The time slice axiom in perturbative QFT on globally hyperbolic - PowerPoint PPT Presentation

The time slice axiom in perturbative QFT on globally hyperbolic spacetimes Bruno Chilian June 7 2008

Overview Introduction: The time slice axiom in AQFT Outline of the proof First step: Wick-Polynomials Second step: Perturbation Theory

Motivation: Predictability in Physics ◮ Requirement on physical theory: Future should be predictable from present ◮ In classical physics: Well-posed initial value problem for field equation ◮ In QFT: Initial value problem not well-posed, status of field equation unclear

How is predictability implemented in AQFT?

How is predictability implemented in AQFT? M A ( M ) ⊂ A ( M )

How is predictability implemented in AQFT? M M ′ A ( M ) ⊃ A ( M ′ )

How is predictability implemented in AQFT? M M ′ A ( M ) ⊂ A ( M ) Cauchy surface

How is predictability implemented in AQFT? The time slice axiom M M ′ A ( M ) ⊂ A ( M ′ ) Cauchy surface

Outline of the proof is organized as follows: First step: Wick-Polynomials 1. Review of Wick polynomial algebra 2. Time slice axiom for Wick polynomials Second step: Perturbation Theory 3. Review of perturbation theory 4. Time slice axiom for interacting theories

The algebra of the free field ◮ Free scalar field ϕ satisfies ( ✷ + m 2 − ξ R ) f � � = 0 and [ ϕ ( f ) , ϕ ( g )] = � f , ∆ g � . ϕ ◮ Algebra of the free field consists of functionals N � � F ( ϕ ) = d x 1 · · · d x n ϕ ( x 1 ) · · · ϕ ( x n ) f n ( x 1 , . . . , x n ) n = 0 ◮ Initial value problem for Klein-Gordon equation is well posed on globally hyperbolic spacetimes ⇒ time slice axiom holds for the free theory.

The algebra of Wick polynomials ◮ To treat interactions, nonlinear functionals in the field must be used. ◮ Smear normally ordered products with test-distributions: � φ ⊗ n ( f ) = d x 1 · · · d x n : ϕ ( x 1 ) · · · ϕ ( x n ) : f ( x 1 , . . . , x n ) ◮ To get a well-defined algebra, test-distributions f must satisfy condition on their wavefront sets: WF ( f ) ∩ V n − ∪ V n + = ∅

Time slice axiom in terms of smearing distributions Let T n ( M ) be the space of compactly supported distributions f on M n with WF ( f ) ∩ V n − ∪ V n + = ∅ . Proposition Let f ∈ T n ( M ) and let N be a neighborhood of a Cauchy surface in the past of supp f. Then there exists a g ∈ T n ( M ) s.th. ◮ g = f + ( ✷ + m 2 − ξ R ) h, where h ∈ T n ( M ) and ◮ supp g ⊂ N .

Time slice axiom in terms of smearing distributions Let T n ( M ) be the space of compactly supported distributions f on M n with WF ( f ) ∩ V n − ∪ V n + = ∅ . Proposition Let f ∈ T n ( M ) and let N be a neighborhood of a Cauchy surface in the past of supp f. Then there exists a g ∈ T n ( M ) s.th. ◮ g = f + ( ✷ + m 2 − ξ R ) h, where h ∈ T n ( M ) and ◮ supp g ⊂ N . Proposition is equivalent to time slice axiom because Wick polynomials have the form � φ ⊗ n ( f ) = d x 1 · · · d x n : ϕ ( x 1 ) · · · ϕ ( x n ) : f ( x 1 , . . . , x n ) and : ϕ ( x 1 ) · · · ϕ ( x n ) : satisfies the Klein-Gordon equation.

Main idea for proof: Cauchy-Evolution of distributions Let f be a distribution with compact support. For two Cauchy surfaces in the past of supp f , let χ be a smooth function which is zero in the past and one in the future. supp f χ = 1 Now define another distribution by g = f − ( ✷ + m 2 − ξ R ) χ ∆ adv f . supp g It has compact support between the Cauchy χ = 0 surfaces.

Adaption of this simple argument Generalize to n variables Use suitable partition of unity to check support property Check wavefront property Use H¨ ormanders theorem on the propagation of singularities

The time slice axiom in perturbation theory ◮ Perturbation theory expresses interacting fields by free fields. ◮ Therefore: A g ( M ) ⊂ A 0 ( M ) . ◮ Using the time slice axiom for the free fields: A g ( M ) ⊂ A 0 ( N ) for a neighborhood N of some Cauchy surface. ◮ If one could express free fields in N by interacting fields in N , one would have A g ( M ) ⊂ A 0 ( N ) ⊂ A g ( N ) which is the time slice axiom for interacting fields. ◮ We need to show that A 0 ( N ) ⊂ A g ( N ) .

Causal perturbation theory ◮ Interacting fields: Constructed from local S-matrices: S g ( f ) = S ( g ) − 1 S ( f + g ) ◮ S-matrices: Power series of Wick polynomials causal factorization property supp f S ( f + g + h ) = S ( f + g ) S ( g ) − 1 S ( g + h ) if supp f is in the future and supp h supp g in the past of some Cauchy surface. supp h

Starting idea for the proof of A 0 ( N ) ⊂ A g ( N ) ◮ Elements S g ( f ) also possess causal factorization property: S g ( f 1 + f 2 + f 3 ) = S g ( f 1 + f 2 ) S g ( f 2 ) − 1 S g ( f 2 + f 3 ) if supp f 1 is in the future and supp f 3 in the past of some Cauchy surface. ◮ Multiply interacting fields in a clever way to cancel out interactions. ◮ Obtain some free field. ◮ Check if every free field can be obtained in this way. ◮ Then: A 0 ( N ) ⊂ A g ( N ) .

Noncompact interaction regions If supp g is past compact, define the interacting field S g ( f ) def = S b ( f ) , where b must coincide with g in the past of some compact region containing supp f . This definition does not supp g depend on the choice of b. Proof uses causal supp b factorization property. supp f

Prove: Definition of S g ( f ) is independent of chosen b . Let ˜ b also coincide with g in the past of the compact region containing supp f . Set c = ˜ b − b . Then supp g supp ˜ b supp b supp f b ( f ) = S (˜ b ) − 1 S (˜ b + f ) = S ( c + b ) − 1 S ( c + b + f ) S ˜ = S ( c + b ) − 1 S ( c + b ) S ( b ) − 1 S ( f + b ) = S b ( f ) .

Prove: Definition of S g ( f ) is independent of chosen b . Let ˜ b also coincide with g in the past of the compact region containing supp f . Set c = ˜ b − b . Then supp g supp c supp f b ( f ) = S (˜ b ) − 1 S (˜ b + f ) = S ( c + b ) − 1 S ( c + b + f ) S ˜ = S ( c + b ) − 1 S ( c + b ) S ( b ) − 1 S ( f + b ) = S b ( f ) .

Compensating interactions ◮ Consider the expression S g ( − b ′ ) − 1 S g ( − b ′ + f ) , where b ′ coincides with g in a neighborhood of supp f . ◮ Using causal factorization, it can be shown that S g ( − b ′ ) − 1 S g ( − b ′ + f ) = S ( b − ) − 1 S ( f ) S ( b − ) , where supp b − is in the past and supp f in the future of some Cauchy surface. 1st observation 2nd observation S ( b − ) − 1 S ( f ) S ( b − ) ∈ A g ( N ) S ( b − ) − 1 S ( f ) S ( b − ) ∈ A 0 ( M )

◮ The map S ( f ) �→ S g ( − b ′ ) − 1 S g ( − b ′ + f ) = S ( b − ) − 1 S ( f ) S ( b − ) defines an endomorphism of A 0 ( M ) , whose image is contained in A g ( N ) . ◮ For each relatively compact region, it is just conjugation by a certain invertible element of A 0 ( M ) . ◮ The last step is to show that this endomorphism is invertible, i.e. S ( b − ) S ( h ) S ( b − ) − 1 does not depend on the chosen b − . ◮ Place additional restriction on the choice of b − .

Summary ◮ The time slice axiom holds in perturbative AQFT. ◮ One only needs the causal factorization property... ◮ and the time slice axiom for Wick polynomials.

supp b supp b ′ N ′ Σ K N Σ 1 S Σ 2 Locally compensating the interaction g with b ′ : S g ( − b ′ ) − 1 S g ( − b ′ + f ) = S ( b − ) − 1 S ( f ) S ( b − )