**4. On Bell's theorem**

Yet, this conclusion is often criticised. To set the stage, let us recapitulate J.S. Bell's theorem [15]: a source is constructed that emits two entangled photons simultaneously. Such sources exist, so no further justification of its properties is asked for. If �*z* is the direction of the photons emitted, then the helicities are in the *xy* direction. Entanglement here means that the 2-photon state is.<sup>3</sup>

$$|\psi\rangle\_{\text{source}} = \frac{1}{\sqrt{2}}(|\mathbf{0}\,\mathbf{0}\rangle + |\mathbf{1}\,\mathbf{1}\rangle)\,,\tag{9}$$

where the 0 stands for the *x* polarisation and 1 stands for the *y* polarisation. Alice and Bob use polarisers to analyse the photons they see. Alice rotates her polariser to an angle *a* and Bob chooses an angle *b*, and these choices are assumed to be totally independent. The two photons "do not know" what the angles *a* and *b* are; they are assumed to emerge with a polarisation angle *λ*. According to the usual view of what hidden variables are, the probability that both Alice and Bob are detecting their photon is written as *P a*ð Þ , *b* ; the probability that a photon with orientation *λ* is detected by Alice is assumed to be *pA*ð Þ *λ*, *a* , and the probability that Bob makes an observation is written as *pB*ð Þ *λ*, *b* . One then writes.

$$P(a,b) \stackrel{?}{=} \int \mathrm{d}\lambda \cdot \rho(\lambda) \cdot p\_A(a,\lambda) \cdot p\_B(b,\lambda). \tag{10}$$

All probabilities *p* and *P* are assumed to be between 0 and 1. *ρ λ*ð Þ is also positive and integrates to one. Bell would argue that this expression should apply to theories such as ours, simply by merging the fast variables *φ<sup>i</sup> x* !� � with the parameter *<sup>λ</sup>*.

**Figure 3** shows what assumptions go in Eq. (10). It seems to be obvious that observers in regions 1 and 2 may choose any setting *a* and *b* to identify their photon.

Writing *<sup>a</sup>* <sup>¼</sup> *<sup>a</sup>* � <sup>90</sup><sup>∘</sup> , photons obey:

$$P(\overline{a}) + P(a) = \mathbf{1} \; . \tag{11}$$

The correlation between Alice's and Bob's measurement is then written as.

<sup>3</sup> Alternatively, a source emitting two spin 1*=*2 particles could be used. The angles of the polarisation will then be twice the angles of the photon orientation discussed here, and there will be other modifications due to the fact that these particles are fermions.

**Figure 3.**

*Bell's definition of locality. 1 and 2 are two small regions of space–time, space-like separated. "Full specification of what happens in region 3 makes events in 2 irrelevant for predictions about 1 if local causality holds".*

$$E(a,b) = P(a,b) + P(\overline{a}, \overline{b}) - P(a, \overline{b}) - P(\overline{a}, b) \ . \tag{12}$$

Standard quantum mechanics allows one to choose the entangled photon state (9) as if it is oriented towards either Alice or Bob, since it is rotation independent. The outcome is then.

$$E\_{\text{quant}}(a,b) = 2\cos^2(a-b) - \mathbf{1} = \cos 2(a-b) \ . \tag{13}$$

In the fashionable hidden variable language, Eq. (10) is assumed to be valid, which implies that the photon must take care of giving Alice and Bob their measurement outcome whatever their choices *a* and *b* are, and these outcomes are found to obey the CHSH inequality [16], derived directly from Eq. (10). One then finds that Eq. (5) conflicts with Eq. (10). There is a mismatch of at least a factor ffiffi 2 <sup>p</sup> , realised when <sup>∣</sup>*<sup>a</sup>* � *<sup>b</sup>*<sup>∣</sup> <sup>¼</sup> <sup>22</sup>*:*5<sup>∘</sup> or 67*:*5<sup>∘</sup> .

Several 'loopholes' were proposed, having to do with the limited accuracy of the experiments, but these will not help us, since we claim that our theory exactly reproduces quantum mechanics, and therefore Eq. (13) should be reproduced by our theory. It violates CHSH. How can this happen?

Our short answer is that we have a classically evolving system that exactly reproduces the probability expressions predicted by the Schrödinger equation, including Eq. (13), in a given basis of Hilbert space. The model is local and allows for any initial state; it does not require any kind of 'conspiracy' or 'retrocausality', or even non-locality.

This should settle the matter, but it is true that the violation of the CHSH inequality is quite surprising.

The difficulty resides in assumption (10). Bell derives it directly from causality. If no signal can travel from the space–time point where Alice does her measurement to the point where Bob does his experiment, and *vice versa*, then Eq. (10) just follows. Nevertheless, an assumption was made.

It amounts to the statement that the variables *λ*, *a*, and *b*, are mutually independent. However, in [9], we computed the minimal non-vanishing correlations between the angles *a*, *b* and *λ* that could reproduce the quantum expression (13) exactly. We found<sup>4</sup> :

$$P(a,b,\lambda) = \mathbb{C}|\sin 2(a+b-2\lambda)|\ \text{ ,}\tag{14}$$

where *C* normalises the total probability to one (its value depends on the integration domain chosen). This expression shows a non-vanishing 3-variable correlation, without any 2-variable correlations as soon as one averages over any of the three variables. An equation such as (14) should replace (10).

<sup>4</sup> This outcome is model dependent, and if we choose the model to be physically more plausible, the correlations become even stronger.

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

One can read this to mean that the settings *a* and *b* have an effect on *λ*, but one can also say that the choice of *λ* made by the photon, affected the settings chosen by Alice and Bob.5 Perhaps the best way to interpret this strange feature is that it be an aspect of information: the fact that the fast variables occupy all states in their orbits with equal probabilities is expressed by saying that they live in their energy ground states. The choice of the phases here is a man-made ambiguity that may propagate backwards in time. It is not an observable 'spooky signal', since nothing propagates backwards in time in the classical formulation.

When we say that the photons (together with the fast variables) 'affect' the settings chosen by Alice and Bob, it implies that Alice and Bob have no 'free will'. Of course they haven't, their actions are completely controlled by the equations. We can't change setting *a* without changing what happens in region 3 of **Figure 3**.

It is important then to realise that our theory is *not* a theory about statistical distributions. If we include the fast variables, everything that happens in region 3, occurs with probability 1, or, if it does not happen, it has probability 0. There is no in-between. Remember that we reproduce the Schrödinger equation *in a given basis of Hilbert space*. The probabilities of the Schrödinger equation emerge exactly, but only if we start with the right basis elements.

We can add to this an important observation when the classical degrees of freedom are considered: *even a minute change of the setting a will require an initial state in* **Figure 3***, region 3 that is orthogonal to what it was before that adjustment.* This is because the settings are classically described. The required rotation of the fast variable erases the information as to where its transition point was located (see **Figure 2**).

We note that this aspect of our scenario implies the absence of 'free will' for Alice and Bob in choosing their settings. Alice and Bob are forced to obey classical laws, such that the rule *ontology in* ¼ *ontology out* is obeyed. The same can be said of Schrödinger's cat. Eventually, what we see when inspecting the cat is its classical behaviour. Only after adding the (in practice invisible) fast variables, we can perform a basis transformation to quantum states to say that the cat is superimposed. The statement belongs in the world of logic generated by the vector representation, but means nothing as long as we hold on to the classical description.
