The thermal time hypothesis: geometrical action of the modular group - PowerPoint PPT Presentation

The thermal time hypothesis: geometrical action of the modular group in 2D conformal field theory with boundary Pierre Martinetti Universit` a di Roma Tor Vergata and CMTP eminaire CALIN, LIPN Paris 13, 8 th February 2011 S´ collaboration R. Longo, K.-H. Rehren Review in Mathematical Physics 22 3 (2010) 1-23

Outline: 1. Modular group as a flow of time 2. Double-cones in 2d boundary conformal field theory 3. Vacuum modular group for free Fermi fields

1. Time flow from the modular group Modular group ”Von Neumann algebras naturally evolve with time” ( Connes ) Let A be a von Neumann algebra equipped with a one parameter group of automorphism { σ s , s ∈ R } . A weight (i.e. positive linear map) ϕ on A satisfies the modular condition iff - ϕ = ϕ ◦ σ s , ∀ s ∈ R , - for every a , b ∈ n ϕ ∩ n ∗ ϕ ( n ϕ = { a ∈ A , ϕ ( a ∗ a ) < + ∞} ) there exists a bounded continuous function F ab , analytic on the strip 0 ≤ Im z < 1 such that F ab ( s ) = ϕ ( σ s ( a ) b ) , F ab ( s + i ) = ϕ ( b σ s ( a )) . ◮ Each weight ϕ satisfies the modular condition with respect to at most one unique group of automorphism σ s .

 - a von Neumann algebra A acting on H  ⇒ Tomita’s operator:  S a Ω → a ∗ Ω - a vector Ω in H cyclic and separating 2 where ∆ = ∆ ∗ > 0 and J is unitary, antilinear. 1 Polar decomposition: S = J ∆ Tomita’s Theorem: ∆ it A ∆ − it = A hence t �→ σ s : a �→ σ s ( a ) . = ∆ is a ∆ − is is a 1 parameter group of automorphism. Moreover the state ω : a �→ � Ω , a Ω � satisfies the modular condition with respect to σ s . ◮ mathematical importance: Ω ′ � = Ω gives the same modular group, modulo inner automorphism. Classification of factors. ◮ physical importance: ω is KMS with respect to σ s , with temperature − 1, ω ( σ s ( a ) b ) = ω ( b σ s − i ( a )) .

Thermal-time hypothesis Can σ s be interpreted as a real physical time flow ? σ s ( a ) = e iHs ae − iHs H = ln ∆ yields or - A carries a representation of a symmetry group G of spacetime (e.g. Poincar´ e), ⇒ geometrical action of the modular group, - σ s is generated by elements of g = - the orbit of a point under this geometric action is timelike. But the tangent vector ∂ s to these orbits must be normalised, � � = ∂ s dt � = | dt ∂ t . with β . � � = � ∂ s � = � ∂ t ds | . � � β ds Writing α − β s . = σ s , ω (( α − β s a ) b ) = ω ( b ( α − β s + i β a )) . ◮ ω is an equilibrium state at temperature β − 1 with respect to the time evolution t = − β s .

Algebraic field theory Haag, Kastler ... Buchholz, Fredenhagen A net of algebras of local observables is a map O ∈ B (Minkovski) → A ( O ) where A ( O )’s are C ∗ -algebras fulfilling - isotony: O 1 ⊂ O 2 = ⇒ A ( O 1 ) ⊂ A ( O 2 ), - locality: O 1 spacelike to O 2 = ⇒ [ A ( O 1 ) , A ( O 2 )] = 0, together with an irreducible representation π on an Hilbert space H such that e covariance: U (Λ) π ( A ( O )) U ∗ (Λ) = π ( A (Λ O )) for a unitary - Poincar´ representation U of the Poincar´ e group G , - vacuum: there exists a vector Ω ∈ H such that U (Λ)Ω = U (Λ) ∀ Λ ∈ G . Ω defines the vacuum state ω : a �→ � Ω , a Ω � . In the associated GNS representation ( the vacuum representation ) one defines M ( O ) = π ( A ( O )) ′′ which is the von Neumann algebra of local observables associated to O .

Wedge and Unruh temperature Bisognano, Wichman, Sewell � algebra of observables M ( W ) W − → vacuum modular group σ W → boosts → geometrical action s uniformly accelerated observer’s trajectory orbit of the modular group = τ ∈ ] − ∞ , + ∞ [ s ∈ ] − ∞ , + ∞ [ T X W β = | d τ ds | = | τ s | = 2 π a = T − 1 Unruh . ◮ The temperature is constant along a given trajectory, and vanishes as a → 0.

Double-cone in Minkowski space Hislop, Longo; P.M., Rovelli � algebra of observables M ( D ) D − → vacuum modular group σ D s D = ϕ ( W ) for a some conformal map ϕ . For a Conformal Field Theory: uniformly accelerated observer’s trajectory = orbit of the modular group τ ∈ ] − τ 0 , + τ 0 [ s ∈ ] − ∞ , + ∞ [ T L X −L β ( τ ) = | d τ ds | = 2 π � 1 + a 2 L 2 − ch a τ ) . La 2 ( ◮ T D . = 1 β is not constant along the orbit, and does not vanish for a = 0: π k b L ≃ 10 − 11 � T D ( L ) a =0 = K → thermal effect for inertial observer. L

Temperature, horizon, conformal factor ◮ Physical argument: for eternal observers, causal horizon ⇐ ⇒ acceleration. For non-eternal observers, whatever a , there is a ”life horizon” � D = future(birth) past(death) . ◮ Mathematical argument: ϕ : W → D induces on W a metric ˜ g , g ( U , V ) = g ( ϕ ∗ U , ϕ ∗ V ) = C 2 g ( U , V ) . ˜ The double-cone temperature is proportional to the inverse of C , β ( x ) = 2 π a ′ C ( ϕ − 1 ( x )) . ϕ shrinks W to D , hence C cannot be infinite.

3. Double-cone in 2d boundary CFT Longo, P. M., Rehren t Boundary CFT I 4/3 CFT on the half plane ( t , x > 0). Conservation 1 of stress energy tensor T with zero-trace imply I u = t+x 2 1 2( T 00 + T 01 ) = T L ( t + x ) , x 0 1 2( T 00 − T 01 ) = T R ( t − x ) . (t,x) Boundary condition (no energy flow across the boundary x = 0) implies T L = T R = T . I v = t ! x 1 T yields a chiral net of local v.Neumann algebras I =( A , B ) ⊂ R �→ A ( I ) := { T ( f ) , T ( f ) ∗ : supp f ⊂ I} , as well as a net of double-cone algebras O = I 1 × I 2 �→ M ( O ) . = M ( I 1 ) ∨ M ( I 2 ) .

From the boundary to the circle A extends to a chiral net over the intervals of the circle, via Cayley transform: z = 1 + ix ⇒ x = ( z − 1) / i 1 − ix ∈ S 1 ⇐ ∈ R ∪ {∞} . z + 1 Square and square root: 2 x x �→ σ ( x ) . z �→ z 2 ⇐ ⇒ = 1 − x 2 , √ 1 + x 2 − 1 z �→ ±√ z x �→ ρ ± ( x ) = ± ⇐ ⇒ . x A pair of symmetric intervals: I 1 , I 2 ⊂ R such that σ ( I 1 ) = σ ( I 2 ) = I . ⇒ I 1 = ( − 1 A , − 1 I 2 = ( A , B ) = B ) .

M¨ obius covariance In Minkowski space, the Poincar´ e group is both the covariance automorphism group and the group of invariance of the vacuum. The net of algebra A ( I ) is covariant under an action of Diff( S 1 ). But the vacuum is only M¨ obius invariant where obius = PSL (2 , R ) = SL (2 , R ) / {− 1 , 1 } M¨ acts on ¯ R as � � x �→ gx = ax + b a b g = : cx + d . c d ◮ Two equivalent points of view: S 1 or ¯ R ; three important one-parameter subgroups of M¨ obius � e � 1 s cos ϕ sin ϕ � � � � 0 t 2 2 2 R ( ϕ ) = , δ ( s ) = , τ ( t ) = , − sin ϕ cos ϕ s 0 e 0 1 2 2 2 acting as δ ( s ) x = e s x on ¯ τ ( t ) x = x + t on ¯ R ( ϕ ) z = e i ϕ z on S 1 , R , R .

Modular group Given a pair of symmetric intervals I 1 , I 2 such that I 1 ∩ I 2 = ∅ . Consider the state ϕ = ( ϕ 1 ⊗ ϕ 2 ) ◦ χ where χ : A ( I 1 ) ∨ A ( I 2 ) → A ( I 1 ) ⊗ A ( I 2 ) (split property) , ϕ k = ω ◦ Ad U ( γ k ) with ω the vacuum and γ k a diffeomorphism of S 1 such that z �→ z 2 on I k . The associated modular group has a geometrical action ( u , v ) ∈ O �→ ( u s , v s ) ∈ O s ∈ R , with orbits ρ + ◦ m ◦ λ s ◦ m − 1 ◦ σ ( u ) ∈ I 2 , u s = ρ − ◦ m ◦ λ s ◦ m − 1 ◦ σ ( v ) ∈ I 1 , v s = where λ s ( x ) = e s x is the dilation of R , and m is a M¨ obius transformation which maps R + to I = σ ( I 1 ) = σ ( I 2 ).

Implicit equation of the orbits: ( u s − B )( Bu s + 1) · ( v s − B )( Bv s + 1) ( u s − A )( Au s + 1) ( v s − A )( Av s + 1) = const , B 1 u A ◮ This equation only depends on the end 0 points of I 2 = ( A , B ) , I 1 = ( − 1 A , − 1 B ). � 1 � 1 ◮ All orbits are time-like, hence β = | d τ ds | B makes sense as a temperature. ◮ One and only one orbit is a boost (const = 1) and thus is the trajectory of a uniformly accelerated observer. v � 1 A

Explicit equation of the orbits: I ∈ R + = ⇒ A = tanh λ A 2 , B = tanh λ B ⇒ I 2 = ( A , B ) ⊂ (0 , 1) = 2 . u ∈ ( A , B ) = tanh λ for λ A < λ < λ B , σ ( u ) = sinh λ, 2 A ) = − coth λ ′ − 1 B , − 1 for λ A < λ ′ < λ B , σ ( v ) = sinh λ ′ . v ∈ ( 2 B 1 √ ( e s k a − k b ) 2 +( e s k ab − k ba ) 2 − ( e s k a − k b ) u u s = , e s k ab − k ba A − √ ba ) 2 − ( e s k ′ ( e s k ′ b ) 2 +( e s k ′ a − k ′ b ) a − k ′ ab − k ′ 0 = v s e s k ′ ab − k ′ ba where k i . = sinh λ − sinh λ i , k ij . � 1 = k i sinh λ j . � 1 B ◮ complicated dynamics (e. g. the sign of the acceleration may change). v ◮ difficult to parametrize such a curve by its proper length τ , hence difficult to find the temperature ds � 1 d τ . A

Temperature on the boost trajectory Constant acceleration: d τ 2 = du dv hence √ β = d τ ds = u ′ v ′ with ′ = d ds . On the boost orbit, v s = − 1 u s hence β = u ′ u = d ⇒ u s = u o e τ ( s ) . ⇒ τ ( s ) = ln u s − ln u 0 = ds ln u s = Knowing = ( u s − A )( Au s + 1)( B − u s )( Bu s + 1) s = f AB ( u s ) . u ′ . ( B − A )(1 + AB ) · (1 + u 2 s ) one finally gets β ( τ ) = f AB ( u o e τ ) . u o e τ