content_image/359012.jpg content_image/359013.jpg content_image/359014.jpg
such that 

<img src='content_image/359012.jpg'>

The | x 〉 states are again off-shell. However, using the well-known relation [36–38] 

in Eq. (14), along with Eq. (13), gives 

<img src='content_image/359013.jpg'>

<img src='content_image/359014.jpg'>

In a path integral approach, the probability amplitude 〈 x $^{′}$| x 〉 0 can be interpreted as the superposition of the probability amplitudes for each possible spacetime path from x to x $^{′}$. One can think of the | x 〉 0 states as representing the position x at which the paths start , while the | x $^{′}$〉 states represent the position x ′ at which the paths end . The difference between the states reflects the directionality of the paths—propagation is always from the start of the path to the end of the path. (The appendix covers in more detail the underlying spacetime path derivation of the formalism used in the main body of the text.) 

The decomposition of Eq. (15) separates the case in which x ′ is in the future of x from that in which x ′ is in the past of x . This shows that, while normal particles propagate into the future, antiparticles can effectively be considered to propagate backwards in time, into the past [11, 39, 40]. This division into particle and antiparticle paths depends, of course, on the choice of a specific coordinate system in which to define the time coordinate. However, if we take the time limit of the end point of the path to infinity for particles and negative infinity for antiparticles, then the particle/antiparticle distinction will be coordinate system independent [25]. 

The off-shell states | x 〉 are particularly useful because they essentially represent virtual particles and the probability amplitudes 〈 x $^{′}$| x 〉 0 are the propagation amplitudes for these particles on the inner edges of Feynman diagrams. They can therefore be used to construct a convenient representation for the scattering amplitudes of interacting particles. 

To do this, it is first necessary to consider multiple fields corresponding to different types of particles that may interact with each other. Then, an individual interaction vertex can be considered an event at which some number of incoming particles are destroyed and some number of outgoing particles are created. Note that the qualifiers “incoming” and “outgoing” 

