Going back to C β , we get that C β is holomorphic in B (0 , δ ) + i V β . R 2 . We want to replace B (0 , δ ) by I Set: • P , L generators of space-time translations in the GNS rep. for ω , • Ω β GNS vector for ω β , so that: C β ( t, x ) = ( ϕ (0) Ω β , e i( tL + xP ) ϕ (0) Ω β ) Then ϕ (0) Ω β is in D (e − ( sL + yP ) / 2 ), ( s, y ) ∈ V β . R 2 + i V β . Hence C β ( t, x ) is holomorphic in I

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