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A process model of quantum mechanics utilizes a combinatorial game to generate a discrete and finite causal space, which can be defined as a self-consistent quantum mechanics. An emergent space-time and continuous wave function arise through a non-uniform interpolation process. Standard non-relativistic quantum mechanics emerges under the limit of infinite information (the causal space grows to infinity) and infinitesimal scale (the separation between points goes to zero). This model has the potential to address several paradoxes in quantum mechanics while remaining computationally powerful.

Questions concerning the completeness of quantum mechanics and the proper interpretation of the wave function date back to its earliest days and remain unresolved to this present day. Recent research into hidden variable theories [

There have been previous attempts to provide a realist, emergent, or process based quantum mechanics: Bohmian mechanics, quantum hydrodynamics, Wolfram’s cellular automata, continuous spontaneous localization (GRW), Finkelstein’s quantum relativity, Noyes’s bit-string physics, Bastin and Kilmister’s combinatorial physics, Hiley’s process physics [

The key idea, based on Whitehead’s Process Theory [

Space-time, and the physical entities that manifest in space-time, are all postulated to be emergent from informons which manifest at a scale that precludes any possibility of direct observation by the physical entities that emerge from them. Ironically, informons must be interpreted in reference to the entities they generate [

The passing of information from one generation to another induces a causal structure. Given two informons, n, n', write _{n} of properties inherited from the process that generates it, the most important being the “strength” or “coupling effectiveness” of the generating process, _{n}, which for convenience is simply referenced to the prior informons that contributed information to its generation. G_{n} will thus be a set of informons from previous generations of informons that forms either an acyclic directed or ordered set. Properties are intrinsic, whereas interpretations are extrinsic but provide the connection to theory.

Although informons are transient, it is useful to exploit the artifice of a history of their appearance, thus interpreting each informon n as a point x_{n} in a causal manifold

Formally, denote an informon by

Informons are understood to be generated by processes. Processes possess only algebraic properties: they generate space-time and so cannot be situated in space-time. A process may be active, in which case it acts in a series of rounds to generate informons, or inactive. Each process

A primitive process is defined as generating a single informon during a single round

The generation of an informon can trigger a coupling between processes or the activation or inactivation of processes, depending upon the compatibility of these informons [

Primitive processes combine to generate multiple informons during a round

A product

The process viewpoint leads to the insight that the proper setting for quantum dynamics is not the Hilbert space

where for two sets of functions A, B the sum

The initial causal tapestry

Interpolation theory shows that given certain choices of the interpolation function g, in the limit

The process covering map (PCM) provides a linkage between the space of processes,

Conservation laws and symmetries applied to the properties of processes provide a set of algebraic constraints upon possible interactions among processes. These are inherited by the wave functions through the process covering map suggesting that processes are primary and wave functions secondary. Quantum mechanics may be best viewed as an effective theory, valid under certain asymptotic limits, but not necessarily the final theory.

Processes are considered to act non-deterministically, a term used in computation theory to mean that actions are described by set-valued maps without any intrinsic probability structure. Probabilities arise through two mechanisms: combinatorial proliferation, similar to the case for iterated function systems, and coupling of processes, in particular through couplings to measurement processes. One cannot do justice to this topic in a short note and the theory of interaction and measurement will be discussed in a separate letter.

The simplest, heuristic representation of process is as a two player, co-operative, combinatorial game [

Let us consider a single primitive non-relativistic process interacting with a potential

Player I propagates information forward to the nascent generation while Player II uses this to construct the new informons. Let

Let

form

translated sinc function_{P} and l_{P} respectively. Depending upon the exact model, t_{P}, l_{P} may be universal, for example the Planck time and length respectively, or they may depend upon the energy and momentum of the system. Note that the dynamics is defined using information from

where

Then

Parzen’s theorem (see [

Now

Let

which for very large r and N is approximately

which, if recursively

which according to Parzen’s theorem and Feynman and Hibbs [

in the case that _{P}, l_{P} can be finite). If this is not the case then one must also take the limit as

The discrepancy between the global Hilbert space interpretation given above and the standard non-relativistic quantum mechanics wave function depends upon the accuracy of the approximation to the integral

rent informons contributing information to any nascent informon as well as the values of t_{P}, l_{P}. This is a difficult problem to assess in general but results are available in special cases. For example, in one dimension, if the

wave function

between each embedding point and its ideal lattice embedding point is less that

where

Hence, _{P} is the Planck length. In three dimensions,

The appropriate strategies and interpolation methods are a matter for future study and comparison to empirical data. Most promising are radiative, action or kernel and non-uniform strategies.

In summary, the adoption of a process point of view results in a novel model of (non-relativistic) quantum mechanics which is discrete, finite and intuitive. The model is self contained, and NRQM appears as an idealization under certain limiting conditions. It involves the propagation of only causally local information, although the discontinuous action of process makes it quasi-local. Informons possess definite properties, but only those properties imparted to them by their generating processes, which can only be determined through an interaction with a measurement process. The model is locally non-contextual but globally contextual, hence quasi-non-contextual. Any type of computation involving the NRQM wave function may be carried out using the global

Thanks are due to Irina Trofimova and Robert Mann for many fruitful discussions.