String-localized fields of higher spin: massless limit and - PowerPoint PPT Presentation

KHRehren Leipzig, June 2017 Higher spin fields 1 / 29 String-localized fields of higher spin: massless limit and stress-energy tensor Karl-Henning Rehren Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen LQP-40, Leipzig, June 24, 2017

KHRehren Leipzig, June 2017 Higher spin fields 2 / 29 Abstract “Lifting” the massless limit of Wigner representations of higher spin to the associated local quantum fields , encounters several obstructions due to the well-known conflicts between Hilbert space positivity, covariance and causality. In a unified setting using “string-localization”, these conflicts can be resolved, and details of the decoupling of the degrees of freedom can be studied. Joint work with Jens Mund, Bert Schroer (arXiv:1703.04407 and 04408)

KHRehren Leipzig, June 2017 Higher spin fields 3 / 29 Plan: The lesson from “spin one” Spin two String-localized potentials Higher spin

KHRehren Leipzig, June 2017 Higher spin fields 4 / 29 THE LESSON FROM “SPIN ONE”

KHRehren Leipzig, June 2017 Higher spin fields 5 / 29 The quantum Maxwell potential “Canonical quantization” produces a conflict between Hilbert space positivity, covariance, and locality: Quantum field A µ such that F µν = ∂ µ A ν − ∂ ν A µ (“curl”)? � � � d 4 k θ ( k 0 ) δ ( k 2 )[ − η µν ] e − ikx : Feynman gauge A µ A ν = indefinite . ξ -gauges [ − η µν δ ( k 2 ) + ( ξ − 1) k µ k ν δ ′ ( k 2 )]: indefinite . � � � � δ ij − k i k j Coulomb gauge A 0 = 0, = : not covariant, A i A j � k 2 non-local . A positive, covariant, and local potential does not exist. Only the field strength (= curl of either of the above) is positive: � � F [ µν ] F [ κλ ] = − η µκ k ν k λ + η νκ k µ k λ + η µλ k ν k κ − η νλ k µ k κ .

KHRehren Leipzig, June 2017 Higher spin fields 6 / 29 Wigner quantization of free fields Wigner rep’ns of the Poincar´ e group = Hilbert space H 1 of one-particle states (induced from unirep of stabilizer gp of k 0 ). massive: (half-)integer spin, 2 s + 1 states (per momentum) massless: (half-)integer helicity, 1 state; or “infinite spin” . Local free fields on Fock space F ( H 1 ) of the form � � u i α ( k ) a α ( k ) e − ikx + v a ∗ e + ikx � Φ i ( x ) = d µ m ( k ) transform covariantly iff u i α ( k ) and v i α ( k ) fulfil an intertwining condition between a matrix representation of the Lorentz group and the given unitary representation of the stabilizer group.

KHRehren Leipzig, June 2017 Higher spin fields 7 / 29 For integer spin, Wigner quantization yields ( m > 0 , s ): symmetric traceless rank s tensor fields A µ 1 ...µ s (generalizing the Proca field). ( m = 0 , h = ± s ): rank 2 s field strength tensors F [ µ 1 ν 1 ] ... [ µ s ν s ] . (Single helicity fields are non-local). Intertwiners for massless potentials A µ 1 ...µ s do not exist. Massless Wigner rep’ns are “contractions” of massive rep’ns (ie, the inducing massless stabilizer group E (2) is a contraction of the massive SO (3)). Apparently, this limit does not lift to the associated quantum fields.

KHRehren Leipzig, June 2017 Higher spin fields 8 / 29 s = 1 : Massive case (Proca) � � m = − η µν + k µ k ν A µ A ν m 2 ≡ − π µν . Positive semi-definite . UV-dim = 2 ⇒ weak interaction non-renormalizable . Limit m → 0 does not exist. Define F µν := ∂ µ A ν − ∂ ν A µ , then � � F [ µν ] F [ κλ ] m = − π µκ k ν k λ ± · · · = − η µκ k ν k λ ± . . . is exactly the same as for m = 0, except that k 2 = m 2 . F [ m > 0] converges to F [ m = 0]. Moreover, ∂ ν F µν = m 2 A µ recovers the privileged (positive, covariant, local, conserved) potential from its field strength.

KHRehren Leipzig, June 2017 Higher spin fields 9 / 29 SPIN TWO

KHRehren Leipzig, June 2017 Higher spin fields 10 / 29 Why higher spin? Gravity (helicity 2) Why should Nature not use it?

KHRehren Leipzig, June 2017 Higher spin fields 11 / 29 “Spin 2” is similar: � � � � F [ µ 1 ν 1 ][ µ 2 ν 2 ] F [ κ 1 λ 1 ][ κ 2 λ 2 ] = curls of A µ 1 µ 2 A κ 1 κ 2 where � � 0 = 1 2 ( η µ 1 κ 1 η µ 2 κ 2 + η µ 2 κ 1 η µ 1 κ 2 ) − 1 m = 0 : A µ 1 µ 2 A κ 1 κ 2 2 η µ 1 µ 2 η κ 1 κ 2 , � � m = 1 2 ( π µ 1 κ 1 π µ 2 κ 2 + π µ 2 κ 1 π µ 1 κ 2 ) − 1 m > 0 : A µ 1 µ 2 A κ 1 κ 2 3 π µ 1 µ 2 π κ 1 κ 2 . Both field strengths are positive, covariant and local , with 2 s + 1 = 5 resp. 2 one-particle states per momentum; but Indefinite Feynman gauge massless potentials do not exist on the Fock space, Coulomb gauge non-covariant & non-local . Massive potential is recovered from its field strength via ∂ ν 1 ∂ ν 2 F [ µ 1 ν 1 ][ µ 2 ν 2 ] = ( m 2 ) 2 A µ 1 µ 2 . Positive, covariant, local, traceless and conserved. No massless limit .

KHRehren Leipzig, June 2017 Higher spin fields 12 / 29 . . . but different: The “curls” do not see the difference between η µν and � � π µν = η µν − k µ k ν in AA ; m 2 – but they see the different coefficients − 1 2 vs − 1 3 of the third term. Therefore also the massive field strength does not converge to the massless field strength . Even in lowest order (where non-renormalizability doesn’t matter), or in indefinite gauges (where the massless potentials can be used), perturbative massive gravity does not converge to massless gravity (vanDam-Veltman–Zakharov 1970). The UV dimension of the massive potential increases with s . Weinberg-Witten (1980): No local stress-energy tensor for m = 0. (Field strengths involve too many derivatives!)

KHRehren Leipzig, June 2017 Higher spin fields 13 / 29 STRING-LOCALIZED POTENTIALS

KHRehren Leipzig, June 2017 Higher spin fields 14 / 29 Some answers in this talk: Identification of potentials of any (integer) spin and any mass m ≥ 0 that live on the respective Wigner Fock spaces, do admit a massless limit, have non-increasing UV dimension 1, quantify the DVZ discontinuity, admit massless stress-energy tensors. The price: a weaker localization property (. . . of the potentials, not of the particles!)

KHRehren Leipzig, June 2017 Higher spin fields 15 / 29 s = 1

KHRehren Leipzig, June 2017 Higher spin fields 16 / 29 For any mass m ≥ 0, define op-val distributions in x and e ∈ R 4 � R + d λ F µν ( x + λ e ) e ν . A µ ( x , e ) := Short hand: A ( e ) = I e Fe = I e curl ( A ) e . These are potentials for their respective field strengths, defined on the respective Fock space, hence positive, regular at m = 0 (because F are), axial gauge potentials: e µ A µ ( e ) = 0, covariant: U (Λ) A ( x , e ) U (Λ ∗ ) = Λ − 1 A (Λ x , Λ e ), UV-tame: dimension 1, ”string-localized”: the commutator vanishes when the two “strings” x + R + e and x ′ + R + e ′ are spacelike separated; Remark: Causality requires spacelike e , WLoG e 2 = − 1.

KHRehren Leipzig, June 2017 Higher spin fields 17 / 29 Correlation functions For any m ≥ 0: � � e ′ µ k ν = − η µν + k µ e ν ( ee ′ ) k µ k ν A µ ( − e ) A ν ( e ′ ) ( ek ) + + ( e ′ k ) + − ( ek ) + ( e ′ k ) + m ≡ − E ( e , e ′ ) µν . The same formula for m > 0 and m = 0, except that k 2 = m 2 . Massless limit exists (as a limit of states on the Borchers algebra: the correlation functions define the fields). The string-localized massive potential converges to the massless potential (not only the field strength).

KHRehren Leipzig, June 2017 Higher spin fields 18 / 29 s = 2 is different

KHRehren Leipzig, June 2017 Higher spin fields 19 / 29 � R + d λ 1 d λ 2 F [ µ 1 ν 1 ][ µ 2 ν 2 ] ( x + λ 1 e + λ 2 e ) e ν 1 e ν 2 A µ 1 µ 2 ( x , e ) := A ( e ) = I e I e Fee = I e I e curl curl ( A ) ee . Again, these are potentials for their respective field strengths, defined on the respective Fock space, hence positive, regular at m = 0 (because F are), covariant: U (Λ) A ( x , e ) U (Λ ∗ ) = (Λ ⊗ Λ) − 1 A (Λ x , Λ e ) UV-tame: non-increasing dimension = 1, string-localized.

KHRehren Leipzig, June 2017 Higher spin fields 20 / 29 But, unlike s = 1 : � � m = 1 2 ( E µ 1 κ 1 E µ 2 κ 2 + E µ 2 κ 1 E µ 1 κ 2 ) − 1 m > 0 : A µ 1 µ 2 A κ 1 κ 2 3 E µ 1 µ 2 E κ 1 κ 2 , � � 0 = 1 2 ( E µ 1 κ 1 E µ 2 κ 2 + E µ 2 κ 1 E µ 1 κ 2 ) − 1 m = 0 : 2 E µ 1 µ 2 E κ 1 κ 2 . A µ 1 µ 2 A κ 1 κ 2 A µ 1 µ 2 ( e ) is regular in the massless limit, but the limit is not the massless string-localized potential (because of “ − 1 3 vs − 1 2 ”). Instead: A (2) µν ( e ) := A µν ( e ) + 1 2 E ( e , e ) µν a ( e ) where a ( e ) = − η µν A µν ( e ) = m − 2 ∂ µ ∂ ν A µν ( e ) and E ( e , e ) µν = η µν + e µ I e ∂ ν + e ν I e ∂ µ + e 2 I e I e ∂ µ ∂ ν is the momentum space kernel of the 2-point function as an operator in x -space. Proposition: The (string-localized) field strengths of the potentials A (2) on the massive spin-2 Fock spaces converge to the massless field strength.