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 Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

Generalised Dirac Equations – History

On flat Minkowski spacetime, [Dirac 1936]:

• ∂A

˙ X0ψAA1...Ak ˙ X1... ˙ Xl + µ ϕA1...Ak ˙ X0... ˙ Xl = 0

∂ ˙

X A0ϕA1...Ak ˙ X ˙ X1... ˙ Xl − ν ψA0...Ak ˙ X1... ˙ Xl = 0 ,

Naive minimal coupling to gravitation:

• ∇A

˙ X0ψAA1...Ak ˙ X1... ˙ Xl + µ ϕA1...Ak ˙ X0... ˙ Xl = 0

∇ ˙

X A0ϕA1...Ak ˙ X ˙ X1... ˙ Xl − ν ψA0...Ak ˙ X1... ˙ Xl = 0 .

Problem [Buchdahl 1962] Inconsistent for k + l > 1.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

Buchdahl’s equation

Buchdahl’s 1982 modification for spin s

2:

( ∇A ˙

Xϕ A1...As−1 ˙ X

− (s−1)(s−2)

µs

ǫA(A1Ψ|PQD|A2ψ

A3...As−1) PQD

− νψAA1...As−1 = 0 ∇ ˙

XAψAA1...As−1 − µϕ A1...As−1 ˙ X

= 0 for symmetric spinor fields ψAA1...As−1 = ψ(AA1...As−1) ϕ

A1...As−1 ˙ X

= ϕ

(A1...As−1) ˙ X

Operator notation: „ − 1

µ − Ps

\ ∇−ν / ∇ −µ « ψ ϕ ! = 0 1st order 0st order

( / ∇ψ) A1...As−1

˙ X

:= / ∇ ˙

XAψAA1...As−1

( \ ∇ψ)AA1...As−1 := \ ∇

A ˙ Xψ A1...As−1 ˙ X

(Psϕ)AA1...As−1 := (s−1)(s−2)

s

ǫA(A1Ψ|PQD|A2ϕ

A3...As−1) PQD Consistent for all s ∈ ◆ and µ = 0 (massive fields) Violation of minimal coupling principal?

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

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 Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

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) = 2j + 1 Symmetric tensor products are irreducible D( 1

2) is the fundamental representation.

D(j) = (D( 1

2 ))∨2j = D( 1 2) ∨ . . . ∨ D( 1 2 )

• n

∆j = (∆j)∨2j .

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

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)

c

⊗ ¯ D(j′)

c

• n

∆j,j′ := ∆j ⊗ ¯ ∆j′ for spin numbers j, j′ ∈ {0, 1

2, 1, 3 2, . . .}. Notice:

dim(∆j,j′) = (2j + 1)(2j′ + 1) . Notation D(j)

c

– extension of D(j) to SL(2, ❈). ¯ D(j)

c

– complex conjugate of D(j)

c

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

SL(2, ❈)-representation theory II

Symmetric tensor products SL(2, ❈) has two fundamental representations: D( 1

2,0)

and D(0, 1

2 )

Symmetrised tensor products are irreducible: D(j,j′) = (D( 1

2,0))∨2j ⊗ (D(0, 1 2 ))∨2j′

• n

∆j,j′ := (∆ 1

2 ,0)∨2j ⊗ (∆0, 1 2 )∨2j′ . Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

SL(2, ❈)-spinors

Abstract index notation for 2-spinors (p, q)-spinors: ϕA1...Ap ˙

X1... ˙ Xq ∈ (∆ 1

2,0)⊗p ⊗ (∆0, 1 2 )⊗q

(p, q)-co-spinors: ψA1...Ap ˙

X1... ˙ Xq ∈ (∆∗

1 2 ,0)⊗p ⊗ (∆∗

0, 1

2 )⊗q

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 Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

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)

a

(T × S) = ∇(p,0)

a

S ⊗ T + S ⊗ ∇(0,q)

a

T ∇aǫAB = 0, ∇aσ A ˙

X b

= 0 Caution (symmetry and irreducibility again) ϕA1...Ap ˙

X1... ˙ Xq is element of an irreducible SL(2, ❈)-representation

⇔ ϕA1...Ap ˙

X1... ˙ Xq = ϕ(A1...Ap)( ˙ X1... ˙ Xq)

Thus: ∆j,j′ (∆ 1

2 ,0)⊗2j ⊗ (∆0, 1 2 )⊗2j′

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

Buchdahl’s equation

Buchdahl’s 1982 modification for spin s

2:

( ∇A ˙

Xϕ A1...As−1 ˙ X

− (s−1)(s−2)

µs

ǫA(A1Ψ|PQD|A2ψ

A3...As−1) PQD

− νψAA1...As−1 = 0 ∇ ˙

XAψAA1...As−1 − µϕ A1...As−1 ˙ X

= 0 for symmetric spinor fields ψAA1...As−1 = ψ(AA1...As−1) ϕ

A1...As−1 ˙ X

= ϕ

(A1...As−1) ˙ X

Operator notation: „ − 1

µ − Ps

\ ∇−ν / ∇ −µ « ψ ϕ ! = 0 1st order 0st order

( / ∇ψ) A1...As−1

˙ X

:= / ∇ ˙

XAψAA1...As−1

( \ ∇ψ)AA1...As−1 := \ ∇

A ˙ Xψ A1...As−1 ˙ X

(Psϕ)AA1...As−1 := (s−1)(s−2)

s

ǫA(A1Ψ|PQD|A2ϕ

A3...As−1) PQD Consistent for all s ∈ ◆ and µ = 0 (massive fields) Violation of minimal coupling principal?

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

Buchdahl-W¨ unsch equation

Equivalent formulation by [W¨

unsch, 1985]

0 = ∇ A ˙

Xϕ A1...As−1 ˙ X

− ν ψAA1...As−1 0 = ∇ ˙

XAψAA1...As−1 − µ ϕ A1...As−1 ˙ X

with µ, ν ∈ ❈, µ = 0, and ψAA1...As−1 = ψ(AA1...As−1) and ϕ A1...As−1

˙ X

= ϕ (A1...As−1)

˙ X

Why symmetrisation? Symmetrisation projects back onto the original irreducible SL(2, ❈)-representation.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary History Spinors and Representation Theory Buchdahl’s Equation and W¨ unsch’s Version of it

Generalised Dirac Equations – History

On flat Minkowski spacetime, [Dirac 1936]:

• ∂A

˙ X0ψAA1...Ak ˙ X1... ˙ Xl + µ ϕA1...Ak ˙ X0... ˙ Xl = 0

∂ ˙

X A0ϕA1...Ak ˙ X ˙ X1... ˙ Xl − ν ψA0...Ak ˙ X1... ˙ Xl = 0 ,

Naive minimal coupling to gravitation:

• ∇A

˙ X0ψAA1...Ak ˙ X1... ˙ Xl + µ ϕA1...Ak ˙ X0... ˙ Xl = 0

∇ ˙

X A0ϕA1...Ak ˙ X ˙ X1... ˙ Xl − ν ψA0...Ak ˙ X1... ˙ Xl = 0 .

Problem [Buchdahl 1962] Inconsistent for k + l > 1.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Green’s Operators

Situation: E any vector bundle on spacetime manifold M P : Γ(E) → Γ(E) first order differential operator looking for advanced/retarded Green’s operators: Definition: advanced/retarded Green’s operators Operators G± : Γ0(E) → Γ(E) are called advanced (+) resp. retarded (−) Green’s operators for P if P ◦ G± = Id on Γ0(E) G± ◦ P = Id on Γ0(E) supp(G±ϕ) ⊆ JM

± (supp(ϕ)) for ϕ ∈ Γ0(E)

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

General solution theorem – Statement

Theorem [Dimock 1982], [R.M. 2007] Let P, Q : Γ(E) → Γ(E) be first order linear differential operators, such that PQ is normally hyperbolic. Then there are unique adv./ret. Green’s operators for P, S± : Γ0(E) → Γ(E) , which do not depend on the choice of Q.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

General solution theorem – Remark

Spin 1

2 special case

P = −i / ∇ + m and Q = i / ∇ + m , was proven in [Dimock 1982].

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Idea of the proof

1 First ingredience:

Lemma PQ normally hyperbolic ⇔ QP normally hyperbolic ⇔ P∗Q∗ normally hyperbolic

2 Using results by [B¨

ar, Ginoux, Pf¨ affle, 2007], there are adv./ret. Green’s operators G± : Γ0(E) → Γ(E) for PQ G ′

± : Γ0(E∗) → Γ(E∗) for P∗Q∗

3 Set:

S± := QG± and S′

± := Q∗G ′ ±

4 Check: they are adv./ret. Green’s ops. for P resp. P∗. Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Technical details I

1 PS± = LG± = idΓ0(E) and P∗S′

± = L′G ′ ± = idΓ0(E∗)

immediately.

2 Support property of S± = QG±:

supp(G±ϕ) ⊆ JM

± (supp ϕ) for all ϕ ∈ Γ0(E)

as G± are Green’s operators. supp(QG±ϕ) ⊆ supp(G±ϕ) on general grounds. thus, supp(S±ϕ) ⊆ JM

± (supp(ϕ)).

similar argument for S′

± = Q∗G ′ ±)

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Technical details II

3 Deduction of S±P|Γ0(E) = idΓ0(E) and S′

±P∗|Γ0(E∗) = idΓ0(E∗):

for ψ ∈ Γ0(E∗) and f ∈ Γ0(E): S′

∓ψ, f = S′ ∓ψ, PS±f (2)

= P∗S′

∓ψ, S±f = ψ, S±f . (∗)

(2) because J±(supp(f )) ∩ J∓(supp(ψ)) is compact as a consequence of global hyperbolicity. Thus, (∗) shows: S± = (S′

∓)∗.

From P∗S′

∓ = idΓ0(E∗)

idΓ0(E) = (S′

∓)∗P = S±P

idΓ0(E∗) = S′

±P∗ by an analogous argument.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Technical details III

4 Uniqueness:

T± = S± second pair of advanced and retarded Green’s

• perators for P.

⇒ PT± = idΓ0(E) Thus, for every ψ ∈ Γ0(E∗) and f ∈ Γ0(E): ψ, S±f

(∗)

= S′

∓ψ, f = S′ ∓ψ, PT±f (3)

= P∗S′

∓ψ, T±f = ψ, T±f

step (3) like step (2) above. Thus, T± = S±. Choice of T± was independent of Q, thus, construction of S± is independent of Q.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Application to Buchdahl’s equation

Lemma

Setting Bs := „ − 1

µ − Ps

\ ∇ − ν / ∇ −µ « and B′

s :=

„ − 1

µ + Ps

\ ∇ − ν / ∇ −µ « , BsB′

s is normally hyperbolic.

Proof:

BsB′

s = 2

ν2 −

1 µ2 P2 s + \

∇ / ∇ −(ν + µ) \ ∇ − 1

µPs \

∇ −(ν + µ) / ∇ + 1

µ /

∇Ps / ∇ \ ∇ + µ2 !

Corollary There are unique advanced and retarded Green’s operators S± for Buchdahl’s operators Bs on (M, g).

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Cauchy Problem – Statement

Notation

Σ ⊆ M smooth spacelike Cauchy hypersurface DB

s M bundle of spin s 2 Buchdahl spinors on M

Φ ∈ Γ(DB

s M)

⇔ Φ = ψ(AA1...As−1) ϕ

(A1...As−1) ˙ X

!

Cauchy Problem for Buchdahl’s equations When does

• BsΦ = 0,

Φ ∈ Γ(DB

s M)

Φ|Σ = Φ0 have a (unique) solution for given cauchy datum Φ0 ∈ Γ0(DB

s Σ)?

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Cauchy Problem – General Results

E vector bundle on M P, Q : Γ(E) → Γ(E) 1st order diff. ops. such that PQ normally hyperbolic Σ ⊆ M spacelike C ∞-Cauchy hypersurface 8 > < > : QPΦ = 0, Φ ∈ Γ(E) Φ|Σ = Φ0 (∇nΦ)|Σ = Ψ0 for given Φ0, Ψ0 ∈ Γ0(E|Σ) solution guaranteed*

∗cf. [B¨

ar, Ginoux, Pf¨ affle, 2007]

( PΦ = 0, Φ ∈ Γ(E) Φ|Σ = Φ0 for given Φ0 ∈ Γ0(E|Σ) 8 > < > : QPΦ = 0, Φ ∈ Γ(E) Φ|Σ = Φ0 (PΦ)|Σ = 0 for given Φ0 ∈ Γ0(E|Σ) solution wanted auxiliary problem

in general

• perator specific constraints

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Cauchy Problem – Constraints

Proposition (Constraints for Bs)

For Φ = `ψ

ϕ

´ ∈ Γ(DB

s M) and Φ0 =

`ψ0

ϕ0

´ ∈ Γ0(DB

s Σ) we have

B′

sBsΦ = 0

Φ|Σ = Φ0 (∇nΦ)|Σ = Ψ0

for a Ψ0 ∈ Γ0(E|Σ)

9 > > = > > ; ⇔ 8 > < > : B′

sBsΦ = 0

Φ|Σ = Φ0 (BsΦ)|Σ = 0 if and only if n

˙ X A1 ( ˜

∇ ˙

XBψ BA1...As−1

− µϕ

A1...As−1 0 ˙ X

) = 0 (only if s ≥ 2) Local version of this was proved by [W¨ unsch, 1985].

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Sketch of the proof

1 One finds that

(BsΦ)|Σ = 0 ⇔ 8 < : (∇nψAA1...As−1)|Σ = −2nA ˙

X( ˜

∇ ˙

XBψ BA1...As−1

− µϕ

A1...As−1 0 ˙ X

) (∇nϕ

A1...As−1 ˙ X

)|Σ = −2n ˙

XB( ˜

∇B ˙

Xϕ A1...As−1 0 ˙ X

− 1

µ Psψ BA1...As−1

− νψ

BA1...As−1

) ,

So Ψ0 and Φ0 = ψ0

ϕ0

• are related by

Ψ0 = (∇nΦ)|Σ = @ −2nA ˙

X( ˜

∇ ˙

XBψ BA1...As−1

− µϕ

A1...As−1 0 ˙ X

) −2n ˙

XB( ˜

∇B ˙

Xϕ A1...As−1 0 ˙ X

− 1

µ Psψ BA1...As−1

− νψ

BA1...As−1

) 1 A

2 Caution: not always admitted Cauchy datum! Symmetry must

be enforced in upper component:

!

= ǫAA1nA ˙

X( ˜

∇ ˙

XBψ BA1...As−1

− µϕ

A1...As−1 0 ˙ X

) = n

˙ X A1 ( ˜

∇ ˙

XBψ BA1...As−1

− µϕ

A1...As−1 0 ˙ X

) .

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Cauchy Problem – Final Result I

Theorem [W¨

unsch 1985] [R.M. 2007]

For s ∈ ◆, the Cauchy problem

• BsΦ = 0,

Φ ∈ Γ(DB

s M)

Φ|Σ = Φ0 for given Φ0 ∈ Γ0(DB

s Σ) has a solution if and only if

Φ0 = (ψ0, ϕ0)tr satisfies the constraint n

˙ X A1 ( ˜

∇ ˙

XBψ BA1...As−1

− µϕ

A1...As−1 0 ˙ X

) = 0 (only if s ≥ 2) . Moreover, solutions are unique.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary A General Solution Theorem Cauchy Problem for Buchdahl’s Equations

Cauchy Problem – Final Result II

Corollary (Compatibility of the constraints) Σ′ ⊆ M second smooth spacelike Cauchy hypersurface with future-directed timelike unit normal field m. Φ solution of the Cauchy problem for Bs with Cauchy datum Φ0 ∈ Γ0(DB

s Σ).

Then Φ is also a solution of the Cauchy problem

• BsΦ = 0,

Φ ∈ Γ(DB

s M)

Φ|Σ′ = Φ′ with Cauchy datum Φ′

0 := Φ|Σ′.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Quantisation – General Idea

Σ ⊆ M spacelike C ∞-Cauchy hypersurface. H s

Σ := {Φ0 ∈ Γ0(DB s Σ) | Φ0 is an admitted Cauchy datum for BsΦ = 0 on Φ}

H s := {Φ ∈ Γ(DB

s M) | BsΦ = 0 and Φ|Σ ∈ H s Σ } ,

Canonical isomorphism of vector spaces: H s → H s

Σ

Φ → Φ|Σ , Σ′ ⊆ M second Cauchy hypersurface. We have canonically: H s

Σ′ ∼

= H s ∼ = H s

Σ .

Big plan Find a suitable Hermitian scalar product ·, · on H s, making (H s, ·, ·) a pre-Hilbert space. Quantise the system by buidling CAR(H s, ·, ·).

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Review of spin 1

2 case

For s = 1 the constraints vanish. Thus, H s

Σ = Γ0(DB 1 Σ) .

Dirac current: ja := Φ+

˜ A γa˜ A ˜ B Φ ˜ B ∈ Γ(TM)

Hermitian scalar product on H 1

Σ :

b1

Σ(Ψ0, Φ0) :=

• Σ

ιna ja(Ψ0, Φ0) dµΣ Pullback of b1

Σ to H s:

b1(Ψ, Φ) := b1

Σ(Ψ|Σ, Φ|Σ)

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Review of spin 1

2 case

Theorem [Dimock, 1982] The scalar product b1(Ψ, Φ) := b1

Σ(Ψ|Σ, Φ|Σ)

does not depend on the choice of Cauchy hypersurface Σ. There are canonical isometric isomorphisms: (H 1, b1) ∼ = (H 1

Σ , b1 Σ) ∼

= (H 1

Σ′, b1 Σ′) ∼

= . . . Question Is there a generalisation to arbitrary spin s

2?

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Generalised scalar product

Theorem (scalar product on H s

Σ )

The sesquilinear form bs

Σ(Φ, Ψ) :=

Z

Σ

(Φ+)

˙ X1... ˙ Xs−1 ˜ A

n

˜ A ˜ B n ˙ X1A1 · · · n ˙ Xs−1As−1Ψ ˜ AA1...As−1 dµΣ

with Φ, Ψ ∈ Γ0(DB

s Σ),

Ψ = (ψ1)AA1...As−1 (ψ2)

A1...As−1 ˙ X

! , Φ = (ϕ1)AA1...As−1 (ϕ2)

A1...As−1 ˙ X

! , Φ+ := ( ¯ ϕ2)

˙ X1... ˙ Xs−1 A

( ¯ ϕ1)

˙ X ˙ X1... ˙ Xs−1

!

is a Hermitian scalar product on H s

Σ .

Hard bit of the proof: Positivity. (Holds even fiberwise.) Even harder/unclear: Independence of Cauchy hypersurface.

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Quantisation using Illges framework [Illge 1993]

Lagrangian density depending on four fields ϕAA1...As−1 ξ

˙ X ˙ X1... ˙ Xs−1

χ

A1...As−1 ˙ X

ϑ

˙ X1... ˙ Xs−1 A

. The Euler-Lagrange equations for the variational principal associated with Lagrangian density are 0 = Bs ϕAA1...As−1 χ

A1...As−1 ˙ X

! , 0 = Bs ¯ ξAA1...As−1 ¯ ϑ

A1...As−1 ˙ X

! (two copies of Buchdahl’s equations).

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Quantisation using Illges framework [Illge 1993]

For Fermionic fields, minimal coupling to electromagnetic field yields a U(1)-invariance of Lagrangian. Current vector: jA ˙

X = ie

ˆ k(ϕAA1...As−1 ¯ ϑ

A1...As−1 ˙ X

+ χ ˙

XA1...As−1 ¯

ξ

A1...As−1 A

) − ¯ k( ¯ ϕ ˙

X ˙ X1... ˙ Xs−1ϑ ˙ X1... ˙ Xs−1 A

+ ¯ χA ˙

X1... ˙ Xs−1ξ ˙ X1... ˙ Xs−1 ˙ X

) ˜ , Equivalently: ja = e 2− s−1

2

· Ψ+

˜ Aa1...as−1 γa˜ A ˜ B Ψ ˜ Ba1...as−1

for Ψ

˜ Aa1...as−1 :=

ϕAA1...As−1ǫ

˙ X1... ˙ Xs−1 + ϑA ˙ X1... ˙ Xs−1ǫA1...As−1

χ

A1...As−1 ˙ X

ǫ ˙

X1... ˙ Xs−1 + ξ ˙ X1... ˙ Xs−1 ˙ X

ǫA1...As−1 ! , Ψ

+ a1...as−1 ˜ A

= ¯ χ

˙ X1... ˙ Xs−1 A

ǫA1...As−1 + ¯ ξ

A1...As−1 A

ǫ

˙ X1... ˙ Xs−1

¯ ϕ ˙

X ˙ X1... ˙ Xs−1εA1...As−1 + ¯

ϑ ˙

XA1...As−1ǫ ˙ X1... ˙ Xs−1

! .

Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary Outline of the Procedure Generalisation of the (Dimock 1982) spin 1

2 construction

Quantisation in Illge’s framework

Quantisation using Illges framework [Illge 1993]

Construction of the scalar product: Current: ja(Ψ, Φ) := e 2− s−1

2

· Ψ+

˜ Aa1...as−1 γa˜ A ˜ B Φ ˜ Ba1...as−1

Sesquilinear form: bs

Σ(Ψ, Φ) :=

Z

Σ

ιna ja(Ψ, Φ) dµΣ . Proposition bs

Σ is Hermitian but not positive definite. Rainer M¨ uhlhoff Higher Spin Fields

Statement of Buchdahl’s Equations Solving the Cauchy Problem Quantisation Summary

Summary

Achievements: Buchdahl’s equations in a mathematically rigorous formalism (vector bundles, abstract index notation) General solution theorem for a big class of first order differential operators on spacetime manifolds Global solution of the Cauchy problem for Buchdahl’s equation Quantisation attempt 1: naive generalisation of [Dimock 1982] – still open questions Quantisation attempt 2: using [Illge 1993] – seems not suitable Open questions: How arbitrary is the scalar product? regarding its construction, regarding the choice of a Cauchy hypersurface.

Rainer M¨ uhlhoff Higher Spin Fields