ˆ G as in Eq. (17) gives a sum of Feynman diagrams for each possible number of interactions. The time-limit momentum states give the correct amplitudes for the truncated external legs of the diagrams [25]. Unfortunately, an interaction operator ˆ G of the form given in Eq. (17) with a vertex operator ˆ V of the form shown in Eq. (16) cannot actually generate all Feynman diagrams. For example, the vertex operator ˆ V from Eq. (16) necessarily has ˆ V | 0 〉 = 0. This means that ˆ G cannot generate vacuum bubble diagrams. Indeed, in general, ˆ V cannot construct a vertex unless all incoming particles already exist in the incoming state, which prevents the construction of loops involving particles incoming from vertexes constructed “later”. The problem is that the directionality of propagation implied by 〈 x $^{′}$| x 〉 0 is essentially arbitrary. We could just as well have defined “reverse” particle states | ¯ x 〉 and | ¯ x 〉 0 such that That is, in terms of the spacetime paths, | ¯ x 〉 now represents the start of the paths, while | ¯ x 〉 0 represents the end . To properly construct all possible Feynman diagrams, it is necessary to include such reverse particle states. This can be easily done in the formalism by doubling the Fock space through the addition of a reverse particle type ¯ n corresponding to each particle type n , whose field operators have the commutation rule Then, define That is, using this operator, the destruction of an n particle is treated as equivalent to the creation of an ¯ n particle (and vice versa). This is similar to the way particle destruction is related to antiparticle creation in the traditional field theory formalism, but the distinction for reverse particles is based on direction of propagation along particle paths (which is Lorentz invariant), not the direction of propagation in time. Now use the new operators ˆ ψ ( n $^{′}$) instead of ˆ ψ ( n ) in the construction of the vertex operator ˆ V . For example, an interaction of the form given in Eq. (16) becomes 12 ˆ V = ∫ d $^{4}$x ˆ V ( x ) , ˆ ψ ( n $^{$^{′}$)}$( x ) ≡ ˆ ψ ( n $^{)}$( x ) + ˆ ψ (¯ n ) $^{‡}$( x ) . [ ˆ ψ (¯ n $^{)}$( x $^{′}$) , ˆ ψ (¯ n ) $^{‡}$( x )] = Δ( x − x $^{′}$) . 〈 ¯ x $^{′}$| ¯ x 〉 0 = 〈 x | x $^{′}$〉 0 = Δ( x − x $^{′}$) .