These wave functions are parametrized probability amplitude functions in the sense first de- fined by Stueckelberg [39, 40]. In this sense, the ψ ( x ; λ ) represent the probability amplitude for a particle to reach position x at the point along its path with parameter value λ . (For other related approaches using an invariant “fifth parameter”, though not necessarily a path evolution parameter, see [43–53].) The functions defined in Eq. (A3) form a Hilbert space over four dimensional spacetime, parametrized by λ , in the same way that traditional non-relativistic wave functions form a Hilbert space over three dimensional space, parametrized by time. We can therefore define a consistent family of position state bases | x ; λ 〉 , such that given a single Hilbert space state vector | ψ 〉 . These position states are normalized such that for each value of λ . Further, it follows from Eqs. (A3) and (A4) that Thus, Δ( x − x$_{0}$ ; λ − λ$_{0}$ ) effectively defines a unitary transformation between the various Hilbert space bases | x ; λ 〉 , indexed by the parameter λ . The overall state for propagation from x$_{0}$ to x is given by the superposition of the states for paths of all intrinsic lengths. If we fix q $^{µ}$( λ$_{0}$ ) = x µ $_{0}$, then | x ; λ 〉 already includes all paths of length λ − λ$_{0}$ . Therefore, the overall state | x 〉 for the particle to arrive at x should be given by the superposition of the states | x ; λ 〉 for all λ > λ$_{0}$ : Then, using Eq. (A5), Since Δ( x − x$_{0}$ ; λ − λ$_{0}$ ) only depends on the difference λ − λ$_{0}$ , the actual starting value λ$_{0}$ of the path parameter can be shifted arbitrarily. (This can be viewed as a gauge invariance of the path parameter λ [14, 25].) Nevertheless, it is convenient to consistently denote the starting value for λ as λ$_{0}$ . The position states | x ; λ$_{0}$ 〉 can then be identified with the states 32 | x 〉 ≡ ∫ ∞ λ 0 d λ | x ; λ 〉 . (A6) 〈 x | x$_{0}$ ; λ$_{0}$ 〉 = ∫ ∞ λ 0 d λ Δ( x − x$_{0}$ ; λ − λ$_{0}$ ) = ∫ ∞ 0 d λ Δ( x − x$_{0}$ ; λ ) = Δ( x − x$_{0}$ ) . Δ( x − x$_{0}$ ; λ − λ$_{0}$ ) = 〈 x ; λ | x$_{0}$ ; λ$_{0}$ 〉 . (A5) 〈 x $^{′}$; λ | x ; λ 〉 = δ $^{4}$( x ′ − x ) . ψ ( x ; λ ) = 〈 x ; λ | ψ 〉 , (A4)