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?

M M′ A(M) ⊂A(M′) Cauchy surface

The time slice axiom

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

ϕ

• (✷ + m2 − ξR)f
• = 0 and [ϕ(f), ϕ(g)] = f, ∆g .

◮ Algebra of the free field consists of functionals

F(ϕ) =

N

• n=0
• dx1 · · · dxn ϕ(x1) · · · ϕ(xn) fn(x1, . . . , xn)

◮ 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) =

• dx1 · · · dxn : ϕ(x1) · · · ϕ(xn) : f(x1, . . . , xn)

◮ 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 Tn(M) be the space of compactly supported distributions f

• n Mn with WF(f) ∩ V n

− ∪ V n + = ∅.

Proposition

Let f ∈ Tn(M) and let N be a neighborhood of a Cauchy surface in the past of supp f. Then there exists a g ∈ Tn(M) s.th.

◮ g = f + (✷ + m2 − ξR)h, where h ∈ Tn(M) and ◮ supp g ⊂ N .

Time slice axiom in terms of smearing distributions

Let Tn(M) be the space of compactly supported distributions f

• n Mn with WF(f) ∩ V n

− ∪ V n + = ∅.

Proposition

Let f ∈ Tn(M) and let N be a neighborhood of a Cauchy surface in the past of supp f. Then there exists a g ∈ Tn(M) s.th.

◮ g = f + (✷ + m2 − ξR)h, where h ∈ Tn(M) and ◮ supp g ⊂ N .

Proposition is equivalent to time slice axiom because Wick polynomials have the form φ⊗n(f) =

• dx1 · · · dxn : ϕ(x1) · · · ϕ(xn) : f(x1, . . . , xn)

and : ϕ(x1) · · · ϕ(xn) : 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. Now define another distribution by g = f −(✷+m2−ξR)χ∆advf . It has compact support between the Cauchy surfaces. supp f χ = 1 χ = 0 supp g

Adaption of this simple argument

Generalize to n variables

Use suitable partition of unity to check support property

Check wavefront property

Use H¨

• rmanders theorem on the propagation of singularities

The time slice axiom in perturbation theory

◮ Perturbation theory expresses interacting fields by free

fields.

◮ Therefore: Ag(M) ⊂ A0(M). ◮ Using the time slice axiom for the free fields:

Ag(M) ⊂ A0(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 Ag(M) ⊂ A0(N) ⊂ Ag(N) which is the time slice axiom for interacting fields.

◮ We need to show that A0(N) ⊂ Ag(N).

Causal perturbation theory

◮ Interacting fields: Constructed from local S-matrices:

Sg(f) = S(g)−1S(f + g)

◮ S-matrices: Power series of Wick polynomials

causal factorization property

S(f+g+h) = S(f+g)S(g)−1S(g+h) if supp f is in the future and supp h in the past of some Cauchy surface. supp f supp g supp h

Starting idea for the proof of A0(N) ⊂ Ag(N)

◮ Elements Sg(f) also possess causal factorization property:

Sg(f1 + f2 + f3) = Sg(f1 + f2)Sg(f2)−1Sg(f2 + f3) if supp f1 is in the future and supp f3 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: A0(N) ⊂ Ag(N).

Noncompact interaction regions

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

Prove: Definition of Sg(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 f supp g supp ˜ b supp b S˜

b(f) = S(˜

b)−1S(˜ b + f) = S(c + b)−1S(c + b + f) = S(c + b)−1S(c + b)S(b)−1S(f + b) = Sb(f) .

Prove: Definition of Sg(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 f supp g supp c S˜

b(f) = S(˜

b)−1S(˜ b + f) = S(c + b)−1S(c + b + f) = S(c + b)−1S(c + b)S(b)−1S(f + b) = Sb(f) .

Compensating interactions

◮ Consider the expression Sg(−b′)−1Sg(−b′ + f), where b′

coincides with g in a neighborhood of supp f.

◮ Using causal factorization, it can be shown that

Sg(−b′)−1Sg(−b′ + f) = S(b−)−1S(f)S(b−) , where supp b− is in the past and supp f in the future of some Cauchy surface.

1st observation

S(b−)−1S(f)S(b−) ∈ Ag(N)

2nd observation

S(b−)−1S(f)S(b−) ∈ A0(M)

◮ The map

S(f) → Sg(−b′)−1Sg(−b′ + f) = S(b−)−1S(f)S(b−) defines an endomorphism of A0(M), whose image is contained in Ag(N).

◮ For each relatively compact region, it is just conjugation by

a certain invertible element of A0(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.

S Σ Σ1 Σ2 K supp b′ supp b N′ N Locally compensating the interaction g with b′: Sg(−b′)−1Sg(−b′ + f) = S(b−)−1S(f)S(b−)