Temperature for double-cones in 2D boundary CFT P. Martinetti G - PowerPoint PPT Presentation

Temperature for double-cones in 2D boundary CFT P. Martinetti G¨ ottingen Universit¨ at Vietri sul Mare, September 2009

Outline: 1. Physical interpretation of the modular group as a flow of time 2. Wedges in Minkowski space-time 3. Double-cones in Minkowski space-time 4. Double-cones in 2d boundary conformal field theory

1. Time flow from the modular group Time, state and temperature A : algebra of observables of a system, α t : time evolution (e.g. α t a = e − iHt ae iHt ). An equilibrium state ω at temperature β − 1 is a state that satisfies the KMS condition: ω (( α t a ) b ) = ω ( b ( α t + i β a )) ∀ a , b ∈ A .

”Von Neumann algebras naturally evolve with time” ( Connes ) Tomita’s operator:  S a Ω → a ∗ Ω - a von Neumann algebra A acting on H   ⇒ yields a 1-parameter group - a vector Ω in H cyclic and separating σ of automorphisms of A (modular group) The state ω : a �→ � Ω , a Ω � is KMS with respect to σ s , ω (( σ s a ) b ) = ω ( b ( σ s − i a )) ∀ a , b ∈ A , s ∈ R . Hence ω is thermal at temperature − 1 with respect to the evolution σ s .

”Von Neumann algebras naturally evolve with time” ( Connes ) Tomita’s operator:  S a Ω → a ∗ Ω - a von Neumann algebra A acting on H   ⇒ yields a 1-parameter group - a vector Ω in H cyclic and separating σ of automorphisms of A (modular group) The state ω : a �→ � Ω , a Ω � is KMS with respect to σ s , ω (( σ s a ) b ) = ω ( b ( σ s − i a )) ∀ a , b ∈ A , s ∈ R . Hence ω is thermal at temperature − 1 with respect to the evolution σ s . Writing α − β s . = σ s , ω (( α − β s a ) b ) = ω ( b ( α − β s + i β a )) An equilibrium state at temperature β − 1 is a faithful state over the algebra of observables whose modular group σ s is the physical time translation, up to rescaling t = − β s .

� time flow α t = = = = = ⇒ equilibrium state ω temperature β − 1 KMS � state ω = = = = = = = = = = ⇒ time flow α − β s temperature β − 1 modular theory The thermal time hypothesis (Connes, Rovelli 1993): assuming the system is in a thermal state at temperature β − 1 , then the physical time t is the modular flow up to rescaling t = − β s . If another notion of time is available (e.g. geometrical time τ ), one should check that τ = t , i.e. β = − τ s . � state = ⇒ temperature time

2. Temperature for the wedge Bisognano, Wichman, Sewell � algebra of observables A ( 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 = | τ s | = 2 π a = T − 1 Unruh . The temperature is constant along a given trajectory.

Same analysis for other open regions O of Minkowski space-time ? ◮ The vacuum modular group σ O s must have a geometrical action. ◮ The orbits must coincide with the trajectories of some observers with proper time τ . ◮ The ratio τ s should be constant along each orbit.

Same analysis for other open regions O of Minkowski space-time ? ◮ The vacuum modular group σ O s must have a geometrical action. ◮ The orbits must coincide with the trajectories of some observers with proper time τ . ◮ The ratio τ s should be constant along each orbit. This last assumption may be relaxed: to identify ∂ s to ∂ τ , one only needs ∂ s to be normalised, = ∂ s ∂ t . with β . = � ∂ s � . β Putting ∂ t = ∂ τ then yields ∂ τ = ∂ s β ⇒ β = | d τ ds | .

Same analysis for other open regions O of Minkowski space-time ? ◮ The vacuum modular group σ O s must have a geometrical action. ◮ The orbits must coincide with the trajectories of some observers with proper time τ . ◮ The ratio τ s should be constant along each orbit. This last assumption may be relaxed: to identify ∂ s to ∂ τ , one only needs ∂ s to be normalised, = ∂ s ∂ t . with β . = � ∂ s � . β Putting ∂ t = ∂ τ then yields ∂ τ = ∂ s β ⇒ β = | d τ ds | . ◮ For wedges, d τ ds = constant along a given orbit = τ s .

Same analysis for other open regions O of Minkowski space-time ? ◮ The vacuum modular group σ O s must have a geometrical action. ◮ The orbits must coincide with the trajectories of some observers with proper time τ . ◮ The ratio τ s should be constant along each orbit. This last assumption may be relaxed: to identify ∂ s to ∂ τ , one only needs ∂ s to be normalised, = ∂ s ∂ t . with β . = � ∂ s � . β Putting ∂ t = ∂ τ then yields ∂ τ = ∂ s β ⇒ β = | d τ ds | . ◮ For wedges, d τ ds = constant along a given orbit = τ s . ◮ β still makes sense when it is no longer constant ⇒ local equilibrium temperature.

3. Temperature for the double-cone P. M., Rovelli � algebra of observables A ( D ) D − → vacuum modular group σ D s D = ϕ ( W ) for a conformal map ϕ . So for conformal qft ( Hislop, Longo ): uniformly accelerated observer’s trajectory = orbit of the modular group τ ∈ ] − τ 0 , + τ 0 [ s ∈ ] − ∞ , + ∞ [ T L X −L Ratio τ s no longer constant, β ( τ ) = | d τ ds | = 2 π � 1 + a 2 L 2 − ch a τ ) . La 2 (

Along a given orbit, the inverse temperature β ( τ ), − τ 0 < τ < τ 0 varies: 7 6 L = 10 5 5 4 L = 10 4 3 2 L = 10 3 1 -10 -5 5 10

Along a given orbit, the inverse temperature β ( τ ), − τ 0 < τ < τ 0 varies: 7 6 L = 10 5 5 4 L = 10 4 3 2 L = 10 3 1 -10 -5 5 10 The conformal map 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 )) where a is the acceleration characterizing the modular orbit of ϕ − 1 ( x ).

work in progress with R. Longo 3. Double-cone in 2d boundary CFT and K. H. Rehren t I 4/3 1 I u = t+x 2 A CFT on the half plane ( t , x > 0) yields a chiral net I �→ A ( I ) , I =] A , B [ ∈ R , x 0 and generates a net of double-cone algebras (t,x) O = I 1 × I 2 �→ A ( O ) . I v = t ! x 1 One can build on A ( O ) a state whose associated modular group has a geometrical action.

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

Modular group For a pair of symmetric intervals I 1 , I 2 , i.e. σ ( I 1 ) = σ ( I 2 ) = I , the 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 ) , u s = ρ − ◦ m ◦ λ s ◦ m − 1 ◦ σ ( v ) , 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 .

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

B 1 u A 0 � 1 � 1 Other orbits have more complicated B dynamics (e. g. the sign of the accelera- tion may change). v � 1 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 ) . ds ln u s = ⇒ τ ( s ) = ln u s − ln u 0 = 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

Contrary to double-cones in Minkowski space-time, the temperature along the boost-orbit does not present any plateau region. 0.5 0.4 0.3 0.2 0.1 � 1.0 � 0.5 0.5 1.0 ◮ What happens far from the boundary (require double-cone defined by a non-symmetric pair of intervals) ?