# Unifying Aspects of Generalized Calculus
## 7 Escort Probabilities and Quantum Mechanical Hidden Variables  
Non-Diophantine arithmetics have several properties that make them analogous to sets of values of incompatible random variables in quantum mechanics. Generalized arithmetics and non-Newtonian calculi have nontrivial consequences for the problem of hidden variables and completeness of quantum mechanics. **Example 7**. : _Pauli matrices \(\sigma_{1}\) and \(\sigma_{2}\) represent random variables whose values are \(s_{1}=\pm 1\) and \(s_{2}=\pm 1\), respectively. However, it is not allowed to assume that \(\sigma_{1}+\sigma_{2}\) represents a random variable whose possible values are \(s_{1}+s_{2}=0,\pm 2\), even though an average of \(\sigma_{1}+\sigma_{2}\) ia a sum of independent averages of \(\sigma_{1}\) and \(\sigma_{2}\). In non-Diophantine arithmetic one encounters a similar problem. In general it makes no sense to perform additions of the form \(x_{\mathbb{X}}+y_{\mathbb{Y}}\) even if \(x_{\mathbb{X}}\in\mathbb{R}\) and \(y_{\mathbb{Y}}\in\mathbb{R}\). One should not be surprised if non-Diophantine probabilities turn out to be analogous to quantum probabilities, at least in some respects._  
Normalization of probability implies  
\[1_{\mathbb{X}}=f^{-1}(1)=f^{-1}\left(\sum_{k}p_{k}\right)=f^{-1}\left(\sum_{k }f(p_{k}^{\prime})\right)=\bigoplus_{k}p_{k}^{\prime}. \tag{81}\]  
In principle, \(1_{\mathbb{X}}\
eq 1\). An interesting and highly nontrivial case occurs if both \(p_{k}\) and \(p_{k}^{\prime}=f^{-1}(p_{k})\) are probabilities in the ordinary sense, i.e., in addition to (81) one finds \(1_{\mathbb{X}}=1\), \(0\leq p_{k}^{\prime}\leq 1\), and \(\ \sum_{k}p_{k}^{\prime}=1\). What can be then said about \(f\)? We can formalize the question as follows. **Problem 1**. : _Find a characterization of those functions \(g:[0,1]\to[0,1]\) that satisfy_  
\[\sum_{k}g(p_{k})=1,\quad\text{for any choice of probabilities $p_{k}$}. \tag{82}\]In analogy to the generalized thermostatistics literature we can term \(p_{k}^{\prime}=g(p_{k})\) the escort probabilities [56; 57; 58]. Notice that we are _not_ in interested in the trivial solution, often employed in the context of Tsallis and Renyi entropies, where \(p_{k}\) is replaced by \(p_{k}^{q}\) and then _renormalized_,  
\[P_{k}=\frac{p_{k}^{q}}{\sum_{j}p_{j}^{q}}=g_{k}(p_{1},\dots,p_{n},\dots) \tag{83}\]  
as \(g_{k}(p_{1},\dots,p_{n},\dots)\
eq g(p_{k})\) for a single function \(g\) of one variable. As we will shortly see, the solution of (82) turns out to have straightforward implications for the quantum mechanical problem of hidden variables, and relations between classical and quantum probabilities. The most nontrivial result is found for binary probabilities, \(p_{1}+p_{2}=1\). **Lemma 1**. : \(g(p_{1})+g(p_{2})=1\) _for all \(p_{1}+p_{2}=1\) if and only if_  
\[g(p)=\frac{1}{2}+h\left(p-\frac{1}{2}\right) \tag{84}\]  
_where \(h(-x)=-h(x)\)._  
Proof. : See Appendix B. The lemma has profound consequences for foundations of quantum mechanics, as it allows to circumvent [PERSON]'s theorem by non-Newtonian hidden variables. For more details the readers are referred to [13; 15], but here just a few examples. **Example 8**. : _The trivial case \(g(p)=p\) implies \(h(x)=x\), where \(0\leq p\leq 1\) and \(-1/2\leq x\leq 1/2\)._  
**Example 9**. : _Consider \(g(p)=\sin^{2}\frac{\pi}{2}p\). Then,_  
\[h(x)=g\left(x+\frac{1}{2}\right)-\frac{1}{2}=\frac{1}{2}\sin\pi x. \tag{85}\]  
_Let us cross-check,_  
\[g(p)+g(1-p)=\sin^{2}\frac{\pi}{2}p+\sin^{2}\frac{\pi}{2}(1-p)=\sin^{2}\frac{ \pi}{2}p+\cos^{2}\frac{\pi}{2}p=1. \tag{86}\]  
_Now let \(p=(\pi-\theta)/\pi\) be the probability of finding a point belonging to the overlap of two half-circles rotated by \(\theta\). Then,_  
\[g(p)=\sin^{2}\frac{\pi}{2}\frac{\pi-\theta}{\pi}=\cos^{2}\frac{\theta}{2} \tag{87}\]  
_is the quantum-mechanical law describing the conditional probability for two successive measurements of spin-1/2 in two Stern-Gerlach devices placed one after another, with relative angle \(\theta\).