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

Spectral theory of automorphism groups in QFT Wojciech Dybalski (G¨ ottingen) A A A 0 0 0 ω X 1

Outline 1. Particle content in QM and QFT. 2. Space translations in QM and QFT. 3. Spectral decomposition: ˆ A = ˆ A pp ⊕ ˆ A pc ⊕ ˆ A ac . 4. Infrared structure. dim ˆ A pc < ∞ . 5. Massive theories. ˆ A pc = { 0 } . Existence of particles. 6. Conclusions. 2

1(a). Particle content of QM. Spectrum of H . Example: two-body scattering, short-range interaction. H = H 0 + V, H = H pp ⊕ H ac ⊕ H sc , Ω ± = t →±∞ e itH e − itH 0 . lim The theory has a complete particle interpretation if Ran Ω ± = H ac , and H sc = { 0 } . 3

1(b). Particle content of QFT. Arveson spectrum. • Def. Sp α x B is the support of the distribution � 1 � d 4 x e − ipx α x ( B ) , B ∈ A . B ( p ) := (2 π ) 2 • Fact: Sp α x B = the energy-momentum transfer of B i.e. BP (∆) H ⊂ P (∆ + Sp α x B ) H , where P (∆) - spectral measure of ( H, � P ). • Def. B is energy-decreasing if Sp α x B ∩ V + = ∅ . 4

• Theorem. [Buchholz 90] If B ∈ A is almost local and energy-decreasing then � B ∗ B � E, 1 < ∞ for all E ≥ 0. � � B ∗ B � E, 1 := sup d 3 x | ω ( α � x ( B ∗ B )) | . ω ∈ S E • Def. Space of particle detectors: A (1) := { C ∈ A | � C � E, 1 < ∞ for all E ≥ 0 } < 8 + 0 1 + 0 1 + 0 1 + 0 1 + 0 1 + ω ω ο ω ο 5

• Asymptotic functionals: Let ω ∈ S E , C ∈ A (1) , � � � σ ( t ) ω ( C ) := α ∗ d 3 x α � t ω x C , σ + ω − limit points as t → ∞ . • Particle content: { σ + ω | ω ∈ S E for some E ≥ 0 } . 0 1 0 1 0 1 0 1 0 1 0 1 + + + + + + ∗ω α t ω ο ω ο • Question: When is the particle content non-trivial? • Strategy: Detailed spectral analysis of α � x . 6

2(a). Space translations (QM). Setting: Ψ , Φ ∈ L 2 ( R 3 , d 3 x ), • supp Φ-compact. • Function: � x → (Ψ | U ( � x )Φ). Φ Φ Φ Ψ x • Facts: � x )Φ) | 2 < ∞ for all Φ. d 3 x | (Ψ | U ( � ( a ) sup � Ψ �≤ 1 � x )Φ) | 2 − ε = ∞ for some Φ. d 3 x | (Ψ | U ( � ( b ) sup � Ψ �≤ 1 • Square integrability is the best possible generic feature. 7

2(b). Space translations (QFT). A := � Setting: ˆ O⊂ R 4 A ( O ), A ∈ ˆ • A , ω ∈ S E . • x → ω ( α � Function: � x A ). A A A 0 0 0 ω X � 1 d 3 x | ω ( α � x A ) | 2 ) 2 . • Def: � A � E, 2 := sup ( ω ∈ S E A (2) := { A ∈ ˆ ˆ A | � A � E, 2 < ∞ for all E ≥ 0 } . A (2) is ’large’ (of finite co-dimension in ˆ QM suggests: ˆ • A ). 8

3. Spectral decomposition: ˆ A = ˆ A pp ⊕ ˆ A pc ⊕ ˆ A ac . A = � • ˆ O⊂ R 4 A ( O ) - α � x -invariant ∗ -algebra. • Step 1: ˆ A = ˆ A pp ⊕ ˆ A c , ˆ { λI | λ ∈ C } , := A pp ˆ { A ∈ ˆ := A | ω 0 ( A ) = 0 } . A c 0 for A ∈ ˆ Phase-space conditions ⇒ ω ( α � x A ) − → A c . | � x |→∞ • Step 2: ˆ A c = ˆ A pc ⊕ ˆ A ac , A ac := ˆ ˆ A (2) , ˆ A pc − direct sum complement. 9

4(a). Infrared structure. dim ˆ A pc < ∞ . A ∗ and Condition L (2) : There exist functionals τ 1 , . . . , τ N ∈ ˆ • pointlike localized fields φ 1 , . . . , φ N s.t. for any A ∈ ˆ A N � + R (2) ( A ) � R (2) ( A ) � E, 2 < ∞ . A = ω 0 ( A ) I + τ i ( A ) φ i , � �� � � �� � i =1 � �� � ˆ ˆ A pp A ac ˆ A pc Implication: ˆ • A ac ⊃ ker ω 0 ∩ ker τ 1 ∩ . . . ∩ ker τ N i.e. dim ˆ A pc < ∞ . 10

4(b). Infrared order. ⇒ regularity of ω ( � • Decay of ω ( α � x A ) ⇐ A ( � p )) � p ) := (2 π ) − 3 � d 3 x e i� p� x α � A ( � x ( A ) . 2 • Def. Infrared order of an operator A is given by � p )) | 2 < ∞ for all E ≥ 0 } . p | β | ω ( � d 3 p | � ord ( A ) := inf { β ≥ 0 | sup A ( � ω ∈ S E • Theorem.[Buchholz 90] ord ( A ) ≤ 4 for A ∈ ˆ A . 11

4(c). Examples. • Massless free field theory: A ∈ ˆ A A = ω 0 ( A ) I + τ 1 ( A ) φ + τ 2 ( A ) : φ 2 : + τ 3 ( A ) : φ 3 : + R (2) ( A ) . Implications: (a) ˆ dim ˆ A pc ∈ { 2 , 3 } , ord ˆ A - full theory: A = { 0 , 1 , 2 } . A e - even part: A e = { 0 , 1 } . (b) ˆ dim ˆ ord ˆ A e pc = 1 , A d - derivatives: (c) ˆ ˆ A d pc = { 0 } . ˆ • Massive free field theory: A pc = { 0 } . 12

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 A c . (b) � A ( g ) � E, 1 ≤ c n,E, O � R n AR n � , A ∈ A ( O ) c , R := (1 + H ) − 1 . Status: Holds in massive free field theory. 13

5(b). Massive theories. ˆ A pc = { 0 } . Theorem. Condition L (1) ( a ) implies that ˆ A pc = { 0 } . Proof: To show: A ∈ ˆ A c ⇒ A ∈ ˆ A (2) • P E AP E = P E { A ( f − ) + A ( g ) + A ( f + ) } P E . ~ ~ ~ − f + g f − − µ µ E E 0 • 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 ). � 14

5(c). Massive theories. T µν . • Approximation properties. Let φ be a pointlike localized field s.t. ω 0 ( φ ) = 0. Then, for some n > 0 , A r ∈ A ( O r ) c (a) [Bostelmann 05] r → 0 � R n ( φ − A r ) R n � = 0 . lim (b) Under Condition L (1) (b), i.e. � A ( g ) � E, 1 ≤ c n,E, O � R n AR n � , r → 0 � φ ( g ) − A r ( g ) � E, 1 = 0 . lim Condition T: There exists a pointlike localized field T 00 , s.t. • ω 0 ( T 00 ) = 0, which satisfies � d 3 x ω ( α � x T 00 ( g )) = ω ( H ) , ω ∈ S E . 15

5(e). Massive theories. Non-trivial particle content. Theorem. L (1) , T ⇒ σ + ω � = 0 for any ω ∈ S E s.t. ω ( H ) > 0. Proof. � � � • To show: σ ( t ) d 3 x ω ω ( C ) = α ( t,� x ) C has non-zero limit points. • Choose C ∈ A (1) s.t. � T 00 ( g ) − C � E, 1 ≤ ε . Then � � � d 3 x ω x ) T 00 ( g ) ≤ ε + | σ ( t ) 0 < ω ( H ) = ω ( C ) | . � α ( t,� 0 1 + 0 1 + 0 1 + 0 1 + 0 1 + 0 1 + ∗ω α t ω ο ω ο 16

6. Conclusions: • We found a decomposition ˆ A = ˆ A pp ⊕ ˆ A pc ⊕ ˆ A ac . ˆ A pc carries information about the infrared structure. • dim ˆ A pc < ∞ in theories satisfying Condition L (2) . A pc is ’at the boundary’ between ˆ ˆ A pp and ˆ A c . • ˆ A pc = { 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. 17