48 Chapter 2: Motivation linear Regge trajectories: Soon afterward Virasoro [316] published an alternative form with full s , t , and u symmetry: There were N -particle generalizations of these formulae, but most surprisingly they were found to be factorizable into a spectrum of single particle states of an infinite number of harmonic oscillators { α µ $_{m}$} , m = 1 , 2 , . . . . This suggested a tantalizing interpretation: these were not just phenomenological fitting func- tions but rather a tree approximation of a full-fledged quantum field theory. Unfortunately the Lorentz transformation properties of these operators im- plied the existence of negative norm states. By imposing algebraic constraints, now named after him, on the oscillator operators Virasoro removed these states [317]. Solving one problem led to another, though: this algebra implied that the leading trajectory associated with Eq. (2.13) had an intercept at α (0) = 1, which meant in turn that in addition to a massless vector the theory admit- ted a tachyonic ground state as well. Additionally, unitarity required not the usual d = 4 dimensions of hadronic physics but rather d = 26. Finally the T = A ( s, t ) + A ( s, u ) + A ( t, u ) (2.12) A ( s, t ) = Γ $^{(}$− α ( s ) $^{)}$Γ$^{(}$− α ( t ) Γ $^{(}$− α ( s ) − α ( t ) . (2.13) T = Γ $^{(}$− 1 $_{2}$α ( s ) $^{)}$Γ$^{(}$− 1 $_{2}$α ( t ) $^{)}$Γ$^{(}$− 1 $_{2}$α ( u ) Γ $^{(}$− 1 $_{2}$α ( s ) − 1 $_{2}$α ( t ) $^{)}$Γ$^{(}$− 1 $_{2}$α ( s ) − 1 $_{2}$α ( u ) $^{)}$Γ$^{(}$− 1 $_{2}$α ( t ) − 1 $_{2}$α ( u ) $_{)}$. (2.14)