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denoted | x 〉 0 in Sec. III, with the states denoted | x 〉 there being the same as those defined in Eq. (A6). 

The position states | x 〉 as defined above make no distinction based on the time-direction of propagation of particles. Normally, particles are considered to propagate from the past to the future. Therefore, we can define normal particle states | x$_{+}$ 〉 such that 

<img src='content_image/349753.jpg'>

On the other hand, antiparticles may be considered to propagate from the future into the past [11, 39, 40]. Therefore, antiparticle states | x$_{−}$ 〉 are such that 

<img src='content_image/349752.jpg'>

Note that the states | x$_{±}$ 〉 defined here differ from the definitions of the similarly notated states in Sec. III in that the Heaviside theta functions are included in the definitions of the states in Eqs. (A7) and (A8) but not in the definitions in Sec. III. This means that the states | x$_{±}$ 〉 defined here are not actually on-shell, but, on the other hand, they clearly capture the fact that particles propagate only into the future and antiparticles propagate only into the past. Nevertheless, as noted in Sec. III, we can recover on-shell states by going to the infinite-time limit. 

In taking the infinite-time limit of a spacetime path, one cannot expect to hold the 3- position of the path end point constant. For a free particle, though, it is reasonable to take the particle 3-momentum as being fixed. In [25] it is shown that, at the time limit of infinity (for particles) or negative infinity (for antiparticles), such 3-momentum states do indeed become on-shell. Thus, the momentum shell constraint is not imposed arbitrarily but, rather, is a natural consequence of the infinite-time limit for free particles—but only holds approximately, otherwise. 

For the purposes of this paper, the 4-dimensional position states | x$_{0}$ ; λ$_{0}$ 〉 (or | x 〉 $_{0}$, as they are denoted in the main body) are more useful then the on-shell particle and antiparticle states. It is, of course, straightforward to construct corresponding momentum states: 

<img src='content_image/349751.jpg'>

But such states are inherently off shell, with no restriction on the value of the energy p 0 relative to the 3-momentum p . Nevertheless, keep in mind that in any scattering-like 

