## The Free Particle

QCI theory determines the behavior for the free particle, resulting in a number of different propagators corresponding, per the present interpretation, to different universes. Due to the presence of the p-norm in the generalized functional integrand, as described in the previous section, the limit at infinity of the complex time-discretization index is not unique, and a spectrum of distinct results are obtained for the behavior of the free particle. Much as different Calabi-Yau manifolds are interpreted to describe different universes in superstring theory, different limits of the generalized index could correspond to alternate realities.

The relevant functional integral involves integration over a complex number of distinct variables, a process novel to the mathematics of QCI Theory. We applied a generalization of variable substitution, as well as the separation of variables for decomposable multiple integrals, to compute this limit for complex index. The result is the standard free particle propagator, with the mass multiplied by a complex number z. The value of z is determined exactly by the values of the index phase and the choice of p-norm, and all possible values of z lie within an interesting region in the complex plane, between two boundary curves. This region is illustrated in our paper.

The relevant functional integral involves integration over a complex number of distinct variables, a process novel to the mathematics of QCI Theory. We applied a generalization of variable substitution, as well as the separation of variables for decomposable multiple integrals, to compute this limit for complex index. The result is the standard free particle propagator, with the mass multiplied by a complex number z. The value of z is determined exactly by the values of the index phase and the choice of p-norm, and all possible values of z lie within an interesting region in the complex plane, between two boundary curves. This region is illustrated in our paper.