Noncommutative Friedmann-Walker spacetimes, quantum field theory - PowerPoint PPT Presentation

Noncommutative Friedmann-Walker spacetimes, quantum field theory and the Einstein equations Luca Tomassini Universit` a di Roma “Tor Vergata” Joint work with G. Morsella June 21, 2014

Introduction ◮ The DFR model: spacetime and fields ◮ Friedmann expanding backgrounds ◮ Friedmann expanding n.c. spacetimes ◮ Quantum fields and Friedmann equations

The DFR proposal DFR (1995): “Grav. stability under localization experiments”: Determining the localization of a quantum field theoretic observable needs concentration of energy in a region of the size of the uncertainty; extreme precision should cause the formation of a black hole. The following program was outlined: ◮ Derive physically meaningful uncertainty relations between coordinates of spacetime events from gravitational stability under localization experiments. ◮ Promote these coordinates to the status of operators and find commutation relations among them from which the uncertainty relations follow. ◮ Construct quantum fields over the resulting noncommutative spacetime. Starting point: fixed classical background, to be recovered by some L P → 0 procedure.

Noncommutative Minkowski space (DFR) Spacetime uncertainty relations (STUR) derived by the linear approximation: ∆ x 1 + ∆ x 2 + ∆ x 3 � ≥ L 2 � c ∆ t P ∆ x 1 ∆ x 2 + ∆ x 1 ∆ x 3 + ∆ x 2 ∆ x 3 ≥ L 2 P From here, commutation relations (in principle, highly non unique!): [ x µ , x ν ] = iL 2 x µ = x ∗ P Q µν , µ One can show that the STUR are satisfied using the “Quantum conditions” ( Q µν ( ∗ Q ) µν ) 2 = 16 I . [ x µ , Q νρ ] = 0 , Q µν Q µν = 0 , The x µ generate a C ∗ -algebra E , (some of) its states are our n.c. Minkowski. Covariance is granted by the following action of the (full) Poincar´ e group P : α (Λ , a ) ( Q µν ) = Λ µ ′ µ Λ ν ′ α (Λ , a ) ( x µ ) = Λ ν µ x ν + a µ I , ν Q µ ′ ν ′ .

Quantum fields on n.c. Minkowski space A quantum field Φ on the quantum spacetime is defined by � R 4 dk e ikx ⊗ ˆ Φ( x ) = Φ( k ) . It is a map from states on E to smeared field operators, � ω → Φ( ω ) = � ω ⊗ I , Φ( x ) � = R 4 dx Φ( x ) ψ ω ( x ) . The r.h.s. is a quantum field on the ordinary spacetime, smeared with ψ ω defined by ˆ ψ ω ( k ) = � ω, e ikx � . If products of fields are evaluated in a state, the r.h.s. will in general involve non-local expressions. One has � [Φ( ω ) , Φ( ω ′ )] = i d 4 xd 4 y ∆( x − y ) ψ ω ( x ) ψ ω ′ ( y ) . Thus (smeared) non commutative quantum fields are functions from a quantum spacetime to a C ∗ -algebra F (analogue to the one generated by ordinary fields) and are described by elements affiliated to E ⊗ F . Knowledge of the classical commutator entails knowledge of its n.c. spacetime counterpart one.

Curved spacetimes and n.c. Einstein’s equations Problem: generalise the above construction to curved spacetimes. Big problem: make sense of “n.c. Einstein equations” R µν − 1 2 Rg µν = 8 π GT µν (Φ) F (Φ) = 0 , [ x µ , x ν ] = iQ µν ( g ) . Friedmann flat expanding spacetimes with metric (comoving coordinates) ds 2 = dt 2 − a ( t ) 2 ( dx 2 1 + dx 2 2 + dx 2 3 ) . Combination of mathematical simplicity (due to symmetry) and physical relevance (cosmological models).

Uncertainty relations for FFE spacetimes (comoving coordinates) ◮ Black holes do not form if the (positive) excess of proper mass-energy δ E inside a two-surface S of proper area ∆ A contained in a slice of constant universal time t 0 satisfies the inequality: √ � � √ 4 √ π + H 1 ∆ A ≥ G ∆ A c 4 δ E . 4 π c where H ( t ) = a ′ ( t ) / a ( t ) is the Hubble parameter ( a , a ′ > 0). For a box-like localisation region with comoving edges ∆ x c 1 , ∆ x c 2 , ∆ x c 3 , ∆ A = a 2 ( t )(∆ x c 1 ∆ x c 2 + ∆ x c 1 ∆ x c 3 + ∆ x c 2 ∆ x 3 ) = a 2 ( t )∆ A c . Estimate δ E making use of Heisenberg’s uncertainty relations and get + a ′ ( t ) √ ∆ A c � 1 ≥ λ 2 � a 2 ( t )∆ A c P √ 2 , 12 c 4 3 � 1 + a ′ ( t ) √ ∆ A c ≥ λ 2 � �� � P c ∆ t · ∆ A c min a ( t ) √ 2 . 12 c 4 3 t ∈ ∆ t

Solve the first inequality with respect to the comoving area ∆ A c gives ∆ A c ≥ f ( a ( t ) , a ′ ( t )) , √ 3 a / a ′ ) 2 and x 0 is the greatest solution of a certain cubic with f 1 = ( x 0 − c equation from which one has ∆ A c ≥ λ 2 t ∈ ∆ t { a ( t )∆ A c } ≥ λ 2 � t ∈ ∆ t { a ( t ) f ( a ( t ) , a ′ ( t )) } . P P c ∆ t · 2 max 2 max The corresponding quantum uncertainty relations are: ∆ ω A c ≥ λ 2 P 2 | ω ( f ) | , c ∆ ω t (∆ ω x 1 + ∆ ω x 2 + ∆ ω x 3 ) ≥ λ 2 P 2 | ω ( af ) | .

Uncertainty relations for FFE spacetimes (conformal coordinates) If we work in conformal coordinates, an analogous procedure and the approximation ∆ t = ∆ t ∆ τ ∆ τ ≃ a ( t ( τ ))∆ τ gives ∆ ω A c ≥ λ 2 P 2 | ω ( f ) | , (1) c ∆ ω τ (∆ ω x 1 + ∆ ω x 2 + ∆ ω x 3 ) ≥ λ 2 P 2 | ω ( f ) | . (2) where the function f is the same as before.

Building n.c. FFE spacetimes Definition. A C ∗ -algebra E of operators with (self adjoint?) generators x µ , µ = 0 , . . . , 3 affiliated to it, is said to be a concrete covariant realisation of the n.c. spacetime M corresponding to the (classical) spacetime M with global isometry group G if: 1) the relevant STUR are satisfied; 2) there is a (strongly continuous) unitary representation of the global isometry group G under which the operators η transform as their classical counterparts (covariance); 3) there is some reasonable classical limit procedure for L P → 0 such that the η µ become in an appropriate sense commutative coordinates on some space containing the manifold M as a factor. For FFE (De Sitter exluded) G = SO (3) ⋉ R 3 . By isotropy and homogeneity, we restrict attention to c.r. of the form ( x 0 = t ,or x 0 = τ , and ι = 1 , . . . , n ) [ x µ , x ν ] = Q ( t , X ι ) µν , [ x µ , X ι ] = 0 , [ X ι , X ι ] = 0 . 4) The generators of the Friedmann spacetime algebra have commutation relations of the form above. 5) We should in some suitable sense recover the DFR model in the limit a → 1. Items 3) and 5) will not be addressed.

The assumption that the Q ’s only depend on t (or τ ), combined with covariance, has far reaching consequences. Proposition. Let the generators t , x , X ι satisfy 1) and 3) and the components of the two-tensor Q ( t , X ι ) be regular functions of the comm. variables ( t , X ι ). Then the corresponding commutation relations are of the form [ t , x ] = g 1 ( t ) e ( X ι ) , [ x , x ] = m ( X ι ) + m ⊥ ( t , X ι ) , (3) with m ⊥ ( t , X ι ) · e ( X ι ) = 0 and some regular function g . Moreover, the operators e ( X ι ) , m ( X ι ) , m ⊥ ( t , X ι ) transform as vectors under the action of the automorphism α R , R ∈ SO (3). We set m ⊥ ( t , X ι ) = g 2 ( t ) m ⊥ and are left with two arbitrary (regular) functions g 1 , g 2 and nine central generators assembled in a triple e , m , m ⊥ of three-vectors, plus one orthogonality condition.

Proposition. Let x µ , e , m , m ⊥ be as above. Suppose the Quantum Conditions e 2 = I , m 2 m = 0 , ⊥ = I , are satisfied and let f the function on the right hand side of the conformal STUR above. Then, for any state ω ∈ E ∗ in the domain of τ, x , e , m ⊥ , requirement 2) is met if g 1 ( τ ) = g 2 ( τ ) = f ( τ ). Sketch of Proof. The operators e , m ⊥ being central with joint spectrum Σ, we perform the corresponding central decomposition of E ∗ . The proof relies on the inequalities � ∆ ω σ ( x µ )∆ ω σ ( x ν ) d µ ω ( σ ) ≥ 1 � ∆ ω ( x µ )∆ ω ( x ν ) ≥ | ω σ ( Q µν ) | d µ ω ( σ ) . (4) 2 Σ Σ which entail, for example, � 3 3 � � � � � � ∆ ω σ ( t ) 2 ∆ ω σ ( x i ) 2 d µ ω ( σ ) ≥ ∆ ω σ ( t )∆ ω σ ( x i ) d µ ω ( σ ) ≥ � Σ Σ k =1 k =1 ≥ 1 � || ω σ ( g 1 ( t ) e ) || d µ ω ( σ ) = 1 � || e ( σ ) || · | ω σ ( g 1 ( t )) | d µ ω ( σ ) ≥ 2 2 Σ Σ ≥ 1 � ω σ ( g 1 ( t )) d µ ω ( σ ) | = 1 2 | 2 | ω ( g 1 ( τ )) | . Σ

Existence of covariant representations To construct irrep. for the conf. coord., set σ 0 = ( e , m ⊥ ) with e = (1 , 0 , 0) , m ⊥ = (0 , 0 , 1). Thus   0 − 1 0 0 1 0 0 0 Q ( τ, σ s td ) = f σ s td = f   Q ( τ, σ 0 ) = f σ 0 ,   0 0 0 0   0 0 0 0 with Q ( τ, σ s td ) = AQ ( τ, σ 0 ) A T for some invertible matrix A . Setting x = ( τ, x ), � � = i λ 2 f ( x σ 0 0 ) σ s td we have ( Ax σ 0 ) µ , ( Ax σ 0 ) ν µν . An irrep. is thus a suitable odinger operators p , q on L 2 ( R ) and two combination of two Schr¨ central operators. Fixing a diffeomorphism γ : R → sp ( τ ), we must have x σ s td = [ γ ′ ( q ) − 1 f ( γ ( q )) , p ] + , γ ( q ) , aI , bI � � .