Higher Spin Fields on Curved Spacetimes Rainer M uhlhoff Institut - PowerPoint PPT Presentation

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Higher Spin Fields on Curved Spacetimes Rainer M¨ uhlhoff Institut f¨ ur Theoretische Physik Universit¨ at Leipzig 22nd Workshop “Foundations and Constructive Aspects of QFT” Hamburg, June 6th–7th, 2008 Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline 1 Statement of Buchdahl’s Equations History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it 2 Solving the Cauchy Problem A General Solution Theorem Cauchy Problem for Buchdahl’s Equations 3 Quantisation Outline of the Procedure Generalisation of the (Dimock 1982) spin 1 2 construction Quantisation in Illge’s framework 4 Summary Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Generalised Dirac Equations – History On flat Minkowski spacetime, [Dirac 1936]: � ∂ A X 0 ψ AA 1 ... A k ˙ X l + µ ϕ A 1 ... A k ˙ X l = 0 X 1 ... ˙ X 0 ... ˙ ˙ ∂ ˙ X A 0 ϕ A 1 ... A k ˙ X l − ν ψ A 0 ... A k ˙ X l = 0 , X ˙ X 1 ... ˙ X 1 ... ˙ Naive minimal coupling to gravitation: � ∇ A X 0 ψ AA 1 ... A k ˙ X l + µ ϕ A 1 ... A k ˙ X l = 0 X 1 ... ˙ X 0 ... ˙ ˙ ∇ ˙ X A 0 ϕ A 1 ... A k ˙ X l − ν ψ A 0 ... A k ˙ X l = 0 . X ˙ X 1 ... ˙ X 1 ... ˙ Problem [Buchdahl 1962] Inconsistent for k + l > 1. Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Buchdahl’s equation Buchdahl’s 1982 modification for spin s 2 : ( ∇ A ˙ A 1 ... A s − 1 A 3 ... A s − 1 ) − νψ AA 1 ... A s − 1 = 0 X ϕ − ( s − 1)( s − 2) ǫ A ( A 1 Ψ | PQD | A 2 ψ ˙ µ s PQD X A 1 ... A s − 1 XA ψ AA 1 ... A s − 1 − µϕ ∇ ˙ = 0 ˙ X Consistent for all s ∈ ◆ and µ � = 0 for symmetric spinor fields (massive fields) ψ AA 1 ... A s − 1 = ψ ( AA 1 ... A s − 1 ) Violation of minimal A 1 ... A s − 1 ( A 1 ... A s − 1 ) ϕ = ϕ coupling principal? ˙ ˙ X X Operator notation: „ − 1 « ! \ µ − P s ∇− ν ψ 1st order = 0 / ∇ − µ ϕ 0st order ∇ ψ ) A 1 ... A s − 1 XA ψ AA 1 ... A s − 1 ( / := / ∇ ˙ ˙ X A ˙ X ψ A 1 ... A s − 1 ∇ ψ ) AA 1 ... A s − 1 := \ ( \ ∇ ˙ X ( P s ϕ ) AA 1 ... A s − 1 := ( s − 1)( s − 2) A 3 ... A s − 1 ) ǫ A ( A 1 Ψ | PQD | A 2 ϕ PQD s Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Geometric Setting Underlying manifold By a spacetime manifold ( M , g ), we mean a time-oriented and space-oriented, globally hyperbolic, 4-dimensional Lorentzian manifold of signature (+ − −− ). This implies that ( M , g ) is oriented and connected, satisfies the strong Causality condition, has a (potentially non unique) spin structure S ( M ). Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary SU (2)-representations Notation Irreducible complex SU (2)-representations: D ( j ) : SU (2) → Aut (∆ j ) with j ∈ { 0 , 1 2 , 1 , 3 2 , . . . } “spin number” ∆ j ❈ -vector space, dim ❈ (∆ j ) = 2 j + 1 Symmetric tensor products are irreducible D ( 1 2 ) is the fundamental representation. D ( j ) = ( D ( 1 2 ) ) ∨ 2 j = D ( 1 2 ) ∨ . . . ∨ D ( 1 2 ) ∆ j = (∆ j ) ∨ 2 j . on Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary SL (2 , ❈ )-representation theory I Theorem (irreducible SL (2 , ❈ )-representations) Finite dimensional complex irreducible SL (2 , ❈ )-representations are (up to equivalence) of the form D ( j , j ′ ) := D ( j ) D ( j ′ ) ⊗ ¯ ∆ j , j ′ := ∆ j ⊗ ¯ on ∆ j ′ c c for spin numbers j , j ′ ∈ { 0 , 1 2 , 1 , 3 2 , . . . } . Notice: dim(∆ j , j ′ ) = (2 j + 1)(2 j ′ + 1) . Notation – extension of D ( j ) to SL (2 , ❈ ). D ( j ) c D ( j ) – complex conjugate of D ( j ) ¯ c c Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary SL (2 , ❈ )-representation theory II Symmetric tensor products SL (2 , ❈ ) has two fundamental representations: D ( 1 D (0 , 1 2 , 0) 2 ) and Symmetrised tensor products are irreducible: D ( j , j ′ ) = ( D ( 1 2 , 0) ) ∨ 2 j ⊗ ( D (0 , 1 2 ) ) ∨ 2 j ′ 2 ) ∨ 2 j ′ . 2 , 0 ) ∨ 2 j ⊗ (∆ 0 , 1 on ∆ j , j ′ := (∆ 1 Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary SL (2 , ❈ )-spinors Abstract index notation for 2-spinors ϕ A 1 ... A p ˙ X 1 ... ˙ 2 , 0 ) ⊗ p ⊗ (∆ 0 , 1 X q ∈ (∆ 1 2 ) ⊗ q ( p , q )-spinors: 2 , 0 ) ⊗ p ⊗ (∆ ∗ X q ∈ (∆ ∗ 2 ) ⊗ q ( p , q )-co-spinors: ψ A 1 ... A p ˙ X 1 ... ˙ 1 0 , 1 Spinors on spacetime manifold ( M , g ) Spin structure S ( M ) is principal SL (2 , ❈ )-bundle with induced connection. We have the following associated vector bundles: Bundle of ( p , q )-spinors: D ( p , q ) M bundle of ( p , q )-co-spinors: D ( p , q ) ∗ M . Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Symmetric Spinors Spinors on spacetime manifold ( M , g ) Covariant derivatives on D ( p , q ) M : ∇ ( p , q ) : Γ( TM ) ⊗ Γ( D ( p , q ) M ) → Γ( D ( p , q ) M ) are compatible, i. e. ∇ ( p , q ) ( T × S ) = ∇ ( p , 0) S ⊗ T + S ⊗ ∇ (0 , q ) T a a a ∇ a ǫ AB = 0 , ∇ a σ A ˙ X = 0 b Caution (symmetry and irreducibility again) ϕ A 1 ... A p ˙ X 1 ... ˙ X q is element of an irreducible SL (2 , ❈ )-representation ϕ A 1 ... A p ˙ X 1 ... ˙ X q = ϕ ( A 1 ... A p )( ˙ X 1 ... ˙ X q ) ⇔ 2 , 0 ) ⊗ 2 j ⊗ (∆ 0 , 1 2 ) ⊗ 2 j ′ Thus: ∆ j , j ′ � (∆ 1 Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Buchdahl’s equation Buchdahl’s 1982 modification for spin s 2 : ( ∇ A ˙ A 1 ... A s − 1 A 3 ... A s − 1 ) − ( s − 1)( s − 2) − νψ AA 1 ... A s − 1 = 0 X ϕ ǫ A ( A 1 Ψ | PQD | A 2 ψ ˙ PQD X µ s XA ψ AA 1 ... A s − 1 − µϕ A 1 ... A s − 1 ∇ ˙ = 0 ˙ X Consistent for all s ∈ ◆ and µ � = 0 for symmetric spinor fields (massive fields) ψ AA 1 ... A s − 1 = ψ ( AA 1 ... A s − 1 ) Violation of minimal A 1 ... A s − 1 ( A 1 ... A s − 1 ) ϕ = ϕ coupling principal? ˙ ˙ X X Operator notation: „ − 1 « ! µ − P s ∇− ν \ ψ 1st order = 0 ∇ / − µ 0st order ϕ ∇ ψ ) A 1 ... A s − 1 ( / := / XA ψ AA 1 ... A s − 1 ∇ ˙ ˙ X A ˙ ∇ ψ ) AA 1 ... A s − 1 := \ X ψ A 1 ... A s − 1 ( \ ∇ ˙ X ( P s ϕ ) AA 1 ... A s − 1 := ( s − 1)( s − 2) A 3 ... A s − 1 ) ǫ A ( A 1 Ψ | PQD | A 2 ϕ s PQD Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Buchdahl-W¨ unsch equation Equivalent formulation by [W¨ unsch, 1985] 0 = ∇ A ˙ X ϕ A 1 ... A s − 1 − ν ψ AA 1 ... A s − 1 ˙ X XA ψ AA 1 ... A s − 1 − µ ϕ A 1 ... A s − 1 0 = ∇ ˙ ˙ X with µ, ν ∈ ❈ , µ � = 0, and = ϕ ( A 1 ... A s − 1 ) ψ AA 1 ... A s − 1 = ψ ( AA 1 ... A s − 1 ) ϕ A 1 ... A s − 1 and ˙ ˙ X X Why symmetrisation? Symmetrisation projects back onto the original irreducible SL (2 , ❈ )-representation. Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations History Solving the Cauchy Problem Spinors and Representation Theory Quantisation Buchdahl’s Equation and W¨ unsch’s Version of it Summary Generalised Dirac Equations – History On flat Minkowski spacetime, [Dirac 1936]: � ∂ A X 0 ψ AA 1 ... A k ˙ X l + µ ϕ A 1 ... A k ˙ X l = 0 X 1 ... ˙ X 0 ... ˙ ˙ ∂ ˙ X A 0 ϕ A 1 ... A k ˙ X l − ν ψ A 0 ... A k ˙ X l = 0 , X ˙ X 1 ... ˙ X 1 ... ˙ Naive minimal coupling to gravitation: � ∇ A X 0 ψ AA 1 ... A k ˙ X l + µ ϕ A 1 ... A k ˙ X l = 0 X 1 ... ˙ X 0 ... ˙ ˙ ∇ ˙ X A 0 ϕ A 1 ... A k ˙ X l − ν ψ A 0 ... A k ˙ X l = 0 . X ˙ X 1 ... ˙ X 1 ... ˙ Problem [Buchdahl 1962] Inconsistent for k + l > 1. Rainer M¨ uhlhoff Higher Spin Fields