Spectral theory of automorphism groups in QFT Wojciech Dybalski (G - - PowerPoint PPT Presentation

Spectral theory of automorphism groups in QFT

Wojciech Dybalski (G¨

• ttingen)

ω

X A A A

1

Outline

• 1. Particle content in QM and QFT.
• 2. Space translations in QM and QFT.
• 3. Spectral decomposition: ˆ

A = ˆ App ⊕ ˆ Apc ⊕ ˆ Aac.

• 4. Infrared structure. dim ˆ

Apc < ∞.

• 5. Massive theories. ˆ

Apc = {0}. Existence of particles.

• 6. Conclusions.

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1(a). Particle content of QM. Spectrum of H. Example: two-body scattering, short-range interaction. H = H0 + V, H = Hpp ⊕ Hac ⊕ Hsc, Ω± = lim

t→±∞ eitHe−itH0.

The theory has a complete particle interpretation if Ran Ω± = Hac, and Hsc = {0}.

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1(b). Particle content of QFT. Arveson spectrum.

• Def. SpαxB is the support of the distribution
• B(p) :=

1 (2π)2

• d4x e−ipxαx(B),

B ∈ A.

• Fact: SpαxB = the energy-momentum transfer of B i.e.

BP(∆)H ⊂ P(∆ + SpαxB)H, where P(∆) - spectral measure of (H, P).

• Def. B is energy-decreasing if SpαxB ∩ V + = ∅.

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• Theorem. [Buchholz 90] If B ∈ A is almost local and

energy-decreasing then B∗BE,1 < ∞ for all E ≥ 0. B∗BE,1 := sup

ω∈SE

• d3x |ω(α

x(B∗B))|.

• Def. Space of particle detectors:

A(1) := { C ∈ A | CE,1 < ∞ for all E ≥ 0 }

8 <

ω ωο

ο

+ + + + + +

1 1 1 1 1

ω

5

• Asymptotic functionals: Let ω ∈ SE, C ∈ A(1),

σ(t)

ω (C) := α∗ t ω

d3x α

xC

• ,

σ+

ω − limit points as t → ∞.

• Particle content: { σ+

ω | ω ∈ SE for some E ≥ 0 }.

ω ωο

ο

+ + + + + +

1

1 1 1 1 1

t α

∗ω

• Question: When is the particle content non-trivial?
• Strategy: Detailed spectral analysis of α

x. 6

2(a). Space translations (QM).

• Setting: Ψ, Φ ∈ L2(R3, d3x),

supp Φ-compact.

• Function:

x → (Ψ|U( x)Φ).

x Φ Φ Φ Ψ

• Facts:

(a) sup

Ψ≤1

• d3x|(Ψ|U(

x)Φ)|2 < ∞ for all Φ. (b) sup

Ψ≤1

• d3x|(Ψ|U(

x)Φ)|2−ε = ∞ for some Φ.

• Square integrability is the best possible generic feature.

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2(b). Space translations (QFT).

• Setting: ˆ

A :=

O⊂R4 A(O), A ∈ ˆ

A, ω ∈ SE.

• Function:

x → ω(α

xA).

ω

X A A A

• Def: AE,2 := sup

ω∈SE

(

• d3x |ω(α

xA)|2)

1 2 .

ˆ A(2) := { A ∈ ˆ A | AE,2 < ∞ for all E ≥ 0}.

• QM suggests: ˆ

A(2) is ’large’ (of finite co-dimension in ˆ A).

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• 3. Spectral decomposition: ˆ

A = ˆ App ⊕ ˆ Apc ⊕ ˆ Aac.

• ˆ

A =

O⊂R4 A(O) - α x-invariant ∗-algebra.

• Step 1: ˆ

A = ˆ App ⊕ ˆ Ac, ˆ App := { λI | λ ∈ C}, ˆ Ac := { A ∈ ˆ A | ω0(A) = 0 }. Phase-space conditions ⇒ ω(α

xA) −

→

| x|→∞

0 for A ∈ ˆ Ac.

• Step 2: ˆ

Ac = ˆ Apc ⊕ ˆ Aac, ˆ Aac := ˆ A(2), ˆ Apc − direct sum complement.

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4(a). Infrared structure. dim ˆ Apc < ∞.

• Condition L(2): There exist functionals τ1, . . . , τN ∈ ˆ

A∗ and pointlike localized fields φ1, . . . , φN s.t. for any A ∈ ˆ A A = ω0(A)I

ˆ App

+

N

• i=1

τi(A)φi

• ˆ

Apc

+ R(2)(A)

ˆ Aac

, R(2)(A)E,2 < ∞.

• Implication: ˆ

Aac ⊃ ker ω0 ∩ ker τ1 ∩ . . . ∩ ker τN i.e. dim ˆ Apc < ∞.

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4(b). Infrared order.

• Decay of ω(α

xA) ⇐

⇒ regularity of ω( A( p))

• A(

p) := (2π)− 3

2

• d3x ei

p xα x(A).

• Def. Infrared order of an operator A is given by
• rd (A) := inf{ β ≥ 0 | sup

ω∈SE

• d3p |

p|β|ω( A( p))|2 < ∞ for all E ≥ 0 }.

• Theorem.[Buchholz 90] ord (A) ≤ 4 for A ∈ ˆ

A.

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4(c). Examples.

• Massless free field theory: A ∈ ˆ

A A = ω0(A)I + τ1(A)φ + τ2(A) :φ2: +τ3(A) :φ3: +R(2)(A). Implications: (a) ˆ A - full theory: dim ˆ Apc ∈ {2, 3}, ord ˆ A = {0, 1, 2}. (b) ˆ Ae - even part: dim ˆ Ae

pc = 1,

• rd ˆ

Ae = {0, 1}. (c) ˆ Ad - derivatives: ˆ Ad

pc = {0}.

• Massive free field theory:

ˆ Apc = {0}.

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5(a). Massive theories. Particle detectors.

• Theorem. [Buchholz 90]

B∗B ∈ A(1) if B ∈ A almost local and energy-decreasing.

• Condition L(1):

There exists µ > 0 s.t. for any g ∈ S(R), supp ˜ g ⊂ [−µ, µ] (a) A(g) ∈ A(1) if A ∈ ˆ Ac. (b) A(g)E,1 ≤ cn,E,ORnARn, A ∈ A(O)c, R := (1 + H)−1. Status: Holds in massive free field theory.

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5(b). Massive theories. ˆ Apc = {0}.

• Theorem. Condition L(1)(a) implies that ˆ

Apc = {0}. Proof: To show: A ∈ ˆ Ac ⇒ A ∈ ˆ A(2)

• PEAPE = PE{A(f −) + A(g) + A(f +)}PE.

E − − E µ µ g f f + ~ ~ ~ −

• A(f −) - energy decreasing and almost local ⇒

A(f −)∗A(f −) ∈ A(1) ⇒ A(f −) ∈ A(2).

• A(g) ∈ A(1) ⊂ A(2) by Condition L(1)(a).

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5(c). Massive theories. T µν.

• Approximation properties.

Let φ be a pointlike localized field s.t. ω0(φ) = 0. Then, for some n > 0, Ar ∈ A(Or)c (a) [Bostelmann 05] lim

r→0 Rn(φ − Ar)Rn = 0.

(b) Under Condition L(1)(b), i.e. A(g)E,1 ≤ cn,E,ORnARn, lim

r→0 φ(g) − Ar(g)E,1 = 0.

• Condition T: There exists a pointlike localized field T 00, s.t.

ω0(T 00) = 0, which satisfies

• d3x ω(α

xT 00(g)) = ω(H),

ω ∈ SE.

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5(e). Massive theories. Non-trivial particle content.

• Theorem. L(1), T ⇒ σ+

ω = 0 for any ω ∈ SE s.t. ω(H) > 0.

Proof.

• To show: σ(t)

ω (C) =

• d3x ω
• α(t,

x)C

• has non-zero limit points.
• Choose C ∈ A(1) s.t. T 00(g) − CE,1 ≤ ε. Then

0 < ω(H) =

• d3x ω
• α(t,

x)T 00(g)

• ≤ ε + |σ(t)

ω (C)|.

ω ωο

ο

+ + + + + +

1

1 1 1 1 1

t α

∗ω

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• 6. Conclusions:
• We found a decomposition ˆ

A = ˆ App ⊕ ˆ Apc ⊕ ˆ Aac. ˆ Apc carries information about the infrared structure.

• dim ˆ

Apc < ∞ in theories satisfying Condition L(2). ˆ Apc is ’at the boundary’ between ˆ App and ˆ Ac.

• ˆ

Apc = {0} in massive theories satisfying Condition L(1). Such theories, admitting T µν, have non-trivial particle content.

• Open problem:

Non-triviality of the particle content in the massless case.

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