content_image/333903.jpg content_image/333902.jpg content_image/333900.jpg content_image/333901.jpg content_image/333899.jpg
quantum field theoretic approach described in Sec. III. The formalism presented here can also be extended to particles of non-zero spin [26], but, for simplicity, this will not explicitly be considered here, since the introduction of spin indices does not fundamentally affect the points to be made in this paper. However, note that the introduction of reverse particles in Sec. III is an extension to the formalism presented in [25] that is necessary to fully reproduce the results of Sec. V and Sec. VI. 

A spacetime path is specified by four functions q $^{µ}$( λ ), for µ = 0 , 1 , 2 , 3, of a path parameter λ . Note that such a path is not constrained to be timelike or even to maintain any particular direction in time. The only requirement is that it must be continuous. And, while there is no a priori requirement for the paths to be differentiable, we can, as usual, treat them as differentiable within the context of a path integral (see the discussion in [25].) 

It is well known that a spacetime path integral of the form 

<img src='content_image/333903.jpg'>

for an appropriate normalization constant η and the Lagrangian function 

<img src='content_image/333902.jpg'>

gives the free-particle Feynman propagator [12, 14, 18, 25]. In the path integral above, the notation D $^{4}$q indicates that the integral is over the four functions q $^{µ}$( λ ) and the delta functions constrain the starting and ending points of the paths integrated over. (See also [25] for a justification of Eq. (A1) from a small number of physically motivated postulates.) 

Consider, however, that Eq. (A1) can be written 

where 

<img src='content_image/333900.jpg'>

The value λ − λ$_{0}$ in Δ( x − x$_{0}$ ; λ − λ$_{0}$ ) can be thought of as fixing a specific intrinsic length for the paths being integrated over. Equation (A2) now has a similar path integral form as the usual non-relativistic propagation kernel [5, 6], except with paths parametrized by λ rather than time. We can, therefore, use the relativistic kernel of Eq. (A2) to define parametrized wave function in a similar fashion to the non-relativistic case: 

<img src='content_image/333901.jpg'>

<img src='content_image/333899.jpg'>

