Summary
Using the operatorial expression for the gauge conditions, we derive an explicit form for the twisting operator and the semi-twisting operator which takes theP states intoV states. The expression for the last operator facilitates the proof of theV states factorization previously found by us. We show that the twisting operator proposed in the literature is inconsistent with multiple factorization,i.e. factorization of amplitudes with extermal spinning particles. The correct twisting operator depends on the integration variables of the twisted line, and automatically satisfies double-twist invariance.
Riassunto
Utilizzando il formalismo operatoriale otteniamo un operatore di rovesciamento (twist) che soddisfa l'invarianza per doppio rovesciamento e che permette la fattorizzazione di ampiezze con particelle esterne con spin. Questo operatore differisce da quello proposto nella letteratura per una trasformazione di misura (gauge). Otteniamo anche l'operatore che diagonalizza l'operatore di rovesciamento e che porta alla fattorizzazione con gli statiV ottenuta in un lavoro precedente con altri metodi.
Резюме
Используя операторное выражение для калибровочных условий, мы выводим точную форму для оператора скручивания и оператора полу-скручивания, который переводитP состояния вV состояния. Выражение для последнего оператора облегчает доказательство факторизацииV состояний, которая была предварительно обнаружена нами. Мы показываем, что оператор скручивания, предложенный в литературе, не соответствует многократной факторизации, т. е. факторизации амплитуд с частицами, вращающимися относительно внещней оси. Правильный оператор скручивания зависит от переменных интегрирования линии скручивания, и это приводит к более вырожденному спектру, чем спектр, полученный ранее.
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References
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Our metric isg 00=−g ii=+1, so that the creation operator,a (n)†0 , acting on the vacuum, gives states of negative norm. The dot product between operators is defined by:a (n)†\a(n)=a (n)†μ gμνa (n)ν .
D. Amati, C. Bouchiat andJ. L. Gervais:Lett. Nuovo Cimento,2, 399 (1969).
Since the two-dimensional representation ofSU 1,1 is not unitary, one has to be careful in using alwaysL + andL −, and not their Hermitian conjugates. Moreover, since we have complex axes for the rotations, we are indeed dealing with a representation of the complex groupSL 2,σ.
This result has been previously obtained byC. B. Chiu, S. Matsuda andC. Rebbi:Phys. Rev. Lett.,23, 1526 (1969), by direct calculation. Further relation with their work is given in the Appendix.
We use the notations and conventions of I: dϕ(ϱ,p), for instance, is the integrand of the VenezianoN-point function. However, ourV (n)'s are divided by a factor 2n, so that we come back to the original definition ofFubini andVeneziano (2).
In the preprint version of I there is an extra factor 4 in the first expression of (3.7) due to a misprint.
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Amati, D., Le Bellac, M. & Olive, D. The twisting operator in multi-Veneziano theory. Nuovo Cimento A (1965-1970) 66, 831–844 (1970). https://doi.org/10.1007/BF02824724
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DOI: https://doi.org/10.1007/BF02824724