The Spectrum of Physical States of the Dual Resonance Model Paolo Di Vecchia Niels Bohr Instituttet, Copenhagen and Nordita, Stockholm Firenze, May 18, 2007 Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 1 / 23
Foreword ◮ This talk is based on P . Di Vecchia, "The Birth of String Theory", arXiv:0704.0101. PdV and A. Schwimmer, "The Beginning of String Theory: a Historical Sketch". ◮ Contributions to the Gabriele Veneziano celebrative volume "String theory and fundamental interactions", Ed.s M. Gasperini and J. Maharana, Springer. Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 2 / 23
Plan of the talk N -point amplitude 1 Factorization 2 Problem with ghosts 3 QED 4 The Virasoro conditions 5 Characterization of physical states 6 Scattering amplitudes for physical states 7 DDF states and no ghosts 8 From DRM to String Theory 9 10 Conclusions Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 3 / 23
N -point amplitude ◮ Following the principle of planar duality and the axioms of S-matrix theory the scattering amplitude B N ( p 1 , p 2 , . . . p N ) for the scattering of N particles was constructed: � ∞ N � N 1 dz i θ ( z i − z i + 1 ) � ( z i − z i + 1 ) α 0 − 1 � � � ( z i − z j ) 2 α ′ p i · p j B N = dV abc −∞ i = 1 j > i ◮ There is a Koba-Nielsen variable z i for each external particle. ◮ Invariance under the projective group : z i → Az i + B Cz i + D . Three of the variables z i can be fixed: z 1 = ∞ , z 2 = 1 , z N = 0. ◮ Only simple poles lying on linearly rising Regge Trajectories: α ( s ) = α 0 + α ′ s ◮ What is the meaning of this amplitude? What is the spectrum of particles? Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 4 / 23
Factorization ◮ Since a particle corresponds to a pole in the scattering amplitude with factorized residue, the "obvious" thing to do was to study the factorization properties of the amplitude at each pole. [ Fubini and Veneziano + Bardaçki and Mandelstam, 1969] ◮ Introduce an infinite set of harmonic oscillators [Fubini, Gordon and Veneziano; Nambu, Susskind, 1969 ] [ a n µ , a † [ˆ q µ , ˆ m ν ] = η µν δ nm ; p ν ] = i η µν , the Fubini-Veneziano operator [Fubini and Veneziano, 1969 and 1970]: Q µ ( z ) = Q (+) ( z ) + Q ( 0 ) µ ( z ) + Q ( − ) ( z ) µ µ where ∞ ∞ √ √ a † a n Q (+) = i √ nz − n ; Q ( − ) = − i � � n √ nz n 2 α ′ 2 α ′ n = 1 n = 1 Q ( 0 ) = ˆ q − 2 i α ′ ˆ p log z Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 5 / 23
◮ and the vertex operator; V ( z ; p ) =: e ip · Q ( z ) : ≡ e ip · Q ( − ) ( z ) e ip ˆ p · p log z e ip · Q (+) ( z ) q e + 2 α ′ ˆ ◮ In terms of them we can rewrite the N -point amplitude using this operator formalism: � ∞ N � N 1 dz i θ ( z i − z i + 1 ) A N ≡ ( 2 π ) d δ ( d ) ( � p i ) B N = × dV abc −∞ i = 1 N N � ( z i − z i + 1 ) α 0 − 1 � � � × � 0 , 0 | V ( z i , p i ) | 0 , 0 � i = 1 i = 1 Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 6 / 23
p 2 + � ∞ n = 1 na † ◮ or introducing the propagator ( L 0 = α ′ ˆ n · a n ) : � 1 1 1 dxx L 0 − 1 − α 0 ( 1 − x ) α 0 − 1 = D = L 0 − 1 = p 2 + R − 1 if α 0 = 1 α ′ ˆ 0 ◮ we get A N ≡ � 0 , p 1 | V ( 1 , p 2 ) D . . . V ( 1 , p M ) DV ( 1 , p M + 1 . . . DV ( 1 , p N − 1 ) | 0 , p N � ◮ that can be rewritten as follows: A N ( p 1 , p 2 . . . p N ) = � p ( 1 , M ) | D | p ( M + 1 , N ) � where � p ( 1 , M ) | = � 0 , p 1 | V ( 1 , p 2 ) DV ( 1 , p 3 ) . . . V ( 1 , p M ) and | p ( M + 1 , N ) � = V ( 1 , p M + 1 ) D . . . V ( 1 , p N − 1 ) | p N , 0 � Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 7 / 23
◮ At the pole the amplitude can be factorized by introducing two complete set of states: 1 � A N = � p ( 1 , M ) | λ, P �� λ, P | R − α ( s ) | µ, P �� µ, P | p ( M + 1 , N ) � λ,µ ◮ The propagator develops a pole when ( R = � ∞ n = 1 na † n · a n ) ∞ α ( s ) ≡ 1 − α ′ P 2 ≡ 1 − α ′ ( p 1 + · · · + p M ) 2 = � na † n · a n = m n = 1 is a non-negative integer ( m ≥ 0). ◮ The residue at the pole α ( s ) = m factorizes in a finite sum of terms corresponding to the states | µ, P � satisfying the condition: ∞ na † � R | µ, P � ≡ n · a n | µ, P � = m | µ, P � n = 1 Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 8 / 23
◮ The lowest state, corresponding to m = 0, is the vacuum of oscillators: | 0 , P � with 1 − α ′ P 2 = 0. This is a tachyon because α 0 = 1. ◮ The next state with m = 1 is the state: a † 1 µ | 0 , P � corresponding to a massless vector. ◮ At the level m = 2 we have the following states ( 1 − α ′ P 2 = 2): a † 1 µ a † 1 ν | 0 , P � ; a † 2 µ | 0 , P � ◮ At the level m = 3 we have the following states ( 1 − α ′ P 2 = 3 ) : a † 1 µ a † 1 ν a † 1 ρ | 0 , P � ; a † 2 µ a † 1 ν | 0 , P � ; a † 3 µ | 0 , P � ◮ and so on Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 9 / 23
Problems with ghosts ◮ The N -point amplitude is Lorentz invariant. ◮ This forces to factorize the amplitude by introducing a space that is not positive definite: [ a n µ , a † m ν ] = η µν δ nm ; η µν = ( − 1 , 1 , . . . , 1 ) ◮ Therefore the states with an odd number of time components have a negative norm. ◮ This is in contradiction with the fact that in a quantum theory the Hilbert space must be positive definite due to the probabilistic interpretation of the norm of a state. ◮ General problem: how to put together Quantum theory ⇔ Special Relativity Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 10 / 23
QED ◮ Consider a scattering amplitude in QED near a photon pole. We can write it as follows: A µ ( p 1 , . . . p M , P ) η µν P 2 B ν ( P , p M + 1 . . . p N ) ; η µν = ( − 1 , 1 , 1 , 1 ) Naively it seems that the residue consists of four terms and one of them is a ghost corresponding to a negative norm state. ◮ But gauge invariance implies: P µ A µ = P µ B µ = 0 ◮ In the frame where P µ = E ( 1 , 0 , 0 , 1 ) gauge invariance implies: A 3 − A 0 = B 3 − B 0 = 0 Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 11 / 23
◮ They imply that the residue at the photon pole has only two terms: 2 A i ( p 1 , . . . p M , P ) δ ij � P 2 B j ( P , p M + 1 . . . p N ) ; i , j = 1 , 2 i , j = 1 corresponding to the two helicities ± 1 of the photon. ◮ In this way QED solves the potential conflict between special relativity and quantum theory. ◮ We can write everything in a covariant way in a space containing negative norm states, ◮ but then we know that gauge invariance eliminates the unwanted states, ◮ and the spectrum of physical states is positive definite. ◮ The physical states are characterized by the "Fermi condition" ∂ µ A (+) µ | Phys . � = 0 ◮ Do we have similar relations in the DRM? Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 12 / 23
The Virasoro conditions ◮ One such condition was immediately found: W 1 | p ( 1 , M ) � = 0 ; W 1 = L 1 − L 0 L 0 and L 1 can be written in terms of harmonic oscillators. ◮ It was used to show that there was no negative norm state at the first excited level [Fubini and Veneziano, 1970]. ◮ But it was not enough to eliminate all the non-positive norm states. ◮ Then Virasoro realized that, if α 0 = 1, one can find an infinite number of such conditions: W n | p 1 ... M � = 0 ; n = 1 . . . ∞ ; W n = L n − L 0 − ( n − 1 ) [ Virasoro , 1969] ◮ and hope that they can cancel all the non-positive norm states. Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 13 / 23
Characterization of physical states ◮ Virasoro found the analogous of the condition imposed by gauge invariance. ◮ But what is the condition that is the analogous of the Fermi condition in QED? ◮ Those conditions were found proceeding as in QED L n | Phys ., P � = ( L 0 − 1 ) | Phys ., P � = 0 ; 1 − α ′ P 2 = m [Del Giudice and PDV, 1970] ◮ At the level m = 1 the analysis reduces to the one in QED. ◮ At the level m = 2 the physical states are a spin 2: d − 1 1 | Phys > 1 = [ a † 1 , i a † � a † 1 , k a † 1 , j − ( d − 1 ) δ ij 1 , k ] | 0 , P � k = 1 with positive norm ( i , j are space indices), Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 14 / 23
◮ and a spin 0 � d − 1 � 1 , i + d − 1 � a † 1 , i a † ( a † 2 1 , 0 − 2 a † | Phys � 2 = 2 , 0 ) | 0 , P � 5 i = 1 ◮ with norm equal to 2 ( d − 1 )( 26 − d ) (1) that is positive if d > 26. ◮ The state decouples from the physical spectrum if d =26. ◮ But the original analysis was done taking for grant that d = 4.... as was...obvious...at that time.... ◮ The absence of ghosts was also shown at the level m = 3, but it was difficult to proceed further. ◮ The remaining question was: Is the DRM free of ghosts? ◮ But we had to wait few years to get an answer. Paolo Di Vecchia (NBI+NO) Physical Spectrum Firenze, May 18, 2007 15 / 23
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