Linear Dependences in the Operator Formalism of Fubini, Veneziano, and Gordon

In this work we consider the Ward identities within the operator formalism of Fubini, Veneziano, and Gordon. We construct an operator $A$ which, when acting on any state of the Hilbert space of oscillators, gives a spurious state. We then show how one can use $A$ to remove the contributions of spurious states in unitary-loop calculations. In particular, we obtain the result of Bardakci, Halpern, and Shapiro for the planar closed loop. No effort is made to prove that $A$ generates the complete set of linear dependences.