Dual Reality

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Abstract. A major issue for modern physics is how reality in terms of general relativity may emerge from quantum mechanics. Based on my previous proposal for an extension of the QM standard model this paper investigates in a emergent Riemannian space. It shows that this extended model enables a stochastic state reduction process that simply follows geodesic curves in a 4-dimensional spacetime.

Two observations motivate these papers and those that probably will follow:

1) General Relativity with Einstein's Field equation is highly recursive in how it is formulated. The energy distribution determines spacetime geometry and vice versa spacetime geometry determines local trajectories and the evolution of the mass-energy distribution. This feature is absent in quantum mechanics. Here the systems state and observables are different things. A state cannot operate on itself. To introduce recursion in QM requires an extended concept of what the state of a system is.

2) Many papers on collapse or decoherence deal with how observed classical reality is determined by QM. Common understanding is that the Universe evolves according to the Schroedinger equation - i.e. within a unitary U-process. I don't know of any work that considers the Universe - as we observe it - evolving according to a badly understood state reduction process (R-process).

This paper focuses on the second item. It proposes a toy model involving a single spin½ particle. The R-process shows exactly the standard behavior under observations, whereas the internal behavior can be modeled into a 4 dimensional Riemannian space.

Introduction

This paper and an earlier one had initially been motivated by looking for proceedings in artificial intelligence during the past decades. Though recent models and implementations perform quite successful on complex tasks the gaps are obvious. The understanding of intelligent reasoning and problem solving has not fundamentally improved over the past 20 years. A similar situation can be found in Biology and Psychology. They also fail to explain what really goes on behind intelligence, conscious behavior and consciousness.

Still insufficient but yet most rigorous are attempts to explain intelligent behavior (and finally how consciousness may evolve) by quantum models.

The major objective here is to show that the approach presented here looks promising enough to continue on that path. Its promise is to finally establish one common model that lets QM and GR emerge. And as a side effect we may find very new perspectives into artificial intelligence and conscious behavior.

The proposed extension originally is a discrete integer based model. Extended to reals it is equivalent to a complex model fully isomorphic to QM standard model. The base concept is a triple (A, B, v) of two complex operators A,B and a complex vector v. The operator A is the classical observable, the operator B represents the system, v is a perspective vector so that Bv represents the classical state vector.

On a first sight the model simply adds complexity that seems to be good for nothing. But indeed it provides some additional internal structure. For simple spin½ particles its state B is a complex vector of 8 real dimensions. This structure supports a simple random walk like stochastic process. It is controlled by the external operator A. This process externally behaves exactly as expected from wave reduction: It leaves the state in one of two eigenstates of the observable with the exact probabilities given by the QM standard model. This process experiences external excitations and attractors imprinted by the observable. It works like a grain of sand moving randomly on a sound board and ultimately approaching a wave node. This empirically developed process is quite amazing as it indeed requires 8 real dimensions to work as a simple random walk. Considering classical states with only 4 real dimensions will not produce the correct probabilities. What externally looks like a simple wave reduction caused by measurement now internally leads to concepts of position, movement, acceleration, forces, attractors, time. The entire process can easily be projected to 3 real dimensions. The projected movements within this toy model follow geodesic lines in a curved 4-dimensional spacetime. To achieve the expected results its crucial to leverage both the special GenI specific mapping to the 2-dimensional Hilbert space as well as its discrete structure.

The first part deals with specifying this stochastic state reduction process and discussing its features to some detail. The second part will focus on a spacetime structure that “absorbs” the process parameters and may be viewed as being imprinted by an external operator. The third part gives some outlook to further investigations in recursion and many particle systems.