**2. The generic realistic model**

The central core of our theory consists of a set of models whose logic is entirely classical and deterministic. Deterministic does not mean *pre-*deterministic: there is no shortcut that would enable one to foresee any special feature of the future without performing extremely complex simulation calculations using the given evolution laws. There is no 'conspiracy'. Also, we do not take our refuge into any form of statistics. The equations determine exactly what is happening. Of course we do not know today exactly what the equations are, but we do assume them to exist.

The equations will be *more precise* even than Newton's equations for the motion of the planets. Newton's equations are given in terms of variables whose values are determined by real numbers. But, in practice, it is impossible to specify these numbers with infinite precision, and consequently, *chaos* takes place: it is fundamentally impossible, for instance, to predict the location of the dwarf planet Pluto, one billion years from now, because such a calculation would require the knowledge of the locations and masses of all planets in more than 20 digits accuracy today [8]. That's a tiny fraction of a micron for Pluto's orbit. Following Pluto during the age of the universe would require accuracies beyond 2000 digits, much tinier margins than the Planck length. To describe Pluto's position exactly would require an infinite number of decimal places to be rigorously defined.

Our deterministic theory will be formulated in terms of integer numbers only, which can be defined exactly without the need of infinitely many decimal places. This kind of precision in defining theories may well be what is needed to understand quantum mechanics.

For simplicity, we imagine a universe with finite size and finite time. As for their mathematical structure, all deterministic models are then very much alike. All finite-size discrete models must have finite Poincaré recursion times. There will be different closed cycles with different periods, see **Figure 1**. Counting these cycles, one finds that the rank of a cycle is physically a conserved quantity, almost by definition. For simplicity, we constrain ourselves to *time reversible* evolution laws, although it is suspected that one might be allowed to relax this rule, but then the mathematics becomes more complex.

We now emphasise that the evolution law of such a deterministic system can be exactly described in terms of a legitimate, conventional Schrödinger equation. We say that quantum mechanics is a *vector representation* of our model: every possible state the system can be in is regarded as a vector in the basis of Hilbert space. This set of vectors is orthonormal. The classical evolution law will send any of these vectors into an other one. Since these vectors are all orthonormal and since the evolution is time-reversible, one can easily prove that the evolution matrix is unitary. It contains only the numbers 1 and 0. There is only one 1 in each row and in each column; all the other entries are 0, from which unitarity follows.

By diagonalising this matrix, one finds all its eigenvectors and eigenvalues. Within one cycle, the eigenvalues of *U t*ð Þ are *<sup>e</sup>*�2*πint=<sup>T</sup>*, where *<sup>t</sup>* is time, *<sup>T</sup>* is the period, and *n* is an integer. The formal expression for the eigenvectors is easily obtained:

$$\left<\mathbf{k}\middle|n\right>^{E} = \frac{\mathbf{1}}{\sqrt{N}} e^{-2\pi ink/N} \quad , \tag{1}$$

where <sup>∣</sup>*k*iont are the ontological states, labelled by the integer *<sup>k</sup>*, and <sup>∣</sup>*n*i*<sup>E</sup>* are the energy eigenvectors. We read off in the basis formed by the states <sup>∣</sup>*n*i*<sup>E</sup>* that the Hamiltonian takes the values.

$$H\_{nm} = 2\pi n \,\delta\_{nm}/T.\tag{2}$$

At first sight, this does not look like quantum mechanics; the series of eigenvalues (2) seems to be too regular. In [9] it was proposed to add arbitrary additive

**Figure 1.**

*Generic evolution law for a realistic model with different periodicities. In this example we see 5 cycles, with ranks 2, 3, 6, 8 and 11.*

energy renormalization terms, depending on the cycle we are in, but the problem is then still that it is difficult to see how this can reproduce Hamiltonians that we are more familiar with. The energy eigenstates seem to consist of large sequences of spectral lines with uniform separations. A more powerful idea has been proposed recently [10, 11]. More use must be made of locality. We wish the Hamiltonian to be the sum of locally defined energy density operators,

$$H = \sum\_{\vec{x}} \mathcal{H}(\vec{x})\ . \tag{3}$$

Now this is really possible. The price to be paid is to add *fast fluctuating, localised variables*, called 'fast variables' for short. They replace the vague 'hidden variables' that were introduced in many earlier proposals [12].

The fast variables, 0 ≤*φi*ð*x* !Þ< 2*π*, are basically fields that rapidly repeat their values with periodicities *Ti*ð*x* !Þ, which we choose all to be large and mostly different. To reproduce *realistic* quantum mechanical models, we need these periods to be considerably shorter in time than the inverse of the highest energy collision processes that are relevant.

To a good approximation, the fast variables will be non-interacting. This means that the energy levels will take the form *E* ¼ 2*π* P *i*,*x* !*ni*ð*x* !Þ*=Ti*, where the *ni* are all integer, and it implies that there is one ground level, *E*<sup>0</sup> ¼ 0, while all excited states have energies *E* ≥2*π=Ti*. Clearly, our conditions on the fast variables were chosen such that their excited energy levels exceed all energy values that can be reached in our experiments.<sup>1</sup>

Note that energy is exactly conserved. Therefor we may assume that, if an initial state is dominated by the state ∣*E* ¼ 0i, it will stay in that state.

Now consider the quantum model that we wish to mimic. Let that have a basis of *N* states, ∣*α*i, ∣*β*i, ⋯, with 1≤*α*, *β*, ⋯ < *N*, to be called the *slow* variables. Their interactions are introduced as *classical* interactions with the fast variables, as follows:

Two states ∣*α*i and *β*i are interchanged whenever the fast variables in the immediate vicinity of states *α* and *β* simultaneously cross a certain pre-defined point on their (fast) orbits.

Here, the 'vicinity' must be a well-defined notion for these states. In the case of non-relativistic particles, it means that we defined the states as the particle(s) in the coordinate representation ∣*x* !ð Þi *t* . In the relativistic case we take the basis of states specified by the fields *ϕi*ð*x* !Þ. This does imply that, in both cases, we regard the particles and/or fields to undergo exchange transitions that eventually will generate the desired Schrödinger equation or field equations.

One can describe these classical interchange transitions in terms of a 'quantum' perturbation Hamiltonian.

$$H^{\rm int} = \frac{\pi}{2} \sum\_{a,\beta,s} \sigma\_{\mathcal{Y}}^{[a,\beta]} \delta\_{\varphi\_a,\rho\_a^{(\flat)}} \delta\_{\varphi\_\beta,\rho\_\beta^{(\flat)}} \,, \tag{4}$$

where *σ*½ � *<sup>α</sup>*,*<sup>β</sup> <sup>y</sup>* is one of the three Pauli matrices *σx*, *σy*, *σz*, acting on the two-dimensional subspace spanned by the two states ∣*α*i and ∣*β*i.

<sup>1</sup> More precisely, we talk of energies that can be associated to single quantum particles at isolated points in space–time.

Some special points on the orbits of the fast variables *φα* and *φβ* will be indicated as *φ*ð Þ*<sup>s</sup> <sup>α</sup>* and *φ*ð Þ*<sup>s</sup> <sup>β</sup>* . If the fast variables *φα* and *φβ* reach their special positions *simultaneously* then the corresponding classical states ∣*α*i and ∣*β*i are interchanged.

In Eq. (4), we used a discretised notation, where the time unit is chosen such that it is the time needed to advance the fast variables by only one step in their (discretised) orbits. One may check that the factor *π=*2 is crucial to guarantee that, if the special point is reached, the equation.

$$e^{-\not\varphi\_{\mathcal{V}}} = -i\sigma\_{\mathcal{V}} = \begin{pmatrix} \mathbf{0} & -\mathbf{1} \\ \mathbf{1} & \mathbf{0} \end{pmatrix} \tag{5}$$

describes a classical interchange, without generating superpositions. The minus sign is unavoidable but causes no harm. We chose the Pauli matrix *σ<sup>y</sup>* because, when combined with the factor *i* in the Schrödinger equation, the wave function will be propagated as a *real*-valued quantity. One might desire to generate one of the other Pauli matrices also using classical physics. This can be done by adding a dummy binary variable, as described in [11] (the binary variable also propagates classically).

The Hamiltonian describing the evolution of the slow variables is now derived by assuming that the fast variables never get enough energy to go to any of their excited energy states. Their lowest energy states are <sup>∣</sup>0i*<sup>E</sup>* obeying

$$\langle \mathbf{k} | \mathbf{0} \rangle^{E} = \frac{\mathbf{1}}{\sqrt{N}} \ , \tag{6}$$

so that the expectation value of a Kronecker delta is

$$
\left< \mathbf{0} \middle| \delta\_{\boldsymbol{\varphi}\_a, \boldsymbol{\varphi}\_a^{(\boldsymbol{\gamma})}} \middle| \mathbf{0} \right> = \frac{1}{N} \quad , \tag{7}
$$

where *N* is the number of points on the fast orbit of this variable.

Eq. (5) could als be used if we had only one Kronecker delta in Eq. (4), but this would cause exactly one transition during one period of the fast variable, which makes the effective Hamiltonian too large to be useful. Choosing two Kronecker deltas causes one transition only to take place after much more time, making the insertion (4) of the desired order of magnitude to serve as a contribution in the effective Hamiltonian of the slow variables.

By adding a large number of similar transition events in the orbits of all fast variables, causing transitions for all pairs of (neighbouring) slow variables, we can now generate any desired contributions to the effective Hamiltonian elements *Hαβ* causing transition among the slow variables. The result will be.

$$H\_{a\beta}^{\text{int}} = \frac{\pi}{2} \sigma\_{\mathcal{Y}}^{[a,\beta]} \frac{N^{[s]}}{N\_{[a]} N\_{[\beta]}} \quad , \tag{8}$$

where the numbers *N*½ � *<sup>α</sup>* and *N*½ � *<sup>β</sup>* are the total numbers of points on the orbits of the fast variables *α* and *β*, and the numbers *N<sup>s</sup>* indicate the numbers of the special transition points on the donut formed by the orbits of the pair *α*, *β*.

We encounter the restriction that the matrix elements will come with rational coefficients in front. The fundamental reason for the coefficients to be rational is that, eventually, all discretised classical models have finite Poincaré recursion times. In practice one may expect that this problem goes away when, for realistic classical systems, the Poincaré recursion times will rapidly go to infinity.

#### *Ontology in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.99852*

We have now achieved the following. Let there be given any Hamiltonian with matrix elements *Hαβ* in a finite-dimensional vector space, and given a suitably chosen basis in this vector space, preferably one where every state can be endowed with coordinates *x* !. Then we have defined slow variables <sup>∣</sup>*α*<sup>i</sup> describing the physical states, and we added fast variables whose excited states are beyond the reach of our experiments. We found classical interactions, prescribed as exchanges between the classical states, such that the effective Hamiltonian will approach the given one.

The system obeys the Schrödinger equation dictated by this Hamiltonian, and, by construction, all probabilities evolve as is mandated by the Copenhagen doctrine. The reader may ask how to obtain the diagonal elements of the Hamiltonian, and how to make it contain complex numbers. The answer is that these can also be generated by using an additional binary degree of freedom as mentioned above.

In principle, one could have used any orthonormal basis of states to be used in our construction, but in practice we would like to recover locality in some way. The demand of locality in the classical system implies that we should demand locality for the fast variables and the slow ones. This appears to be straightforward. For nonrelativistic particles, one may use the basis of states defined by the position opera-

tors *x* !. In the relativistic case, one needs the field operators *φ<sup>i</sup> x* ! and their quantum eigen states to start off with.

The theory we arrive at appears to be closely related to Nelson's 'stochastic quantum mechanics' [13]. We think our construction has a more solid mathematical foundation, explaining how the quantum entanglement arises naturally from the energy conservation law, associated to time translation invariance.
