On the Localization Properties of Quantum Field Theories with - PowerPoint PPT Presentation

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: Little group construction choose reference momentum q ∈ sp P little group G q := stab q = { R ∈ SL (2 , C ) : q Λ( R ) = q } representation D on Hilbert space H q m > 0: massive representations q m := ( m ,� 0) ∈ H + m (rest frame) stab q m = SU (2) D : spin s representation, H q m = C 2 s +1 m = 0: massless representations q 0 := ( 1 2 , 1 e 3 ) ∈ ∂ V + 2 � stab q 0 = � λ E (2) → E (2) (covering of 2d Euclidean group) � [ ϕ, a ] ∈ SL (2 , C ) : ϕ ∈ R , a ∈ R 2 � � E (2) = � e i ϕ � [ ϕ, a ] = e − i ϕ a D [([ ϕ, a ]) v ]( k ) = e − i k · a v ( k λ ( ϕ )) ∀ v ∈ H q 0 := L 2 ( κ S 1 ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 6

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: Little group construction choose reference momentum q ∈ sp P little group G q := stab q = { R ∈ SL (2 , C ) : q Λ( R ) = q } representation D on Hilbert space H q m > 0: massive representations q m := ( m ,� 0) ∈ H + m (rest frame) stab q m = SU (2) D : spin s representation, H q m = C 2 s +1 m = 0: massless representations q 0 := ( 1 2 , 1 e 3 ) ∈ ∂ V + 2 � stab q 0 = � λ E (2) → E (2) (covering of 2d Euclidean group) � [ ϕ, a ] ∈ SL (2 , C ) : ϕ ∈ R , a ∈ R 2 � � E (2) = � e i ϕ � [ ϕ, a ] = e − i ϕ a D [([ ϕ, a ]) v ]( k ) = e − i k · a v ( k λ ( ϕ )) ∀ v ∈ H q 0 := L 2 ( κ S 1 ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 6

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: Little group construction choose reference momentum q ∈ sp P little group G q := stab q = { R ∈ SL (2 , C ) : q Λ( R ) = q } representation D on Hilbert space H q m > 0: massive representations q m := ( m ,� 0) ∈ H + m (rest frame) stab q m = SU (2) D : spin s representation, H q m = C 2 s +1 m = 0: massless representations q 0 := ( 1 2 , 1 e 3 ) ∈ ∂ V + 2 � stab q 0 = � λ E (2) → E (2) (covering of 2d Euclidean group) � [ ϕ, a ] ∈ SL (2 , C ) : ϕ ∈ R , a ∈ R 2 � � E (2) = � e i ϕ � [ ϕ, a ] = e − i ϕ a D [([ ϕ, a ]) v ]( k ) = e − i k · a v ( k λ ( ϕ )) ∀ v ∈ H q 0 := L 2 ( κ S 1 ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 6

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: Little group construction choose reference momentum q ∈ sp P little group G q := stab q = { R ∈ SL (2 , C ) : q Λ( R ) = q } representation D on Hilbert space H q m > 0: massive representations q m := ( m ,� 0) ∈ H + m (rest frame) stab q m = SU (2) D : spin s representation, H q m = C 2 s +1 m = 0: massless representations q 0 := ( 1 2 , 1 e 3 ) ∈ ∂ V + 2 � stab q 0 = � λ E (2) → E (2) (covering of 2d Euclidean group) � [ ϕ, a ] ∈ SL (2 , C ) : ϕ ∈ R , a ∈ R 2 � � E (2) = � e i ϕ � [ ϕ, a ] = e − i ϕ a D [([ ϕ, a ]) v ]( k ) = e − i k · a v ( k λ ( ϕ )) ∀ v ∈ H q 0 := L 2 ( κ S 1 ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 6

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: One-particle space Wigner boost B p with q Λ( B p ) = p :  �  p �  m > 0  m � � B p := p − p  1 m = 0  √ p −  1 Wigner rotation R ( A , p ) = B p AB − 1 p Λ( A ) ∈ stab q representation of SL (2 , C ) on H 1 := L 2 ( sp P ) ⊗ H q [ U 1 ( A , a ) ψ ]( p ) = e i pa D ( R ( A , p )) ψ ( p Λ( A )) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 7

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: One-particle space Wigner boost B p with q Λ( B p ) = p :  �  p �  m > 0  m � � B p := p − p  1 m = 0  √ p −  1 Wigner rotation R ( A , p ) = B p AB − 1 p Λ( A ) ∈ stab q representation of SL (2 , C ) on H 1 := L 2 ( sp P ) ⊗ H q [ U 1 ( A , a ) ψ ]( p ) = e i pa D ( R ( A , p )) ψ ( p Λ( A )) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 7

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Infinite Spin Representations Irreducible representations: One-particle space Wigner boost B p with q Λ( B p ) = p :  �  p �  m > 0  m � � B p := p − p  1 m = 0  √ p −  1 Wigner rotation R ( A , p ) = B p AB − 1 p Λ( A ) ∈ stab q representation of SL (2 , C ) on H 1 := L 2 ( sp P ) ⊗ H q [ U 1 ( A , a ) ψ ]( p ) = e i pa D ( R ( A , p )) ψ ( p Λ( A )) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 7

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Tomita operator for wedges Standard wedge W 0 := { x ∈ M : ± x ± > 0 } ∆ i t := U 1 ( e − πσ 3 t ) subgroup of boosts preserving W 0 reflection ( R W 0 x ) ± = − x ± , J := U ( R W 0 ) complex conjugation 1 Tomita operator S W 0 := J ∆ 2 (domain restricted by required analytic continuation) real subspace for the standard wedge 1 2 : S W 0 ψ = ψ } K 1 ( W 0 ) := { ψ ∈ dom ∆ extension to arbitrary wedges by covariance: K 1 ( W ) := U 1 ( A , a ) K 1 ( W 0 ) for W = Λ( A ) W 0 + x C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 8

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Tomita operator for wedges Standard wedge W 0 := { x ∈ M : ± x ± > 0 } ∆ i t := U 1 ( e − πσ 3 t ) subgroup of boosts preserving W 0 reflection ( R W 0 x ) ± = − x ± , J := U ( R W 0 ) complex conjugation 1 Tomita operator S W 0 := J ∆ 2 (domain restricted by required analytic continuation) real subspace for the standard wedge 1 2 : S W 0 ψ = ψ } K 1 ( W 0 ) := { ψ ∈ dom ∆ extension to arbitrary wedges by covariance: K 1 ( W ) := U 1 ( A , a ) K 1 ( W 0 ) for W = Λ( A ) W 0 + x C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 8

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Tomita operator for wedges Standard wedge W 0 := { x ∈ M : ± x ± > 0 } ∆ i t := U 1 ( e − πσ 3 t ) subgroup of boosts preserving W 0 reflection ( R W 0 x ) ± = − x ± , J := U ( R W 0 ) complex conjugation 1 Tomita operator S W 0 := J ∆ 2 (domain restricted by required analytic continuation) real subspace for the standard wedge 1 2 : S W 0 ψ = ψ } K 1 ( W 0 ) := { ψ ∈ dom ∆ extension to arbitrary wedges by covariance: K 1 ( W ) := U 1 ( A , a ) K 1 ( W 0 ) for W = Λ( A ) W 0 + x C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 8

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Modular Localization Real subspaces for arbitrary regions subspace for O ⊂ M � K ( O ) := K ( W ) W ⊃O wedge real subspace K ⊂ H 1 is standard iff K ∩ i K = 0 (separating) K + i K = H (cyclic) O ⊂ O ′ := { ˜ x − x ) 2 < 0 ∀ x ∈ O} ˜ x ∈ M : (˜ ⇒ K ( ˜ O ) ⊥K ( O ) wrt. ℑ ◦ �· , ·� K ( C ) is standard for C ⊂ M a spacelike cone [Brunetti, Guido, Longo ’02] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 9

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Definition Let H ( c ) = { e ∈ M ( c ) | e 2 = − 1 } the manifold of spacelike directions. m /∂ V + × H → H q u : H + is called an intertwiner , if D ( R ( A , p )) u ( p Λ( A ) , e ) = u ( p , Λ( A ) e ) (intertwiner eq) loc & pol. bounded in p , analytic for e ∈ H c with ℑ ( e ) ∈ V + L 2 and bounded by an inverse power at the boundary: || u ( p , e ) || H q ≤ M ( p ) |ℑ ( e ) | − N with M pol., N ∈ N Two ways of constructing intertwiners: 1 pullback representation on G q -orbits [Mund, Schroer, Yngvason ’06] 2 characterization using the intertwiner equation C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 10

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Definition Let H ( c ) = { e ∈ M ( c ) | e 2 = − 1 } the manifold of spacelike directions. m /∂ V + × H → H q u : H + is called an intertwiner , if D ( R ( A , p )) u ( p Λ( A ) , e ) = u ( p , Λ( A ) e ) (intertwiner eq) loc & pol. bounded in p , analytic for e ∈ H c with ℑ ( e ) ∈ V + L 2 and bounded by an inverse power at the boundary: || u ( p , e ) || H q ≤ M ( p ) |ℑ ( e ) | − N with M pol., N ∈ N Two ways of constructing intertwiners: 1 pullback representation on G q -orbits [Mund, Schroer, Yngvason ’06] 2 characterization using the intertwiner equation C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 10

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Definition Let H ( c ) = { e ∈ M ( c ) | e 2 = − 1 } the manifold of spacelike directions. m /∂ V + × H → H q u : H + is called an intertwiner , if D ( R ( A , p )) u ( p Λ( A ) , e ) = u ( p , Λ( A ) e ) (intertwiner eq) loc & pol. bounded in p , analytic for e ∈ H c with ℑ ( e ) ∈ V + L 2 and bounded by an inverse power at the boundary: || u ( p , e ) || H q ≤ M ( p ) |ℑ ( e ) | − N with M pol., N ∈ N Two ways of constructing intertwiners: 1 pullback representation on G q -orbits [Mund, Schroer, Yngvason ’06] 2 characterization using the intertwiner equation C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 10

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Definition Let H ( c ) = { e ∈ M ( c ) | e 2 = − 1 } the manifold of spacelike directions. m /∂ V + × H → H q u : H + is called an intertwiner , if D ( R ( A , p )) u ( p Λ( A ) , e ) = u ( p , Λ( A ) e ) (intertwiner eq) loc & pol. bounded in p , analytic for e ∈ H c with ℑ ( e ) ∈ V + L 2 and bounded by an inverse power at the boundary: || u ( p , e ) || H q ≤ M ( p ) |ℑ ( e ) | − N with M pol., N ∈ N Two ways of constructing intertwiners: 1 pullback representation on G q -orbits [Mund, Schroer, Yngvason ’06] 2 characterization using the intertwiner equation C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 10

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields String-localized one-particle states conjugate intertwiner: u c ( p , h ) := Ju ( − pR W 0 , ( R W 0 ) ∗ h ) u has distributional boundary value in e . Single particle vectors ψ ( c ) ( f , h ) ∈ H 1 are defined by ψ ( c ) ( f , h )( p ) = � f ( p ) u ( c ) ( p , h ) for f ∈ S ( M ) , D ( H ). R + supp h x 0 e 0 H supp f supp h � x e � covariance under P c : U ((Λ( A ) , a )) ψ ( c ) ( f , e ) = ψ ( c ) ((Λ( A ) , a ) ∗ f , Λ( A ) ∗ e ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 11

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields String-localized one-particle states conjugate intertwiner: u c ( p , h ) := Ju ( − pR W 0 , ( R W 0 ) ∗ h ) u has distributional boundary value in e . Single particle vectors ψ ( c ) ( f , h ) ∈ H 1 are defined by ψ ( c ) ( f , h )( p ) = � f ( p ) u ( c ) ( p , h ) for f ∈ S ( M ) , D ( H ). R + supp h x 0 e 0 H supp f supp h � x e � covariance under P c : U ((Λ( A ) , a )) ψ ( c ) ( f , e ) = ψ ( c ) ((Λ( A ) , a ) ∗ f , Λ( A ) ∗ e ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 11

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields String-localized one-particle states conjugate intertwiner: u c ( p , h ) := Ju ( − pR W 0 , ( R W 0 ) ∗ h ) u has distributional boundary value in e . Single particle vectors ψ ( c ) ( f , h ) ∈ H 1 are defined by ψ ( c ) ( f , h )( p ) = � f ( p ) u ( c ) ( p , h ) for f ∈ S ( M ) , D ( H ). R + supp h x 0 e 0 H supp f supp h � x e � covariance under P c : U ((Λ( A ) , a )) ψ ( c ) ( f , e ) = ψ ( c ) ((Λ( A ) , a ) ∗ f , Λ( A ) ∗ e ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 11

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields One-particle vectors are localized in spacelike truncated cones ψ ( f , h ) + ψ c ( f , h ) ∈ K 1 ( supp f + R + supp h ) supp f + R + supp h x 0 � x Bosonic Fock space H := � ∞ n =0 Sym ( H ⊗ n 1 ), H 0 = C Ω CCR: [ a ( ϕ ) , a † ( ψ )] = � ϕ, ψ � H 1 1 , [ a ( ϕ ) , a ( ψ )] = [ a † ( ϕ ) , a † ( ψ )] = 0 C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 12

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields One-particle vectors are localized in spacelike truncated cones ψ ( f , h ) + ψ c ( f , h ) ∈ K 1 ( supp f + R + supp h ) supp f + R + supp h x 0 � x Bosonic Fock space H := � ∞ n =0 Sym ( H ⊗ n 1 ), H 0 = C Ω CCR: [ a ( ϕ ) , a † ( ψ )] = � ϕ, ψ � H 1 1 , [ a ( ϕ ) , a ( ψ )] = [ a † ( ϕ ) , a † ( ψ )] = 0 C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 12

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Field operators are defined by � � � � ˆ f ( p ) u ( p , h ) ◦ a † ( p ) + ˆ Φ( f , h ) = d p f ( − p ) u c ( p , h ) ◦ a ( p ) for f ∈ S ( M ), h ∈ D ( H ). ( ◦ : scalar product in H q ) cov. U ( A , a )Φ( f , h ) U † ( A , a ) = Φ((Λ( A ) , a ) ∗ f ) , Λ( A ) ∗ h ) and PCT U ( j 0 )Φ( f , h ) U † ( j 0 ) = Φ(( j 0 ) ∗ f , ( j 0 ) ∗ h ) † lead to supp f + R + supp h String-localization: x 0 W ′ [Φ( f , h ) , Φ( f ′ , h ′ ) † ] = 0 if supp f + R + supp h � x and supp f ′ + R + supp h ′ are spacelike separated. W construction possible for all supp f ′ + R + supp h ′ positive energy representations C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 13

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Field operators are defined by � � � � ˆ f ( p ) u ( p , h ) ◦ a † ( p ) + ˆ Φ( f , h ) = d p f ( − p ) u c ( p , h ) ◦ a ( p ) for f ∈ S ( M ), h ∈ D ( H ). ( ◦ : scalar product in H q ) cov. U ( A , a )Φ( f , h ) U † ( A , a ) = Φ((Λ( A ) , a ) ∗ f ) , Λ( A ) ∗ h ) and PCT U ( j 0 )Φ( f , h ) U † ( j 0 ) = Φ(( j 0 ) ∗ f , ( j 0 ) ∗ h ) † lead to supp f + R + supp h String-localization: x 0 W ′ [Φ( f , h ) , Φ( f ′ , h ′ ) † ] = 0 if supp f + R + supp h � x and supp f ′ + R + supp h ′ are spacelike separated. W construction possible for all supp f ′ + R + supp h ′ positive energy representations C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 13

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook String-Localized Fields Field operators are defined by � � � � ˆ f ( p ) u ( p , h ) ◦ a † ( p ) + ˆ Φ( f , h ) = d p f ( − p ) u c ( p , h ) ◦ a ( p ) for f ∈ S ( M ), h ∈ D ( H ). ( ◦ : scalar product in H q ) cov. U ( A , a )Φ( f , h ) U † ( A , a ) = Φ((Λ( A ) , a ) ∗ f ) , Λ( A ) ∗ h ) and PCT U ( j 0 )Φ( f , h ) U † ( j 0 ) = Φ(( j 0 ) ∗ f , ( j 0 ) ∗ h ) † lead to supp f + R + supp h String-localization: x 0 W ′ [Φ( f , h ) , Φ( f ′ , h ′ ) † ] = 0 if supp f + R + supp h � x and supp f ′ + R + supp h ′ are spacelike separated. W construction possible for all supp f ′ + R + supp h ′ positive energy representations C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 13

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook 1 Introduction 2 Compact Localization Two-Particle States Candidates for Two-Particle Observables 3 No-Go Theorem 4 Limit of Representations 5 Summary & Outlook C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 14

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Two-Particle States Dependency on semi-infinite string-direction is intrinsic for infinite spin-case → No-Go Thm. [Yngvason ’70] [Longo, Morinelli, Rehren ’15] Construction of two-particle intertwiners [MSY ’06] Let F ∈ S ( R ) and define u 2 : ( ∂ V + ) × 2 → H ⊗ 2 by q � � z e i ˜ p )( k , ˜ d 2 z e i kz d 2 ˜ k ˜ z F ( A ( p , ˜ u 2 ( p , ˜ k ) := p , z , ˜ z )), z ) := ξ ( z )Λ( B p B − 1 where A ( p , ˜ p , z , ˜ p ) ξ (˜ z ) ˜ and ξ is a parametrization of stab q . u 2 fulfils the two-particle intertwiner equation D ( R ( A , p )) ⊗ D ( R ( A , ˜ p )) u 2 ( p Λ( A ) , ˜ p Λ( A )) = u 2 ( p , ˜ p ). C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 15

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Two-Particle States Dependency on semi-infinite string-direction is intrinsic for infinite spin-case → No-Go Thm. [Yngvason ’70] [Longo, Morinelli, Rehren ’15] Construction of two-particle intertwiners [MSY ’06] Let F ∈ S ( R ) and define u 2 : ( ∂ V + ) × 2 → H ⊗ 2 by q � � z e i ˜ p )( k , ˜ d 2 z e i kz d 2 ˜ k ˜ z F ( A ( p , ˜ u 2 ( p , ˜ k ) := p , z , ˜ z )), z ) := ξ ( z )Λ( B p B − 1 where A ( p , ˜ p , z , ˜ p ) ξ (˜ z ) ˜ and ξ is a parametrization of stab q . u 2 fulfils the two-particle intertwiner equation D ( R ( A , p )) ⊗ D ( R ( A , ˜ p )) u 2 ( p Λ( A ) , ˜ p Λ( A )) = u 2 ( p , ˜ p ). C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 15

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Two-Particle States Dependency on semi-infinite string-direction is intrinsic for infinite spin-case → No-Go Thm. [Yngvason ’70] [Longo, Morinelli, Rehren ’15] Construction of two-particle intertwiners [MSY ’06] Let F ∈ S ( R ) and define u 2 : ( ∂ V + ) × 2 → H ⊗ 2 by q � � z e i ˜ p )( k , ˜ d 2 z e i kz d 2 ˜ k ˜ z F ( A ( p , ˜ u 2 ( p , ˜ k ) := p , z , ˜ z )), z ) := ξ ( z )Λ( B p B − 1 where A ( p , ˜ p , z , ˜ p ) ξ (˜ z ) ˜ and ξ is a parametrization of stab q . u 2 fulfils the two-particle intertwiner equation D ( R ( A , p )) ⊗ D ( R ( A , ˜ p )) u 2 ( p Λ( A ) , ˜ p Λ( A )) = u 2 ( p , ˜ p ). C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 15

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Two-Particle States Localized two-particle wavefunctions (cf. MSY ’06, ) Let O ⊂ M compact and g ∈ S ( M × 2 ) real-valued with loc ⊗ H ⊗ 2 is polynomially bounded, i.e. supp g ⊂ O × 2 . If u 2 ∈ L 2 || u 2 ( p , ˜ p ) || H ⊗ 2 ≤ M ( p , ˜ p ) q with M a polynomial, then the function p , ˜ p )( k , ˜ ψ ( p , k , ˜ k ) := ˜ g ( p , ˜ p ) u 2 ( p , ˜ k ) is modular localized in O , which means ψ ∈ K 2 ( O ) with the two-particle subspace K 2 defined via second quantization of the operators S W . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 16

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Candidates for Two-Particle Observables Proposed construction of two-particle observables [MSY ’06] Candidate observables are of the form � � � � � � d ν (˜ p )( k , ˜ B ( g ) := d p d ν ( k ) d ˜ p k ) ˆ g ( p , ˜ p ) u 2 ( p , ˜ k ) a † ( p , k ) a † (˜ p , ˜ k ) + . . . such that B ( g )Ω ∈ H 2 is a two-particle wavefunction given by p , k , ˜ p )( k , ˜ ( p , ˜ k ) �→ ˆ g ( p , ˜ p ) u 2 ( p , ˜ k ). Locality in the vacuum expectation value � Ω , [ B ( g ) , B (˜ g )]Ω � = 0 if ( x − x ′ ) 2 < 0 ∀ x ∈ supp g , x ′ ∈ supp ˜ g . Relative locality wrt. string-field Φ( f , h )? C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 17

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Candidates for Two-Particle Observables Proposed construction of two-particle observables [MSY ’06] Candidate observables are of the form � � � � � � d ν (˜ p )( k , ˜ B ( g ) := d p d ν ( k ) d ˜ p k ) ˆ g ( p , ˜ p ) u 2 ( p , ˜ k ) a † ( p , k ) a † (˜ p , ˜ k ) + . . . such that B ( g )Ω ∈ H 2 is a two-particle wavefunction given by p , k , ˜ p )( k , ˜ ( p , ˜ k ) �→ ˆ g ( p , ˜ p ) u 2 ( p , ˜ k ). Locality in the vacuum expectation value � Ω , [ B ( g ) , B (˜ g )]Ω � = 0 if ( x − x ′ ) 2 < 0 ∀ x ∈ supp g , x ′ ∈ supp ˜ g . Relative locality wrt. string-field Φ( f , h )? C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 17

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Candidates for Two-Particle Observables Proposed construction of two-particle observables [MSY ’06] Candidate observables are of the form � � � � � � d ν (˜ p )( k , ˜ B ( g ) := d p d ν ( k ) d ˜ p k ) ˆ g ( p , ˜ p ) u 2 ( p , ˜ k ) a † ( p , k ) a † (˜ p , ˜ k ) + . . . such that B ( g )Ω ∈ H 2 is a two-particle wavefunction given by p , k , ˜ p )( k , ˜ ( p , ˜ k ) �→ ˆ g ( p , ˜ p ) u 2 ( p , ˜ k ). Locality in the vacuum expectation value � Ω , [ B ( g ) , B (˜ g )]Ω � = 0 if ( x − x ′ ) 2 < 0 ∀ x ∈ supp g , x ′ ∈ supp ˜ g . Relative locality wrt. string-field Φ( f , h )? C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 17

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook 1 Introduction 2 Compact Localization 3 No-Go Theorem Assumptions & Statement Characterization of Intertwiners Relative Commutator Restriction of the Integrals Analysis of Singularities 4 Limit of Representations 5 Summary & Outlook C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 18

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Assumptions & Statement Q: Existence of nontriv. operators with compact localization? → Negative result for the following class of operators on F , motivated by the suggestions in [YMS ’06], [Schroer ’08]. Definition An operator-valued distribution B on S ( M × 2 ) of the form � � � � � � d ν (˜ B ( g ) = d p d ˜ p d ν ( k ) k ) p )( k , ˜ k ) a † ( p , k ) a † (˜ p , ˜ g ( p , ˜ ˆ p ) u 2 ( p , ˜ k ) p )( k , ˜ p , ˜ +ˆ g ( − p , − ˜ p ) u 2 c ( p , ˜ k ) a ( p , k ) a (˜ k ) p )( k , ˜ p , ˜ k ) a † ( p , k ) a (˜ +ˆ g ( p , − ˜ p ) u 0 ( p , ˜ k ) p )( k , ˜ p , ˜ k ) a † (˜ +ˆ g ( − p , ˜ p ) u 0 c ( p , ˜ k ) a ( p , k ) with fixed coefficient functions u 2 , u 2 c , u 0 , u 0 c is called a Two-particle observable if... [cf. Streater, Wightman ’64, chap. 3] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 19

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Assumptions & Statement 1 Domain and Continuity For all g ∈ S ( M × 2 ), B ( g ) is defined on the domain D of vectors which is spanned by products of the String fields Φ( f , h ) applied to the vacuum Ω. By the Reeh-Schlieder Thm., D is dense in the Fock space F . For fixed vectors φ, ψ ∈ H , the assignment g ∈ S ( M × 2 ) �→ � φ | B ( g ) | ψ � ∈ C is a tempered distribution, i.e. g �→ B ( g ) is an operator-valued distribution . B ( g ) = B ( g ) † C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 20

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Assumptions & Statement 2 Transformation Law p ∈ ∂ V + and A ∈ SL (2 , C ), the two-particle intertwiner For p , ˜ equation holds almost everywhere in the sense of d p � � p d ν ( k ) d ν (˜ d ˜ k ): D ( R ( A , p )) ⊗ D ( R ( A , ˜ p )) u 2 ( p Λ( A ) , ˜ p Λ( A )) = u 2 ( p , ˜ p ). u 2 , u 2 c , u 0 , u 0 c are locally square-integrable and polynomially bounded. 3 Relative locality Let f ∈ S ( M ), h ∈ D ( H ) and g ∈ S ( M × 2 ) such, that ( x + λ e − y 1 , 2 ) 2 < 0 ∀ x ∈ supp f , e ∈ supp h , λ ∈ R + , ( y 1 , y 2 ) ∈ supp g . Then the associated fields commute: [Φ( f , h ) , B ( g )] = 0 C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 21

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners One-particle string-intertwiners Lemma Let u 1 ( p , e )( k ) a solution of the one-particle intertwinereq. Then there is a function F 1 , defined on the interior of the upper half-plane, such that: 1 The intertwiner u 1 is given by e − e − p − p u 1 ( p , e )( k ) = e i k · 2 p · e F 1 ( p · e ). 2 A choice of the function F 1 can be made in such a way that u 1 is polynomially bounded in p , analytic in e for ℑ ( e ) ∈ V + and bounded by an inverse power at the boundary. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 22

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 1 e A = B p ∈ SL (2 , C ) p R ( B − 1 p , p ) = 1 k u 1 ( q , Λ( B p ) e ) = u 1 ( p , e ) f := Λ( B p ) e κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 1 e A = B p ∈ SL (2 , C ) p f R ( B − 1 p , p ) = 1 k q u 1 ( q , Λ( B p ) e ) = u 1 ( p , e ) f := Λ( B p ) e κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 1 A = B p ∈ SL (2 , C ) f R ( B − 1 p , p ) = 1 k q u 1 ( q , Λ( B p ) e ) = u 1 ( p , e ) f := Λ( B p ) e κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 2 q · e = cst . A = [0 , f / f + ] ∈ G q f ⇒ R ( A , q ) = A k q − i k · f f + u 1 ( q , f ) = u 1 ( q , f + ) e κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 2 q · e = cst . A = [0 , f / f + ] ∈ G q f ⇒ R ( A , q ) = A k q f + − i k · f f + u 1 ( q , f ) = u 1 ( q , f + ) e κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 2 q · e = cst . A = [0 , f / f + ] ∈ G q ⇒ R ( A , q ) = A k q f + − i k · f f + u 1 ( q , f ) = u 1 ( q , f + ) e κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 3 A = [ ϕ, 0]: q and f + invariant, F 1 ( f + / 2) := u 1 ( q , f + )( k ) k q f + κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 3 A = [ ϕ, 0]: q and f + invariant, F 1 ( f + / 2) := u 1 ( q , f + )( k ) k q f + l κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners ∂ V + Step 3 A = [ ϕ, 0]: q and f + invariant, F 1 ( f + / 2) := u 1 ( q , f + )( k ) q f + l κ S 1 H cf. uniqueness proof for string-localized fields [MSY ’06, Lemma B 3 ii)] C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 23

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners Substitution of the intertwiner equations yields the first part e − e − p − p u 1 ( p , e )( k ) = e i k · 2 p · e F 1 ( p · e ). 2 p · e in exponent produces essential singularities at the boundary ℑ ( e ) = 0. � �� � � � e − e − At any singularity one can show � k · p − p � ≤ κ . u 1 is therefore an intertwiner iff F 1 r in F 1 ( p · e ) = e − i κ 2 p · e F 1 r ( p · e ) is pol. bounded distributional boundary value of analytic function on H + . F 1 r ( p · e ) := 1 yields the candidate � e − e − � p − p k · − κ u 1 ( p , e )( k ) = e i . 2 p · e C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 24

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners Two-particle scalar intertwiners Similar result for the two-particle intertwiner u 2 : Lemma p )( k , ˜ Let u 2 ( p , ˜ k ) the function given in assumption 2 , which is a solution of D ( R ( A , p )) ⊗ D ( R ( A , ˜ p )) u 2 ( p Λ( A ) , ˜ p Λ( A )) = u 2 ( p , ˜ p ) loc -function F 2 : R 2 → C such, that Then there is a L 2 − i ˜ 1 1 k · − i k · p p − p − p ˜ p − p − ˜ p )( k , ˜ ˜ p − e ˜ u 2 ( p , ˜ k ) = e p − � � � � �� p p − p − p˜ p − ( k ˜ k ) − 1 F 2 p − ˜ ˜ p − ˜ p − C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 25

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Characterization of Intertwiners Extension of the characterization for u 2 to the coefficient functions u 2 c , u 0 and u 0 c : Lemma There are L 2 loc -functions F 0 and F 0 c , such that the following equations hold: + i ˜ 1 1 k · + i k · p p − p − p ˜ p − p )( k , ˜ p − ˜ ˜ p − e u 2 c ( p , ˜ k ) = e ˜ p − � � � � �� p p − p − p˜ p − ( k ˜ k ) − 1 F 2 p − ˜ ˜ p − ˜ p − + i ˜ 1 1 k · − i k · p p − p − p ˜ p − p − ˜ p )( k , ˜ ˜ p − F 0 ( . . . ) p − e u 0 ( p , ˜ k ) = e ˜ − i ˜ 1 1 k · + i k · p p − p − p ˜ p − p − ˜ p )( k , ˜ ˜ p − F 0 c ( . . . ) p − e u 0 c ( p , ˜ k ) = e ˜ C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 26

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Consider the function γ ( a ) = � φ, [ B ( g ) , Φ( f a , h )]Ω � , where f s := ( 1 , sn ) ∗ f Proof strategy: supp f a + R + supp h γ evaluates nontrivial W ′ matrix elements a B tempered distribution O ⇒ pol. bounded x 0 rel. locality to Φ ⇒ half-sided support x � W supp f + R + supp h dist. FT of γ is S ′ -BV of an analytic function incompatible with singularities in u 2 , u 0 , ... C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 27

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Consider the function γ ( a ) = � φ, [ B ( g ) , Φ( f a , h )]Ω � , where f s := ( 1 , sn ) ∗ f Proof strategy: supp f a + R + supp h γ evaluates nontrivial W ′ matrix elements a B tempered distribution O ⇒ pol. bounded x 0 rel. locality to Φ ⇒ half-sided support x � W supp f + R + supp h dist. FT of γ is S ′ -BV of an analytic function incompatible with singularities in u 2 , u 0 , ... C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 27

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Consider the function γ ( a ) = � φ, [ B ( g ) , Φ( f a , h )]Ω � , where f s := ( 1 , sn ) ∗ f Proof strategy: supp f a + R + supp h γ evaluates nontrivial W ′ matrix elements a B tempered distribution O ⇒ pol. bounded x 0 rel. locality to Φ ⇒ half-sided support x � W supp f + R + supp h dist. FT of γ is S ′ -BV of an analytic function incompatible with singularities in u 2 , u 0 , ... C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 27

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Lemma (regularity of γ ) The function γ has the following properties: 1 Support: supp γ ⊆ ( −∞ , − b ] 2 Boundedness: There are constants C , L > 0 and N ∈ N , such that � 1 � L χ [ − L , 0] − b ( a ) + | a + b | N − 1 | γ ( a ) | ≤ C ∀ a < − b . 3 Continuity: γ is a continuous function. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 28

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Lemma (holomorphic FT) The holomorphic Fourier transform of a continuous polynomially bounded function γ : R → C with supp γ ⊆ ( −∞ , − b ] for some b > 0, which is defined by � d a e − i za γ ( a ) ∀ z ∈ H + , ˆ γ ( z ) = where H + := { z ∈ C : ℑ ( z ) > 0 } is the upper half-plane, has the following properties: γ is an analytic function on H + . 1 Analyticity: ˆ 2 Boundedness: There are constants C > 0 , N ∈ N , such that γ ( z ) | ≤ C e − b ℑ ( z ) (1 + ℑ ( z ) − N ) ∀ z ∈ H + | ˆ 3 Distributional boundary value: . . . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 29

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Lemma (holomorphic FT) The holomorphic Fourier transform of a continuous polynomially bounded function γ : R → C with supp γ ⊆ ( −∞ , − b ] for some b > 0, which is defined by � d a e − i za γ ( a ) ∀ z ∈ H + , ˆ γ ( z ) = where H + := { z ∈ C : ℑ ( z ) > 0 } is the upper half-plane, has the following properties: γ is an analytic function on H + . 1 Analyticity: ˆ 2 Boundedness: There are constants C > 0 , N ∈ N , such that γ ( z ) | ≤ C e − b ℑ ( z ) (1 + ℑ ( z ) − N ) ∀ z ∈ H + | ˆ 3 Distributional boundary value: . . . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 29

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator Lemma (holomorphic FT, part II) 3 Distributional boundary value: The sequence of γ t ∈ S ′ ( R ), given by the restrictions of ˆ distributions ˆ γ to horizontal lines of constant imaginary part t > 0, � ˆ γ t : S ( R ) �→ C , ϕ �→ d s γ ( s + i t ) ϕ ( s ), converges for t → 0 to the distributional FT of γ , � γ : S ( R ) → C , ϕ �→ ˆ d a γ ( a ) ˆ ϕ ( a ) � d s e − i sa ϕ ( s ) the FT on S ( R ), with ˆ ϕ ( a ) := in the sense of S ′ ( R ): lim t → 0 ˆ γ t ( ϕ ) = ˆ γ ( ϕ ) ∀ ϕ ∈ S ( R ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 30

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator γ ( a ) can be stated in terms of functions of p − ∈ R Ψ( p , k ) := ˆ f ( p )˜ u 1 ( p , h )( k ) with � for p ∈ ∂ V + u 1 ( p , h )( k ) u 1 ( p , h )( k ) := ˜ for p ∈ ∂ V − u 1 c ( − p , h )( k ) − i ˜ 1 1 k · + i k · p p − p − p ˜ p − p , k , ˜ p − ˜ ˜ p − e p − S ( p , ˜ I ( p , ˜ k ) := e ˜ p , ψ ) with p e i ψ /κ 2 ) S ( p , ˜ p , ψ ) := Θ( p ˜ p )[ˆ g (˜ p , − p ) F 0 (2 p ˜ p e i ψ /κ 2 )] +ˆ g ( − p , ˜ p ) F 0 c (2 p ˜ p e i ψ /κ 2 ) +Θ( − p ˜ p )[ˆ g (˜ p , − p ) F 2 (2 p ˜ p e i ψ /κ 2 )], +ˆ g ( − p , ˜ p ) F 2 (2 p ˜ coordinate ψ is stable under k , ˜ k �→ λ k , λ − 1 ˜ k for λ ∈ SO (2) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 31

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Relative Commutator γ ( a ) can be stated in terms of functions of p − ∈ R Ψ( p , k ) := ˆ f ( p )˜ u 1 ( p , h )( k ) with � for p ∈ ∂ V + u 1 ( p , h )( k ) u 1 ( p , h )( k ) := ˜ for p ∈ ∂ V − u 1 c ( − p , h )( k ) − i ˜ 1 1 k · + i k · p p − p − p ˜ p − p , k , ˜ p − ˜ ˜ p − e p − S ( p , ˜ I ( p , ˜ k ) := e ˜ p , ψ ) with p e i ψ /κ 2 ) S ( p , ˜ p , ψ ) := Θ( p ˜ p )[ˆ g (˜ p , − p ) F 0 (2 p ˜ p e i ψ /κ 2 )] +ˆ g ( − p , ˜ p ) F 0 c (2 p ˜ p e i ψ /κ 2 ) +Θ( − p ˜ p )[ˆ g (˜ p , − p ) F 2 (2 p ˜ p e i ψ /κ 2 )], +ˆ g ( − p , ˜ p ) F 2 (2 p ˜ coordinate ψ is stable under k , ˜ k �→ λ k , λ − 1 ˜ k for λ ∈ SO (2) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 31

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Restriction of the Integrals p , k , ˜ With abbreviation q := (p , ˜ k ) (measure µ ), one obtains � d p − � p , ˜ p , ˜ e i p − a / 2 γ ( a ) = d µ ( q ) φ (˜ k )Ψ( p , k ) I ( p , k , ˜ k ) p − Singularities contained in I can be exposed: replacing φ and ψ by 1 p , ˜ χ B ǫ (˜ k 0 ) (˜ k ) p 0 , ˜ p , ˜ φ ˜ k 0 ,ǫ (˜ k ) := p 0 , ˜ p 0 , ˜ µ ( B ǫ (˜ k 0 )) → valid choice for φ ∈ H 1 2 � � p − , | p | 2 Ψ p 0 , k 0 ,ǫ := ˆ f δ p 0 ,ǫ ( p ) δ k 0 ,ǫ ( k ) p − → Ψ is determined by Φ( f , h ), limiting procedure necessary. Resulting sequence of functions denoted by ( γ q 0 ,ǫ ) ǫ> 0 . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 32

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Restriction of the Integrals p , k , ˜ With abbreviation q := (p , ˜ k ) (measure µ ), one obtains � d p − � p , ˜ p , ˜ e i p − a / 2 γ ( a ) = d µ ( q ) φ (˜ k )Ψ( p , k ) I ( p , k , ˜ k ) p − Singularities contained in I can be exposed: replacing φ and ψ by 1 p , ˜ χ B ǫ (˜ k 0 ) (˜ k ) p 0 , ˜ p , ˜ φ ˜ k 0 ,ǫ (˜ k ) := p 0 , ˜ p 0 , ˜ µ ( B ǫ (˜ k 0 )) → valid choice for φ ∈ H 1 2 � � p − , | p | 2 Ψ p 0 , k 0 ,ǫ := ˆ f δ p 0 ,ǫ ( p ) δ k 0 ,ǫ ( k ) p − → Ψ is determined by Φ( f , h ), limiting procedure necessary. Resulting sequence of functions denoted by ( γ q 0 ,ǫ ) ǫ> 0 . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 32

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Restriction of the Integrals p , k , ˜ With abbreviation q := (p , ˜ k ) (measure µ ), one obtains � d p − � p , ˜ p , ˜ e i p − a / 2 γ ( a ) = d µ ( q ) φ (˜ k )Ψ( p , k ) I ( p , k , ˜ k ) p − Singularities contained in I can be exposed: replacing φ and ψ by 1 p , ˜ χ B ǫ (˜ k 0 ) (˜ k ) p 0 , ˜ p , ˜ φ ˜ k 0 ,ǫ (˜ k ) := p 0 , ˜ p 0 , ˜ µ ( B ǫ (˜ k 0 )) → valid choice for φ ∈ H 1 2 � � p − , | p | 2 Ψ p 0 , k 0 ,ǫ := ˆ f δ p 0 ,ǫ ( p ) δ k 0 ,ǫ ( k ) p − → Ψ is determined by Φ( f , h ), limiting procedure necessary. Resulting sequence of functions denoted by ( γ q 0 ,ǫ ) ǫ> 0 . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 32

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Restriction of the Integrals Let p 0 ∈ R 2 , k 0 ∈ κ S 1 such, that p 0 ∦ k 0 . For ǫ > 0, consider the function � � p − , | p | 2 Ψ p 0 , k 0 ,ǫ : ∂ V × κ S 1 → C , ( p , k ) �→ ˆ f δ p 0 ,ǫ ( p ) δ k 0 ,ǫ ( k ). p − There is a sequence of sets of finitely many functions � � ( f i ǫ, N , h i ǫ, N ) ∈ S ( M ) × D ( H ) , i = 1 , ..., M ǫ, N N ∈ N which conserve the support properties of Φ( f , h ), i.e. supp f i ǫ, N ⊂ W , supp h i ǫ, N ⊂ W ∩ H ∀ i = 1 , ..., M ǫ, N , N ∈ N , which converge to Ψ p 0 , k 0 ,ǫ in the sense of L 2 up to a continuous function c (p , k ): � � � d p − � � M ǫ, N 2 � � i =1 ˆ | p − | d 2 p f i u 1 ( p , h i d ν ( k ) � ǫ, N ( p )˜ ǫ, N )( k ) − c (p , k )Ψ p 0 , k 0 ,ǫ ( p , k ) � converges to 0. The function c is has the property c (p , k 0 ) = 1. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 33

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Restriction of the Integrals Let p 0 ∈ R 2 , k 0 ∈ κ S 1 such, that p 0 ∦ k 0 . For ǫ > 0, consider the function � � p − , | p | 2 Ψ p 0 , k 0 ,ǫ : ∂ V × κ S 1 → C , ( p , k ) �→ ˆ f δ p 0 ,ǫ ( p ) δ k 0 ,ǫ ( k ). p − There is a sequence of sets of finitely many functions � � ( f i ǫ, N , h i ǫ, N ) ∈ S ( M ) × D ( H ) , i = 1 , ..., M ǫ, N N ∈ N which conserve the support properties of Φ( f , h ), i.e. supp f i ǫ, N ⊂ W , supp h i ǫ, N ⊂ W ∩ H ∀ i = 1 , ..., M ǫ, N , N ∈ N , which converge to Ψ p 0 , k 0 ,ǫ in the sense of L 2 up to a continuous function c (p , k ): � � � d p − � � M ǫ, N 2 � � i =1 ˆ | p − | d 2 p f i u 1 ( p , h i d ν ( k ) � ǫ, N ( p )˜ ǫ, N )( k ) − c (p , k )Ψ p 0 , k 0 ,ǫ ( p , k ) � converges to 0. The function c is has the property c (p , k 0 ) = 1. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 33

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Analysis of Singularities The analyticity of each ˆ γ q 0 ,ǫ is preserved in the limit ǫ → 0: Lemma (compact convergence) The set of sequences of functions � γ q 0 ,ǫ : H + → C , z �→ d a e − i za γ q 0 ,ǫ ( a ) ˆ has the following property: For µ -almost all q 0 ∃ analytic function γ q 0 on H + such, that ˆ γ q 0 ( z ) ∀ z ∈ H + ǫ → 0 ˆ lim γ q 0 ,ǫ ( z ) = ˆ in the sense of compact convergence. Consider the difference ˆ γ ( z ) := ˆ γ q 1 ( z ) − P ( z , q 1 , q 0 )ˆ γ q 0 ( z ), with q 0 �→ q 1 by ( k 0 , ˜ k 0 ) �→ ( λ k 0 , λ − 1 ˜ k 0 ), P relative phase C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 34

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Analysis of Singularities Lemma (Uniform convergence) Let ( γ ǫ ) ǫ> 0 a sequence of analytic functions on H + with the following properties: lim ǫ → 0 γ ǫ = γ exists in the sense of compact convergence, 1 with γ an analytic function on H + . The sequence fulfils the uniform bound | γ ǫ ( z ) | < C ℑ ( z ) − 1 ∀ z ∈ H + , ǫ > 0 for some C > 0. 2 For ǫ > 0, the (boundary-) lim t ց 0 γ ǫ ( · + i t ) = g ǫ exists and is given by a function g ǫ ∈ L 1 ( R ), where convergence is understood in the weak-* topology. 3 The corresponding sequence of boundary functions ( g ǫ ) ǫ> 0 fulfils lim ǫ → 0 g ǫ = 0 in L 1 ( R ) . . . . C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 35

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Analysis of Singularities Lemma (Uniform convergence, part II) t ց 0, weak-* g ǫ γ ǫ ( · + i t ) ǫ → 0 +uniform bound ǫ → 0 L 1 t ց 0, weak-* γ ( · + i t ) 0 Then γ = 0 on all of H + . (using [SW ’64, Thm. 2-17]) ⇒ ˆ γ q 1 has a singularity, which is a contradiction! C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 36

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook 1 Introduction 2 Compact Localization 3 No-Go Theorem 4 Limit of Representations Reference Momenta & Little Groups Little Group Representations Construction of Intertwiners 5 Summary & Outlook C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 37

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups Pauli-Lubanski spin-vector S µ = 1 2 ǫ µνλκ M νλ P κ M νλ : Lie-Algebra of generators of L ↑ + m > 0 interpretation: “angular momentum” in particle’s rest frame S 2 = S µµ defines another Casimir operator. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 38

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups Pauli-Lubanski spin-vector S µ = 1 2 ǫ µνλκ M νλ P κ M νλ : Lie-Algebra of generators of L ↑ + m > 0 interpretation: “angular momentum” in particle’s rest frame S 2 = S µµ defines another Casimir operator. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 38

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups Comparison of the massive and massless case Important distinction between massive and massless case: m 2 − 1 4 S 2 sp P q B p G q H q � (1 ,� H + C 2 s m 2 s ( s + 1) 1 0) � p / m SU (2) � p − � p (1 ,� e ) � 1 ∂ V + L 2 ( S 1 ) κ 2 0 E (2) √ p − 2 1 Construction of the previous objects is usually done separately for m > 0 and m = 0. Fundamentally different properties in the case m = 0 , κ > 0 How do these difficulties arise in the limit κ = const . , m → 0? Idea: Comparison between massive and massless fields is simplified, if construction is unified. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 39

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups Comparison of the massive and massless case Important distinction between massive and massless case: m 2 − 1 4 S 2 sp P q B p G q H q � (1 ,� H + C 2 s m 2 s ( s + 1) 1 0) � p / m SU (2) � p − � p (1 ,� e ) � 1 ∂ V + L 2 ( S 1 ) κ 2 0 E (2) √ p − 2 1 Construction of the previous objects is usually done separately for m > 0 and m = 0. Fundamentally different properties in the case m = 0 , κ > 0 How do these difficulties arise in the limit κ = const . , m → 0? Idea: Comparison between massive and massless fields is simplified, if construction is unified. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 39

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups m -parametrized approach Reference momentum q m is given by H + m q 1 ∂ V + q m q 0 � 1 � ( m ,� 0) � q m = m 2 with q m − independent of m . Usual choice for q is ( m ,� 0), switching between conventions amounts to the Lorentz transform: � √ m � , since q m Λ( B m ) = ( m ,� √ m − 1 B m := 0). C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 40

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups m -parametrized approach Reference momentum q m is given by H + m q 1 ∂ V + q m q 0 � 1 � ( m ,� 0) � q m = m 2 with q m − independent of m . Usual choice for q is ( m ,� 0), switching between conventions amounts to the Lorentz transform: � √ m � , since q m Λ( B m ) = ( m ,� √ m − 1 B m := 0). C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 40

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups m -dependence of Wigner rotations Massless form of the Wigner boost B p is still valid for all m , q m Λ( B p ) = p ∀ p ∈ H + m , result depends on m only via q m . Wigner rotation in m -parametrized form: � a � b B − 1 p Λ( A ) = CB − 1 R = B p A q m Λ( C ) , C =: c d ���� =: C with C independent of m . Explicit form: � � � a − m 2 c 1 ∈ SU (2) m = 1 � R = ∈ � | a | 2 + m 2 | c | 2 c a E (2) m = 0 C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 41

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups m -dependence of Wigner rotations Massless form of the Wigner boost B p is still valid for all m , q m Λ( B p ) = p ∀ p ∈ H + m , result depends on m only via q m . Wigner rotation in m -parametrized form: � a � b B − 1 p Λ( A ) = CB − 1 R = B p A q m Λ( C ) , C =: c d ���� =: C with C independent of m . Explicit form: � � � a − m 2 c 1 ∈ SU (2) m = 1 � R = ∈ � | a | 2 + m 2 | c | 2 c a E (2) m = 0 C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 41

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups Special cases For m = 1, G 1 = SU (2), there is a correspondence between R rotating the sphere and R acting as M¨ obius transform on the complex plane - stereographic projection. � a � . z = az + c b [ D ( R ) f ]( z ) = f ( R − 1 . z ) where c d bz + d For m = 0, G 0 = � E (2), the M¨ obius transforms become rotations/shifts on the plane. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 42

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Reference Momenta & Little Groups Special cases For m = 1, G 1 = SU (2), there is a correspondence between R rotating the sphere and R acting as M¨ obius transform on the complex plane - stereographic projection. � a � . z = az + c b [ D ( R ) f ]( z ) = f ( R − 1 . z ) where c d bz + d For m = 0, G 0 = � E (2), the M¨ obius transforms become rotations/shifts on the plane. C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 42

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Little Group Representations Stereographic projection: identification between z ∈ C and n ∈ S 2 given by � n 3 = d 2 − | z | 2 2 zd d 2 + | z | 2 , n 1 + i n 2 = d 2 + | z | 2 z C � n S 2 d R corresponding to the usual choice ( m ,� 0) can be obtained by conjugation with B m : � � 1 a − mc R m := B − 1 � m RB m = ∈ SU (2) | a | 2 + m 2 | c | 2 mc a Compatible with stereographic projection if md = 1: R m � n ( z ) = � n ( R . z ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 43

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Little Group Representations Stereographic projection: identification between z ∈ C and n ∈ S 2 given by � n 3 = d 2 − | z | 2 2 zd d 2 + | z | 2 , n 1 + i n 2 = d 2 + | z | 2 z C � n S 2 d R corresponding to the usual choice ( m ,� 0) can be obtained by conjugation with B m : � � 1 a − mc R m := B − 1 � m RB m = ∈ SU (2) | a | 2 + m 2 | c | 2 mc a Compatible with stereographic projection if md = 1: R m � n ( z ) = � n ( R . z ) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 43

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Little Group Representations Representation spaces C 2 l +1 of SU (2) are spanned by spherical harmonics Y l n ( z )) = e i h arg z P l h ( � h ( n 3 ( z )) with � d � h 2 3 ) d (1 − n 2 P l + l ( l + 1) − h ( n 3 ) = 0. 1 − n 2 d n 3 d n 3 3 (Legendre polynomials) Stereographic projection transforms the equation into   � � 2   κ 2 | z | 2  | z | d  � 2 � 2 − h 2  P l + h ( n 3 ( | z | )) = 0 ,   � �  d | z | | z | 1 + d with κ 2 := 4 l ( l + 1) / d 2 . Solutions J h ( κ | z | ) in the limit d → ∞ , κ = const span representation spaces L 2 ( κ S 1 ) of � E (2): (Bessel functions) C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 44

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Construction of Intertwiners Once m is chosen, one can construct the following parametrization of Γ q m : � | z | 2 � d 2 z ξ d : R 2 → Γ q , [ ξ d ( z )] � = d 2 + | z | 2 z 1 Crucial property: ξ d ( R . z ) = ξ d ( z )Λ( R ) Γ 0 H + H + ∂ V + 1 m Γ m q m q 1 ∂ V + Γ 1 ( m , 0) q 0 Parametrization can also be given in terms of the usual choice for m = 1: � = ( B − 1 m ) † ( 1 + � n ) B − 1 [ ξ ( z )] σ · � m Intuition: Lorentz-boosted “celestial sphere” C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 45

Introduction Compact Localization No-Go Theorem Limit of Representations Summary & Outlook Construction of Intertwiners Once m is chosen, one can construct the following parametrization of Γ q m : � | z | 2 � d 2 z ξ d : R 2 → Γ q , [ ξ d ( z )] � = d 2 + | z | 2 z 1 Crucial property: ξ d ( R . z ) = ξ d ( z )Λ( R ) Γ 0 H + H + ∂ V + 1 m Γ m q m q 1 ∂ V + Γ 1 ( m , 0) q 0 Parametrization can also be given in terms of the usual choice for m = 1: � = ( B − 1 m ) † ( 1 + � n ) B − 1 [ ξ ( z )] σ · � m Intuition: Lorentz-boosted “celestial sphere” C. K¨ ohler Localization of QFTs with Infinite Spin 2015-05-29 LQP36 45